name
stringlengths 3
112
| file
stringlengths 21
116
| statement
stringlengths 17
8.64k
| state
stringlengths 7
205k
| tactic
stringlengths 3
4.55k
| result
stringlengths 7
205k
| id
stringlengths 16
16
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|---|---|---|---|---|---|---|
MeasureTheory.Supermartingale.setIntegral_le
|
Mathlib/Probability/Martingale/Basic.lean
|
theorem setIntegral_le [SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : Supermartingale f ℱ μ)
{i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) :
∫ ω in s, f j ω ∂μ ≤ ∫ ω in s, f i ω ∂μ
|
Ω : Type u_1
ι : Type u_3
inst✝¹ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ι m0
inst✝ : SigmaFiniteFiltration μ ℱ
f : ι → Ω → ℝ
hf : Supermartingale f ℱ μ
i j : ι
hij : i ≤ j
s : Set Ω
hs : MeasurableSet s
⊢ μ[f j|↑ℱ i] ≤ᶠ[ae μ] f i
|
filter_upwards [hf.2.1 i j hij] with _ heq using heq
|
no goals
|
d9882fe29a1eccc9
|
lowerCentralSeries_antitone
|
Mathlib/GroupTheory/Nilpotent.lean
|
theorem lowerCentralSeries_antitone : Antitone (lowerCentralSeries G)
|
case intro.intro.intro
G : Type u_1
inst✝ : Group G
n : ℕ
x : G
hx : x ∈ closure {x | ∃ p ∈ lowerCentralSeries G n, ∃ q, p * q * p⁻¹ * q⁻¹ = x}
y z : G
hz : z ∈ lowerCentralSeries G n
a : G
ha : z * a * z⁻¹ * a⁻¹ = y
⊢ z * (a * z⁻¹ * a⁻¹) ∈ lowerCentralSeries G n
|
exact mul_mem hz (Normal.conj_mem (lowerCentralSeries_normal n) z⁻¹ (inv_mem hz) a)
|
no goals
|
db060d010fc2a818
|
SeminormFamily.basisSets_smul_left
|
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
theorem basisSets_smul_left (x : 𝕜) (U : Set E) (hU : U ∈ p.basisSets) :
∃ V ∈ p.addGroupFilterBasis.sets, V ⊆ (fun y : E => x • y) ⁻¹' U
|
case pos
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : Nonempty ι
x : 𝕜
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : x ≠ 0
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (s.sup p).ball 0 (r / ‖x‖)
|
use (s.sup p).ball 0 (r / ‖x‖)
|
case h
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : Nonempty ι
x : 𝕜
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : x ≠ 0
⊢ (s.sup p).ball 0 (r / ‖x‖) ∈ AddGroupFilterBasis.toFilterBasis.sets ∧
(s.sup p).ball 0 (r / ‖x‖) ⊆ (s.sup p).ball 0 (r / ‖x‖)
|
746096ac124c9ebb
|
CategoryTheory.Functor.IsCoverDense.Types.naturality_apply
|
Mathlib/CategoryTheory/Sites/DenseSubsite/Basic.lean
|
theorem naturality_apply [G.IsLocallyFull K] {X Y : C} (i : G.obj X ⟶ G.obj Y) (x) :
ℱ'.1.map i.op (α.app _ x) = α.app _ (ℱ.map i.op x)
|
C : Type u_1
inst✝² : Category.{u_5, u_1} C
D : Type u_2
inst✝¹ : Category.{u_6, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
ℱ : Dᵒᵖ ⥤ Type v
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val
inst✝ : G.IsLocallyFull K
X Y : C
i : G.obj X ⟶ G.obj Y
x : (G.op ⋙ ℱ).obj (op Y)
⊢ ℱ'.val.map i.op (α.app (op Y) x) = α.app (op X) (ℱ.map i.op x)
|
have {X Y} (i : X ⟶ Y) (x) :
ℱ'.1.map (G.map i).op (α.app _ x) = α.app _ (ℱ.map (G.map i).op x) := by
exact congr_fun (α.naturality i.op).symm x
|
C : Type u_1
inst✝² : Category.{u_5, u_1} C
D : Type u_2
inst✝¹ : Category.{u_6, u_2} D
K : GrothendieckTopology D
G : C ⥤ D
ℱ : Dᵒᵖ ⥤ Type v
ℱ' : Sheaf K (Type v)
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val
inst✝ : G.IsLocallyFull K
X Y : C
i : G.obj X ⟶ G.obj Y
x : (G.op ⋙ ℱ).obj (op Y)
this :
∀ {X Y : C} (i : X ⟶ Y) (x : (G.op ⋙ ℱ).obj (op Y)),
ℱ'.val.map (G.map i).op (α.app (op Y) x) = α.app (op X) (ℱ.map (G.map i).op x)
⊢ ℱ'.val.map i.op (α.app (op Y) x) = α.app (op X) (ℱ.map i.op x)
|
0e18cd3a80d5b906
|
MeasureTheory.integral_Ioi_deriv_mul_eq_sub
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
theorem integral_Ioi_deriv_mul_eq_sub
(hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x) (hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x)
(huv : IntegrableOn (u' * v + u * v') (Ioi a))
(h_zero : Tendsto (u * v) (𝓝[>] a) (𝓝 a')) (h_infty : Tendsto (u * v) atTop (𝓝 b')) :
∫ (x : ℝ) in Ioi a, u' x * v x + u x * v' x = b' - a'
|
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
⊢ u * v =ᶠ[atTop] f
|
filter_upwards [eventually_ne_atTop a] with x hx
|
case h
A : Type u_1
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra ℝ A
a : ℝ
a' b' : A
u v u' v' : ℝ → A
inst✝ : CompleteSpace A
hu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x
hv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x
huv : IntegrableOn (u' * v + u * v') (Ioi a) volume
h_zero : Tendsto (u * v) (𝓝[Ici a \ {a}] a) (𝓝 a')
h_infty : Tendsto (u * v) atTop (𝓝 b')
f : ℝ → A := Function.update (u * v) a a'
hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x
x : ℝ
hx : x ≠ a
⊢ (u * v) x = f x
|
c1989a63bc42f58a
|
minpoly.mem_range_of_degree_eq_one
|
Mathlib/FieldTheory/Minpoly/Basic.lean
|
theorem mem_range_of_degree_eq_one (hx : (minpoly A x).degree = 1) :
x ∈ (algebraMap A B).range
|
A : Type u_1
B : Type u_2
inst✝² : CommRing A
inst✝¹ : Ring B
inst✝ : Algebra A B
x : B
hx : (minpoly A x).degree = 1
h : IsIntegral A x
key : x = (algebraMap A B) (-(minpoly A x).coeff 0)
⊢ x ∈ (algebraMap A B).range
|
exact ⟨-(minpoly A x).coeff 0, key.symm⟩
|
no goals
|
f3de3c81a71d387c
|
IsSelfAdjoint.hasEigenvector_of_isMaxOn
|
Mathlib/Analysis/InnerProductSpace/Rayleigh.lean
|
theorem hasEigenvector_of_isMaxOn (hT : IsSelfAdjoint T) {x₀ : E} (hx₀ : x₀ ≠ 0)
(hextr : IsMaxOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
HasEigenvector (T : E →ₗ[𝕜] E) (↑(⨆ x : { x : E // x ≠ 0 }, T.rayleighQuotient x)) x₀
|
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hx₀ : x₀ ≠ 0
hextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
hx₀' : 0 < ‖x₀‖
hx₀'' : x₀ ∈ sphere 0 ‖x₀‖
x : E
hx : x ∈ sphere 0 ‖x₀‖
⊢ ‖x‖ = ‖x₀‖
|
simpa using hx
|
no goals
|
ab8f63176b60c4a5
|
MixedCharZero.reduce_to_p_prime
|
Mathlib/Algebra/CharP/MixedCharZero.lean
|
theorem reduce_to_p_prime {P : Prop} :
(∀ p > 0, MixedCharZero R p → P) ↔ ∀ p : ℕ, p.Prime → MixedCharZero R p → P
|
R : Type u_1
inst✝ : CommRing R
P : Prop
h : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
q : ℕ
q_pos : q > 0
q_mixedChar : MixedCharZero R q
I : Ideal R
hI_ne_top : I ≠ ⊤
right✝ : CharP (R ⧸ I) q
M : Ideal R
hM_max : M.IsMaximal
h_IM : I ≤ M
r : ℕ := ringChar (R ⧸ M)
q_zero : ↑q = 0
⊢ r ≠ 0
|
apply ne_zero_of_dvd_ne_zero (ne_of_gt q_pos)
|
R : Type u_1
inst✝ : CommRing R
P : Prop
h : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P
q : ℕ
q_pos : q > 0
q_mixedChar : MixedCharZero R q
I : Ideal R
hI_ne_top : I ≠ ⊤
right✝ : CharP (R ⧸ I) q
M : Ideal R
hM_max : M.IsMaximal
h_IM : I ≤ M
r : ℕ := ringChar (R ⧸ M)
q_zero : ↑q = 0
⊢ r ∣ q
|
c9cd4718e7a6656f
|
ModelWithCorners.interior_disjointUnion
|
Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean
|
lemma interior_disjointUnion :
ModelWithCorners.interior (I := I) (M ⊕ M') =
Sum.inl '' (ModelWithCorners.interior (I := I) M)
∪ Sum.inr '' (ModelWithCorners.interior (I := I) M')
|
case pos
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
hp : p ∈ Sum.inl '' ModelWithCorners.interior M ∪ Sum.inr '' ModelWithCorners.interior M'
h : p.isLeft = true
⊢ p ∈ ModelWithCorners.interior (M ⊕ M')
|
set x := Sum.getLeft p h with x_eq
|
case pos
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H M'
p : M ⊕ M'
hp : p ∈ Sum.inl '' ModelWithCorners.interior M ∪ Sum.inr '' ModelWithCorners.interior M'
h : p.isLeft = true
x : M := p.getLeft h
x_eq : x = p.getLeft h
⊢ p ∈ ModelWithCorners.interior (M ⊕ M')
|
a37cb5a73686bbc7
|
Lean.Grind.eqNDRec_heq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Grind/Lemmas.lean
|
theorem eqNDRec_heq.{u_1, u_2} {α : Sort u_2} {a : α}
{motive : α → Sort u_1} (v : motive a) {b : α} (h : a = b)
: HEq (@Eq.ndrec α a motive v b h) v
|
α : Sort u_2
a : α
motive : α → Sort u_1
v : motive a
⊢ HEq (⋯ ▸ v) v
|
rfl
|
no goals
|
1a1ced6def61e073
|
Subgroup.isOpen_of_mem_nhds
|
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
theorem isOpen_of_mem_nhds [ContinuousMul G] (H : Subgroup G) {g : G} (hg : (H : Set G) ∈ 𝓝 g) :
IsOpen (H : Set G)
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
⊢ ↑H ∈ 𝓝 x
|
have hg' : g ∈ H := SetLike.mem_coe.1 (mem_of_mem_nhds hg)
|
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
hg' : g ∈ H
⊢ ↑H ∈ 𝓝 x
|
b78c6ecd314e40e6
|
IsPrimitiveRoot.arg
|
Mathlib/RingTheory/RootsOfUnity/Complex.lean
|
theorem IsPrimitiveRoot.arg {n : ℕ} {ζ : ℂ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :
∃ i : ℤ, ζ.arg = i / n * (2 * Real.pi) ∧ IsCoprime i n ∧ i.natAbs < n
|
case neg.convert_2
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ -Real.pi < (↑i - ↑n) * (2 * Real.pi) / ↑n ∧ (↑i - ↑n) * (2 * Real.pi) / ↑n ≤ Real.pi
|
refine ⟨?_, le_trans ?_ Real.pi_pos.le⟩
|
case neg.convert_2.refine_1
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ -Real.pi < (↑i - ↑n) * (2 * Real.pi) / ↑n
case neg.convert_2.refine_2
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ (↑i - ↑n) * (2 * Real.pi) / ↑n ≤ 0
|
27e9c2564262c07b
|
LocallyFinite.finite_nonempty_of_compact
|
Mathlib/Topology/Compactness/Compact.lean
|
theorem LocallyFinite.finite_nonempty_of_compact [CompactSpace X] {f : ι → Set X}
(hf : LocallyFinite f) : { i | (f i).Nonempty }.Finite
|
X : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
f : ι → Set X
hf : LocallyFinite f
⊢ {i | (f i).Nonempty}.Finite
|
simpa only [inter_univ] using hf.finite_nonempty_inter_compact isCompact_univ
|
no goals
|
91a62b7c98a05f57
|
Int.isUnit_sq
|
Mathlib/Data/Int/Order/Units.lean
|
theorem isUnit_sq {a : ℤ} (ha : IsUnit a) : a ^ 2 = 1
|
a : ℤ
ha : IsUnit a
⊢ a ^ 2 = 1
|
rw [sq, isUnit_mul_self ha]
|
no goals
|
5411e649b7b09c66
|
List.set_getElem_succ_eraseIdx_succ
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Erase.lean
|
theorem set_getElem_succ_eraseIdx_succ
{l : List α} {i : Nat} (h : i + 1 < l.length) :
(l.eraseIdx (i + 1)).set i l[i + 1] = l.eraseIdx i
|
case h.isFalse.isTrue.isTrue
α : Type u_1
l : List α
i : Nat
h : i + 1 < l.length
n : Nat
h₁ : n < ((l.eraseIdx (i + 1)).set i l[i + 1]).length
h₂ : n < (l.eraseIdx i).length
h✝² : ¬i = n
h✝¹ : n < i + 1
h✝ : n < i
⊢ l[n] = l[n]
|
rfl
|
no goals
|
31983ea669fb242b
|
Subalgebra.finrank_sup_le_of_free
|
Mathlib/RingTheory/Adjoin/Dimension.lean
|
theorem finrank_sup_le_of_free : finrank R ↥(A ⊔ B) ≤ finrank R A * finrank R B
|
case pos
R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : StrongRankCondition R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
inst✝¹ : Free R ↥A
inst✝ : Free R ↥B
h : Module.Finite R ↥A ∧ Module.Finite R ↥B
⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B
|
obtain ⟨_, _⟩ := h
|
case pos.intro
R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : StrongRankCondition R
inst✝³ : CommRing S
inst✝² : Algebra R S
A B : Subalgebra R S
inst✝¹ : Free R ↥A
inst✝ : Free R ↥B
left✝ : Module.Finite R ↥A
right✝ : Module.Finite R ↥B
⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B
|
9b7b5a8cbdf91750
|
LinearMap.isClosed_or_dense_ker
|
Mathlib/Topology/Algebra/Module/Simple.lean
|
theorem LinearMap.isClosed_or_dense_ker (l : M →ₗ[R] N) :
IsClosed (LinearMap.ker l : Set M) ∨ Dense (LinearMap.ker l : Set M)
|
case inr
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
⊢ IsClosed ↑⊤ ∨ Dense ↑⊤
|
left
|
case inr.h
R : Type u
M : Type v
N : Type w
inst✝⁹ : Ring R
inst✝⁸ : TopologicalSpace R
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R M
inst✝³ : ContinuousSMul R M
inst✝² : Module R N
inst✝¹ : ContinuousAdd M
inst✝ : IsSimpleModule R N
⊢ IsClosed ↑⊤
|
8a89eedc2ab854fa
|
Associates.eq_factors_of_eq_counts
|
Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean
|
theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
(h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
a.factors = b.factors
|
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → p.count ↑sa = p.count ↑sb
h_sb : b.factors = ↑sb
⊢ ∀ (p : Associates α) (hp : Irreducible p), Multiset.count ⟨p, hp⟩ sa = Multiset.count ⟨p, hp⟩ sb
|
intro p hp
|
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p // Irreducible p }
h : ∀ (p : Associates α), Irreducible p → p.count ↑sa = p.count ↑sb
h_sb : b.factors = ↑sb
p : Associates α
hp : Irreducible p
⊢ Multiset.count ⟨p, hp⟩ sa = Multiset.count ⟨p, hp⟩ sb
|
92d336a2c52c647d
|
Associates.eq_pow_count_factors_of_dvd_pow
|
Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean
|
theorem eq_pow_count_factors_of_dvd_pow {p a : Associates α}
(hp : Irreducible p) {n : ℕ} (h : a ∣ p ^ n) : a = p ^ p.count a.factors
|
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
a✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
⊢ ∀ (p_1 : Associates α), Irreducible p_1 → p_1.count a.factors = p_1.count (p ^ p.count a.factors).factors
|
have eq_zero_of_ne : ∀ q : Associates α, Irreducible q → q ≠ p → _ = 0 := fun q hq h' =>
Nat.eq_zero_of_le_zero <| by
convert count_le_count_of_le hph hq h
symm
rw [count_pow hp.ne_zero hq, count_eq_zero_of_ne hq hp h', mul_zero]
|
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
p a : Associates α
hp : Irreducible p
n : ℕ
h : a ∣ p ^ n
a✝ : Nontrivial α
hph : p ^ n ≠ 0
ha : a ≠ 0
eq_zero_of_ne : ∀ (q : Associates α), Irreducible q → q ≠ p → q.count a.factors = 0
⊢ ∀ (p_1 : Associates α), Irreducible p_1 → p_1.count a.factors = p_1.count (p ^ p.count a.factors).factors
|
db8650aa84ceeb2f
|
Batteries.RBNode.Balanced.append
|
Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/WF.lean
|
theorem Balanced.append {l r : RBNode α}
(hl : l.Balanced c₁ n) (hr : r.Balanced c₂ n) :
(l.append r).RedRed (c₁ = black → c₂ ≠ black) n
|
α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝² x✝¹ a✝ : RBNode α
x✝ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node red a✝ x✝ b).Balanced c₁ n
hr : (node red c y✝ d✝).Balanced c₂ n
ha : a✝.Balanced black n
hb : b.Balanced black n
hc : c.Balanced black n
hd : d✝.Balanced black n
w✝ : RBColor
IH : (b.append c).Balanced w✝ n
⊢ RedRed (red = black → red ≠ black)
(match b.append c with
| node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)
| bc => node red a✝ x✝ (node red bc y✝ d✝))
n
|
split
|
case h_1
α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝³ x✝² a✝¹ : RBNode α
x✝¹ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node red a✝¹ x✝¹ b).Balanced c₁ n
hr : (node red c y✝ d✝).Balanced c₂ n
ha : a✝¹.Balanced black n
hb : b.Balanced black n
hc : c.Balanced black n
hd : d✝.Balanced black n
w✝ : RBColor
IH : (b.append c).Balanced w✝ n
l✝ a✝ : RBNode α
x✝ : α
b✝ : RBNode α
heq✝ : b.append c = node red a✝ x✝ b✝
⊢ RedRed (red = black → red ≠ black) (node red (node red a✝¹ x✝¹ a✝) x✝ (node red b✝ y✝ d✝)) n
case h_2
α : Type u_1
c₁ : RBColor
n : Nat
c₂ : RBColor
x✝³ x✝² a✝ : RBNode α
x✝¹ : α
b c : RBNode α
y✝ : α
d✝ : RBNode α
hl : (node red a✝ x✝¹ b).Balanced c₁ n
hr : (node red c y✝ d✝).Balanced c₂ n
ha : a✝.Balanced black n
hb : b.Balanced black n
hc : c.Balanced black n
hd : d✝.Balanced black n
w✝ : RBColor
IH : (b.append c).Balanced w✝ n
l✝ : RBNode α
x✝ : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), b.append c = node red a x b_1 → False
⊢ RedRed (red = black → red ≠ black) (node red a✝ x✝¹ (node red (b.append c) y✝ d✝)) n
|
b3ecc8a07b6c9ec6
|
LinearMap.mapsTo_biSup_of_mapsTo
|
Mathlib/Algebra/DirectSum/LinearMap.lean
|
lemma mapsTo_biSup_of_mapsTo {ι : Type*} {N : ι → Submodule R M}
(s : Set ι) {f : Module.End R M} (hf : ∀ i, MapsTo f (N i) (N i)) :
MapsTo f ↑(⨆ i ∈ s, N i) ↑(⨆ i ∈ s, N i)
|
R : Type u_2
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
ι : Type u_4
N : ι → Submodule R M
s : Set ι
f : Module.End R M
hf : ∀ (i : ι), Submodule.map f (N i) ≤ N i
⊢ Submodule.map f (⨆ i ∈ s, N i) ≤ ⨆ i ∈ s, N i
|
simpa only [Submodule.map_iSup] using iSup₂_mono <| fun i _ ↦ hf i
|
no goals
|
b3d6df199b89a4ff
|
CategoryTheory.Adjunction.localization_counit_app
|
Mathlib/CategoryTheory/Localization/Adjunction.lean
|
@[simp]
lemma localization_counit_app (X₂ : C₂) :
(adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) =
G'.map ((CatCommSq.iso F L₂ L₁ F').inv.app X₂) ≫
(CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂) ≫
L₂.map (adj.counit.app X₂)
|
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝⁷ : Category.{u_8, u_1} C₁
inst✝⁶ : Category.{u_7, u_2} C₂
inst✝⁵ : Category.{u_6, u_3} D₁
inst✝⁴ : Category.{u_5, u_4} D₂
G : C₁ ⥤ C₂
F : C₂ ⥤ C₁
adj : G ⊣ F
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G' : D₁ ⥤ D₂
F' : D₂ ⥤ D₁
inst✝¹ : CatCommSq G L₁ L₂ G'
inst✝ : CatCommSq F L₂ L₁ F'
X₂ : C₂
⊢ (adj.localization L₁ W₁ L₂ W₂ G' F').counit.app (L₂.obj X₂) =
G'.map ((CatCommSq.iso F L₂ L₁ F').inv.app X₂) ≫
(CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂) ≫ L₂.map (adj.counit.app X₂)
|
apply Localization.η_app
|
no goals
|
c65ffa4546fd9d0a
|
CategoryTheory.Limits.colim.exact_mapShortComplex
|
Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Colim.lean
|
/-- Assuming `HasExactColimitsOfShape J C`, this lemma rephrases the exactness
of the functor `colim : (J ⥤ C) ⥤ C` by saying that if `S : ShortComplex (J ⥤ C)`
is exact, then the short complex obtained by taking the colimits is exact,
where we allow the replacement of the chosen colimit cocones of the
colimit API by arbitrary colimit cocones. -/
lemma colim.exact_mapShortComplex :
(mapShortComplex S hc₁ c₂ c₃ f g hf hg).Exact
|
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : HasExactColimitsOfShape J C
inst✝ : HasZeroMorphisms C
S : ShortComplex (J ⥤ C)
hS : S.Exact
c₁ : Cocone S.X₁
hc₁ : IsColimit c₁
c₂ : Cocone S.X₂
hc₂ : IsColimit c₂
c₃ : Cocone S.X₃
hc₃ : IsColimit c₃
f : c₁.pt ⟶ c₂.pt
g : c₂.pt ⟶ c₃.pt
hf : ∀ (j : J), c₁.ι.app j ≫ f = S.f.app j ≫ c₂.ι.app j
hg : ∀ (j : J), c₂.ι.app j ≫ g = S.g.app j ≫ c₃.ι.app j
⊢ (mapShortComplex S hc₁ c₂ c₃ f g hf hg).Exact
|
refine (ShortComplex.exact_iff_of_iso ?_).2 (hS.map colim)
|
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : HasExactColimitsOfShape J C
inst✝ : HasZeroMorphisms C
S : ShortComplex (J ⥤ C)
hS : S.Exact
c₁ : Cocone S.X₁
hc₁ : IsColimit c₁
c₂ : Cocone S.X₂
hc₂ : IsColimit c₂
c₃ : Cocone S.X₃
hc₃ : IsColimit c₃
f : c₁.pt ⟶ c₂.pt
g : c₂.pt ⟶ c₃.pt
hf : ∀ (j : J), c₁.ι.app j ≫ f = S.f.app j ≫ c₂.ι.app j
hg : ∀ (j : J), c₂.ι.app j ≫ g = S.g.app j ≫ c₃.ι.app j
⊢ mapShortComplex S hc₁ c₂ c₃ f g hf hg ≅ S.map colim
|
dc1c71373d793112
|
gramSchmidt_mem_span
|
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
theorem gramSchmidt_mem_span (f : ι → E) :
∀ {j i}, i ≤ j → gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j)
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 (f '' Set.Iic j)
|
rw [gramSchmidt_def 𝕜 f i]
|
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
j i : ι
hij : i ≤ j
⊢ f i - ∑ i_1 ∈ Finset.Iio i, ↑((orthogonalProjection (span 𝕜 {gramSchmidt 𝕜 f i_1})) (f i)) ∈ span 𝕜 (f '' Set.Iic j)
|
2db3f8db7f4c82a7
|
ContinuousLinearMap.exists_preimage_norm_le
|
Mathlib/Analysis/Normed/Operator/Banach.lean
|
theorem exists_preimage_norm_le (surj : Surjective f) :
∃ C > 0, ∀ y, ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖
|
case succ
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y => y - f (g y)
hle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖
y : F
n : ℕ
IH : ‖h^[n] y‖ ≤ (1 / 2) ^ n * ‖y‖
⊢ ‖h^[n + 1] y‖ ≤ (1 / 2) ^ (n + 1) * ‖y‖
|
rw [iterate_succ']
|
case succ
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y => y - f (g y)
hle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖
y : F
n : ℕ
IH : ‖h^[n] y‖ ≤ (1 / 2) ^ n * ‖y‖
⊢ ‖(h ∘ h^[n]) y‖ ≤ (1 / 2) ^ (n + 1) * ‖y‖
|
fbb58cd4816e6d13
|
LinearIndependent.map_pow_expChar_pow_of_isSeparable
|
Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean
|
theorem LinearIndependent.map_pow_expChar_pow_of_isSeparable [Algebra.IsSeparable F E]
(h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n)
|
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝ : Algebra.IsSeparable F E
h : ∀ (s : Finset ι), LinearIndependent F (v ∘ Subtype.val)
halg : Algebra.IsAlgebraic F E
s : Finset ι
⊢ LinearIndependent F ((fun x => v x ^ q ^ n) ∘ Subtype.val)
|
let E' := adjoin F (s.image v : Set E)
|
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝ : Algebra.IsSeparable F E
h : ∀ (s : Finset ι), LinearIndependent F (v ∘ Subtype.val)
halg : Algebra.IsAlgebraic F E
s : Finset ι
E' : IntermediateField F E := adjoin F ↑(Finset.image v s)
⊢ LinearIndependent F ((fun x => v x ^ q ^ n) ∘ Subtype.val)
|
8836e43fdea16b48
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastArithShiftRight.go_denote_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean
|
theorem go_denote_eq (aig : AIG α) (distance : AIG.RefVec aig n) (curr : Nat)
(hcurr : curr ≤ n - 1) (acc : AIG.RefVec aig w)
(lhs : BitVec w) (rhs : BitVec n) (assign : α → Bool)
(hacc : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, acc.get idx hidx, assign⟧ = (BitVec.sshiftRightRec lhs rhs curr).getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < n), ⟦aig, distance.get idx hidx, assign⟧ = rhs.getLsbD idx) :
∀ (idx : Nat) (hidx : idx < w),
⟦
(go aig distance curr acc).aig,
(go aig distance curr acc).vec.get idx hidx,
assign
⟧
=
(BitVec.sshiftRightRec lhs rhs (n - 1)).getLsbD idx
|
case isTrue.hright
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
n w : Nat
aig : AIG α
distance : aig.RefVec n
curr : Nat
hcurr : curr ≤ n - 1
acc : aig.RefVec w
lhs : BitVec w
rhs : BitVec n
assign : α → Bool
hacc :
∀ (idx : Nat) (hidx : idx < w),
⟦assign, { aig := aig, ref := acc.get idx hidx }⟧ = (lhs.sshiftRightRec rhs curr).getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < n), ⟦assign, { aig := aig, ref := distance.get idx hidx }⟧ = rhs.getLsbD idx
idx✝ : Nat
hidx✝ : idx✝ < w
res : RefVecEntry α w
h✝ : curr < n - 1
hgo :
go (twoPowShift aig { n := n, lhs := acc, rhs := distance, pow := curr + 1 }).aig (distance.cast ⋯) (curr + 1)
(twoPowShift aig { n := n, lhs := acc, rhs := distance, pow := curr + 1 }).vec =
res
idx : Nat
hidx : idx < n
⊢ ((distance.cast ⋯).get idx hidx).gate < aig.decls.size
|
simp [Ref.hgate]
|
no goals
|
a62c971c970b9dfe
|
RCLike.nonUnitalContinuousFunctionalCalculus
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean
|
theorem RCLike.nonUnitalContinuousFunctionalCalculus :
NonUnitalContinuousFunctionalCalculus 𝕜 (p : A → Prop) where
predicate_zero
|
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ContinuousFunctionalCalculus 𝕜 p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
(↑(inrRangeEquiv 𝕜 A).symm).comp
(codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
⊢ IsClosedEmbedding (Unitization.inr ∘ ⇑ψ)
|
have : inr ∘ ψ = cfcₙAux hp₁ a ha := by ext1; rw [Function.comp_apply, coe_ψ]
|
𝕜 : Type u_1
A : Type u_2
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace 𝕜 A
inst✝⁵ : IsScalarTower 𝕜 A A
inst✝⁴ : SMulCommClass 𝕜 A A
inst✝³ : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
inst✝² : ContinuousFunctionalCalculus 𝕜 p₁
inst✝¹ : CompleteSpace A
inst✝ : CStarRing A
a : A
ha : p a
ψ : C(↑(σₙ 𝕜 a), 𝕜)₀ →⋆ₙₐ[𝕜] A :=
(↑(inrRangeEquiv 𝕜 A).symm).comp
(codRestrict (cfcₙAux ⋯ a ha) (NonUnitalStarAlgHom.range (inrNonUnitalStarAlgHom 𝕜 A)) ⋯)
coe_ψ : ∀ (f : C(↑(σₙ 𝕜 a), 𝕜)₀), ↑(ψ f) = (cfcₙAux ⋯ a ha) f
this : Unitization.inr ∘ ⇑ψ = ⇑(cfcₙAux ⋯ a ha)
⊢ IsClosedEmbedding (Unitization.inr ∘ ⇑ψ)
|
95b319408dc7d905
|
integral_bernoulliFun_eq_zero
|
Mathlib/NumberTheory/ZetaValues.lean
|
theorem integral_bernoulliFun_eq_zero {k : ℕ} (hk : k ≠ 0) :
∫ x : ℝ in (0)..1, bernoulliFun k x = 0
|
k : ℕ
hk : k ≠ 0
⊢ ∫ (x : ℝ) in 0 ..1, bernoulliFun k x = 0
|
rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)
((Polynomial.continuous _).intervalIntegrable _ _)]
|
k : ℕ
hk : k ≠ 0
⊢ bernoulliFun (k + 1) 1 / (↑k + 1) - bernoulliFun (k + 1) 0 / (↑k + 1) = 0
|
6e6ae5d2aa4d445c
|
Std.Tactic.BVDecide.BVExpr.bitblast.go_decl_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Expr.lean
|
theorem bitblast.go_decl_eq (aig : AIG BVBit) (expr : BVExpr w) :
∀ (idx : Nat) (h1) (h2), (go aig expr).val.aig.decls[idx]'h2 = aig.decls[idx]'h1
|
case arithShiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.size),
(go aig rhs).val.aig.decls[idx] = aig.decls[idx]
aig : AIG BVBit
h1 : idx < aig.decls.size
h2 : idx < (go aig (lhs.arithShiftRight rhs)).val.aig.decls.size
this✝¹ this✝ : aig.decls.size ≤ (go aig lhs).val.aig.decls.size
this : (go aig lhs).val.aig.decls.size ≤ (go (go aig lhs).val.aig rhs).val.aig.decls.size
⊢ (blastArithShiftRight (go (go aig lhs).1.aig rhs).1.aig
{ n := n✝, target := (go aig lhs).1.vec.cast ⋯,
distance := (go (go aig lhs).1.aig rhs).1.vec }).aig.decls[idx] =
aig.decls[idx]
|
rw [AIG.LawfulVecOperator.decl_eq (f := blastArithShiftRight)]
|
case arithShiftRight
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.size),
(go aig rhs).val.aig.decls[idx] = aig.decls[idx]
aig : AIG BVBit
h1 : idx < aig.decls.size
h2 : idx < (go aig (lhs.arithShiftRight rhs)).val.aig.decls.size
this✝¹ this✝ : aig.decls.size ≤ (go aig lhs).val.aig.decls.size
this : (go aig lhs).val.aig.decls.size ≤ (go (go aig lhs).val.aig rhs).val.aig.decls.size
⊢ (go (go aig lhs).1.aig rhs).1.aig.decls[idx] = aig.decls[idx]
case arithShiftRight.h1
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.size),
(go aig rhs).val.aig.decls[idx] = aig.decls[idx]
aig : AIG BVBit
h1 : idx < aig.decls.size
h2 : idx < (go aig (lhs.arithShiftRight rhs)).val.aig.decls.size
this✝¹ this✝ : aig.decls.size ≤ (go aig lhs).val.aig.decls.size
this : (go aig lhs).val.aig.decls.size ≤ (go (go aig lhs).val.aig rhs).val.aig.decls.size
⊢ idx < (go (go aig lhs).1.aig rhs).1.aig.decls.size
|
f92988d5a1595b39
|
Set.exists_subset_encard_eq
|
Mathlib/Data/Set/Card.lean
|
theorem exists_subset_encard_eq {k : ℕ∞} (hk : k ≤ s.encard) : ∃ t, t ⊆ s ∧ t.encard = k
|
α : Type u_1
s : Set α
k : ℕ∞
x✝ : 0 ≤ s.encard
⊢ ∅.encard = 0
|
simp
|
no goals
|
05b71bd015f46ce9
|
MeasureTheory.AEMeasurable.ae_eq_of_forall_setLIntegral_eq
|
Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean
|
theorem AEMeasurable.ae_eq_of_forall_setLIntegral_eq {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
(hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) (hgi : ∫⁻ x, g x ∂μ ≠ ∞)
(hfg : ∀ ⦃s⦄, MeasurableSet s → μ s < ∞ → ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ) :
f =ᵐ[μ] g
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
this : f =ᶠ[ae (μ.restrict (s ∪ t))] g
⊢ ∀ᵐ (x : α) ∂μ.restrict (s ∪ t)ᶜ, f x = g x
|
simp only [Set.compl_union]
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hgi : ∫⁻ (x : α), g x ∂μ ≠ ⊤
hfg : ∀ ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → ∫⁻ (x : α) in s, f x ∂μ = ∫⁻ (x : α) in s, g x ∂μ
hf' : AEFinStronglyMeasurable f μ
hg' : AEFinStronglyMeasurable g μ
s : Set α := hf'.sigmaFiniteSet
t : Set α := hg'.sigmaFiniteSet
this : f =ᶠ[ae (μ.restrict (s ∪ t))] g
⊢ ∀ᵐ (x : α) ∂μ.restrict (sᶜ ∩ tᶜ), f x = g x
|
928eae5949838171
|
HomologicalComplex₂.D₂_D₁
|
Mathlib/Algebra/Homology/TotalComplex.lean
|
@[reassoc (attr := simp)]
lemma D₂_D₁ (i₁₂ i₁₂' i₁₂'' : I₁₂) :
K.D₂ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = - K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂''
|
case pos
C : Type u_1
inst✝⁴ : Category.{u_5, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₂.Rel i₂ (c₂.next i₂)
h₅ : c₁.π c₂ c₁₂ (i₁, c₂.next i₂) = i₁₂'
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' =
-(c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ (c₁.next i₁)).f i₂ ≫ K.d₂ c₁₂ (c₁.next i₁) i₂ i₁₂'')
|
have h₆ : ComplexShape.π c₁ c₂ c₁₂ (c₁.next i₁, c₂.next i₂) = i₁₂'' := by
rw [← c₁₂.next_eq' h₂, ← ComplexShape.next_π₁ c₂ c₁₂ h₃, h₅]
|
case pos
C : Type u_1
inst✝⁴ : Category.{u_5, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₂.Rel i₂ (c₂.next i₂)
h₅ : c₁.π c₂ c₁₂ (i₁, c₂.next i₂) = i₁₂'
h₆ : c₁.π c₂ c₁₂ (c₁.next i₁, c₂.next i₂) = i₁₂''
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' =
-(c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ (c₁.next i₁)).f i₂ ≫ K.d₂ c₁₂ (c₁.next i₁) i₂ i₁₂'')
|
2d728f63029305d6
|
intervalIntegral.intervalIntegrable_cpow'
|
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
theorem intervalIntegrable_cpow' {r : ℂ} (h : -1 < r.re) :
IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) volume a b
|
case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
⊢ IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
|
rw [IntervalIntegrable.iff_comp_neg, neg_zero]
|
case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
⊢ IntervalIntegrable (fun x => ↑(-x) ^ r) volume 0 (-c)
|
e33db8d106f7d67d
|
NormedAddGroupHom.mkNormedAddGroupHom_norm_le'
|
Mathlib/Analysis/Normed/Group/Hom.lean
|
theorem mkNormedAddGroupHom_norm_le' (f : V₁ →+ V₂) {C : ℝ} (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mkNormedAddGroupHom C h‖ ≤ max C 0 :=
opNorm_le_bound _ (le_max_right _ _) fun x =>
(h x).trans <| by gcongr; apply le_max_left
|
case h
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : V₁ →+ V₂
C : ℝ
h : ∀ (x : V₁), ‖f x‖ ≤ C * ‖x‖
x : V₁
⊢ C ≤ C ⊔ 0
|
apply le_max_left
|
no goals
|
9103a32bfa4bec78
|
PartENat.le_of_lt_add_one
|
Mathlib/Data/Nat/PartENat.lean
|
theorem le_of_lt_add_one {x y : PartENat} (h : x < y + 1) : x ≤ y
|
case a.intro
n m : ℕ
h : ↑m < ↑n + 1
⊢ m ≤ n
|
apply Nat.le_of_lt_succ
|
case a.intro.a
n m : ℕ
h : ↑m < ↑n + 1
⊢ m < n.succ
|
941bfcc29fd6ae31
|
Subfield.mem_iSup_of_directed
|
Mathlib/Algebra/Field/Subfield/Basic.lean
|
theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {S : ι → Subfield K} (hS : Directed (· ≤ ·) S)
{x : K} : (x ∈ ⨆ i, S i) ↔ ∃ i, x ∈ S i
|
K : Type u
inst✝ : DivisionRing K
ι : Sort u_1
hι : Nonempty ι
S : ι → Subfield K
hS : Directed (fun x1 x2 => x1 ≤ x2) S
x : K
s : Subfield K :=
let __spread.0 := (⨆ i, (S i).toSubring).copy (⋃ i, ↑(S i).toSubring) ⋯;
{ toSubring := __spread.0, inv_mem' := ⋯ }
this : iSup S = s
⊢ x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i
|
exact this ▸ Set.mem_iUnion
|
no goals
|
e62260f3c816ead1
|
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet
|
Mathlib/Analysis/Complex/AbelLimit.lean
|
theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l)
|
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M
F : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖
z : ℂ
zn : ‖z‖ < 1
zm : ‖1 - z‖ < M * (1 - ‖z‖)
zd : ‖z - 1‖ < ε / 4 / (F + 1)
B₂ : ℕ
hB₂ : ‖l - ∑' (n : ℕ), f n * z ^ n - (1 - z) * ∑ i ∈ range (B₁ ⊔ B₂), (l - ∑ j ∈ range (i + 1), f j) * z ^ i‖ < ε / 2
S₁ : ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4
⊢ ‖1 - z‖ * (ε / 4 / M) * ∑' (i : ℕ), ‖z‖ ^ i = ‖1 - z‖ * (ε / 4 / M) / (1 - ‖z‖)
|
rw [tsum_geometric_of_lt_one (by positivity) zn, ← div_eq_mul_inv]
|
no goals
|
e1f8d437480849f9
|
PowerBasis.repr_pow_isIntegral
|
Mathlib/RingTheory/Adjoin/PowerBasis.lean
|
theorem repr_pow_isIntegral [IsDomain S] (hB : IsIntegral R B.gen) {x : A}
(hx : ∀ i, IsIntegral R (B.basis.repr x i))
(hmin : minpoly S B.gen = (minpoly R B.gen).map (algebraMap R S)) (n : ℕ) :
∀ i, IsIntegral R (B.basis.repr (x ^ n) i)
|
S : Type u_2
inst✝⁷ : CommRing S
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
B : PowerBasis S A
inst✝ : IsDomain S
hB : IsIntegral R B.gen
x : A
hx : ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n : ℕ
a✝ : Nontrivial A
⊢ ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr (x ^ n)) i)
|
revert hx
|
S : Type u_2
inst✝⁷ : CommRing S
R : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : Algebra R S
A : Type u_4
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : IsScalarTower R S A
B : PowerBasis S A
inst✝ : IsDomain S
hB : IsIntegral R B.gen
x : A
hmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)
n : ℕ
a✝ : Nontrivial A
⊢ (∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr x) i)) → ∀ (i : Fin B.dim), IsIntegral R ((B.basis.repr (x ^ n)) i)
|
d2285bbb358047c1
|
LinearMap.nilRank_le_natTrailingDegree_charpoly
|
Mathlib/Algebra/Module/LinearMap/Polynomial.lean
|
lemma nilRank_le_natTrailingDegree_charpoly (x : L) :
nilRank φ ≤ (φ x).charpoly.natTrailingDegree
|
case h
R : Type u_1
L : Type u_2
M : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup L
inst✝⁷ : Module R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
φ : L →ₗ[R] End R M
inst✝⁴ : Free R M
inst✝³ : Module.Finite R M
inst✝² : Module.Finite R L
inst✝¹ : Free R L
inst✝ : Nontrivial R
x : L
h : (φ.polyCharpoly (chooseBasis R L)).coeff (charpoly (φ x)).natTrailingDegree = 0
⊢ False
|
apply_fun (MvPolynomial.eval ((chooseBasis R L).repr x)) at h
|
case h
R : Type u_1
L : Type u_2
M : Type u_3
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup L
inst✝⁷ : Module R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
φ : L →ₗ[R] End R M
inst✝⁴ : Free R M
inst✝³ : Module.Finite R M
inst✝² : Module.Finite R L
inst✝¹ : Free R L
inst✝ : Nontrivial R
x : L
h :
(MvPolynomial.eval ⇑((chooseBasis R L).repr x))
((φ.polyCharpoly (chooseBasis R L)).coeff (charpoly (φ x)).natTrailingDegree) =
(MvPolynomial.eval ⇑((chooseBasis R L).repr x)) 0
⊢ False
|
7d44b5dff95aaa8c
|
AList.insertRec_insert
|
Mathlib/Data/List/AList.lean
|
theorem insertRec_insert {C : AList β → Sort*} (H0 : C ∅)
(IH : ∀ (a : α) (b : β a) (l : AList β), a ∉ l → C l → C (l.insert a b)) {c : Sigma β}
{l : AList β} (h : c.1 ∉ l) :
@insertRec α β _ C H0 IH (l.insert c.1 c.2) = IH c.1 c.2 l h (@insertRec α β _ C H0 IH l)
|
α : Type u
β : α → Type v
inst✝ : DecidableEq α
C : AList β → Sort u_1
H0 : C ∅
IH : (a : α) → (b : β a) → (l : AList β) → a ∉ l → C l → C (insert a b l)
c : Sigma β
l : AList β
h : c.fst ∉ l
⊢ insertRec H0 IH (insert c.fst c.snd l) = IH c.fst c.snd l h (insertRec H0 IH l)
|
obtain ⟨l, hl⟩ := l
|
case mk
α : Type u
β : α → Type v
inst✝ : DecidableEq α
C : AList β → Sort u_1
H0 : C ∅
IH : (a : α) → (b : β a) → (l : AList β) → a ∉ l → C l → C (insert a b l)
c : Sigma β
l : List (Sigma β)
hl : l.NodupKeys
h : c.fst ∉ { entries := l, nodupKeys := hl }
⊢ insertRec H0 IH (insert c.fst c.snd { entries := l, nodupKeys := hl }) =
IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl })
|
ac5fbba4f95f506f
|
Real.Angle.sign_two_nsmul_eq_sign_iff
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
theorem sign_two_nsmul_eq_sign_iff {θ : Angle} :
((2 : ℕ) • θ).sign = θ.sign ↔ θ = π ∨ |θ.toReal| < π / 2
|
case neg.refine_1
θ : Angle
hpi : ¬θ = ↑π
h : (2 • θ).sign = θ.sign
hle : π / 2 ≤ θ.toReal ∨ θ.toReal ≤ -(π / 2)
⊢ False
|
have hpi' : θ.toReal ≠ π := by simpa using hpi
|
case neg.refine_1
θ : Angle
hpi : ¬θ = ↑π
h : (2 • θ).sign = θ.sign
hle : π / 2 ≤ θ.toReal ∨ θ.toReal ≤ -(π / 2)
hpi' : θ.toReal ≠ π
⊢ False
|
796c2e838b8abfa4
|
List.findIdx?_go_eq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
findIdx?.go p xs (i+1) = (findIdx?.go p xs 0).map fun k => k + (i + 1)
|
case cons
α : Type u_1
p : α → Bool
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ {i : Nat}, findIdx?.go p tail✝ (i + 1) = Option.map (fun k => k + (i + 1)) (findIdx?.go p tail✝ 0)
i : Nat
⊢ Option.map (fun i => i + 1) (if p head✝ = true then some i else Option.map (fun i => i + 1) (findIdx?.go p tail✝ i)) =
Option.map (fun k => k + (i + 1))
(if p head✝ = true then some 0 else Option.map (fun i => i + 1) (findIdx?.go p tail✝ 0))
|
split
|
case cons.isTrue
α : Type u_1
p : α → Bool
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ {i : Nat}, findIdx?.go p tail✝ (i + 1) = Option.map (fun k => k + (i + 1)) (findIdx?.go p tail✝ 0)
i : Nat
h✝ : p head✝ = true
⊢ Option.map (fun i => i + 1) (some i) = Option.map (fun k => k + (i + 1)) (some 0)
case cons.isFalse
α : Type u_1
p : α → Bool
head✝ : α
tail✝ : List α
tail_ih✝ : ∀ {i : Nat}, findIdx?.go p tail✝ (i + 1) = Option.map (fun k => k + (i + 1)) (findIdx?.go p tail✝ 0)
i : Nat
h✝ : ¬p head✝ = true
⊢ Option.map (fun i => i + 1) (Option.map (fun i => i + 1) (findIdx?.go p tail✝ i)) =
Option.map (fun k => k + (i + 1)) (Option.map (fun i => i + 1) (findIdx?.go p tail✝ 0))
|
00b11065e9d67637
|
VitaliFamily.exists_measurable_supersets_limRatio
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
{x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
A :
ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B :
↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
⊢ ↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) <
↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
|
gcongr
|
case hinf
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
A :
ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
↑p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
B :
↑q * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
h : ¬μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0
⊢ μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≠ ⊤
|
f2acb3dc8c3a0a55
|
Ideal.matricesOver_strictMono_of_nonempty
|
Mathlib/LinearAlgebra/Matrix/Ideal.lean
|
theorem matricesOver_strictMono_of_nonempty [Nonempty n] :
StrictMono (matricesOver (R := R) n) :=
matricesOver_monotone n |>.strictMono_of_injective <| fun I J eq => by
ext x
have : (∀ _ _, x ∈ I) ↔ (∀ _ _, x ∈ J) := congr((Matrix.of fun _ _ => x) ∈ $eq)
simpa only [forall_const] using this
|
R : Type u_1
inst✝³ : Semiring R
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Nonempty n
I J : Ideal R
eq : matricesOver n I = matricesOver n J
⊢ I = J
|
ext x
|
case h
R : Type u_1
inst✝³ : Semiring R
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Nonempty n
I J : Ideal R
eq : matricesOver n I = matricesOver n J
x : R
⊢ x ∈ I ↔ x ∈ J
|
0208c0bde6369ee2
|
List.forIn_eq_foldlM
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean
|
theorem forIn_eq_foldlM [Monad m] [LawfulMonad m]
(f : α → β → m (ForInStep β)) (init : β) (l : List α) :
forIn l init f = ForInStep.value <$>
l.foldlM (fun b a => match b with
| .yield b => f a b
| .done b => pure (.done b)) (ForInStep.yield init)
|
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m (ForInStep β)
a : α
as : List α
ih :
∀ (init : β),
forIn as init f =
ForInStep.value <$>
List.foldlM
(fun b a =>
match b with
| ForInStep.yield b => f a b
| ForInStep.done b => pure (ForInStep.done b))
(ForInStep.yield init) as
init : β
x : ForInStep β
b : β
⊢ (match ForInStep.yield b with
| ForInStep.done b => pure b
| ForInStep.yield b => forIn as b f) =
ForInStep.value <$>
List.foldlM
(fun b a =>
match b with
| ForInStep.yield b => f a b
| ForInStep.done b => pure (ForInStep.done b))
(ForInStep.yield b) as
|
simp [ih]
|
no goals
|
5803c92344e4da54
|
List.map_attachWith
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Attach.lean
|
theorem map_attachWith {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
(f : { x // P x } → β) :
(l.attachWith P H).map f =
l.pmap (fun a (h : a ∈ l ∧ P a) => f ⟨a, H _ h.1⟩) (fun a h => ⟨h, H a h⟩)
|
case cons.h
α : Type u_1
β : Type u_2
P : α → Prop
f : { x // P x } → β
x : α
xs : List α
ih : ∀ {H : ∀ (a : α), a ∈ xs → P a}, map f (xs.attachWith P H) = pmap (fun a h => f ⟨a, ⋯⟩) xs ⋯
H : ∀ (a : α), a ∈ x :: xs → P a
⊢ ∀ (a : α), a ∈ xs → ∀ (h₁ : a ∈ xs ∧ P a) (h₂ : a ∈ x :: xs ∧ P a), f ⟨a, ⋯⟩ = f ⟨a, ⋯⟩
|
simp
|
no goals
|
5c127c9a27e15920
|
Ideal.sup_pow_add_le_pow_sup_pow
|
Mathlib/RingTheory/Ideal/Operations.lean
|
lemma sup_pow_add_le_pow_sup_pow {n m : ℕ} : (I ⊔ J) ^ (n + m) ≤ I ^ n ⊔ J ^ m
|
case neg
R : Type u
inst✝ : CommSemiring R
I J : Ideal R
n m i : ℕ
hi : i ∈ Finset.range (n + m + 1)
hn : ¬n ≤ i
⊢ m ≤ n + m - i
|
omega
|
no goals
|
690ef13be270f2c6
|
NonUnitalRing.ext
|
Mathlib/Algebra/Ring/Ext.lean
|
theorem ext ⦃inst₁ inst₂ : NonUnitalRing R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂
|
case mk.mk
R : Type u
toNonUnitalNonAssocRing✝¹ : NonUnitalNonAssocRing R
mul_assoc✝¹ : ∀ (a b c : R), a * b * c = a * (b * c)
toNonUnitalNonAssocRing✝ : NonUnitalNonAssocRing R
mul_assoc✝ : ∀ (a b c : R), a * b * c = a * (b * c)
h_add : HAdd.hAdd = HAdd.hAdd
h_mul : HMul.hMul = HMul.hMul
this : toNonUnitalNonAssocRing = toNonUnitalNonAssocRing
⊢ mk mul_assoc✝¹ = mk mul_assoc✝
|
congr
|
no goals
|
3a34c023b5fdf73d
|
Finset.image₂_left_comm
|
Mathlib/Data/Finset/NAry.lean
|
theorem image₂_left_comm {γ : Type*} {u : Finset γ} {f : α → δ → ε} {g : β → γ → δ}
{f' : α → γ → δ'} {g' : β → δ' → ε} (h_left_comm : ∀ a b c, f a (g b c) = g' b (f' a c)) :
image₂ f s (image₂ g t u) = image₂ g' t (image₂ f' s u) :=
coe_injective <| by
push_cast
exact image2_left_comm h_left_comm
|
α : Type u_1
β : Type u_3
δ : Type u_7
δ' : Type u_8
ε : Type u_9
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
s : Finset α
t : Finset β
inst✝ : DecidableEq δ
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ ↑(image₂ f s (image₂ g t u)) = ↑(image₂ g' t (image₂ f' s u))
|
push_cast
|
α : Type u_1
β : Type u_3
δ : Type u_7
δ' : Type u_8
ε : Type u_9
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
s : Finset α
t : Finset β
inst✝ : DecidableEq δ
γ : Type u_14
u : Finset γ
f : α → δ → ε
g : β → γ → δ
f' : α → γ → δ'
g' : β → δ' → ε
h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c)
⊢ image2 f (↑s) (image2 g ↑t ↑u) = image2 g' (↑t) (image2 f' ↑s ↑u)
|
e951e1d0d2442dd3
|
map_wittPolynomial
|
Mathlib/RingTheory/WittVector/WittPolynomial.lean
|
theorem map_wittPolynomial (f : R →+* S) (n : ℕ) : map f (W n) = W n
|
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
n : ℕ
⊢ ∑ x ∈ range (n + 1), (map f) ((monomial (single x (p ^ (n - x)))) (↑p ^ x)) =
∑ i ∈ range (n + 1), (monomial (single i (p ^ (n - i)))) (↑p ^ i)
|
refine sum_congr rfl fun i _ => ?_
|
p : ℕ
R : Type u_1
inst✝¹ : CommRing R
S : Type u_2
inst✝ : CommRing S
f : R →+* S
n i : ℕ
x✝ : i ∈ range (n + 1)
⊢ (map f) ((monomial (single i (p ^ (n - i)))) (↑p ^ i)) = (monomial (single i (p ^ (n - i)))) (↑p ^ i)
|
812afc8e6e227a9a
|
CategoryTheory.isCardinalPresentable_of_equivalence
|
Mathlib/CategoryTheory/Presentable/Basic.lean
|
lemma isCardinalPresentable_of_equivalence
{C' : Type u₃} [Category.{v₃} C'] [IsCardinalPresentable X κ] (e : C ≌ C') :
IsCardinalPresentable (e.functor.obj X) κ
|
case h.up
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
inst✝ : IsCardinalPresentable X κ
e : C ≌ C'
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
Y : J ⥤ C'
this : PreservesColimitsOfShape J (coyoneda.obj (op X))
X✝ Y✝ : C'
f : X✝ ⟶ Y✝
g : (coyoneda.obj (op (e.functor.obj X))).obj X✝
⊢ ((coyoneda.obj (op (e.functor.obj X)) ⋙ uliftFunctor.{v₁, v₃}).map f ≫
((fun Z => (Equiv.ulift.trans ((e.toAdjunction.homEquiv X Z).trans Equiv.ulift.symm)).toIso) Y✝).hom)
{ down := g } =
(((fun Z => (Equiv.ulift.trans ((e.toAdjunction.homEquiv X Z).trans Equiv.ulift.symm)).toIso) X✝).hom ≫
(e.inverse ⋙ coyoneda.obj (op X) ⋙ uliftFunctor.{v₃, v₁}).map f)
{ down := g }
|
apply Equiv.ulift.injective
|
case h.up.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X : C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
C' : Type u₃
inst✝¹ : Category.{v₃, u₃} C'
inst✝ : IsCardinalPresentable X κ
e : C ≌ C'
J : Type w
x✝¹ : SmallCategory J
x✝ : IsCardinalFiltered J κ
Y : J ⥤ C'
this : PreservesColimitsOfShape J (coyoneda.obj (op X))
X✝ Y✝ : C'
f : X✝ ⟶ Y✝
g : (coyoneda.obj (op (e.functor.obj X))).obj X✝
⊢ Equiv.ulift
(((coyoneda.obj (op (e.functor.obj X)) ⋙ uliftFunctor.{v₁, v₃}).map f ≫
((fun Z => (Equiv.ulift.trans ((e.toAdjunction.homEquiv X Z).trans Equiv.ulift.symm)).toIso) Y✝).hom)
{ down := g }) =
Equiv.ulift
((((fun Z => (Equiv.ulift.trans ((e.toAdjunction.homEquiv X Z).trans Equiv.ulift.symm)).toIso) X✝).hom ≫
(e.inverse ⋙ coyoneda.obj (op X) ⋙ uliftFunctor.{v₃, v₁}).map f)
{ down := g })
|
fd601d9188daee0d
|
List.append_eq_appendTR
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Basic.lean
|
theorem append_eq_appendTR : @List.append = @appendTR
|
case h.h.h.cons
α : Type u_1
bs : List α
a : α
as : List α
ih : as.append bs = (as.reverseAux nil).reverseAux bs
⊢ (a :: as).append bs = (a :: nil).reverseAux ((as.reverseAux nil).reverseAux bs)
|
simp [List.append, ih, reverseAux]
|
no goals
|
aae20de0c8888126
|
Finset.kruskal_katona_lovasz_form
|
Mathlib/Combinatorics/SetFamily/KruskalKatona.lean
|
theorem kruskal_katona_lovasz_form (hir : i ≤ r) (hrk : r ≤ k) (hkn : k ≤ n)
(h₁ : (𝒜 : Set (Finset (Fin n))).Sized r) (h₂ : k.choose r ≤ #𝒜) :
k.choose (r - i) ≤ #(∂^[i] 𝒜)
|
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ #(powersetCard (r - i) range'k) = #(∂ ^[i] 𝒞)
|
congr!
|
case h.e'_2
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ powersetCard (r - i) range'k = ∂ ^[i] 𝒞
|
edb5f2f186d17abf
|
WeierstrassCurve.b₆_of_isCharTwoJNeZeroNF
|
Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean
|
theorem b₆_of_isCharTwoJNeZeroNF : W.b₆ = 4 * W.a₆
|
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJNeZeroNF
⊢ W.b₆ = 4 * W.a₆
|
rw [b₆, a₃_of_isCharTwoJNeZeroNF]
|
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJNeZeroNF
⊢ 0 ^ 2 + 4 * W.a₆ = 4 * W.a₆
|
62cfcc7187a72866
|
MeasureTheory.setToFun_congr_measure_of_integrable
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem setToFun_congr_measure_of_integrable {μ' : Measure α} (c' : ℝ≥0∞) (hc' : c' ≠ ∞)
(hμ'_le : μ' ≤ c' • μ) (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ' T C') (f : α → E) (hfμ : Integrable f μ) :
setToFun μ T hT f = setToFun μ' T hT' f
|
case h_ind
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ setToFun μ T hT (s.indicator fun x => c) = setToFun μ' T hT' (s.indicator fun x => c)
|
have hμ's : μ' s ≠ ∞ := by
refine ((hμ'_le s).trans_lt ?_).ne
rw [Measure.smul_apply, smul_eq_mul]
exact ENNReal.mul_lt_top hc'.lt_top hμs
|
case h_ind
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ⊤
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
hμ's : μ' s ≠ ⊤
⊢ setToFun μ T hT (s.indicator fun x => c) = setToFun μ' T hT' (s.indicator fun x => c)
|
65e35cd00eda3e5b
|
Algebra.Presentation.aux_surjective
|
Mathlib/RingTheory/Presentation.lean
|
private lemma aux_surjective : Function.Surjective (Q.aux P) := fun p ↦ by
induction' p using MvPolynomial.induction_on with a p q hp hq p i h
· use rename Sum.inr <| P.σ a
simp only [aux, aeval_rename, Sum.elim_comp_inr]
have (p : MvPolynomial P.vars R) :
aeval (C ∘ P.val) p = (C (aeval P.val p) : MvPolynomial Q.vars S)
|
case h_add.intro
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : CommRing T
inst✝ : Algebra S T
Q : Presentation S T
P : Presentation R S
q : MvPolynomial Q.vars S
hq : ∃ a, (Algebra.Presentation.aux Q P) a = q
a : MvPolynomial (Q.vars ⊕ P.vars) R
⊢ ∃ a_1, (Algebra.Presentation.aux Q P) a_1 = (Algebra.Presentation.aux Q P) a + q
|
obtain ⟨b, rfl⟩ := hq
|
case h_add.intro.intro
R : Type u
S : Type v
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
T : Type u_3
inst✝¹ : CommRing T
inst✝ : Algebra S T
Q : Presentation S T
P : Presentation R S
a b : MvPolynomial (Q.vars ⊕ P.vars) R
⊢ ∃ a_1, (Algebra.Presentation.aux Q P) a_1 = (Algebra.Presentation.aux Q P) a + (Algebra.Presentation.aux Q P) b
|
6a819377477f7ce9
|
smul_eq_self_of_preimage_zpow_eq_self
|
Mathlib/Data/Set/Pointwise/Iterate.lean
|
theorem smul_eq_self_of_preimage_zpow_eq_self {G : Type*} [CommGroup G] {n : ℤ} {s : Set G}
(hs : (fun x => x ^ n) ⁻¹' s = s) {g : G} {j : ℕ} (hg : g ^ n ^ j = 1) : g • s = s
|
case intro.intro
G : Type u_1
inst✝ : CommGroup G
n : ℤ
s : Set G
hs : (fun x => x ^ n) ⁻¹' s = s
g : G
j : ℕ
hg : g ^ n ^ j = 1
g' : G
hg' : g' ^ n ^ j = 1
y : G
hy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s
⊢ (⇑(zpowGroupHom n))^[j] (g' * y) ∈ s
|
replace hg' : (zpowGroupHom n)^[j] g' = 1 := by simpa [zpowGroupHom]
|
case intro.intro
G : Type u_1
inst✝ : CommGroup G
n : ℤ
s : Set G
hs : (fun x => x ^ n) ⁻¹' s = s
g : G
j : ℕ
hg : g ^ n ^ j = 1
g' y : G
hy : y ∈ (fun x => x ^ n)^[j] ⁻¹' s
hg' : (⇑(zpowGroupHom n))^[j] g' = 1
⊢ (⇑(zpowGroupHom n))^[j] (g' * y) ∈ s
|
c18d60c68b83e353
|
isAlgebraic_of_isLocalization
|
Mathlib/RingTheory/Localization/Integral.lean
|
lemma isAlgebraic_of_isLocalization {R} [CommRing R] (M : Submonoid R) (S) [CommRing S]
[Nontrivial R] [Algebra R S] [IsLocalization M S] : Algebra.IsAlgebraic R S
|
case isAlgebraic.intro.intro
R : Type u_5
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_6
inst✝³ : CommRing S
inst✝² : Nontrivial R
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : R
s : ↥M
⊢ IsAlgebraic R (mk' S x s)
|
by_cases hs : (s : R) = 0
|
case pos
R : Type u_5
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_6
inst✝³ : CommRing S
inst✝² : Nontrivial R
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : R
s : ↥M
hs : ↑s = 0
⊢ IsAlgebraic R (mk' S x s)
case neg
R : Type u_5
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_6
inst✝³ : CommRing S
inst✝² : Nontrivial R
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
x : R
s : ↥M
hs : ¬↑s = 0
⊢ IsAlgebraic R (mk' S x s)
|
5abd58f488e94823
|
Convex.convex_remove_iff_not_mem_convexHull_remove
|
Mathlib/Analysis/Convex/Hull.lean
|
theorem Convex.convex_remove_iff_not_mem_convexHull_remove {s : Set E} (hs : Convex 𝕜 s) (x : E) :
Convex 𝕜 (s \ {x}) ↔ x ∉ convexHull 𝕜 (s \ {x})
|
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
x : E
hx : x ∉ (convexHull 𝕜) (s \ {x})
y : E
hy : y ∈ (convexHull 𝕜) (s \ {x})
⊢ y ∉ {x}
|
rintro (rfl : y = x)
|
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedSemiring 𝕜
inst✝¹ : AddCommMonoid E
inst✝ : Module 𝕜 E
s : Set E
hs : Convex 𝕜 s
y : E
hx : y ∉ (convexHull 𝕜) (s \ {y})
hy : y ∈ (convexHull 𝕜) (s \ {y})
⊢ False
|
2f0adfe15b540bae
|
AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι
|
Mathlib/AlgebraicGeometry/IdealSheaf.lean
|
lemma ideal_le_ker_glueDataObjι (U V : X.affineOpens) :
I.ideal V ≤ RingHom.ker (U.1.ι.app V.1 ≫ (I.glueDataObjι U).app _).hom
|
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
⊢ I.ideal V ≤ RingHom.ker (CommRingCat.Hom.hom (Hom.app (↑U).ι ↑V ≫ Hom.app (I.glueDataObjι U) ((↑U).ι ⁻¹ᵁ ↑V)))
|
intro x hx
|
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑Γ(X, ↑V)
hx : x ∈ I.ideal V
⊢ x ∈ RingHom.ker (CommRingCat.Hom.hom (Hom.app (↑U).ι ↑V ≫ Hom.app (I.glueDataObjι U) ((↑U).ι ⁻¹ᵁ ↑V)))
|
9da7a9bdc95919a0
|
le_hasProd
|
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
theorem le_hasProd (hf : HasProd f a) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) : f i ≤ a :=
calc
f i = ∏ i ∈ {i}, f i
|
ι : Type u_1
α : Type u_3
inst✝² : OrderedCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
f : ι → α
a : α
hf : HasProd f a
i : ι
hb : ∀ (j : ι), j ≠ i → 1 ≤ f j
⊢ f i = ∏ i ∈ {i}, f i
|
rw [prod_singleton]
|
no goals
|
19ef94c7e446040f
|
Complex.norm_one_add_mul_inv_le
|
Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean
|
/-- Give a bound on `‖(1 + t * z)⁻¹‖` for `0 ≤ t ≤ 1` and `‖z‖ < 1`. -/
lemma norm_one_add_mul_inv_le {t : ℝ} (ht : t ∈ Set.Icc 0 1) {z : ℂ} (hz : ‖z‖ < 1) :
‖(1 + t * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹
|
t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ ‖1 + ↑t * z‖⁻¹ ≤ (1 - ‖z‖)⁻¹
|
refine inv_anti₀ (by linarith) ?_
|
t : ℝ
ht : 0 ≤ t ∧ t ≤ 1
z : ℂ
hz : ‖z‖ < 1
⊢ 1 - ‖z‖ ≤ ‖1 + ↑t * z‖
|
3995f690a9b2d2b3
|
Module.End.isNilpotent_restrict_of_le
|
Mathlib/RingTheory/Nilpotent/Lemmas.lean
|
lemma isNilpotent_restrict_of_le {f : End R M} {p q : Submodule R M}
{hp : MapsTo f p p} {hq : MapsTo f q q} (h : p ≤ q) (hf : IsNilpotent (f.restrict hq)) :
IsNilpotent (f.restrict hp)
|
case h.h.mk.a
R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : End R M
p q : Submodule R M
hp : MapsTo ⇑f ↑p ↑p
hq : MapsTo ⇑f ↑q ↑q
h : p ≤ q
n : ℕ
x : M
hx : x ∈ p
hn : (LinearMap.restrict f hq ^ n) ⟨x, ⋯⟩ = 0 ⟨x, ⋯⟩
⊢ ↑((LinearMap.restrict f hp ^ n) ⟨x, hx⟩) = ↑(0 ⟨x, hx⟩)
|
simp_rw [LinearMap.zero_apply, ZeroMemClass.coe_zero, ZeroMemClass.coe_eq_zero] at hn ⊢
|
case h.h.mk.a
R : Type u_1
M : Type u_3
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : End R M
p q : Submodule R M
hp : MapsTo ⇑f ↑p ↑p
hq : MapsTo ⇑f ↑q ↑q
h : p ≤ q
n : ℕ
x : M
hx : x ∈ p
hn : (LinearMap.restrict f hq ^ n) ⟨x, ⋯⟩ = 0
⊢ (LinearMap.restrict f hp ^ n) ⟨x, hx⟩ = 0
|
0c6b0966b16b1e56
|
sSup_eq_bot'
|
Mathlib/Order/CompleteLattice.lean
|
lemma sSup_eq_bot' {s : Set α} : sSup s = ⊥ ↔ s = ∅ ∨ s = {⊥}
|
α : Type u_1
inst✝ : CompleteLattice α
s : Set α
⊢ sSup s = ⊥ ↔ s = ∅ ∨ s = {⊥}
|
rw [sSup_eq_bot, ← subset_singleton_iff_eq, subset_singleton_iff]
|
no goals
|
1cefef35bb8747b5
|
FiberBundle.totalSpaceMk_isClosedEmbedding
|
Mathlib/Topology/FiberBundle/Basic.lean
|
theorem totalSpaceMk_isClosedEmbedding [T1Space B] (x : B) :
IsClosedEmbedding (@TotalSpace.mk B F E x) :=
⟨totalSpaceMk_isEmbedding F E x, by
rw [TotalSpace.range_mk]
exact isClosed_singleton.preimage <| continuous_proj F E⟩
|
B : Type u_2
F : Type u_3
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace F
E : B → Type u_5
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (b : B) → TopologicalSpace (E b)
inst✝¹ : FiberBundle F E
inst✝ : T1Space B
x : B
⊢ IsClosed (range (TotalSpace.mk x))
|
rw [TotalSpace.range_mk]
|
B : Type u_2
F : Type u_3
inst✝⁵ : TopologicalSpace B
inst✝⁴ : TopologicalSpace F
E : B → Type u_5
inst✝³ : TopologicalSpace (TotalSpace F E)
inst✝² : (b : B) → TopologicalSpace (E b)
inst✝¹ : FiberBundle F E
inst✝ : T1Space B
x : B
⊢ IsClosed (TotalSpace.proj ⁻¹' {x})
|
afde15389e7a6a5b
|
MeasureTheory.exists_lt_lowerSemicontinuous_integral_gt_nnreal
|
Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
|
theorem exists_lt_lowerSemicontinuous_integral_gt_nnreal [SigmaFinite μ] (f : α → ℝ≥0)
(fint : Integrable (fun x => (f x : ℝ)) μ) {ε : ℝ} (εpos : 0 < ε) :
∃ g : α → ℝ≥0∞,
(∀ x, (f x : ℝ≥0∞) < g x) ∧
LowerSemicontinuous g ∧
(∀ᵐ x ∂μ, g x < ⊤) ∧
Integrable (fun x => (g x).toReal) μ ∧ (∫ x, (g x).toReal ∂μ) < (∫ x, ↑(f x) ∂μ) + ε
|
case intro.intro.intro.intro.intro.intro.refine_2.hfm
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable (fun x => ↑(f x)) μ
fmeas : AEMeasurable f μ
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤
g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤
Ig : ∫⁻ (a : α), ENNReal.ofReal (g a).toReal ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ AEStronglyMeasurable (fun x => (g x).toReal) μ
|
apply gcont.measurable.ennreal_toReal.aemeasurable.aestronglyMeasurable
|
no goals
|
3f39b00d374e3aed
|
Finset.eq_one_of_prod_eq_one
|
Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
theorem eq_one_of_prod_eq_one {s : Finset α} {f : α → β} {a : α} (hp : ∏ x ∈ s, f x = 1)
(h1 : ∀ x ∈ s, x ≠ a → f x = 1) : ∀ x ∈ s, f x = 1
|
case pos
α : Type u_3
β : Type u_4
inst✝ : CommMonoid β
s : Finset α
f : α → β
a : α
hp : ∏ x ∈ s, f x = 1
h1 : ∀ x ∈ s, x ≠ a → f x = 1
x : α
hx : x ∈ s
h : x = a
⊢ f x = 1
|
rw [h]
|
case pos
α : Type u_3
β : Type u_4
inst✝ : CommMonoid β
s : Finset α
f : α → β
a : α
hp : ∏ x ∈ s, f x = 1
h1 : ∀ x ∈ s, x ≠ a → f x = 1
x : α
hx : x ∈ s
h : x = a
⊢ f a = 1
|
15f6948a1ae4c501
|
Real.hasSum_log_sub_log_of_abs_lt_one
|
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
theorem hasSum_log_sub_log_of_abs_lt_one {x : ℝ} (h : |x| < 1) :
HasSum (fun k : ℕ => (2 : ℝ) * (1 / (2 * k + 1)) * x ^ (2 * k + 1))
(log (1 + x) - log (1 - x))
|
x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n => -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x => 2 * x) = fun k => 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
⊢ HasSum term (log (1 + x) - log (1 - x))
|
have h₁ := (hasSum_pow_div_log_of_abs_lt_one (Eq.trans_lt (abs_neg x) h)).mul_left (-1)
|
x : ℝ
h : |x| < 1
term : ℕ → ℝ := fun n => -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)
h_term_eq_goal : (term ∘ fun x => 2 * x) = fun k => 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)
h₁ : HasSum (fun i => -1 * ((-x) ^ (i + 1) / (↑i + 1))) (-1 * -log (1 - -x))
⊢ HasSum term (log (1 + x) - log (1 - x))
|
ad64e8a77fc6c38e
|
AlgebraicGeometry.LocallyRingedSpace.HasCoequalizer.coequalizer_π_app_isLocalHom
|
Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
|
theorem coequalizer_π_app_isLocalHom
(U : TopologicalSpace.Opens (coequalizer f.toShHom g.toShHom).carrier) :
IsLocalHom ((coequalizer.π f.toShHom g.toShHom :).c.app (op U)).hom
|
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace
this✝¹ :
coequalizer.π (SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom f))
(SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom g)) ≫
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f) (Hom.toShHom g)).hom =
SheafedSpace.forgetToPresheafedSpace.map (coequalizer.π (Hom.toShHom f) (Hom.toShHom g))
this✝ : IsIso (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f) (Hom.toShHom g)).hom.c
this :
IsLocalHom
(CommRingCat.Hom.hom
(limit.π
(PresheafedSpace.componentwiseDiagram
(parallelPair (SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom f))
(SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom g)))
((Opens.map
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f)
(Hom.toShHom g)).hom.base).obj
(unop (op U))))
(op WalkingParallelPair.one)))
⊢ IsLocalHom
(CommRingCat.Hom.hom
(((PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f) (Hom.toShHom g)).hom.c.app
(op U) ≫
(PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit
(parallelPair (SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom f))
(SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom g)))
((Opens.map
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f)
(Hom.toShHom g)).hom.base).obj
(unop (op U)))).hom ≫
limit.π
(PresheafedSpace.componentwiseDiagram
(parallelPair (SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom f))
(SheafedSpace.forgetToPresheafedSpace.map (Hom.toShHom g)))
((Opens.map
(PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f)
(Hom.toShHom g)).hom.base).obj
(unop (op U))))
(op WalkingParallelPair.one)) ≫
Y.presheaf.map (eqToHom ⋯)))
|
infer_instance
|
no goals
|
f9571eebfdddf194
|
CoxeterSystem.getElem_leftInvSeq_alternatingWord
|
Mathlib/GroupTheory/Coxeter/Inversion.lean
|
theorem getElem_leftInvSeq_alternatingWord
(i j : B) (p k : ℕ) (h : k < 2 * p) :
(lis (alternatingWord i j (2 * p)))[k]'(by simp; omega) =
π alternatingWord j i (2 * k + 1)
|
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
p : ℕ
i j : B
h : 0 < 2 * p
⊢ 0 < (alternatingWord i j (2 * p)).length
|
simp [h]
|
no goals
|
d39ee0dd3fb7776c
|
Std.Sat.CNF.any_not_isEmpty_iff_exists_mem
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Basic.lean
|
theorem any_not_isEmpty_iff_exists_mem {f : CNF α} :
(List.any f fun c => !List.isEmpty c) = true ↔ ∃ v, Mem v f
|
case mpr
α : Type u_1
f : CNF α
⊢ (∃ v c, c ∈ f ∧ ((v, false) ∈ c ∨ (v, true) ∈ c)) → ∃ x, x ∈ f ∧ ∃ x_1, x_1 ∈ x
|
intro h
|
case mpr
α : Type u_1
f : CNF α
h : ∃ v c, c ∈ f ∧ ((v, false) ∈ c ∨ (v, true) ∈ c)
⊢ ∃ x, x ∈ f ∧ ∃ x_1, x_1 ∈ x
|
cfda4c463f240c75
|
UniformConcaveOn.neg
|
Mathlib/Analysis/Convex/Strong.lean
|
lemma UniformConcaveOn.neg (hf : UniformConcaveOn s φ f) : UniformConvexOn s φ (-f)
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
φ : ℝ → ℝ
s : Set E
f : E → ℝ
hf : UniformConcaveOn s φ f
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ -(a • (-f) x + b • (-f) y - a * b * φ ‖x - y‖) ≤ -(-f) (a • x + b • y)
|
simpa [add_comm, -neg_le_neg_iff, ← le_sub_iff_add_le', sub_eq_add_neg, neg_add]
using hf.2 hx hy ha hb hab
|
no goals
|
4dfbf23a0715fa0c
|
Geometry.SimplicialComplex.vertex_mem_convexHull_iff
|
Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
|
theorem vertex_mem_convexHull_iff (hx : x ∈ K.vertices) (hs : s ∈ K.faces) :
x ∈ convexHull 𝕜 (s : Set E) ↔ x ∈ s
|
𝕜 : Type u_1
E : Type u_2
inst✝² : OrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
K : SimplicialComplex 𝕜 E
s : Finset E
x : E
hx : x ∈ K.vertices
hs : s ∈ K.faces
h : x ∈ (convexHull 𝕜) ↑s
⊢ x ∈ (convexHull 𝕜) ↑{x}
|
simp
|
no goals
|
cac78c48089373e1
|
Rat.fract_inv_num_lt_num_of_pos
|
Mathlib/Data/Rat/Floor.lean
|
theorem fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) : (fract q⁻¹).num < q.num
|
q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
⊢ ↑q.den - q.num * ⌊q_inv⌋ < q.num
|
have q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * q⁻¹.den < q⁻¹.den := by
have : q⁻¹.num < (⌊q⁻¹⌋ + 1) * q⁻¹.den := Rat.num_lt_succ_floor_mul_den q⁻¹
have : q⁻¹.num < ⌊q⁻¹⌋ * q⁻¹.den + q⁻¹.den := by rwa [right_distrib, one_mul] at this
rwa [← sub_lt_iff_lt_add'] at this
|
q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den
⊢ ↑q.den - q.num * ⌊q_inv⌋ < q.num
|
d2870a8c7e295972
|
LinearEquiv.charpoly_conj
|
Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean
|
@[simp]
lemma LinearEquiv.charpoly_conj (e : M₁ ≃ₗ[R] M₂) (φ : Module.End R M₁) :
(e.conj φ).charpoly = φ.charpoly
|
R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : Nontrivial R
inst✝⁷ : AddCommGroup M₁
inst✝⁶ : Module R M₁
inst✝⁵ : Module.Finite R M₁
inst✝⁴ : Module.Free R M₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Module.Free R M₂
e : M₁ ≃ₗ[R] M₂
φ : Module.End R M₁
⊢ LinearMap.charpoly (e.conj φ) = LinearMap.charpoly φ
|
let b := chooseBasis R M₁
|
R : Type u_1
M₁ : Type u_3
M₂ : Type u_4
inst✝⁹ : CommRing R
inst✝⁸ : Nontrivial R
inst✝⁷ : AddCommGroup M₁
inst✝⁶ : Module R M₁
inst✝⁵ : Module.Finite R M₁
inst✝⁴ : Module.Free R M₁
inst✝³ : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : Module.Finite R M₂
inst✝ : Module.Free R M₂
e : M₁ ≃ₗ[R] M₂
φ : Module.End R M₁
b : Basis (ChooseBasisIndex R M₁) R M₁ := chooseBasis R M₁
⊢ LinearMap.charpoly (e.conj φ) = LinearMap.charpoly φ
|
b91ddefebec43a56
|
Finsupp.single_smul
|
Mathlib/Data/Finsupp/SMul.lean
|
theorem single_smul (a b : α) (f : α → M) (r : R) : single a r b • f a = single a (r • f b) b
|
α : Type u_1
M : Type u_3
R : Type u_6
inst✝² : Zero M
inst✝¹ : MonoidWithZero R
inst✝ : MulActionWithZero R M
a b : α
f : α → M
r : R
⊢ (single a r) b • f a = (single a (r • f b)) b
|
by_cases h : a = b <;> simp [h]
|
no goals
|
01076ce47b0d8e31
|
CoalgebraCat.MonoidalCategoryAux.rightUnitor_hom_toLinearMap
|
Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
|
theorem rightUnitor_hom_toLinearMap :
(ρ_ (CoalgebraCat.of R M)).hom.1.toLinearMap = (TensorProduct.rid R M).toLinearMap :=
TensorProduct.ext <| by ext; rfl
|
R : Type u
inst✝³ : CommRing R
M : Type u
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Coalgebra R M
⊢ (TensorProduct.mk R
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (of R M)))).X)
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (𝟙_ (CoalgebraCat R))))).X)).compr₂
(ρ_ (of R M)).hom.toCoalgHom'.toLinearMap =
(TensorProduct.mk R
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (of R M)))).X)
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (𝟙_ (CoalgebraCat R))))).X)).compr₂
↑(TensorProduct.rid R M)
|
ext
|
case h.h
R : Type u
inst✝³ : CommRing R
M : Type u
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Coalgebra R M
x✝ :
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (of R M)))).X)
⊢ (((TensorProduct.mk R
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (of R M)))).X)
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (𝟙_ (CoalgebraCat R))))).X)).compr₂
(ρ_ (of R M)).hom.toCoalgHom'.toLinearMap)
x✝)
1 =
(((TensorProduct.mk R
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (of R M)))).X)
↑(Opposite.unop
(Opposite.unop
((Comon_.Comon_EquivMon_OpOp (ModuleCat R)).symm.inverse.obj
((comonEquivalence R).symm.inverse.obj (𝟙_ (CoalgebraCat R))))).X)).compr₂
↑(TensorProduct.rid R M))
x✝)
1
|
2cd05fc798c201cd
|
exists_norm_eq_iInf_of_complete_convex
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
(h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by
let δ := ⨅ w : K, ‖u - w‖
letI : Nonempty K := ne.to_subtype
have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _
have δ_le : ∀ w : K, δ ≤ ‖u - w‖ := ciInf_le ⟨0, Set.forall_mem_range.2 fun _ => norm_nonneg _⟩
have δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖ := fun w hw => δ_le ⟨w, hw⟩
-- Step 1: since `δ` is the infimum, can find a sequence `w : ℕ → K` in `K`
-- such that `‖u - w n‖ < δ + 1 / (n + 1)` (which implies `‖u - w n‖ --> δ`);
-- maybe this should be a separate lemma
have exists_seq : ∃ w : ℕ → K, ∀ n, ‖u - w n‖ < δ + 1 / (n + 1)
|
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n => ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n => 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
⊢ ‖u + u - (wq + wp)‖ * ‖u + u - (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = ‖a + b‖ * ‖a + b‖ + ‖a - b‖ * ‖a - b‖
|
have eq₁ : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm
|
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
ne : K.Nonempty
h₁ : IsComplete K
h₂ : Convex ℝ K
u : F
δ : ℝ := ⨅ w, ‖u - ↑w‖
this : Nonempty ↑K := Set.Nonempty.to_subtype ne
zero_le_δ : 0 ≤ δ
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
w : ℕ → ↑K
hw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1 / (↑n + 1)
norm_tendsto : Tendsto (fun n => ‖u - ↑(w n)‖) atTop (𝓝 δ)
b✝ : ℕ → ℝ := fun n => 8 * δ * (1 / (↑n + 1)) + 4 * (1 / (↑n + 1)) * (1 / (↑n + 1))
p q N : ℕ
hp : N ≤ p
hq : N ≤ q
wp : F := ↑(w p)
wq : F := ↑(w q)
a : F := u - wq
b : F := u - wp
half : ℝ := 1 / 2
div : ℝ := 1 / (↑N + 1)
eq₁ : wp - wq = a - b
⊢ ‖u + u - (wq + wp)‖ * ‖u + u - (wq + wp)‖ + ‖wp - wq‖ * ‖wp - wq‖ = ‖a + b‖ * ‖a + b‖ + ‖a - b‖ * ‖a - b‖
|
1aed30586c16c1ed
|
Array.any_toList
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem any_toList {p : α → Bool} (as : Array α) : as.toList.any p = as.any p
|
α : Type u_1
p : α → Bool
as : Array α
⊢ (∃ x, (∃ i h, as[i] = x) ∧ p x = true) ↔ ∃ i x, p as[i] = true
|
exact ⟨fun ⟨_, ⟨i, w, rfl⟩, h⟩ => ⟨i, w, h⟩, fun ⟨i, w, h⟩ => ⟨_, ⟨i, w, rfl⟩, h⟩⟩
|
no goals
|
d0770c9f684b3b7f
|
Con.comap_conGen_equiv
|
Mathlib/GroupTheory/Congruence/Basic.lean
|
theorem comap_conGen_equiv {M N : Type*} [Mul M] [Mul N] (f : MulEquiv M N) (rel : N → N → Prop) :
Con.comap f (map_mul f) (conGen rel) = conGen (fun x y ↦ rel (f x) (f y))
|
M : Type u_4
N : Type u_5
inst✝¹ : Mul M
inst✝ : Mul N
f : M ≃* N
rel : N → N → Prop
a✝² b✝ : M
h : (conGen rel) (f a✝²) (f b✝)
n1 n2 w x y z : N
a✝¹ : ConGen.Rel rel w x
a✝ : ConGen.Rel rel y z
ih : ∀ (a b : M), f a = w → f b = x → (conGen fun x y => rel (f x) (f y)) a b
ih1 : ∀ (a b : M), f a = y → f b = z → (conGen fun x y => rel (f x) (f y)) a b
a b : M
fa : a = f.symm w * f.symm y
fb : b = f.symm x * f.symm z
⊢ f (f.symm x) = x
|
simp
|
no goals
|
d315948435eb07b2
|
Polynomial.bernoulli_three_eval_one_quarter
|
Mathlib/NumberTheory/ZetaValues.lean
|
theorem Polynomial.bernoulli_three_eval_one_quarter :
(Polynomial.bernoulli 3).eval (1 / 4) = 3 / 64
|
⊢ 2 ≠ 1
|
decide
|
no goals
|
729db16dc9d12f16
|
isJacobsonRing_iff_prime_eq
|
Mathlib/RingTheory/Jacobson/Ring.lean
|
theorem isJacobsonRing_iff_prime_eq :
IsJacobsonRing R ↔ ∀ P : Ideal R, IsPrime P → P.jacobson = P
|
R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
⊢ x ∈ I
|
rw [← hI.radical, radical_eq_sInf I, mem_sInf]
|
R : Type u_1
inst✝ : CommRing R
h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P
I : Ideal R
hI : I.IsRadical
x : R
hx : x ∈ I.jacobson
⊢ ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsPrime} → x ∈ I_1
|
4be15f6834fa58b0
|
Finset.sum_Ico_by_parts
|
Mathlib/Algebra/BigOperators/Module.lean
|
theorem sum_Ico_by_parts (hmn : m < n) :
∑ i ∈ Ico m n, f i • g i =
f (n - 1) • G n - f m • G m - ∑ i ∈ Ico m (n - 1), (f (i + 1) - f i) • G (i + 1)
|
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
h₁ : ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range i, g i = ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i
h₂ :
∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range (i + 1), g i =
∑ i ∈ Ico m (n - 1), f i • ∑ i ∈ range (i + 1), g i + f (n - 1) • ∑ i ∈ range n, g i -
f m • ∑ i ∈ range (m + 1), g i
i : ℕ
⊢ f i • ∑ i ∈ range (i + 1), g i - f (i + 1) • ∑ i ∈ range (i + 1), g i =
-(f (i + 1) • ∑ i ∈ range (i + 1), g i - f i • ∑ i ∈ range (i + 1), g i)
|
abel
|
no goals
|
35456e90ec4d4807
|
Cubic.c_eq_three_roots
|
Mathlib/Algebra/CubicDiscriminant.lean
|
theorem c_eq_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.c = φ P.a * (x * y + x * z + y * z)
|
F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ φ P.c = φ P.a * (x * y + x * z + y * z)
|
injection eq_sum_three_roots ha h3
|
no goals
|
bbbbb08af3a215b6
|
MonomialOrder.degree_lt_iff
|
Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean
|
theorem degree_lt_iff {f : MvPolynomial σ R} {d : σ →₀ ℕ} (hd : 0 ≺[m] d) :
m.degree f ≺[m] d ↔ ∀ c ∈ f.support, c ≺[m] d
|
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
d : σ →₀ ℕ
hd : 0 < m.toSyn d
⊢ m.toSyn (m.degree f) < m.toSyn d ↔ ∀ c ∈ f.support, m.toSyn c < m.toSyn d
|
unfold degree
|
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
d : σ →₀ ℕ
hd : 0 < m.toSyn d
⊢ m.toSyn (m.toSyn.symm (f.support.sup ⇑m.toSyn)) < m.toSyn d ↔ ∀ c ∈ f.support, m.toSyn c < m.toSyn d
|
6c45fc57d2fc6548
|
AList.empty_lookupFinsupp
|
Mathlib/Data/Finsupp/AList.lean
|
theorem empty_lookupFinsupp : lookupFinsupp (∅ : AList fun _x : α => M) = 0
|
α : Type u_1
M : Type u_2
inst✝ : Zero M
⊢ ∅.lookupFinsupp = 0
|
classical
ext
simp
|
no goals
|
b378aa077b932512
|
FixedPoints.minpoly.eval₂
|
Mathlib/FieldTheory/Fixed.lean
|
theorem eval₂ :
Polynomial.eval₂ (Subring.subtype <| (FixedPoints.subfield G F).toSubring) x (minpoly G F x) =
0
|
G : Type u
inst✝³ : Group G
F : Type v
inst✝² : Field F
inst✝¹ : MulSemiringAction G F
inst✝ : Fintype G
x : F
⊢ Polynomial.eval₂ (subfield G F).subtype x (minpoly G F x) = 0
|
rw [← prodXSubSMul.eval G F x, Polynomial.eval₂_eq_eval_map]
|
G : Type u
inst✝³ : Group G
F : Type v
inst✝² : Field F
inst✝¹ : MulSemiringAction G F
inst✝ : Fintype G
x : F
⊢ Polynomial.eval x (Polynomial.map (subfield G F).subtype (minpoly G F x)) = Polynomial.eval x (prodXSubSMul G F x)
|
a7d25a02e432b675
|
CategoryTheory.Functor.pi'_eval
|
Mathlib/CategoryTheory/Pi/Basic.lean
|
theorem pi'_eval (f : ∀ i, A ⥤ C i) (i : I) : pi' f ⋙ Pi.eval C i = f i
|
case h_map
I : Type w₀
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
⊢ autoParam (∀ (X Y : A) (f_1 : X ⟶ Y), (pi' f ⋙ Pi.eval C i).map f_1 = eqToHom ⋯ ≫ (f i).map f_1 ≫ eqToHom ⋯) _auto✝
|
intro _ _ _
|
case h_map
I : Type w₀
C : I → Type u₁
inst✝¹ : (i : I) → Category.{v₁, u₁} (C i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
X✝ Y✝ : A
f✝ : X✝ ⟶ Y✝
⊢ (pi' f ⋙ Pi.eval C i).map f✝ = eqToHom ⋯ ≫ (f i).map f✝ ≫ eqToHom ⋯
|
44f316420514ba3a
|
TendstoUniformly.comp
|
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
theorem TendstoUniformly.comp (h : TendstoUniformly F f p) (g : γ → α) :
TendstoUniformly (fun n => F n ∘ g) (f ∘ g) p
|
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
h : TendstoUniformlyOnFilter F f p ⊤
g : γ → α
⊢ TendstoUniformlyOnFilter (fun n => F n ∘ g) (f ∘ g) p ⊤
|
simpa [principal_univ, comap_principal] using h.comp g
|
no goals
|
b08bd153e5f5b4a4
|
AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime
|
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
|
theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
(carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
(carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
show (∀ _, _ ∈ _) ∨ ∀ _, _ ∈ _ by
rw [← and_forall_ne nx, and_iff_left, ← and_forall_ne ny, and_iff_left]
· apply q.2.mem_or_mem; convert hxy (nx + ny) using 1
dsimp
simp_rw [decompose_of_mem_same 𝒜 hnx, decompose_of_mem_same 𝒜 hny,
decompose_of_mem_same 𝒜 (SetLike.GradedMonoid.toGradedMul.mul_mem hnx hny),
mul_pow, pow_add]
simp only [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
HomogeneousLocalization.val_mul, Localization.mk_mul]
simp only [Submonoid.mk_mul_mk, mk_eq_monoidOf_mk']
all_goals
intro n hn; convert q.1.zero_mem using 1
rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
HomogeneousLocalization.val_zero]; simp_rw [proj_apply]
convert mk_zero (S := Submonoid.powers f) _
rw [decompose_of_mem_ne 𝒜 _ hn.symm, zero_pow hm.ne']
· first | exact hnx | exact hny
|
case h.e'_5
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } = 0
|
rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
HomogeneousLocalization.val_zero]
|
case h.e'_5
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
n : ℕ
hn : n ≠ nx
⊢ Localization.mk ↑{ deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }.num
⟨↑{ deg := m * n, num := ⟨(proj 𝒜 n) x ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ =
0
|
5ba35f6f859f61e0
|
IsCompact.image_of_continuousOn
|
Mathlib/Topology/Compactness/Compact.lean
|
theorem IsCompact.image_of_continuousOn {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s) :
IsCompact (f '' s)
|
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsCompact s
hf : ContinuousOn f s
⊢ IsCompact (f '' s)
|
intro l lne ls
|
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
hs : IsCompact s
hf : ContinuousOn f s
l : Filter Y
lne : l.NeBot
ls : l ≤ 𝓟 (f '' s)
⊢ ∃ x ∈ f '' s, ClusterPt x l
|
2cb55c7533cef93b
|
fderivWithin_fderivWithin_eq_of_mem_nhdsWithin
|
Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean
|
lemma fderivWithin_fderivWithin_eq_of_mem_nhdsWithin (h : t ∈ 𝓝[s] x)
(hf : ContDiffWithinAt 𝕜 2 f t x) (hs : UniqueDiffOn 𝕜 s) (ht : UniqueDiffOn 𝕜 t) (hx : x ∈ s) :
fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x = fderivWithin 𝕜 (fderivWithin 𝕜 f t) t x
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s t : Set E
f : E → F
x : E
h : t ∈ 𝓝[s] x
hf : ContDiffWithinAt 𝕜 2 f t x
hs : UniqueDiffOn 𝕜 s
ht : UniqueDiffOn 𝕜 t
hx : x ∈ s
A : ∀ᶠ (y : E) in 𝓝[s] x, fderivWithin 𝕜 f s y = fderivWithin 𝕜 f t y
⊢ fderivWithin 𝕜 f s x = fderivWithin 𝕜 f t x
|
exact fderivWithin_of_mem_nhdsWithin h (hs x hx) (hf.differentiableWithinAt one_le_two)
|
no goals
|
338e0eff11ac573d
|
MeasureTheory.IsStoppingTime.measurableSet_inter_le
|
Mathlib/Probability/Process/Stopping.lean
|
theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι] [OrderTopology ι]
[MeasurableSpace ι] [BorelSpace ι] (hτ : IsStoppingTime f τ) (hπ : IsStoppingTime f π)
(s : Set Ω) (hs : MeasurableSet[hτ.measurableSpace] s) :
MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω})
|
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
this :
s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ⊓ π ω ≤ i} = s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ⊓ π ω ≤ i} ∩ {ω | τ ω ⊓ i ≤ τ ω ⊓ π ω ⊓ i}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ⊓ π ω ≤ i})
|
rw [this]
|
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : LinearOrder ι
f : Filtration ι m
τ π : Ω → ι
inst✝⁴ : TopologicalSpace ι
inst✝³ : SecondCountableTopology ι
inst✝² : OrderTopology ι
inst✝¹ : MeasurableSpace ι
inst✝ : BorelSpace ι
hτ : IsStoppingTime f τ
hπ : IsStoppingTime f π
s : Set Ω
hs : ∀ (i : ι), MeasurableSet (s ∩ {ω | τ ω ≤ i})
i : ι
this :
s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ⊓ π ω ≤ i} = s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ⊓ π ω ≤ i} ∩ {ω | τ ω ⊓ i ≤ τ ω ⊓ π ω ⊓ i}
⊢ MeasurableSet (s ∩ {ω | τ ω ≤ i} ∩ {ω | τ ω ⊓ π ω ≤ i} ∩ {ω | τ ω ⊓ i ≤ τ ω ⊓ π ω ⊓ i})
|
93587e3bd5de43f7
|
CategoryTheory.Limits.IsColimit.OfNatIso.coconeOfHom_fac
|
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
theorem coconeOfHom_fac {Y : C} (f : X ⟶ Y) : coconeOfHom h f = (colimitCocone h).extend f
|
case e_ι.w.h
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₃
inst✝ : Category.{v₃, u₃} C
F : J ⥤ C
X : C
h : coyoneda.obj (op X) ⋙ uliftFunctor.{u₁, v₃} ≅ F.cocones
Y : C
f : X ⟶ Y
j : J
t : h.hom.app Y { down := 𝟙 X ≫ f } = F.cocones.map f (h.hom.app X { down := 𝟙 X })
⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j
|
simp only [id_comp] at t
|
case e_ι.w.h
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
C : Type u₃
inst✝ : Category.{v₃, u₃} C
F : J ⥤ C
X : C
h : coyoneda.obj (op X) ⋙ uliftFunctor.{u₁, v₃} ≅ F.cocones
Y : C
f : X ⟶ Y
j : J
t : h.hom.app Y { down := f } = F.cocones.map f (h.hom.app X { down := 𝟙 X })
⊢ (h.hom.app Y { down := f }).app j = (h.hom.app X { down := 𝟙 X } ≫ (const J).map f).app j
|
4193177dc0262cb9
|
le_inv_iff_mul_le_one_right
|
Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
|
theorem le_inv_iff_mul_le_one_right : a ≤ b⁻¹ ↔ a * b ≤ 1 :=
(mul_le_mul_iff_right b).symm.trans <| by rw [inv_mul_cancel]
|
α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : MulRightMono α
a b : α
⊢ a * b ≤ b⁻¹ * b ↔ a * b ≤ 1
|
rw [inv_mul_cancel]
|
no goals
|
b5a254400e7116b9
|
ContinuousLinearMap.hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear
|
Mathlib/Analysis/Analytic/CPolynomial.lean
|
theorem hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear :
HasFiniteFPowerSeriesOnBall (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2)
f.toFormalMultilinearSeriesOfMultilinear 0 (Fintype.card (Option ι) + 1) ⊤
|
𝕜 : Type u_1
F : Type u_3
G : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
ι : Type u_5
Em : ι → Type u_6
inst✝² : (i : ι) → NormedAddCommGroup (Em i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)
inst✝ : Fintype ι
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 Em F
y : G × ((i : ι) → Em i)
a✝ : y ∈ EMetric.ball 0 ⊤
⊢ ((f.toFormalMultilinearSeriesOfMultilinear (Fintype.card (Option ι))) fun x => y) = (f y.1) y.2
|
rw [toFormalMultilinearSeriesOfMultilinear, dif_pos rfl]
|
𝕜 : Type u_1
F : Type u_3
G : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜 F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
ι : Type u_5
Em : ι → Type u_6
inst✝² : (i : ι) → NormedAddCommGroup (Em i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)
inst✝ : Fintype ι
f : G →L[𝕜] ContinuousMultilinearMap 𝕜 Em F
y : G × ((i : ι) → Em i)
a✝ : y ∈ EMetric.ball 0 ⊤
⊢ ((ContinuousMultilinearMap.domDomCongr (Fintype.equivFinOfCardEq ⋯) f.continuousMultilinearMapOption) fun x => y) =
(f y.1) y.2
|
9e79f3aed59fcb73
|
HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn
|
Mathlib/Analysis/Analytic/Basic.lean
|
theorem HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn
(hf : HasFPowerSeriesWithinOnBall f p s x r) :
TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
((x + ·)⁻¹' (insert x s) ∩ EMetric.ball (0 : E) r)
|
case intro.intro.refine_2
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinOnBall f p s x r
u : Set (F × F)
hu : u ∈ uniformity F
y : E
hy : y ∈ (fun x_1 => x + x_1) ⁻¹' insert x s ∩ EMetric.ball 0 r
r' : ℝ≥0
yr' : edist y 0 < ↑r'
hr' : ↑r' < r
this : EMetric.ball 0 ↑r' ∈ 𝓝 y
⊢ ∀ᶠ (n : ℕ) in atTop,
∀ y ∈ (fun x_1 => x + x_1) ⁻¹' insert x s ∩ EMetric.ball 0 ↑r',
((fun y => f (x + y)) y, (fun n y => p.partialSum n y) n y) ∈ u
|
simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
|
no goals
|
2a880b1d843248d7
|
MeasureTheory.eLpNorm_le_of_ae_nnnorm_bound
|
Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
|
theorem eLpNorm_le_of_ae_nnnorm_bound {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ C) :
eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹
|
case pos
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
C : ℝ≥0
hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C
hμ : NeZero μ
hp : p = 0
⊢ eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹
|
simp [hp]
|
no goals
|
e13c88a1505d5b94
|
IsDiscreteValuationRing.iff_pid_with_one_nonzero_prime
|
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
IsDiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P
|
case mpr.intro
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
RPID : IsPrincipalIdealRing R
Punique : ∃! P, P ≠ ⊥ ∧ P.IsPrime
this : IsLocalRing R
⊢ IsDiscreteValuationRing R
|
refine { not_a_field' := ?_ }
|
case mpr.intro
R : Type u
inst✝¹ : CommRing R
inst✝ : IsDomain R
RPID : IsPrincipalIdealRing R
Punique : ∃! P, P ≠ ⊥ ∧ P.IsPrime
this : IsLocalRing R
⊢ maximalIdeal R ≠ ⊥
|
f7521cb0caad85a8
|
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