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stringlengths 3
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stringlengths 21
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stringlengths 17
8.64k
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stringlengths 7
205k
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MeasureTheory.contDiffOn_convolution_right_with_param
|
Mathlib/Analysis/Convolution.lean
|
theorem contDiffOn_convolution_right_with_param {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F)
{g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
(hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) :
ContDiffOn 𝕜 n (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
this✝¹ : MeasurableSpace eG := borel eG
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ULift.{max uG uF uP, uE'} E'
eF : Type (max uF uG uE' uP) := ULift.{max uG uE' uP, uF} F
eP : Type (max uP uG uE' uF) := ULift.{max uG uE' uF, uP} P
isoG : eG ≃L[𝕜] G := ContinuousLinearEquiv.ulift
isoE' : eE' ≃L[𝕜] E' := ContinuousLinearEquiv.ulift
isoF : eF ≃L[𝕜] F := ContinuousLinearEquiv.ulift
isoP : eP ≃L[𝕜] P := ContinuousLinearEquiv.ulift
ef : eG → E := f ∘ ⇑isoG
eμ : Measure eG := Measure.map (⇑isoG.symm) μ
eg : eP → eG → eE' := fun ep ex => isoE'.symm (g (isoP ep) (isoG ex))
eL : E →L[𝕜] eE' →L[𝕜] eF := (↑(isoE'.arrowCongr isoF).symm).comp L
R : eP × eG → eF := fun q => (ef ⋆[eL, eμ] eg q.1) q.2
R_contdiff : ContDiffOn 𝕜 (↑n) R ((⇑isoP ⁻¹' s) ×ˢ univ)
A : ContDiffOn 𝕜 (↑n) (⇑isoF ∘ R ∘ ⇑(isoP.prod isoG).symm) (s ×ˢ univ)
⊢ ⇑isoF ∘ R ∘ ⇑(isoP.prod isoG).symm = fun q => (f ⋆[L, μ] g q.1) q.2
|
apply funext
|
case h
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
this✝¹ : MeasurableSpace eG := borel eG
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ULift.{max uG uF uP, uE'} E'
eF : Type (max uF uG uE' uP) := ULift.{max uG uE' uP, uF} F
eP : Type (max uP uG uE' uF) := ULift.{max uG uE' uF, uP} P
isoG : eG ≃L[𝕜] G := ContinuousLinearEquiv.ulift
isoE' : eE' ≃L[𝕜] E' := ContinuousLinearEquiv.ulift
isoF : eF ≃L[𝕜] F := ContinuousLinearEquiv.ulift
isoP : eP ≃L[𝕜] P := ContinuousLinearEquiv.ulift
ef : eG → E := f ∘ ⇑isoG
eμ : Measure eG := Measure.map (⇑isoG.symm) μ
eg : eP → eG → eE' := fun ep ex => isoE'.symm (g (isoP ep) (isoG ex))
eL : E →L[𝕜] eE' →L[𝕜] eF := (↑(isoE'.arrowCongr isoF).symm).comp L
R : eP × eG → eF := fun q => (ef ⋆[eL, eμ] eg q.1) q.2
R_contdiff : ContDiffOn 𝕜 (↑n) R ((⇑isoP ⁻¹' s) ×ˢ univ)
A : ContDiffOn 𝕜 (↑n) (⇑isoF ∘ R ∘ ⇑(isoP.prod isoG).symm) (s ×ˢ univ)
⊢ ∀ (x : P × G), (⇑isoF ∘ R ∘ ⇑(isoP.prod isoG).symm) x = (f ⋆[L, μ] g x.1) x.2
|
36b6ddfb418542d6
|
HomologicalComplex.mapBifunctor₁₂.ι_D₃
|
Mathlib/Algebra/Homology/BifunctorAssociator.lean
|
@[reassoc (attr := simp)]
lemma ι_D₃ :
ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h ≫ D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' =
d₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'
|
case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝¹⁹ : Category.{u_17, u_1} C₁
inst✝¹⁸ : Category.{u_16, u_2} C₂
inst✝¹⁷ : Category.{u_14, u_5} C₃
inst✝¹⁶ : Category.{u_13, u_6} C₄
inst✝¹⁵ : Category.{u_15, u_3} C₁₂
inst✝¹⁴ : HasZeroMorphisms C₁
inst✝¹³ : HasZeroMorphisms C₂
inst✝¹² : HasZeroMorphisms C₃
inst✝¹¹ : Preadditive C₁₂
inst✝¹⁰ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝⁹ : F₁₂.PreservesZeroMorphisms
inst✝⁸ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁷ : G.Additive
inst✝⁶ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁵ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₁₂ c₃ c₄
inst✝² : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝¹ : DecidableEq ι₁₂
inst✝ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₃.Rel i₃ (c₃.next i₃)
⊢ (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)).app (K₃.X i₃) ≫
mapBifunctor.d₂ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ (c₁.π c₂ c₁₂ (i₁, i₂)) i₃ j' =
d₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'
|
rw [d₃_eq _ _ _ _ _ _ _ _ _ h₁]
|
case pos
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝¹⁹ : Category.{u_17, u_1} C₁
inst✝¹⁸ : Category.{u_16, u_2} C₂
inst✝¹⁷ : Category.{u_14, u_5} C₃
inst✝¹⁶ : Category.{u_13, u_6} C₄
inst✝¹⁵ : Category.{u_15, u_3} C₁₂
inst✝¹⁴ : HasZeroMorphisms C₁
inst✝¹³ : HasZeroMorphisms C₂
inst✝¹² : HasZeroMorphisms C₃
inst✝¹¹ : Preadditive C₁₂
inst✝¹⁰ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝⁹ : F₁₂.PreservesZeroMorphisms
inst✝⁸ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁷ : G.Additive
inst✝⁶ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁵ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁴ : TotalComplexShape c₁ c₂ c₁₂
inst✝³ : TotalComplexShape c₁₂ c₃ c₄
inst✝² : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝¹ : DecidableEq ι₁₂
inst✝ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₃.Rel i₃ (c₃.next i₃)
⊢ (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)).app (K₃.X i₃) ≫
mapBifunctor.d₂ (K₁.mapBifunctor K₂ F₁₂ c₁₂) K₃ G c₄ (c₁.π c₂ c₁₂ (i₁, i₂)) i₃ j' =
c₁₂.ε₂ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃) •
(G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).map (K₃.d i₃ (c₃.next i₃)) ≫
ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ (c₃.next i₃) j'
|
2c6f8763db2c1a24
|
MeasureTheory.SimpleFunc.mem_image_of_mem_range_restrict
|
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
theorem mem_image_of_mem_range_restrict {r : β} {s : Set α} {f : α →ₛ β}
(hr : r ∈ (restrict f s).range) (h0 : r ≠ 0) : r ∈ f '' s :=
if hs : MeasurableSet s then by simpa [mem_restrict_range hs, h0, -mem_range] using hr
else by
rw [restrict_of_not_measurable hs] at hr
exact (h0 <| eq_zero_of_mem_range_zero hr).elim
|
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
r : β
s : Set α
f : α →ₛ β
hr : r ∈ (f.restrict s).range
h0 : r ≠ 0
hs : ¬MeasurableSet s
⊢ r ∈ ⇑f '' s
|
rw [restrict_of_not_measurable hs] at hr
|
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : Zero β
r : β
s : Set α
f : α →ₛ β
hr : r ∈ SimpleFunc.range 0
h0 : r ≠ 0
hs : ¬MeasurableSet s
⊢ r ∈ ⇑f '' s
|
9a16ed464d2ab632
|
Function.hfunext
|
Mathlib/Logic/Function/Basic.lean
|
lemma hfunext {α α' : Sort u} {β : α → Sort v} {β' : α' → Sort v} {f : ∀a, β a} {f' : ∀a, β' a}
(hα : α = α') (h : ∀a a', HEq a a' → HEq (f a) (f' a')) : HEq f f'
|
α : Sort u
β : α → Sort v
f : (a : α) → β a
β' : α → Sort v
f' : (a : α) → β' a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
⊢ β = β'
|
funext a
|
case h
α : Sort u
β : α → Sort v
f : (a : α) → β a
β' : α → Sort v
f' : (a : α) → β' a
h : ∀ (a a' : α), HEq a a' → HEq (f a) (f' a')
this : ∀ (a : α), HEq (f a) (f' a)
a : α
⊢ β a = β' a
|
7bf359655506b660
|
Algebra.norm_eq_prod_embeddings
|
Mathlib/RingTheory/Norm/Basic.lean
|
theorem norm_eq_prod_embeddings [FiniteDimensional K L] [Algebra.IsSeparable K L] [IsAlgClosed E]
(x : L) : algebraMap K E (norm K x) = ∏ σ : L →ₐ[K] E, σ x
|
K : Type u_4
L : Type u_5
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Algebra K L
E : Type u_7
inst✝⁴ : Field E
inst✝³ : Algebra K E
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsAlgClosed E
x : L
⊢ (algebraMap K E) ((norm K) x) = ∏ σ : L →ₐ[K] E, σ x
|
have hx := Algebra.IsSeparable.isIntegral K x
|
K : Type u_4
L : Type u_5
inst✝⁷ : Field K
inst✝⁶ : Field L
inst✝⁵ : Algebra K L
E : Type u_7
inst✝⁴ : Field E
inst✝³ : Algebra K E
inst✝² : FiniteDimensional K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : IsAlgClosed E
x : L
hx : IsIntegral K x
⊢ (algebraMap K E) ((norm K) x) = ∏ σ : L →ₐ[K] E, σ x
|
92ee720d9e04e443
|
CategoryTheory.Limits.Types.binaryCofan_isColimit_iff
|
Mathlib/CategoryTheory/Limits/Shapes/Types.lean
|
theorem binaryCofan_isColimit_iff {X Y : Type u} (c : BinaryCofan X Y) :
Nonempty (IsColimit c) ↔
Injective c.inl ∧ Injective c.inr ∧ IsCompl (Set.range c.inl) (Set.range c.inr)
|
case mp.intro
X Y : Type u
c : BinaryCofan X Y
h : IsColimit c
⊢ Injective (Sum.inl ≫ (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv) ∧
Injective (Sum.inr ≫ (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv) ∧
IsCompl (Set.range (Sum.inl ≫ (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv))
(Set.range (Sum.inr ≫ (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv))
|
refine
⟨(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp
Sum.inl_injective,
(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp
Sum.inr_injective, ?_⟩
|
case mp.intro
X Y : Type u
c : BinaryCofan X Y
h : IsColimit c
⊢ IsCompl (Set.range (Sum.inl ≫ (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv))
(Set.range (Sum.inr ≫ (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).inv))
|
c9e635615c45e53b
|
LieSubalgebra.normalizer_eq_self_of_engel_le
|
Mathlib/Algebra/Lie/EngelSubalgebra.lean
|
/-- A Lie-subalgebra of an Artinian Lie algebra is self-normalizing
if it contains an Engel subalgebra.
See `LieSubalgebra.normalizer_engel` for a proof that Engel subalgebras are self-normalizing,
avoiding the Artinian condition. -/
lemma normalizer_eq_self_of_engel_le [IsArtinian R L]
(H : LieSubalgebra R L) (x : L) (h : engel R x ≤ H) :
normalizer H = H
|
case h
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsArtinian R L
H : LieSubalgebra R L
x : L
h : engel R x ≤ H
N : LieSubalgebra R L := H.normalizer
aux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H
aux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N
dx : ↥N →ₗ[R] ↥N := LinearMap.restrict ((ad R L) x) aux₂
k : ℕ
y : ↥N
⊢ y ∈ LinearMap.ker (dx ^ (k + 1)) → ((ad R L) x ^ (k + 1)) (N.subtype y) = 0
|
generalize k+1 = k
|
case h
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsArtinian R L
H : LieSubalgebra R L
x : L
h : engel R x ≤ H
N : LieSubalgebra R L := H.normalizer
aux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H
aux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N
dx : ↥N →ₗ[R] ↥N := LinearMap.restrict ((ad R L) x) aux₂
k✝ : ℕ
y : ↥N
k : ℕ
⊢ y ∈ LinearMap.ker (dx ^ k) → ((ad R L) x ^ k) (N.subtype y) = 0
|
253d8e16fa48a5a4
|
ENNReal.mul_div_cancel
|
Mathlib/Data/ENNReal/Inv.lean
|
/-- See `ENNReal.mul_div_cancel'` for a stronger version. -/
protected lemma mul_div_cancel (ha₀ : a ≠ 0) (ha : a ≠ ∞) : a * (b / a) = b :=
ENNReal.mul_div_cancel' (by simp [ha₀]) (by simp [ha])
|
a b : ℝ≥0∞
ha₀ : a ≠ 0
ha : a ≠ ⊤
⊢ a = ⊤ → b = 0
|
simp [ha]
|
no goals
|
92e5070331a0a3d9
|
List.DecEq_eq
|
Mathlib/Data/List/Permutation.lean
|
theorem DecEq_eq [DecidableEq α] :
List.instBEq = @instBEqOfDecidableEq (List α) instDecidableEqList :=
congr_arg BEq.mk <| by
funext l₁ l₂
show (l₁ == l₂) = _
rw [Bool.eq_iff_iff, @beq_iff_eq _ (_), decide_eq_true_iff]
|
case h.h
α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
⊢ (l₁ == l₂) = decide (l₁ = l₂)
|
rw [Bool.eq_iff_iff, @beq_iff_eq _ (_), decide_eq_true_iff]
|
no goals
|
93818e51a19b4d80
|
Module.finrank_tensorProduct
|
Mathlib/LinearAlgebra/Dimension/Constructions.lean
|
theorem Module.finrank_tensorProduct :
finrank R (M ⊗[S] M') = finrank R M * finrank S M'
|
R : Type u
S : Type u'
M : Type v
M' : Type v'
inst✝¹² : Semiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : AddCommMonoid M'
inst✝⁸ : Module R M
inst✝⁷ : StrongRankCondition R
inst✝⁶ : StrongRankCondition S
inst✝⁵ : Module S M
inst✝⁴ : Module S M'
inst✝³ : Free S M'
inst✝² : Algebra S R
inst✝¹ : IsScalarTower S R M
inst✝ : Free R M
⊢ finrank R (M ⊗[S] M') = finrank R M * finrank S M'
|
simp [finrank]
|
no goals
|
1ef66bf41e6e4850
|
Int.subNatNat_eq_coe
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean
|
theorem subNatNat_eq_coe {m n : Nat} : subNatNat m n = ↑m - ↑n
|
case hn
m n✝ i n : Nat
⊢ -↑i + -↑1 = ↑n + -↑n + -↑i + -↑1
|
rw [← @Int.sub_eq_add_neg n, ← ofNat_sub, Nat.sub_self, ofNat_zero, Int.zero_add]
|
case hn
m n✝ i n : Nat
⊢ n ≤ n
|
6cca7cab87ea9b22
|
IsPrimitiveRoot.norm_sub_one_two
|
Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
|
theorem norm_sub_one_two {k : ℕ} (hζ : IsPrimitiveRoot ζ (2 ^ k)) (hk : 2 ≤ k)
[H : IsCyclotomicExtension {(2 : ℕ+) ^ k} K L] (hirr : Irreducible (cyclotomic (2 ^ k) K)) :
norm K (ζ - 1) = 2
|
K : Type u
L : Type v
inst✝² : Field L
ζ : L
inst✝¹ : Field K
inst✝ : Algebra K L
k : ℕ
hζ : IsPrimitiveRoot ζ (2 ^ k)
hk : 2 ≤ k
H : IsCyclotomicExtension {2 ^ k} K L
hirr : Irreducible (cyclotomic (2 ^ k) K)
⊢ ↑(2 ^ 1) < ↑2 ^ k
|
exact Nat.pow_lt_pow_right one_lt_two (lt_of_lt_of_le one_lt_two hk)
|
no goals
|
06a533966d791d57
|
AlgebraicGeometry.exists_eq_pow_mul_of_isAffineOpen
|
Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean
|
theorem exists_eq_pow_mul_of_isAffineOpen (X : Scheme) (U : X.Opens) (hU : IsAffineOpen U)
(f : Γ(X, U)) (x : Γ(X, X.basicOpen f)) :
∃ (n : ℕ) (y : Γ(X, U)), y |_ X.basicOpen f = (f |_ X.basicOpen f) ^ n * x
|
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
⊢ ∃ n y, (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x
|
have := (hU.isLocalization_basicOpen f).2
|
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
x : ↑Γ(X, X.basicOpen f)
this :
∀ (z : ↑Γ(X, X.basicOpen f)),
∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1
⊢ ∃ n y, (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * x
|
3f9284003bd52695
|
AddCircle.volume_closedBall
|
Mathlib/MeasureTheory/Integral/Periodic.lean
|
theorem volume_closedBall {x : AddCircle T} (ε : ℝ) :
volume (Metric.closedBall x ε) = ENNReal.ofReal (min T (2 * ε))
|
case intro
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hT' : |T| = T
I : Set ℝ := Ioc (-(T / 2)) (T / 2)
hε : ε < T / 2
y : ℝ
hy₁ : -ε ≤ y
hy₂ : y ≤ ε
⊢ y ∈ I
|
constructor <;> linarith
|
no goals
|
7eec94214cffe2c7
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
theorem isPullback_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {X : ι → C}
{c : Cofan X} (hc : IsVanKampenColimit c) (i j : ι) [DecidableEq ι] :
IsPullback (P := (if j = i then X i else ⊥_ C))
(if h : j = i then eqToHom (if_pos h) else eqToHom (if_neg h) ≫ initial.to (X i))
(if h : j = i then eqToHom ((if_pos h).trans (congr_arg X h.symm))
else eqToHom (if_neg h) ≫ initial.to (X j))
(Cofan.inj c i) (Cofan.inj c j)
|
case refine_2.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ ∀ (t : Cofan fun k => if k = i then X i else ⊥_ C) (j : ι),
(Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫
(fun t => eqToHom ⋯ ≫ t.inj i) t =
t.inj j
|
intro t j
|
case refine_2.refine_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
⊢ (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫
(fun t => eqToHom ⋯ ≫ t.inj i) t =
t.inj j
|
23c34aefc3417657
|
BitVec.getElem_shiftLeftZeroExtend
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem getElem_shiftLeftZeroExtend {x : BitVec m} {n : Nat} (h : i < m + n) :
(shiftLeftZeroExtend x n)[i] = ((! decide (i < n)) && getLsbD x (i - n))
|
m i : Nat
x : BitVec m
n : Nat
h : i < m + n
⊢ (x.shiftLeftZeroExtend n)[i] = (!decide (i < n) && x.getLsbD (i - n))
|
rw [shiftLeftZeroExtend_eq, getLsbD]
|
m i : Nat
x : BitVec m
n : Nat
h : i < m + n
⊢ (setWidth (m + n) x <<< n)[i] = (!decide (i < n) && x.toNat.testBit (i - n))
|
2dc2db1c2633db68
|
MeasureTheory.Measure.measure_toMeasurable_inter_of_cover
|
Mathlib/MeasureTheory/Measure/Typeclasses.lean
|
theorem measure_toMeasurable_inter_of_cover {s : Set α} (hs : MeasurableSet s) {t : Set α}
{v : ℕ → Set α} (hv : t ⊆ ⋃ n, v n) (h'v : ∀ n, μ (t ∩ v n) ≠ ∞) :
μ (toMeasurable μ t ∩ s) = μ (t ∩ s)
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ⊤
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
|
let w n := toMeasurable μ (t ∩ v n)
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
t : Set α
v : ℕ → Set α
hv : t ⊆ ⋃ n, v n
h'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ⊤
w : ℕ → Set α := fun n => toMeasurable μ (t ∩ v n)
⊢ ∃ t' ⊇ t, MeasurableSet t' ∧ ∀ (u : Set α), MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u)
|
0be0e6e2d4665783
|
SimpleGraph.IsIndepSet.nonempty_mem_compl_mem_edge
|
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
/-- If `s` is an independent set, its complement meets every edge of `G`. -/
lemma IsIndepSet.nonempty_mem_compl_mem_edge
[Fintype α] [DecidableEq α] {s : Finset α} (indA : G.IsIndepSet s) {e} (he : e ∈ G.edgeSet) :
{ b ∈ sᶜ | b ∈ e }.Nonempty
|
case neg
α : Type u_1
G : SimpleGraph α
inst✝¹ : Fintype α
inst✝ : DecidableEq α
s : Finset α
indA : (↑s).Pairwise fun v w => ¬G.Adj v w
e : Sym2 α
v w : α
he : G.Adj v w
c : ∀ ⦃x : α⦄, x ∉ s → ¬x = v ∧ ¬x = w
vins : v ∉ s
⊢ False
|
exact (c vins).left rfl
|
no goals
|
68972c8d9751a792
|
List.cons_sublist_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem cons_sublist_iff {a : α} {l l'} :
a :: l <+ l' ↔ ∃ r₁ r₂, l' = r₁ ++ r₂ ∧ a ∈ r₁ ∧ l <+ r₂
|
α : Type u_1
a : α
l : List α
a' : α
r₁ r₂ : List α
ih : a :: l <+ r₁ ++ r₂ ↔ ∃ r₁_1 r₂_1, r₁ ++ r₂ = r₁_1 ++ r₂_1 ∧ a ∈ r₁_1 ∧ l <+ r₂_1
w : a :: l <+ r₁ ++ r₂
h₁ : a ∈ r₁
h₂ : l <+ r₂
⊢ a' :: (r₁ ++ r₂) = a' :: r₁ ++ r₂
|
simp
|
no goals
|
ea83f7b99e603a26
|
CategoryTheory.Limits.biprod.decomp_hom_to
|
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
lemma biprod.decomp_hom_to (f : Z ⟶ X ⊞ Y) :
∃ f₁ f₂, f = f₁ ≫ biprod.inl + f₂ ≫ biprod.inr :=
⟨f ≫ biprod.fst, f ≫ biprod.snd, by aesop⟩
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
X Y : C
inst✝ : HasBinaryBiproduct X Y
Z : C
f : Z ⟶ X ⊞ Y
⊢ f = (f ≫ fst) ≫ inl + (f ≫ snd) ≫ inr
|
aesop
|
no goals
|
10ef7a4ce3cfeb98
|
Std.Tactic.BVDecide.BVPred.mkUlt_denote_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Ult.lean
|
theorem mkUlt_denote_eq (aig : AIG α) (lhs rhs : BitVec w) (input : BinaryRefVec aig w)
(assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.rhs.get idx hidx, assign⟧ = rhs.getLsbD idx) :
⟦(mkUlt aig input).aig, (mkUlt aig input).ref, assign⟧ = BitVec.ult lhs rhs
|
case e_a.hleft.hidx
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
input : aig.BinaryRefVec w
assign : α → Bool
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.rhs.get idx hidx }⟧ = rhs.getLsbD idx
idx : Nat
hidx : idx < w
⊢ idx < w
|
assumption
|
no goals
|
71c501afc68211e5
|
List.reduceOption_length_eq
|
Mathlib/Data/List/ReduceOption.lean
|
theorem reduceOption_length_eq {l : List (Option α)} :
l.reduceOption.length = (l.filter Option.isSome).length
|
case nil
α : Type u_1
⊢ [].reduceOption.length = (filter Option.isSome []).length
|
simp_rw [reduceOption_nil, filter_nil, length]
|
no goals
|
79e24c01619dd30e
|
Std.DHashMap.Internal.Raw₀.getKey_insert_self
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean
|
theorem getKey_insert_self [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {k : α} {v : β k} :
(m.insert k v).getKey k (contains_insert_self _ h) = k
|
α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
k : α
v : β k
⊢ (m.insert k v).getKey k ⋯ = k
|
simp_to_model [insert] using List.getKey_insertEntry_self
|
no goals
|
51ba6d008afe0f68
|
BitVec.iunfoldr_getLsbD'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Folds.lean
|
theorem iunfoldr_getLsbD' {f : Fin w → α → α × Bool} (state : Nat → α)
(ind : ∀(i : Fin w), (f i (state i.val)).fst = state (i.val+1)) :
(∀ i : Fin w, getLsbD (iunfoldr f (state 0)).snd i.val = (f i (state i.val)).snd)
∧ (iunfoldr f (state 0)).fst = state w
|
w : Nat
α : Type u_1
f : Fin w → α → α × Bool
state : Nat → α
ind : ∀ (i : Fin w), (f i (state ↑i)).fst = state (↑i + 1)
⊢ (∀ (i : Fin w),
(Fin.hIterate (fun i => α × BitVec i) (state 0, nil) fun i q =>
(fun p => (p.fst, cons p.snd q.snd)) (f i q.fst)).snd.getLsbD
↑i =
(f i (state ↑i)).snd) ∧
(Fin.hIterate (fun i => α × BitVec i) (state 0, nil) fun i q =>
(fun p => (p.fst, cons p.snd q.snd)) (f i q.fst)).fst =
state w
|
simp
|
w : Nat
α : Type u_1
f : Fin w → α → α × Bool
state : Nat → α
ind : ∀ (i : Fin w), (f i (state ↑i)).fst = state (↑i + 1)
⊢ (∀ (i : Fin w),
(Fin.hIterate (fun i => α × BitVec i) (state 0, 0#0) fun i q =>
((f i q.fst).fst, cons (f i q.fst).snd q.snd)).snd.getLsbD
↑i =
(f i (state ↑i)).snd) ∧
(Fin.hIterate (fun i => α × BitVec i) (state 0, 0#0) fun i q => ((f i q.fst).fst, cons (f i q.fst).snd q.snd)).fst =
state w
|
2142a073bff43190
|
Multiset.Nodup.pi
|
Mathlib/Data/Multiset/Pi.lean
|
theorem Nodup.pi {s : Multiset α} {t : ∀ a, Multiset (β a)} :
Nodup s → (∀ a ∈ s, Nodup (t a)) → Nodup (pi s t) :=
Multiset.induction_on s (fun _ _ => nodup_singleton _)
(by
intro a s ih hs ht
have has : a ∉ s
|
α : Type u_1
inst✝ : DecidableEq α
β : α → Type u_2
s✝ : Multiset α
t : (a : α) → Multiset (β a)
a : α
s : Multiset α
ih : s.Nodup → (∀ a ∈ s, (t a).Nodup) → (s.pi t).Nodup
ht : ∀ a_1 ∈ a ::ₘ s, (t a_1).Nodup
has : a ∉ s
hs : a ∉ s ∧ s.Nodup
⊢ s.Nodup
|
exact hs.2
|
no goals
|
4440b8044ebd270b
|
contDiffWithinAtProp_id
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
theorem contDiffWithinAtProp_id (x : H) : ContDiffWithinAtProp I I n id univ x
|
case refine_2
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : WithTop ℕ∞
x : H
this : ContDiffWithinAt 𝕜 n id (range ↑I) (↑I x)
⊢ (↑I ∘ ↑I.symm) (↑I x) = id (↑I x)
|
simp only [mfld_simps]
|
no goals
|
0f8203e20f594cf8
|
Polynomial.revAtFun_invol
|
Mathlib/Algebra/Polynomial/Reverse.lean
|
theorem revAtFun_invol {N i : ℕ} : revAtFun N (revAtFun N i) = i
|
case neg
N i : ℕ
h : ¬i ≤ N
⊢ i = i
|
rfl
|
no goals
|
4fb112ed36f820a3
|
RingTheory.Sequence.IsWeaklyRegular.isWeaklyRegular_rTensor
|
Mathlib/RingTheory/Regular/RegularSequence.lean
|
lemma IsWeaklyRegular.isWeaklyRegular_rTensor [Module.Flat R M₂]
{rs : List R} (h : IsWeaklyRegular M rs) :
IsWeaklyRegular (M ⊗[R] M₂) rs
|
case cons
R : Type u_1
M : Type u_3
M₂ : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R M₂
inst✝² : Module.Flat R M₂
rs : List R
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
r : R
rs' : List R
h1 : IsSMulRegular N r
h2✝ : IsWeaklyRegular (QuotSMulTop r N) rs'
ih : IsWeaklyRegular (QuotSMulTop r N ⊗[R] M₂) rs'
e : QuotSMulTop r N ⊗[R] M₂ ≃ₗ[R] QuotSMulTop r (N ⊗[R] M₂) := quotSMulTopTensorEquivQuotSMulTop r M₂ N
⊢ IsWeaklyRegular (N ⊗[R] M₂) (r :: rs')
|
exact ((e.isWeaklyRegular_congr rs').mp ih).cons (h1.rTensor M₂)
|
no goals
|
1b6a5a70d5be14c9
|
List.findIdx?_eq_some_le_of_findIdx?_eq_some
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Find.lean
|
theorem findIdx?_eq_some_le_of_findIdx?_eq_some {xs : List α} {p q : α → Bool} (w : ∀ x ∈ xs, p x → q x) {i : Nat}
(h : xs.findIdx? p = some i) : ∃ j, j ≤ i ∧ xs.findIdx? q = some j
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
p q : α → Bool
h₁ : α
b : Nat
hb : p h₁ = true
l₁' : List α
h₃ : ∀ (a : α) (b : Nat), (a, b) ∈ l₁'.zipIdx → p a = false
a : α
as : List α
w : ∀ (x : α), x ∈ l₁' ++ a :: as → p x = true → q x = true
h₂ : h₁ = a ∧ b = l₁'.length
⊢ ∃ j, j ≤ b ∧ findIdx? q (l₁' ++ a :: as) = some j
|
obtain ⟨rfl, rfl⟩ := h₂
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
p q : α → Bool
h₁ : α
hb : p h₁ = true
l₁' : List α
h₃ : ∀ (a : α) (b : Nat), (a, b) ∈ l₁'.zipIdx → p a = false
as : List α
w : ∀ (x : α), x ∈ l₁' ++ h₁ :: as → p x = true → q x = true
⊢ ∃ j, j ≤ l₁'.length ∧ findIdx? q (l₁' ++ h₁ :: as) = some j
|
ed8a2a2eec4f9e74
|
Sigma.isConnected_iff
|
Mathlib/Topology/Connected/Clopen.lean
|
theorem Sigma.isConnected_iff [∀ i, TopologicalSpace (π i)] {s : Set (Σi, π i)} :
IsConnected s ↔ ∃ i t, IsConnected t ∧ s = Sigma.mk i '' t
|
case refine_1.intro.mk
ι : Type u_1
π : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (π i)
s : Set ((i : ι) × π i)
hs : IsConnected s
i : ι
x : π i
hx : ⟨i, x⟩ ∈ s
⊢ ∃ i t, IsConnected t ∧ s = mk i '' t
|
have : s ⊆ range (Sigma.mk i) :=
hs.isPreconnected.subset_isClopen isClopen_range_sigmaMk ⟨⟨i, x⟩, hx, x, rfl⟩
|
case refine_1.intro.mk
ι : Type u_1
π : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (π i)
s : Set ((i : ι) × π i)
hs : IsConnected s
i : ι
x : π i
hx : ⟨i, x⟩ ∈ s
this : s ⊆ range (mk i)
⊢ ∃ i t, IsConnected t ∧ s = mk i '' t
|
1455e050525a61d2
|
LieModule.nontrivial_lowerCentralSeriesLast
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
theorem nontrivial_lowerCentralSeriesLast [LieModule R L M] [Nontrivial M] [IsNilpotent L M] :
Nontrivial (lowerCentralSeriesLast R L M)
|
R : Type u
L : Type v
M : Type w
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : Nontrivial M
inst✝ : IsNilpotent L M
⊢ Nontrivial ↥(lowerCentralSeriesLast R L M)
|
rw [LieSubmodule.nontrivial_iff_ne_bot, lowerCentralSeriesLast]
|
R : Type u
L : Type v
M : Type w
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : Nontrivial M
inst✝ : IsNilpotent L M
⊢ (match nilpotencyLength L M with
| 0 => ⊥
| k.succ => lowerCentralSeries R L M k) ≠
⊥
|
f3cd833338422bff
|
Mathlib.Tactic.Rify.ratCast_ne
|
Mathlib/Tactic/Rify.lean
|
@[rify_simps] lemma ratCast_ne (a b : ℚ) : a ≠ b ↔ (a : ℝ) ≠ (b : ℝ)
|
a b : ℚ
⊢ a ≠ b ↔ ↑a ≠ ↑b
|
simp
|
no goals
|
c04a13c651c78e44
|
Multiset.prod_map_add
|
Mathlib/Algebra/BigOperators/Ring/Multiset.lean
|
lemma prod_map_add {s : Multiset ι} {f g : ι → α} :
prod (s.map fun i ↦ f i + g i) =
sum ((antidiagonal s).map fun p ↦ (p.1.map f).prod * (p.2.map g).prod)
|
ι : Type u_1
α : Type u_2
inst✝ : CommSemiring α
s : Multiset ι
f g : ι → α
⊢ (map (fun i => f i + g i) s).prod = (map (fun p => (map f p.1).prod * (map g p.2).prod) s.antidiagonal).sum
|
refine s.induction_on ?_ fun a s ih ↦ ?_
|
case refine_1
ι : Type u_1
α : Type u_2
inst✝ : CommSemiring α
s : Multiset ι
f g : ι → α
⊢ (map (fun i => f i + g i) 0).prod = (map (fun p => (map f p.1).prod * (map g p.2).prod) (antidiagonal 0)).sum
case refine_2
ι : Type u_1
α : Type u_2
inst✝ : CommSemiring α
s✝ : Multiset ι
f g : ι → α
a : ι
s : Multiset ι
ih : (map (fun i => f i + g i) s).prod = (map (fun p => (map f p.1).prod * (map g p.2).prod) s.antidiagonal).sum
⊢ (map (fun i => f i + g i) (a ::ₘ s)).prod =
(map (fun p => (map f p.1).prod * (map g p.2).prod) (a ::ₘ s).antidiagonal).sum
|
f0e10135f530bf98
|
coeSubmodule_differentIdeal
|
Mathlib/RingTheory/DedekindDomain/Different.lean
|
lemma coeSubmodule_differentIdeal [NoZeroSMulDivisors A B] :
coeSubmodule L (differentIdeal A B) = 1 / Submodule.traceDual A K 1
|
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : Algebra.IsSeparable K L
inst✝⁴ : IsIntegralClosure B A L
inst✝³ : IsIntegrallyClosed A
inst✝² : IsDedekindDomain B
inst✝¹ : IsFractionRing B L
inst✝ : NoZeroSMulDivisors A B
this : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H :
(algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv =
(↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)
⊢ Submodule.map (↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B))
(differentIdeal A B) =
1 / Submodule.traceDual A K 1
|
have : Algebra.IsSeparable (FractionRing A) (FractionRing B) :=
Algebra.IsSeparable.of_equiv_equiv _ _ H
|
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁹ : CommRing A
inst✝¹⁸ : Field K
inst✝¹⁷ : CommRing B
inst✝¹⁶ : Field L
inst✝¹⁵ : Algebra A K
inst✝¹⁴ : Algebra B L
inst✝¹³ : Algebra A B
inst✝¹² : Algebra K L
inst✝¹¹ : Algebra A L
inst✝¹⁰ : IsScalarTower A K L
inst✝⁹ : IsScalarTower A B L
inst✝⁸ : IsDomain A
inst✝⁷ : IsFractionRing A K
inst✝⁶ : FiniteDimensional K L
inst✝⁵ : Algebra.IsSeparable K L
inst✝⁴ : IsIntegralClosure B A L
inst✝³ : IsIntegrallyClosed A
inst✝² : IsDedekindDomain B
inst✝¹ : IsFractionRing B L
inst✝ : NoZeroSMulDivisors A B
this✝ : ↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B) = Algebra.linearMap B L
H :
(algebraMap (FractionRing A) (FractionRing B)).comp ↑(FractionRing.algEquiv A K).symm.toRingEquiv =
(↑(FractionRing.algEquiv B L).symm.toRingEquiv).comp (algebraMap K L)
this : Algebra.IsSeparable (FractionRing A) (FractionRing B)
⊢ Submodule.map (↑(FractionRing.algEquiv B L).toLinearEquiv ∘ₗ Algebra.linearMap B (FractionRing B))
(differentIdeal A B) =
1 / Submodule.traceDual A K 1
|
de2215dd565ee658
|
Real.summable_abs_int_rpow
|
Mathlib/Analysis/PSeries.lean
|
theorem summable_abs_int_rpow {b : ℝ} (hb : 1 < b) :
Summable fun n : ℤ => |(n : ℝ)| ^ (-b)
|
case hf₁
b : ℝ
hb : 1 < b
⊢ Summable fun n => |↑↑n| ^ (-b)
case hf₂
b : ℝ
hb : 1 < b
⊢ Summable fun n => |↑(-↑n)| ^ (-b)
|
on_goal 2 => simp_rw [Int.cast_neg, abs_neg]
|
case hf₁
b : ℝ
hb : 1 < b
⊢ Summable fun n => |↑↑n| ^ (-b)
case hf₂
b : ℝ
hb : 1 < b
⊢ Summable fun n => |↑↑n| ^ (-b)
|
20eea2c25b52f015
|
Nat.minFac_has_prop
|
Mathlib/Data/Nat/Prime/Defs.lean
|
theorem minFac_has_prop {n : ℕ} (n1 : n ≠ 1) : minFacProp n (minFac n)
|
n : ℕ
⊢ n ≠ 1 → ¬n = 0 → 2 ≤ n
|
rcases n with (_ | _ | _) <;> simp [succ_le_succ]
|
no goals
|
e3c16eee640cf2e2
|
Nat.Partrec.Code.hG
|
Mathlib/Computability/PartrecCode.lean
|
theorem hG : Primrec G
|
a : Primrec fun a => ofNat (ℕ × Code) a.length
k✝ : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1
n✝ : Primrec Prod.snd
k : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).1
n : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).2
⊢ Primrec fun p =>
(fun p k' =>
Code.recOn (ofNat (ℕ × Code) p.1.length).2 (some 0) (some p.2.succ) (some (unpair p.2).1) (some (unpair p.2).2)
(fun cf cg x x => do
let x ← Nat.Partrec.Code.lup p.1 ((ofNat (ℕ × Code) p.1.length).1, cf) p.2
let y ← Nat.Partrec.Code.lup p.1 ((ofNat (ℕ × Code) p.1.length).1, cg) p.2
some (Nat.pair x y))
(fun cf cg x x => do
let x ← Nat.Partrec.Code.lup p.1 ((ofNat (ℕ × Code) p.1.length).1, cg) p.2
Nat.Partrec.Code.lup p.1 ((ofNat (ℕ × Code) p.1.length).1, cf) x)
(fun cf cg x x =>
let z := (unpair p.2).1;
Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (ℕ × Code) p.1.length).1, cf) z) fun y => do
let i ← Nat.Partrec.Code.lup p.1 (k', (ofNat (ℕ × Code) p.1.length).2) (Nat.pair z y)
Nat.Partrec.Code.lup p.1 ((ofNat (ℕ × Code) p.1.length).1, cg) (Nat.pair z (Nat.pair y i)))
fun cf x =>
let z := (unpair p.2).1;
let m := (unpair p.2).2;
do
let x ← Nat.Partrec.Code.lup p.1 ((ofNat (ℕ × Code) p.1.length).1, cf) (Nat.pair z m)
Nat.casesOn x (some m) fun x =>
Nat.Partrec.Code.lup p.1 (k', (ofNat (ℕ × Code) p.1.length).2) (Nat.pair z (m + 1)))
p.1 p.2
|
apply
Nat.Partrec.Code.rec_prim c
(_root_.Primrec.const (some 0))
(Primrec.option_some.comp (_root_.Primrec.succ.comp n))
(Primrec.option_some.comp (Primrec.fst.comp <| Primrec.unpair.comp n))
(Primrec.option_some.comp (Primrec.snd.comp <| Primrec.unpair.comp n))
|
case hpr
a : Primrec fun a => ofNat (ℕ × Code) a.length
k✝ : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1
n✝ : Primrec Prod.snd
k : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).1
n : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).2
⊢ Primrec fun a => do
let x ← Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) a.1.1.2
let y ← Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) a.1.1.2
some (Nat.pair x y)
case hco
a : Primrec fun a => ofNat (ℕ × Code) a.length
k✝ : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1
n✝ : Primrec Prod.snd
k : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).1
n : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).2
⊢ Primrec fun a => do
let x ← Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) a.1.1.2
Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) x
case hpc
a : Primrec fun a => ofNat (ℕ × Code) a.length
k✝ : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1
n✝ : Primrec Prod.snd
k : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).1
n : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).2
⊢ Primrec fun a =>
let z := (unpair a.1.1.2).1;
Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) z)
fun y => do
let i ← Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) a.1.1.1.length).2) (Nat.pair z y)
Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.2.1) (Nat.pair z (Nat.pair y i))
case hrf
a : Primrec fun a => ofNat (ℕ × Code) a.length
k✝ : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1
n✝ : Primrec Prod.snd
k : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).1
n : Primrec fun a => a.1.2
k' : Primrec Prod.snd
c : Primrec fun a => (ofNat (ℕ × Code) a.1.1.length).2
⊢ Primrec fun a =>
let z := (unpair a.1.1.2).1;
let m := (unpair a.1.1.2).2;
do
let x ← Nat.Partrec.Code.lup a.1.1.1 ((ofNat (ℕ × Code) a.1.1.1.length).1, a.2.1) (Nat.pair z m)
Nat.casesOn x (some m) fun x =>
Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (ℕ × Code) a.1.1.1.length).2) (Nat.pair z (m + 1))
|
4867ddf124efd4e0
|
RootPairing.isCompl_rootSpan_ker_rootForm
|
Mathlib/LinearAlgebra/RootSystem/Finite/Nondegenerate.lean
|
lemma isCompl_rootSpan_ker_rootForm :
IsCompl P.rootSpan (LinearMap.ker P.RootForm)
|
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Field R
inst✝² : Module R M
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : P.IsAnisotropic
_iM : IsReflexive R M
_iN : IsReflexive R N
aux : finrank R M = finrank R ↥P.rootSpan + finrank R ↥P.corootSpan.dualAnnihilator
⊢ finrank R M ≤ finrank R ↥P.rootSpan + finrank R ↥(LinearMap.ker P.RootForm)
|
rw [aux, add_le_add_iff_left]
|
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : Fintype ι
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : Field R
inst✝² : Module R M
inst✝¹ : Module R N
P : RootPairing ι R M N
inst✝ : P.IsAnisotropic
_iM : IsReflexive R M
_iN : IsReflexive R N
aux : finrank R M = finrank R ↥P.rootSpan + finrank R ↥P.corootSpan.dualAnnihilator
⊢ finrank R ↥P.corootSpan.dualAnnihilator ≤ finrank R ↥(LinearMap.ker P.RootForm)
|
cd3d1642fc0b7df4
|
FreeGroup.injective_lift_of_ping_pong
|
Mathlib/GroupTheory/CoprodI.lean
|
theorem _root_.FreeGroup.injective_lift_of_ping_pong : Function.Injective (FreeGroup.lift a)
|
case hpp
ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i
i j : ι
hij : i ≠ j
⊢ ∀ (b : ℤ), FreeGroup.freeGroupUnitEquivInt.symm b ≠ 1 → (f i) (FreeGroup.freeGroupUnitEquivInt.symm b) • X' j ⊆ X' i
|
intro n hne1
|
case hpp
ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (i : ι), a⁻¹ i • (X i)ᶜ ⊆ Y i
H : ι → Type := fun _i => FreeGroup Unit
f : (i : ι) → H i →* G := fun i => FreeGroup.lift fun x => a i
X' : ι → Set α := fun i => X i ∪ Y i
i j : ι
hij : i ≠ j
n : ℤ
hne1 : FreeGroup.freeGroupUnitEquivInt.symm n ≠ 1
⊢ (f i) (FreeGroup.freeGroupUnitEquivInt.symm n) • X' j ⊆ X' i
|
7d73ea3c881ec9f6
|
CategoryTheory.classifier_isSheaf
|
Mathlib/CategoryTheory/Sites/Closed.lean
|
theorem classifier_isSheaf : Presieve.IsSheaf J₁ (Functor.closedSieves J₁)
|
case refine_1.mk.mk.a.h.mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.unop (Opposite.op X))
hM : J₁.IsClosed M
N : Sieve (Opposite.unop (Opposite.op X))
hN : J₁.IsClosed N
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve.pullback f N ∈ J₁ Y
|
apply J₁.superset_covering (Sieve.pullback_monotone f inf_le_left)
|
case refine_1.mk.mk.a.h.mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.unop (Opposite.op X))
hM : J₁.IsClosed M
N : Sieve (Opposite.unop (Opposite.op X))
hN : J₁.IsClosed N
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve.pullback f (N ⊓ ?m.12869) ∈ J₁ Y
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁) S.arrows
M : Sieve (Opposite.unop (Opposite.op X))
hM : J₁.IsClosed M
N : Sieve (Opposite.unop (Opposite.op X))
hN : J₁.IsClosed N
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve X
|
902347477e841054
|
ProbabilityTheory.Kernel.indep_iSup_of_directed_le
|
Mathlib/Probability/Independence/Kernel.lean
|
theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] (h_indep : ∀ i, Indep (m i) m' κ μ)
(h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) :
Indep (⨆ i, m i) m' κ μ
|
α : Type u_1
ι : Type u_3
_mα : MeasurableSpace α
Ω : Type u_4
m : ι → MeasurableSpace Ω
m' m0 : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
h_indep : ∀ (i : ι), Indep (m i) m' κ μ
h_le : ∀ (i : ι), m i ≤ m0
h_le' : m' ≤ m0
hm : Directed (fun x1 x2 => x1 ≤ x2) m
p : ι → Set (Set Ω) := fun n => {t | MeasurableSet t}
hp : ∀ (n : ι), IsPiSystem (p n)
h_gen_n : ∀ (n : ι), m n = generateFrom (p n)
hp_supr_pi : IsPiSystem (⋃ n, p n)
p' : Set (Set Ω) := {t | MeasurableSet t}
hp'_pi : IsPiSystem p'
h_gen' : m' = generateFrom p'
⊢ Indep (⨆ i, m i) m' κ μ
|
have h_pi_system_indep : IndepSets (⋃ n, p n) p' κ μ := by
refine IndepSets.iUnion ?_
conv at h_indep =>
intro i
rw [h_gen_n i, h_gen']
exact fun n => (h_indep n).indepSets
|
α : Type u_1
ι : Type u_3
_mα : MeasurableSpace α
Ω : Type u_4
m : ι → MeasurableSpace Ω
m' m0 : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
inst✝ : IsZeroOrMarkovKernel κ
h_indep : ∀ (i : ι), Indep (m i) m' κ μ
h_le : ∀ (i : ι), m i ≤ m0
h_le' : m' ≤ m0
hm : Directed (fun x1 x2 => x1 ≤ x2) m
p : ι → Set (Set Ω) := fun n => {t | MeasurableSet t}
hp : ∀ (n : ι), IsPiSystem (p n)
h_gen_n : ∀ (n : ι), m n = generateFrom (p n)
hp_supr_pi : IsPiSystem (⋃ n, p n)
p' : Set (Set Ω) := {t | MeasurableSet t}
hp'_pi : IsPiSystem p'
h_gen' : m' = generateFrom p'
h_pi_system_indep : IndepSets (⋃ n, p n) p' κ μ
⊢ Indep (⨆ i, m i) m' κ μ
|
d26524d956b58c02
|
AffineBasis.basisOf_reindex
|
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
theorem basisOf_reindex (i : ι') :
(b.reindex e).basisOf i =
(b.basisOf <| e.symm i).reindex (e.subtypeEquiv fun _ => e.eq_symm_apply.not)
|
case a
ι : Type u_1
ι' : Type u_2
k : Type u_5
V : Type u_6
P : Type u_7
inst✝³ : AddCommGroup V
inst✝² : AffineSpace V P
inst✝¹ : Ring k
inst✝ : Module k V
b : AffineBasis ι k P
e : ι ≃ ι'
i : ι'
j : { j // j ≠ i }
⊢ ((b.reindex e).basisOf i) j = ((b.basisOf (e.symm i)).reindex (e.subtypeEquiv ⋯)) j
|
simp
|
no goals
|
36ed7e14a20f44da
|
LieSubmodule.toSubmodule_mk
|
Mathlib/Algebra/Lie/Submodule.lean
|
theorem toSubmodule_mk (p : Submodule R M) (h) :
(({ p with lie_mem := h } : LieSubmodule R L M) : Submodule R M) = p
|
case mk
R : Type u
L : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
toAddSubmonoid✝ : AddSubmonoid M
smul_mem'✝ : ∀ (c : R) {x : M}, x ∈ toAddSubmonoid✝.carrier → c • x ∈ toAddSubmonoid✝.carrier
h :
∀ {x : L} {m : M},
m ∈ { toAddSubmonoid := toAddSubmonoid✝, smul_mem' := smul_mem'✝ }.carrier →
⁅x, m⁆ ∈ { toAddSubmonoid := toAddSubmonoid✝, smul_mem' := smul_mem'✝ }.carrier
⊢ ↑{ toAddSubmonoid := toAddSubmonoid✝, smul_mem' := smul_mem'✝, lie_mem := h } =
{ toAddSubmonoid := toAddSubmonoid✝, smul_mem' := smul_mem'✝ }
|
rfl
|
no goals
|
dbdc8c5b0dcce17f
|
MonomialOrder.div
|
Mathlib/RingTheory/MvPolynomial/Groebner.lean
|
theorem div {ι : Type*} {b : ι → MvPolynomial σ R}
(hb : ∀ i, IsUnit (m.leadingCoeff (b i))) (f : MvPolynomial σ R) :
∃ (g : ι →₀ (MvPolynomial σ R)) (r : MvPolynomial σ R),
f = Finsupp.linearCombination _ b g + r ∧
(∀ i, m.degree (b i * (g i)) ≼[m] m.degree f) ∧
(∀ c ∈ r.support, ∀ i, ¬ (m.degree (b i) ≤ c))
|
case h.left
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
(∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
⊢ f =
(Finsupp.linearCombination (MvPolynomial σ R) b)
(g' + Finsupp.single i ((monomial (m.degree f - m.degree (b i))) (↑⋯.unit⁻¹ * m.leadingCoeff f))) +
r'
|
rw [map_add, add_assoc, add_comm _ r', ← add_assoc, ← H'.1]
|
case h.left
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b i) ≠ 0
hf0 : ¬f = 0
i : ι
hf : m.degree (b i) ≤ m.degree f
deg_reduce : m.toSyn (m.degree (m.reduce ⋯ f)) < m.toSyn (m.degree f)
g' : ι →₀ MvPolynomial σ R
r' : MvPolynomial σ R
H' :
m.reduce ⋯ f = (Finsupp.linearCombination (MvPolynomial σ R) b) g' + r' ∧
(∀ (i_1 : ι), m.toSyn (m.degree (b i_1 * g' i_1)) ≤ m.toSyn (m.degree (m.reduce ⋯ f))) ∧
∀ c ∈ r'.support, ∀ (i : ι), ¬m.degree (b i) ≤ c
⊢ f =
m.reduce ⋯ f +
(Finsupp.linearCombination (MvPolynomial σ R) b)
(Finsupp.single i ((monomial (m.degree f - m.degree (b i))) (↑⋯.unit⁻¹ * m.leadingCoeff f)))
|
f995acdac7645677
|
eVariationOn.add_point
|
Mathlib/Topology/EMetricSpace/BoundedVariation.lean
|
theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ → α) (hu : Monotone u)
(us : ∀ i, u i ∈ s) (n : ℕ) :
∃ (v : ℕ → α) (m : ℕ), Monotone v ∧ (∀ i, v i ∈ s) ∧ x ∈ v '' Iio m ∧
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) ≤
∑ j ∈ Finset.range m, edist (f (v (j + 1))) (f (v j))
|
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
x : α
hx : x ∈ s
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈ s
n : ℕ
h : x < u n
exists_N : ∃ N ≤ n, x < u N
N : ℕ := Nat.find exists_N
hN : N ≤ n ∧ x < u N
w : ℕ → α := fun i => if i < N then u i else if i = N then x else u (i - 1)
ws : ∀ (i : ℕ), w i ∈ s
i : ℕ
hi : i + 1 = N
A : i < N
⊢ ¬i + 1 < i + 1
|
exact fun h => h.ne rfl
|
no goals
|
760e94573f7a043d
|
Fin.findSome?_isNone_iff
|
Mathlib/.lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean
|
theorem findSome?_isNone_iff {f : Fin n → Option α} :
(findSome? f).isNone ↔ ∀ i, (f i).isNone
|
n : Nat
α : Type u_1
f : Fin n → Option α
⊢ (findSome? f).isNone = true ↔ ∀ (i : Fin n), (f i).isNone = true
|
simp
|
no goals
|
ee41793be8df69f6
|
Polynomial.multiset_prod_X_sub_C_coeff_card_pred
|
Mathlib/Algebra/Polynomial/BigOperators.lean
|
theorem multiset_prod_X_sub_C_coeff_card_pred (t : Multiset R) (ht : 0 < Multiset.card t) :
(t.map fun x => X - C x).prod.coeff ((Multiset.card t) - 1) = -t.sum
|
case h.e'_2.hnc
R : Type u
inst✝ : CommRing R
t : Multiset R
ht : 0 < t.card
a✝ : Nontrivial R
⊢ ¬(Multiset.map (fun x => X - C x) t).prod.natDegree = 0
|
rw [natDegree_multiset_prod_of_monic]
|
case h.e'_2.hnc
R : Type u
inst✝ : CommRing R
t : Multiset R
ht : 0 < t.card
a✝ : Nontrivial R
⊢ ¬(Multiset.map natDegree (Multiset.map (fun x => X - C x) t)).sum = 0
case h.e'_2.hnc.h
R : Type u
inst✝ : CommRing R
t : Multiset R
ht : 0 < t.card
a✝ : Nontrivial R
⊢ ∀ f ∈ Multiset.map (fun x => X - C x) t, f.Monic
|
19e5692677d27aaa
|
PythagoreanTriple.isPrimitiveClassified_of_coprime_of_odd_of_pos
|
Mathlib/NumberTheory/PythagoreanTriples.lean
|
theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (hyo : y % 2 = 1)
(hzpos : 0 < z) : h.IsPrimitiveClassified
|
case neg.inl.inl
x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm ⟨(v, w), hp⟩
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
n : ℤ := q.num
hm0 : m ≠ 0
hq2 : q = ↑n / ↑m
hm2n2 : 0 < m ^ 2 + n ^ 2
hm2n20 : ↑m ^ 2 + ↑n ^ 2 ≠ 0
hx1 : ∀ {j k : ℚ}, k ≠ 0 → k ^ 2 + j ^ 2 ≠ 0 → (1 - (j / k) ^ 2) / (1 + (j / k) ^ 2) = (k ^ 2 - j ^ 2) / (k ^ 2 + j ^ 2)
hw2 : w = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
hx2 : ∀ {j k : ℚ}, k ≠ 0 → k ^ 2 + j ^ 2 ≠ 0 → 2 * (j / k) / (1 + (j / k) ^ 2) = 2 * k * j / (k ^ 2 + j ^ 2)
hv2 : v = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hnmcp : n.gcd m = 1
hmncp : m.gcd n = 1
hm2 : m % 2 = 0
hn2 : n % 2 = 0
⊢ False
|
have h1 : 2 ∣ (Int.gcd n m : ℤ) :=
Int.dvd_gcd (Int.dvd_of_emod_eq_zero hn2) (Int.dvd_of_emod_eq_zero hm2)
|
case neg.inl.inl
x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm ⟨(v, w), hp⟩
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
n : ℤ := q.num
hm0 : m ≠ 0
hq2 : q = ↑n / ↑m
hm2n2 : 0 < m ^ 2 + n ^ 2
hm2n20 : ↑m ^ 2 + ↑n ^ 2 ≠ 0
hx1 : ∀ {j k : ℚ}, k ≠ 0 → k ^ 2 + j ^ 2 ≠ 0 → (1 - (j / k) ^ 2) / (1 + (j / k) ^ 2) = (k ^ 2 - j ^ 2) / (k ^ 2 + j ^ 2)
hw2 : w = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
hx2 : ∀ {j k : ℚ}, k ≠ 0 → k ^ 2 + j ^ 2 ≠ 0 → 2 * (j / k) / (1 + (j / k) ^ 2) = 2 * k * j / (k ^ 2 + j ^ 2)
hv2 : v = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hnmcp : n.gcd m = 1
hmncp : m.gcd n = 1
hm2 : m % 2 = 0
hn2 : n % 2 = 0
h1 : 2 ∣ ↑(n.gcd m)
⊢ False
|
e128c623fada3541
|
MeasureTheory.Measure.hausdorffMeasure_zero_or_top
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem hausdorffMeasure_zero_or_top {d₁ d₂ : ℝ} (h : d₁ < d₂) (s : Set X) :
μH[d₂] s = 0 ∨ μH[d₁] s = ∞
|
case h.intro.intro.inl
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ⊤
c : ℝ≥0
hc : c ≠ 0
this : 0 < ↑c ^ (d₂ - d₁)⁻¹
hr₀ : 0 ≤ ↑0
hrc : ↑0 < ↑c ^ (d₂ - d₁)⁻¹
⊢ ↑0 ^ d₂ / ↑0 ^ d₁ ≤ ↑c
|
rcases lt_or_le 0 d₂ with (h₂ | h₂)
|
case h.intro.intro.inl.inl
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ⊤
c : ℝ≥0
hc : c ≠ 0
this : 0 < ↑c ^ (d₂ - d₁)⁻¹
hr₀ : 0 ≤ ↑0
hrc : ↑0 < ↑c ^ (d₂ - d₁)⁻¹
h₂ : 0 < d₂
⊢ ↑0 ^ d₂ / ↑0 ^ d₁ ≤ ↑c
case h.intro.intro.inl.inr
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ⊤
c : ℝ≥0
hc : c ≠ 0
this : 0 < ↑c ^ (d₂ - d₁)⁻¹
hr₀ : 0 ≤ ↑0
hrc : ↑0 < ↑c ^ (d₂ - d₁)⁻¹
h₂ : d₂ ≤ 0
⊢ ↑0 ^ d₂ / ↑0 ^ d₁ ≤ ↑c
|
b6aef1c477f972ca
|
IsLocalHomeomorphOn.mk
|
Mathlib/Topology/IsLocalHomeomorph.lean
|
theorem mk (h : ∀ x ∈ s, ∃ e : PartialHomeomorph X Y, x ∈ e.source ∧ Set.EqOn f e e.source) :
IsLocalHomeomorphOn f s
|
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
h : ∀ x ∈ s, ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source
x : X
hx✝¹ : x ∈ s
e : PartialHomeomorph X Y
hx✝ : x ∈ e.source
he : Set.EqOn f (↑e) e.source
_x : X
hx : _x ∈ e.source
⊢ e.invFun (f _x) = _x
|
rw [he hx]
|
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
h : ∀ x ∈ s, ∃ e, x ∈ e.source ∧ Set.EqOn f (↑e) e.source
x : X
hx✝¹ : x ∈ s
e : PartialHomeomorph X Y
hx✝ : x ∈ e.source
he : Set.EqOn f (↑e) e.source
_x : X
hx : _x ∈ e.source
⊢ e.invFun (↑e _x) = _x
|
56b0f859baebd663
|
Nat.bitwise_of_ne_zero
|
Mathlib/Data/Nat/Bitwise.lean
|
lemma bitwise_of_ne_zero {n m : Nat} (hn : n ≠ 0) (hm : m ≠ 0) :
bitwise f n m = bit (f (bodd n) (bodd m)) (bitwise f (n / 2) (m / 2))
|
f : Bool → Bool → Bool
n m : ℕ
hn : n ≠ 0
hm : m ≠ 0
mod_two_iff_bod : ∀ (x : ℕ), decide (x % 2 = 1) = x.bodd
⊢ (if n = 0 then if f false true = true then m else 0
else
if m = 0 then if f true false = true then n else 0
else
let n' := n / 2;
let m' := m / 2;
let b₁ := n % 2 = 1;
let b₂ := m % 2 = 1;
let r := bitwise f n' m';
if f (decide b₁) (decide b₂) = true then r + r + 1 else r + r) =
bit (f n.bodd m.bodd) (bitwise f (n / 2) (m / 2))
|
simp only [hn, hm, mod_two_iff_bod, ite_false, bit, two_mul, Bool.cond_eq_ite]
|
f : Bool → Bool → Bool
n m : ℕ
hn : n ≠ 0
hm : m ≠ 0
mod_two_iff_bod : ∀ (x : ℕ), decide (x % 2 = 1) = x.bodd
⊢ (if f n.bodd m.bodd = true then bitwise f (n / 2) (m / 2) + bitwise f (n / 2) (m / 2) + 1
else bitwise f (n / 2) (m / 2) + bitwise f (n / 2) (m / 2)) =
(if f n.bodd m.bodd = true then fun x => x + x + 1 else fun x => x + x) (bitwise f (n / 2) (m / 2))
|
a3308a86ed421729
|
Complex.mul_cpow_ofReal_nonneg
|
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r
|
case inl
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
⊢ (↑a * ↑b) ^ 0 = ↑a ^ 0 * ↑b ^ 0
|
simp only [cpow_zero, mul_one]
|
no goals
|
385cc79a0a30c890
|
mellin_comp_mul_left
|
Mathlib/Analysis/MellinTransform.lean
|
theorem mellin_comp_mul_left (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
mellin (fun t => f (a * t)) s = (a : ℂ) ^ (-s) • mellin f s
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
t : ℝ
ht : t ∈ Ioi 0
⊢ 1 - s = -(s - 1)
|
ring
|
no goals
|
eba6520b29ea7f1f
|
MeasureTheory.setIntegral_abs_condExp_le
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
|
theorem setIntegral_abs_condExp_le {s : Set α} (hs : MeasurableSet[m] s) (f : α → ℝ) :
∫ x in s, |(μ[f|m]) x| ∂μ ≤ ∫ x in s, |f x| ∂μ
|
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
⊢ ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : ¬m ≤ m0
⊢ ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
|
swap
|
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : ¬m ≤ m0
⊢ ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
case pos
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
⊢ ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
|
f75c894a42ad0f54
|
Cardinal.ciSup_mul
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem ciSup_mul (c : Cardinal.{v}) : (⨆ i, f i) * c = ⨆ i, f i * c
|
ι : Type u
f : ι → Cardinal.{v}
c : Cardinal.{v}
⊢ (⨆ i, f i) * c = ⨆ i, f i * c
|
cases isEmpty_or_nonempty ι
|
case inl
ι : Type u
f : ι → Cardinal.{v}
c : Cardinal.{v}
h✝ : IsEmpty ι
⊢ (⨆ i, f i) * c = ⨆ i, f i * c
case inr
ι : Type u
f : ι → Cardinal.{v}
c : Cardinal.{v}
h✝ : Nonempty ι
⊢ (⨆ i, f i) * c = ⨆ i, f i * c
|
079631a7d791f82b
|
AlgebraicGeometry.Scheme.range_fromSpecStalk
|
Mathlib/AlgebraicGeometry/Stalk.lean
|
@[stacks 01J7]
lemma range_fromSpecStalk {x : X} :
Set.range (X.fromSpecStalk x).base = { y | y ⤳ x }
|
X : Scheme
x : ↑↑X.toPresheafedSpace
⊢ Set.range ⇑(ConcreteCategory.hom (X.fromSpecStalk x).base) = {y | y ⤳ x}
|
ext y
|
case h
X : Scheme
x y : ↑↑X.toPresheafedSpace
⊢ y ∈ Set.range ⇑(ConcreteCategory.hom (X.fromSpecStalk x).base) ↔ y ∈ {y | y ⤳ x}
|
2c1b30ac23bb0ca4
|
exists_jacobiSum_eq_neg_one_add
|
Mathlib/NumberTheory/JacobiSum/Basic.lean
|
/-- If `χ` and `ψ` are multiplicative characters of order dividing `n` on a finite field `F`
with values in an integral domain `R` and `μ` is a primitive `n`th root of unity in `R`,
then `J(χ,ψ) = -1 + z*(μ - 1)^2` for some `z ∈ ℤ[μ] ⊆ R`. (We assume that `#F ≡ 1 mod n`.)
Note that we do not state this as a divisibility in `R`, as this would give a weaker statement. -/
lemma exists_jacobiSum_eq_neg_one_add {n : ℕ} (hn : 2 < n) {χ ψ : MulChar F R}
{μ : R} (hχ : χ ^ n = 1) (hψ : ψ ^ n = 1) (hn' : n ∣ Fintype.card F - 1)
(hμ : IsPrimitiveRoot μ n) :
∃ z ∈ Algebra.adjoin ℤ {μ}, jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2
|
case neg
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
H : ∀ (x : F), ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ},
0 + 0 - ↑(n * q + 1) + ∑ x ∈ univ \ {0, 1}, (χ x - 1) * (ψ (1 - x) - 1) = -1 + z * (μ - 1) ^ 2
|
have Hcs x := (H x).choose_spec
|
case neg
F : Type u_1
R : Type u_2
inst✝³ : Fintype F
inst✝² : Field F
inst✝¹ : CommRing R
inst✝ : IsDomain R
n : ℕ
hn : 2 < n
χ ψ : MulChar F R
μ : R
hχ : χ ^ n = 1
hψ : ψ ^ n = 1
hμ : IsPrimitiveRoot μ n
q : ℕ
hq : Fintype.card F = n * q + 1
z₁ : R
hz₁ : z₁ ∈ Algebra.adjoin ℤ {μ}
Hz₁ : ↑n = z₁ * (μ - 1) ^ 2
hχ₀ : ¬χ = 1
hψ₀ : ¬ψ = 1
this : NeZero n
H : ∀ (x : F), ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2
Hcs : ∀ (x : F), ⋯.choose ∈ Algebra.adjoin ℤ {μ} ∧ (χ x - 1) * (ψ (1 - x) - 1) = ⋯.choose * (μ - 1) ^ 2
⊢ ∃ z ∈ Algebra.adjoin ℤ {μ},
0 + 0 - ↑(n * q + 1) + ∑ x ∈ univ \ {0, 1}, (χ x - 1) * (ψ (1 - x) - 1) = -1 + z * (μ - 1) ^ 2
|
ae2162c182ec889a
|
MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq_inner
|
Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
|
theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner {u : E → F'}
(hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
{p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
(hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
eLpNorm u p' μ ≤ eLpNormLESNormFDerivOfEqInnerConst μ p * eLpNorm (fderiv ℝ u) p μ
|
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).IsConjExponent n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).IsConjExponent q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h0p' : p' ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
⊢ 1 < ↑γ
|
rwa [h0γ, one_lt_div hnp, mul_sub, mul_one, sub_lt_sub_iff_right, lt_mul_iff_one_lt_left]
|
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).IsConjExponent n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).IsConjExponent q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h0p' : p' ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := ⟨↑p * (↑n - 1) / (↑n - ↑p), ⋯⟩
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
⊢ 0 < ↑n
|
728bbe5fb5258103
|
Equiv.Perm.fin_5_not_solvable
|
Mathlib/GroupTheory/Solvable.lean
|
theorem Equiv.Perm.fin_5_not_solvable : ¬IsSolvable (Equiv.Perm (Fin 5))
|
x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }
y : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }
z : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_inv := ⋯, right_inv := ⋯ }
⊢ x = z * ⁅x, y * x * y⁻¹⁆ * z⁻¹
|
unfold x y z
|
x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }
y : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }
z : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_inv := ⋯, right_inv := ⋯ }
⊢ { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ } =
{ toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_inv := ⋯, right_inv := ⋯ } *
⁅{ toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ },
{ toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ } *
{ toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ } *
{ toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }⁻¹⁆ *
{ toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_inv := ⋯, right_inv := ⋯ }⁻¹
|
2cf02f2116026673
|
CategoryTheory.IsPushout.inl_snd'
|
Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
|
theorem inl_snd' {b : BinaryBicone X Y} (h : b.IsBilimit) :
IsPushout b.inl (0 : X ⟶ 0) b.snd (0 : 0 ⟶ Y)
|
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y : C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
b : BinaryBicone X Y
h : b.IsBilimit
⊢ IsPushout (0 ≫ 0) 0 0 (b.inr ≫ b.snd)
|
simp
|
no goals
|
1ea37e8a4cd6e9ec
|
Std.Tactic.BVDecide.Reflect.unsat_of_verifyBVExpr_eq_true
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Reflect.lean
|
theorem unsat_of_verifyBVExpr_eq_true (bv : BVLogicalExpr) (c : String)
(h : verifyBVExpr bv c = true) :
bv.Unsat
|
case a.a
bv : BVLogicalExpr
c : String
h✝ : verifyBVExpr bv c = true
h : verifyCert (AIG.toCNF bv.bitblast.relabelNat) c = true
⊢ verifyCert (AIG.toCNF bv.bitblast.relabelNat) ?a.cert✝ = true
case a.cert
bv : BVLogicalExpr
c : String
h : verifyBVExpr bv c = true
⊢ String
|
assumption
|
no goals
|
b70561da56bd753e
|
singleton_span_mem_normalizedFactors_of_mem_normalizedFactors
|
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
{a b : R} (ha : a ∈ normalizedFactors b) :
Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R))
|
case intro.intro
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
inst✝ : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (span {a})
c : Ideal R
hc : c ∈ normalizedFactors (span {b})
hc' : Associated (span {a}) c
⊢ span {a} ∈ normalizedFactors (span {b})
|
rwa [associated_iff_eq.mp hc']
|
no goals
|
c68751871b882c67
|
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
hz : ↑(factors' a ⊓ factors' b) ≠ 0
p : α
p0_irr : Irreducible (Associates.mk p)
p0_mem : ⟨Associates.mk p, p0_irr⟩ ∈ factors' a ∩ factors' b
⊢ b ≠ 0
|
apply hb
|
no goals
|
4b4dfcc04fa14351
|
Real.summable_log_one_add_of_summable
|
Mathlib/Analysis/SpecialFunctions/Log/Summable.lean
|
lemma Real.summable_log_one_add_of_summable {f : ι → ℝ} (hf : Summable f) :
Summable (fun i : ι => log (1 + |f i|))
|
ι : Type u_1
f : ι → ℝ
hf : Summable f
this : Summable fun n => ofRealCLM (log (1 + |f n|))
⊢ Summable fun i => log (1 + |f i|)
|
convert Complex.reCLM.summable this
|
no goals
|
4db2ca3d7bda0ab0
|
unitary.star_mem
|
Mathlib/Algebra/Star/Unitary.lean
|
theorem star_mem {U : R} (hU : U ∈ unitary R) : star U ∈ unitary R :=
⟨by rw [star_star, mul_star_self_of_mem hU], by rw [star_star, star_mul_self_of_mem hU]⟩
|
R : Type u_1
inst✝¹ : Monoid R
inst✝ : StarMul R
U : R
hU : U ∈ unitary R
⊢ star (star U) * star U = 1
|
rw [star_star, mul_star_self_of_mem hU]
|
no goals
|
0304e8c706770550
|
BitVec.toFin_not
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem toFin_not (x : BitVec w) :
(~~~x).toFin = x.toFin.rev
|
w : Nat
x : BitVec w
⊢ ↑(~~~x).toFin = ↑x.toFin.rev
|
simp only [val_toFin, toNat_not, Fin.val_rev]
|
w : Nat
x : BitVec w
⊢ 2 ^ w - 1 - x.toNat = 2 ^ w - (x.toNat + 1)
|
0ef361a5f5be1853
|
LaurentSeries.powerSeries_ext_subring
|
Mathlib/RingTheory/LaurentSeries.lean
|
theorem powerSeries_ext_subring :
Subring.map (LaurentSeriesRingEquiv K).toRingHom (powerSeries_as_subring K) =
((idealX K).adicCompletionIntegers (RatFunc K)).toSubring
|
K : Type u_2
inst✝ : Field K
⊢ Subring.map (LaurentSeriesRingEquiv K).toRingHom (powerSeries_as_subring K) =
(adicCompletionIntegers (RatFunc K) (idealX K)).toSubring
|
ext x
|
case h
K : Type u_2
inst✝ : Field K
x : RatFuncAdicCompl K
⊢ x ∈ Subring.map (LaurentSeriesRingEquiv K).toRingHom (powerSeries_as_subring K) ↔
x ∈ (adicCompletionIntegers (RatFunc K) (idealX K)).toSubring
|
db804111fcd80f5d
|
EquicontinuousAt.tendsto_of_mem_closure
|
Mathlib/Topology/UniformSpace/Equicontinuity.lean
|
theorem EquicontinuousAt.tendsto_of_mem_closure {l : Filter ι} {F : ι → X → α} {f : X → α}
{s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Tendsto f (𝓝[s] x) (𝓝 z))
(hs : ∀ y ∈ s, Tendsto (F · y) l (𝓝 (f y))) (hx : x ∈ closure s) :
Tendsto (F · x) l (𝓝 z)
|
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
F : ι → X → α
f : X → α
s : Set X
x : X
z : α
hF : EquicontinuousAt F x
hf : ∀ i ∈ 𝓤 α, ∀ᶠ (x : X) in 𝓝[s] x, f x ∈ {y | (y, z) ∈ id i}
hs : ∀ y ∈ s, Tendsto (fun x => F x y) l (𝓝 (f y))
hx : x ∈ closure s
⊢ ∀ i ∈ 𝓤 α, ∀ᶠ (x_1 : ι) in l, F x_1 x ∈ {y | (y, z) ∈ id i}
|
intro U hU
|
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
F : ι → X → α
f : X → α
s : Set X
x : X
z : α
hF : EquicontinuousAt F x
hf : ∀ i ∈ 𝓤 α, ∀ᶠ (x : X) in 𝓝[s] x, f x ∈ {y | (y, z) ∈ id i}
hs : ∀ y ∈ s, Tendsto (fun x => F x y) l (𝓝 (f y))
hx : x ∈ closure s
U : Set (α × α)
hU : U ∈ 𝓤 α
⊢ ∀ᶠ (x_1 : ι) in l, F x_1 x ∈ {y | (y, z) ∈ id U}
|
b75bfbc553aec3e3
|
MeasureTheory.integral_integral_swap_of_hasCompactSupport
|
Mathlib/MeasureTheory/Integral/Prod.lean
|
/-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
lemma integral_integral_swap_of_hasCompactSupport
{f : X → Y → E} (hf : Continuous f.uncurry) (h'f : HasCompactSupport f.uncurry)
{μ : Measure X} {ν : Measure Y} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν] :
∫ x, (∫ y, f x y ∂ν) ∂μ = ∫ y, (∫ x, f x y ∂μ) ∂ν
|
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this : Fact (ν V < ⊤)
y : Y
x : X
hy : f x y ≠ 0
⊢ y ∈ V
|
have : (x, y) ∈ Function.support f.uncurry := hy
|
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
X : Type u_5
Y : Type u_6
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace Y
inst✝⁵ : MeasurableSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : OpensMeasurableSpace X
inst✝² : OpensMeasurableSpace Y
f : X → Y → E
hf : Continuous (uncurry f)
h'f : HasCompactSupport (uncurry f)
μ : Measure X
ν : Measure Y
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : IsFiniteMeasureOnCompacts ν
U : Set X := Prod.fst '' tsupport (uncurry f)
this✝¹ : Fact (μ U < ⊤)
V : Set Y := Prod.snd '' tsupport (uncurry f)
this✝ : Fact (ν V < ⊤)
y : Y
x : X
hy : f x y ≠ 0
this : (x, y) ∈ support (uncurry f)
⊢ y ∈ V
|
ab5aab42565633fa
|
IsLocalizedModule.subsingleton_iff_ker_eq_top
|
Mathlib/Algebra/Module/LocalizedModule/Basic.lean
|
lemma subsingleton_iff_ker_eq_top (S : Submonoid R) (g : M →ₗ[R] M')
[IsLocalizedModule S g] :
Subsingleton M' ↔ LinearMap.ker g = ⊤
|
R : Type u_1
inst✝⁵ : CommSemiring R
M : Type u_2
M' : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M'
inst✝² : Module R M
inst✝¹ : Module R M'
S : Submonoid R
g : M →ₗ[R] M'
inst✝ : IsLocalizedModule S g
H : ⊤ ≤ LinearMap.ker g
x : M'
⊢ x = 0
|
obtain ⟨⟨x, s⟩, rfl⟩ := IsLocalizedModule.mk'_surjective S g x
|
case intro.mk
R : Type u_1
inst✝⁵ : CommSemiring R
M : Type u_2
M' : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M'
inst✝² : Module R M
inst✝¹ : Module R M'
S : Submonoid R
g : M →ₗ[R] M'
inst✝ : IsLocalizedModule S g
H : ⊤ ≤ LinearMap.ker g
x : M
s : ↥S
⊢ Function.uncurry (mk' g) (x, s) = 0
|
f664477d3de20596
|
LinearIndependent.not_mem_span_image
|
Mathlib/LinearAlgebra/LinearIndependent/Basic.lean
|
theorem LinearIndependent.not_mem_span_image [Nontrivial R] (hv : LinearIndependent R v) {s : Set ι}
{x : ι} (h : x ∉ s) : v x ∉ Submodule.span R (v '' s)
|
ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
hv : LinearIndependent R v
s : Set ι
x : ι
h : x ∉ s
⊢ v x ∈ span R {v x}
|
exact mem_span_singleton_self (v x)
|
no goals
|
340d08fc4e0e9f77
|
Zsqrtd.nonnegg_neg_pos
|
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
theorem nonnegg_neg_pos {c d} : ∀ {a b : ℕ}, Nonnegg c d (-a) b ↔ SqLe a d b c
| 0, b => ⟨by simp [SqLe, Nat.zero_le], fun _ => trivial⟩
| a + 1, b => by rw [← Int.negSucc_coe]; rfl
|
c d a b : ℕ
⊢ Nonnegg c d (-↑(a + 1)) ↑b ↔ SqLe (a + 1) d b c
|
rw [← Int.negSucc_coe]
|
c d a b : ℕ
⊢ Nonnegg c d (Int.negSucc a) ↑b ↔ SqLe (a + 1) d b c
|
3c378fb918c4dd58
|
MeasureTheory.Lp.cauchy_tendsto_of_tendsto
|
Mathlib/MeasureTheory/Function/LpSpace/Basic.lean
|
theorem cauchy_tendsto_of_tendsto {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
(f_lim : α → E) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N)
(h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
atTop.Tendsto (fun n => eLpNorm (f n - f_lim) p μ) (𝓝 0)
|
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))
⊢ ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - f_lim) p μ ≤ ε
|
intro ε hε
|
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))
ε : ℝ≥0∞
hε : ε > 0
⊢ ∃ N, ∀ n ≥ N, eLpNorm (f n - f_lim) p μ ≤ ε
|
4d90eb848aa97c8b
|
HasDerivAt.lhopital_zero_atTop_on_Ioi
|
Mathlib/Analysis/Calculus/LHopital.lean
|
theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioi a, g' x ≠ 0)
(hftop : Tendsto f atTop (𝓝 0)) (hgtop : Tendsto g atTop (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l
|
case intro.intro
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
⊢ Tendsto (fun x => f x / g x) atTop l
|
have fact2 (x) (hx : x ∈ Ioo 0 a'⁻¹) : a < x⁻¹ := lt_trans haa' ((lt_inv_comm₀ ha' hx.1).mpr hx.2)
|
case intro.intro
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x
hgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x
hg' : ∀ x ∈ Ioi a, g' x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) atTop l
a' : ℝ
haa' : a < a'
ha' : 0 < a'
fact1 : ∀ x ∈ Ioo 0 a'⁻¹, x ≠ 0
fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹
⊢ Tendsto (fun x => f x / g x) atTop l
|
fe483e021246704d
|
Real.lipschitzWith_toNNReal
|
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
lemma _root_.Real.lipschitzWith_toNNReal : LipschitzWith 1 Real.toNNReal
|
⊢ LipschitzWith 1 Real.toNNReal
|
refine lipschitzWith_iff_dist_le_mul.mpr (fun x y ↦ ?_)
|
x y : ℝ
⊢ dist x.toNNReal y.toNNReal ≤ ↑1 * dist x y
|
b68dc70022d0a9e5
|
Finset.dens_lt_dens
|
Mathlib/Data/Finset/Density.lean
|
lemma dens_lt_dens (h : s ⊂ t) : dens s < dens t :=
div_lt_div_of_pos_right (mod_cast card_strictMono h) <| by
cases isEmpty_or_nonempty α
· simp [Subsingleton.elim s t, ssubset_irrfl] at h
· exact mod_cast Fintype.card_pos
|
case inl
α : Type u_2
inst✝ : Fintype α
s t : Finset α
h : s ⊂ t
h✝ : IsEmpty α
⊢ 0 < ↑(Fintype.card α)
|
simp [Subsingleton.elim s t, ssubset_irrfl] at h
|
no goals
|
0a359651e498d6b7
|
HurwitzZeta.hurwitzZeta_neg_nat
|
Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean
|
theorem hurwitzZeta_neg_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
hurwitzZeta x (-k) =
-1 / (k + 1) * ((Polynomial.bernoulli (k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ)
|
case intro.inr
x : ℝ
hx : x ∈ Icc 0 1
n : ℕ
hk : 2 * n + 1 ≠ 0
⊢ hurwitzZeta (↑x) (-↑(2 * n + 1)) =
-1 / (↑(2 * n + 1) + 1) *
Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * n + 1 + 1)))
|
exact_mod_cast hurwitzZeta_one_sub_two_mul_nat (by omega : n + 1 ≠ 0) hx
|
no goals
|
fe75c9bfe836a847
|
MvQPF.wEquiv.abs'
|
Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
|
theorem wEquiv.abs' {α : TypeVec n} (x y : q.P.W α)
(h : MvQPF.abs (q.P.wDest' x) = MvQPF.abs (q.P.wDest' y)) :
WEquiv x y
|
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x y : (P F).W α
a₀ : (P F).A
f'₀ : (P F).drop.B a₀ ⟹ α
f₀ : (P F).last.B a₀ → (P F).W α
⊢ ∀ (a : (P F).A) (f' : (P F).drop.B a ⟹ α) (f : (P F).last.B a → (P F).W α),
abs ((P F).wDest' ((P F).wMk a₀ f'₀ f₀)) = abs ((P F).wDest' ((P F).wMk a f' f)) →
WEquiv ((P F).wMk a₀ f'₀ f₀) ((P F).wMk a f' f)
|
intro a₁ f'₁ f₁
|
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x y : (P F).W α
a₀ : (P F).A
f'₀ : (P F).drop.B a₀ ⟹ α
f₀ : (P F).last.B a₀ → (P F).W α
a₁ : (P F).A
f'₁ : (P F).drop.B a₁ ⟹ α
f₁ : (P F).last.B a₁ → (P F).W α
⊢ abs ((P F).wDest' ((P F).wMk a₀ f'₀ f₀)) = abs ((P F).wDest' ((P F).wMk a₁ f'₁ f₁)) →
WEquiv ((P F).wMk a₀ f'₀ f₀) ((P F).wMk a₁ f'₁ f₁)
|
83e1f4ec524846bd
|
SimpleGraph.Walk.support_tail_of_not_nil
|
Mathlib/Combinatorics/SimpleGraph/Walk.lean
|
lemma support_tail_of_not_nil (p : G.Walk u v) (hnp : ¬p.Nil) :
p.tail.support = p.support.tail
|
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hnp : ¬nil.Nil
⊢ nil.tail.support = nil.support.tail
|
simp only [nil_nil, not_true_eq_false] at hnp
|
no goals
|
048835719ff55772
|
PiNat.exists_lipschitz_retraction_of_isClosed
|
Mathlib/Topology/MetricSpace/PiNat.lean
|
theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
(hne : s.Nonempty) :
∃ f : (∀ n, E n) → ∀ n, E n, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f
|
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
⊢ ∃ f, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f
|
have fs : ∀ x ∈ s, f x = x := fun x xs => by simp [f, xs]
|
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsClosed s
hne : s.Nonempty
f : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x => if x ∈ s then x else ⋯.some
fs : ∀ x ∈ s, f x = x
⊢ ∃ f, (∀ x ∈ s, f x = x) ∧ range f = s ∧ LipschitzWith 1 f
|
bd92edd65dec8c2a
|
Ordinal.iSup_iterate_eq_nfp
|
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
theorem iSup_iterate_eq_nfp (f : Ordinal.{u} → Ordinal.{u}) (a : Ordinal.{u}) :
⨆ n : ℕ, f^[n] a = nfp f a
|
case a.a
f : Ordinal.{u} → Ordinal.{u}
a : Ordinal.{u}
l : List Unit
⊢ f^[l.length] a ≤ ⨆ n, f^[n] a
|
exact Ordinal.le_iSup _ _
|
no goals
|
c976a13d93c931d3
|
Ideal.iUnion_minimalPrimes
|
Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean
|
theorem Ideal.iUnion_minimalPrimes :
⋃ p ∈ I.minimalPrimes, p = { x | ∃ y ∉ I.radical, x * y ∈ I.radical }
|
case h
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
⊢ (∃ i ∈ I.minimalPrimes, x ∈ i) ↔ ∃ y ∉ I.radical, x * y ∈ I.radical
|
constructor
|
case h.mp
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
⊢ (∃ i ∈ I.minimalPrimes, x ∈ i) → ∃ y ∉ I.radical, x * y ∈ I.radical
case h.mpr
R : Type u_1
inst✝ : CommSemiring R
I : Ideal R
x : R
⊢ (∃ y ∉ I.radical, x * y ∈ I.radical) → ∃ i ∈ I.minimalPrimes, x ∈ i
|
c2bb79363c5fb210
|
List.idxOf_lt_length
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem idxOf_lt_length [BEq α] [LawfulBEq α] {l : List α} (h : a ∈ l) : l.idxOf a < l.length
|
case cons.inr
α : Type u_1
a : α
inst✝¹ : BEq α
inst✝ : LawfulBEq α
x : α
xs : List α
ih : a ∈ xs → idxOf a xs < xs.length
h : a ∈ xs
⊢ idxOf a (x :: xs) < (x :: xs).length
|
simp only [idxOf_cons, cond_eq_if, beq_iff_eq, length_cons]
|
case cons.inr
α : Type u_1
a : α
inst✝¹ : BEq α
inst✝ : LawfulBEq α
x : α
xs : List α
ih : a ∈ xs → idxOf a xs < xs.length
h : a ∈ xs
⊢ (if (x == a) = true then 0 else idxOf a xs + 1) < xs.length + 1
|
c1e63f121277a338
|
Turing.mem_eval
|
Mathlib/Computability/PostTuringMachine.lean
|
theorem mem_eval {σ} {f : σ → Option σ} {a b} : b ∈ eval f a ↔ Reaches f a b ∧ f b = none
|
case refine_2.refine_1
σ : Type u_1
f : σ → Option σ
a b : σ
x✝ : Reaches f a b ∧ f b = none
h₁ : Reaches f a b
h₂ : f b = none
⊢ Sum.inl b ∈ Part.some (none.elim (Sum.inl b) Sum.inr)
|
apply Part.mem_some
|
no goals
|
f40442f1b60c5272
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.nodup_insertRatUnits
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddResult.lean
|
theorem nodup_insertRatUnits {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ f.assignments.size = n) (units : CNF.Clause (PosFin n)) :
∀ i : Fin (f.insertRatUnits units).1.ratUnits.size, ∀ j : Fin (f.insertRatUnits units).1.ratUnits.size,
i ≠ j → (f.insertRatUnits units).1.ratUnits[i] ≠ (f.insertRatUnits units).1.ratUnits[j]
|
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
units : CNF.Clause (PosFin n)
i j : Fin (f.insertRatUnits units).fst.ratUnits.size
i_ne_j : i ≠ j
⊢ (f.insertRatUnits units).fst.ratUnits[i] ≠ (f.insertRatUnits units).fst.ratUnits[j]
|
rcases hi : (insertRatUnits f units).fst.ratUnits[i] with ⟨li, bi⟩
|
case mk
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.assignments.size = n
units : CNF.Clause (PosFin n)
i j : Fin (f.insertRatUnits units).fst.ratUnits.size
i_ne_j : i ≠ j
li : PosFin n
bi : Bool
hi : (f.insertRatUnits units).fst.ratUnits[i] = (li, bi)
⊢ (li, bi) ≠ (f.insertRatUnits units).fst.ratUnits[j]
|
17810904ccd7a629
|
UniformSpace.Completion.coe_inv
|
Mathlib/Topology/Algebra/UniformField.lean
|
theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K)
|
case pos
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : x = 0
⊢ (if ↑0 = 0 then 0 else (↑0).hatInv) = ↑0
|
norm_cast
|
case pos
K : Type u_1
inst✝³ : Field K
inst✝² : UniformSpace K
inst✝¹ : TopologicalDivisionRing K
inst✝ : CompletableTopField K
x : K
h : x = 0
⊢ (if 0 = 0 then 0 else hatInv 0) = 0
|
5b9fafd0a1cb698a
|
Representation.asModuleEquiv_symm_map_smul
|
Mathlib/RepresentationTheory/Basic.lean
|
theorem asModuleEquiv_symm_map_smul (r : k) (x : V) :
ρ.asModuleEquiv.symm (r • x) = algebraMap k (MonoidAlgebra k G) r • ρ.asModuleEquiv.symm x
|
k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommSemiring k
inst✝² : Monoid G
inst✝¹ : AddCommMonoid V
inst✝ : Module k V
ρ : Representation k G V
r : k
x : V
⊢ ρ.asModuleEquiv.symm (r • x) = (algebraMap k (MonoidAlgebra k G)) r • ρ.asModuleEquiv.symm x
|
apply_fun ρ.asModuleEquiv
|
k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommSemiring k
inst✝² : Monoid G
inst✝¹ : AddCommMonoid V
inst✝ : Module k V
ρ : Representation k G V
r : k
x : V
⊢ ρ.asModuleEquiv (ρ.asModuleEquiv.symm (r • x)) =
ρ.asModuleEquiv ((algebraMap k (MonoidAlgebra k G)) r • ρ.asModuleEquiv.symm x)
|
a6bfa9f1544d58ca
|
LightCondensed.hom_ext
|
Mathlib/Condensed/Light/Basic.lean
|
@[ext]
lemma hom_ext {X Y : LightCondensed.{u} C} (f g : X ⟶ Y) (h : ∀ S, f.val.app S = g.val.app S) :
f = g
|
case h
C : Type w
inst✝ : Category.{v, w} C
X Y : LightCondensed C
f g : X ⟶ Y
h : ∀ (S : LightProfiniteᵒᵖ), f.val.app S = g.val.app S
⊢ f.val = g.val
|
ext
|
case h.w.h
C : Type w
inst✝ : Category.{v, w} C
X Y : LightCondensed C
f g : X ⟶ Y
h : ∀ (S : LightProfiniteᵒᵖ), f.val.app S = g.val.app S
x✝ : LightProfiniteᵒᵖ
⊢ f.val.app x✝ = g.val.app x✝
|
e7235b0140c3a65c
|
bdd_le_mul_tendsto_zero
|
Mathlib/Topology/Algebra/Order/Field.lean
|
theorem bdd_le_mul_tendsto_zero {f g : α → 𝕜} {b B : 𝕜} (hb : ∀ᶠ x in l, b ≤ f x)
(hB : ∀ᶠ x in l, f x ≤ B) (hg : Tendsto g l (𝓝 0)) :
Tendsto (fun x ↦ f x * g x) l (𝓝 0)
|
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
b B : 𝕜
hb : ∀ᶠ (x : α) in l, b ≤ f x
hB : ∀ᶠ (x : α) in l, f x ≤ B
hg : Tendsto g l (𝓝 0)
C : 𝕜 := |b| ⊔ |B|
hbC : -C ≤ b
⊢ Tendsto (fun x => f x * g x) l (𝓝 0)
|
have hBC : B ≤ C := le_max_of_le_right (le_abs_self B)
|
𝕜 : Type u_1
α : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
l : Filter α
f g : α → 𝕜
b B : 𝕜
hb : ∀ᶠ (x : α) in l, b ≤ f x
hB : ∀ᶠ (x : α) in l, f x ≤ B
hg : Tendsto g l (𝓝 0)
C : 𝕜 := |b| ⊔ |B|
hbC : -C ≤ b
hBC : B ≤ C
⊢ Tendsto (fun x => f x * g x) l (𝓝 0)
|
bf00d207f0c8be3d
|
Subalgebra.induction_on_adjoin
|
Mathlib/RingTheory/Adjoin/FG.lean
|
theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥)
(ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x S)))
(S : Subalgebra R A) : P S
|
case intro.refine_2
R : Type u
A : Type v
inst✝³ : CommSemiring R
inst✝² : Semiring A
inst✝¹ : Algebra R A
inst✝ : IsNoetherian R A
P : Subalgebra R A → Prop
base : P ⊥
ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x ↑S))
t✝ : Finset A
x : A
t : Finset A
a✝ : x ∉ t
h : P (Algebra.adjoin R ↑t)
⊢ P (Algebra.adjoin R (insert x ↑t))
|
simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h
|
no goals
|
4474d702e34f7b7b
|
nndist_div_left
|
Mathlib/Topology/MetricSpace/IsometricSMul.lean
|
theorem nndist_div_left [Group G] [PseudoMetricSpace G] [IsometricSMul G G]
[IsometricSMul Gᵐᵒᵖ G] (a b c : G) : nndist (a / b) (a / c) = nndist b c
|
G : Type v
inst✝³ : Group G
inst✝² : PseudoMetricSpace G
inst✝¹ : IsometricSMul G G
inst✝ : IsometricSMul Gᵐᵒᵖ G
a b c : G
⊢ nndist (a / b) (a / c) = nndist b c
|
simp [div_eq_mul_inv]
|
no goals
|
dc1791f46d7fd205
|
MeasureTheory.mul_le_addHaar_image_of_lt_det
|
Mathlib/MeasureTheory/Function/Jacobian.lean
|
theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
(hm : (m : ℝ≥0∞) < ENNReal.ofReal |A.det|) :
∀ᶠ δ in 𝓝[>] (0 : ℝ≥0),
∀ (s : Set E) (f : E → E), ApproximatesLinearOn f A s δ → (m : ℝ≥0∞) * μ s ≤ μ (f '' s)
|
case neg.h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
m : ℝ≥0
hm : ↑m < ENNReal.ofReal |A.det|
mpos : 0 < m
hA : A.det ≠ 0
B : E ≃L[ℝ] E := A.toContinuousLinearEquivOfDetNeZero hA
I : ENNReal.ofReal |(↑B.symm).det| < ↑m⁻¹
δ₀ : ℝ≥0
δ₀pos : 0 < δ₀
hδ₀ : ∀ (t : Set E) (g : E → E), ApproximatesLinearOn g (↑B.symm) t δ₀ → μ (g '' t) ≤ ↑m⁻¹ * μ t
h : ¬Subsingleton E
⊢ 0 < ‖↑B.symm‖₊⁻¹
|
simpa only [h, false_or, inv_pos] using B.subsingleton_or_nnnorm_symm_pos
|
no goals
|
22f8afc7492fca52
|
Behrend.ceil_lt_mul
|
Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
|
theorem ceil_lt_mul {x : ℝ} (hx : 50 / 19 ≤ x) : (⌈x⌉₊ : ℝ) < 1.38 * x
|
x : ℝ
hx : 50 / 19 ≤ x
this : 1.38 = 69 / 50
⊢ 0 < 50 / 19
|
norm_num1
|
no goals
|
8877374224ca2a05
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.unsat_of_encounteredBoth
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
|
theorem unsat_of_encounteredBoth {n : Nat} (c : DefaultClause n)
(assignment : Array Assignment) :
reduce c assignment = encounteredBoth → Unsatisfiable (PosFin n) assignment
|
case h_3
n : Nat
c : DefaultClause n
assignment : Array Assignment
hb : reducedToEmpty = encounteredBoth → Unsatisfiable (PosFin n) assignment
l : Literal (PosFin n)
x✝ : l ∈ c.clause
acc✝ : ReduceResult (PosFin n)
l'✝ : Literal (PosFin n)
ih : reducedToUnit l'✝ = encounteredBoth → Unsatisfiable (PosFin n) assignment
h :
(match assignment[l.fst.val]! with
| pos => if l.snd = true then reducedToNonunit else reducedToUnit l'✝
| neg => if (!l.snd) = true then reducedToNonunit else reducedToUnit l'✝
| both => encounteredBoth
| unassigned => reducedToNonunit) =
encounteredBoth
⊢ Unsatisfiable (PosFin n) assignment
|
split at h
|
case h_3.h_1
n : Nat
c : DefaultClause n
assignment : Array Assignment
hb : reducedToEmpty = encounteredBoth → Unsatisfiable (PosFin n) assignment
l : Literal (PosFin n)
x✝¹ : l ∈ c.clause
acc✝ : ReduceResult (PosFin n)
l'✝ : Literal (PosFin n)
ih : reducedToUnit l'✝ = encounteredBoth → Unsatisfiable (PosFin n) assignment
x✝ : Assignment
heq✝ : assignment[l.fst.val]! = pos
h : (if l.snd = true then reducedToNonunit else reducedToUnit l'✝) = encounteredBoth
⊢ Unsatisfiable (PosFin n) assignment
case h_3.h_2
n : Nat
c : DefaultClause n
assignment : Array Assignment
hb : reducedToEmpty = encounteredBoth → Unsatisfiable (PosFin n) assignment
l : Literal (PosFin n)
x✝¹ : l ∈ c.clause
acc✝ : ReduceResult (PosFin n)
l'✝ : Literal (PosFin n)
ih : reducedToUnit l'✝ = encounteredBoth → Unsatisfiable (PosFin n) assignment
x✝ : Assignment
heq✝ : assignment[l.fst.val]! = neg
h : (if (!l.snd) = true then reducedToNonunit else reducedToUnit l'✝) = encounteredBoth
⊢ Unsatisfiable (PosFin n) assignment
case h_3.h_3
n : Nat
c : DefaultClause n
assignment : Array Assignment
hb : reducedToEmpty = encounteredBoth → Unsatisfiable (PosFin n) assignment
l : Literal (PosFin n)
x✝¹ : l ∈ c.clause
acc✝ : ReduceResult (PosFin n)
l'✝ : Literal (PosFin n)
ih : reducedToUnit l'✝ = encounteredBoth → Unsatisfiable (PosFin n) assignment
x✝ : Assignment
heq✝ : assignment[l.fst.val]! = both
h : encounteredBoth = encounteredBoth
⊢ Unsatisfiable (PosFin n) assignment
case h_3.h_4
n : Nat
c : DefaultClause n
assignment : Array Assignment
hb : reducedToEmpty = encounteredBoth → Unsatisfiable (PosFin n) assignment
l : Literal (PosFin n)
x✝¹ : l ∈ c.clause
acc✝ : ReduceResult (PosFin n)
l'✝ : Literal (PosFin n)
ih : reducedToUnit l'✝ = encounteredBoth → Unsatisfiable (PosFin n) assignment
x✝ : Assignment
heq✝ : assignment[l.fst.val]! = unassigned
h : reducedToNonunit = encounteredBoth
⊢ Unsatisfiable (PosFin n) assignment
|
a1d564233a90b7cf
|
ProbabilityTheory.Kernel.iIndepFun.indepFun_finset
|
Mathlib/Probability/Independence/Kernel.lean
|
theorem iIndepFun.indepFun_finset (S T : Finset ι) (hST : Disjoint S T)
(hf_Indep : iIndepFun m f κ μ) (hf_meas : ∀ i, Measurable (f i)) :
IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) κ μ
|
case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᶠ[ae μ] ⇑η
hη : IsMarkovKernel η
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := univ.pi '' univ.pi fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) MeasurableSpace.pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := univ.pi '' univ.pi fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) MeasurableSpace.pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : univ.pi sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : univ.pi sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
⊢ ∀ᵐ (a : α) ∂μ,
(η a) ((fun a i => f (↑i) a) ⁻¹' univ.pi sets_s ∩ (fun a i => f (↑i) a) ⁻¹' univ.pi sets_t) =
(η a) ((fun a i => f (↑i) a) ⁻¹' univ.pi sets_s) * (η a) ((fun a i => f (↑i) a) ⁻¹' univ.pi sets_t)
|
let sets_s' : ∀ i : ι, Set (β i) := fun i =>
dite (i ∈ S) (fun hi => sets_s ⟨i, hi⟩) fun _ => Set.univ
|
case inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
β : ι → Type u_8
m : (i : ι) → MeasurableSpace (β i)
f : (i : ι) → Ω → β i
S T : Finset ι
hST : Disjoint S T
hf_Indep : iIndepFun m f κ μ
hf_meas : ∀ (i : ι), Measurable (f i)
hμ : μ ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᶠ[ae μ] ⇑η
hη : IsMarkovKernel η
πSβ : Set (Set ((i : { x // x ∈ S }) → β ↑i)) := univ.pi '' univ.pi fun i => {s | MeasurableSet s}
πS : Set (Set Ω) := {s | ∃ t ∈ πSβ, (fun a i => f (↑i) a) ⁻¹' t = s}
hπS_pi : IsPiSystem πS
hπS_gen : MeasurableSpace.comap (fun a i => f (↑i) a) MeasurableSpace.pi = generateFrom πS
πTβ : Set (Set ((i : { x // x ∈ T }) → β ↑i)) := univ.pi '' univ.pi fun i => {s | MeasurableSet s}
πT : Set (Set Ω) := {s | ∃ t ∈ πTβ, (fun a i => f (↑i) a) ⁻¹' t = s}
hπT_pi : IsPiSystem πT
hπT_gen : MeasurableSpace.comap (fun a i => f (↑i) a) MeasurableSpace.pi = generateFrom πT
s : Set ((i : { x // x ∈ S }) → β ↑i)
sets_s : (i : { x // x ∈ S }) → Set (β ↑i)
hs2 : univ.pi sets_s = s
t : Set ((i : { x // x ∈ T }) → β ↑i)
sets_t : (i : { x // x ∈ T }) → Set (β ↑i)
ht2 : univ.pi sets_t = t
hs1 : ∀ (i : { x // x ∈ S }), MeasurableSet (sets_s i)
ht1 : ∀ (i : { x // x ∈ T }), MeasurableSet (sets_t i)
sets_s' : (i : ι) → Set (β i) := fun i => if hi : i ∈ S then sets_s ⟨i, hi⟩ else univ
⊢ ∀ᵐ (a : α) ∂μ,
(η a) ((fun a i => f (↑i) a) ⁻¹' univ.pi sets_s ∩ (fun a i => f (↑i) a) ⁻¹' univ.pi sets_t) =
(η a) ((fun a i => f (↑i) a) ⁻¹' univ.pi sets_s) * (η a) ((fun a i => f (↑i) a) ⁻¹' univ.pi sets_t)
|
0da746b38698ee90
|
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
⊢ g x - g a = ∫ (y : ℝ) in a..id x, g' y
|
symm
|
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) in a..id x, g' y = g x - g a
|
1f10c323da5db2cd
|
Finset.mem_disjUnion
|
Mathlib/Data/Finset/Disjoint.lean
|
theorem mem_disjUnion {α s t h a} : a ∈ @disjUnion α s t h ↔ a ∈ s ∨ a ∈ t
|
α : Type u_4
s t : Finset α
h : Disjoint s t
a : α
⊢ a ∈ s.disjUnion t h ↔ a ∈ s ∨ a ∈ t
|
rcases s with ⟨⟨s⟩⟩
|
case mk.mk
α : Type u_4
t : Finset α
a : α
val✝ : Multiset α
s : List α
nodup✝ : Nodup (Quot.mk (⇑(List.isSetoid α)) s)
h : Disjoint { val := Quot.mk (⇑(List.isSetoid α)) s, nodup := nodup✝ } t
⊢ a ∈ { val := Quot.mk (⇑(List.isSetoid α)) s, nodup := nodup✝ }.disjUnion t h ↔
a ∈ { val := Quot.mk (⇑(List.isSetoid α)) s, nodup := nodup✝ } ∨ a ∈ t
|
fc34f1adb93d2bd5
|
List.Sorted.rel_of_mem_take_of_mem_drop
|
Mathlib/Data/List/Sort.lean
|
theorem Sorted.rel_of_mem_take_of_mem_drop {l : List α} (h : List.Sorted r l) {k : ℕ} {x y : α}
(hx : x ∈ List.take k l) (hy : y ∈ List.drop k l) : r x y
|
α : Type u
r : α → α → Prop
l : List α
h : Sorted r l
k : ℕ
x y : α
hx : x ∈ take k l
hy : y ∈ drop k l
⊢ r x y
|
obtain ⟨iy, hiy, rfl⟩ := getElem_of_mem hy
|
case intro.intro
α : Type u
r : α → α → Prop
l : List α
h : Sorted r l
k : ℕ
x : α
hx : x ∈ take k l
iy : ℕ
hiy : iy < (drop k l).length
hy : (drop k l)[iy] ∈ drop k l
⊢ r x (drop k l)[iy]
|
4811c6bb66aed68d
|
AkraBazziRecurrence.eventually_atTop_sumTransform_ge
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma eventually_atTop_sumTransform_ge :
∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, c * g n ≤ sumTransform (p a b) g (r i n) n
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
⊢ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * g ↑n ≤ sumTransform (p a b) g (r i n) n
|
obtain ⟨c₁, hc₁_mem, hc₁⟩ := R.exists_eventually_const_mul_le_r
|
case intro.intro
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
⊢ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * g ↑n ≤ sumTransform (p a b) g (r i n) n
|
aff66f2e59f80837
|
Grp.SurjectiveOfEpiAuxs.agree
|
Mathlib/Algebra/Category/Grp/EpiMono.lean
|
theorem agree : f.hom.range = { x | h x = g x }
|
case refine_1
A B : Grp
f : A ⟶ B
b : ↑B
⊢ b ∈ ↑(Hom.hom f).range → b ∈ {x | h x = g x}
|
rintro ⟨a, rfl⟩
|
case refine_1.intro
A B : Grp
f : A ⟶ B
a : ↑A
⊢ (Hom.hom f) a ∈ {x | h x = g x}
|
7120fb23e969ffac
|
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