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
|
|---|---|---|---|---|---|---|
exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt
|
Mathlib/MeasureTheory/Function/Jacobian.lean
|
theorem exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
(f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
(r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
(∀ n, IsClosed (t n)) ∧
(s ⊆ ⋃ n, t n) ∧
(∀ n, ApproximatesLinearOn f (A n) (s ∩ t n) (r (A n))) ∧
(s.Nonempty → ∀ n, ∃ y ∈ s, A n = f' y)
|
case e_a
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : SecondCountableTopology F
f : E → F
s : Set E
f' : E → E →L[ℝ] F
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
r : (E →L[ℝ] F) → ℝ≥0
rpos : ∀ (A : E →L[ℝ] F), r A ≠ 0
hs : s.Nonempty
T : Set ↑s
T_count : T.Countable
hT : ⋃ x ∈ T, ball (f' ↑x) ↑(r (f' ↑x)) = ⋃ x, ball (f' ↑x) ↑(r (f' ↑x))
u : ℕ → ℝ
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
M : ℕ → ↑T → Set E :=
fun n z => {x | x ∈ s ∧ ∀ y ∈ s ∩ ball x (u n), ‖f y - f x - (f' ↑↑z) (y - x)‖ ≤ ↑(r (f' ↑↑z)) * ‖y - x‖}
x : E
xs : x ∈ s
z : ↑s
zT : z ∈ T
hz : f' x ∈ ball (f' ↑z) ↑(r (f' ↑z))
ε : ℝ
εpos : 0 < ε
hε : ‖f' x - f' ↑z‖ + ε ≤ ↑(r (f' ↑z))
δ : ℝ
δpos : 0 < δ
hδ : ball x δ ∩ s ⊆ {y | ‖f y - f x - (f' x) (y - x)‖ ≤ ε * ‖y - x‖}
n : ℕ
hn : u n < δ
y : E
hy : y ∈ s ∩ ball x (u n)
⊢ f y - f x - ((f' ↑z) y - (f' ↑z) x) =
f y - f x - ((f' x) y - (f' x) x) + ((f' x) y - (f' ↑z) y - ((f' x) x - (f' ↑z) x))
|
abel
|
no goals
|
6721258aaeea8779
|
AkraBazziRecurrence.GrowsPolynomially.eventually_zero_of_frequently_zero
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
∀ᶠ x in atTop, f x = 0
|
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ x ⊔ 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ 1 < 2
|
norm_num
|
no goals
|
b2c5c4c5d5df705a
|
IsAdicComplete.le_jacobson_bot
|
Mathlib/RingTheory/AdicCompletion/Basic.lean
|
theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson
|
case h.a
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
⊢ ∀ (n : ℕ), (1 + -(x * y)) * L - 1 ≡ 0 [SMOD I ^ n • ⊤]
|
intro n
|
case h.a
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun n => ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]
L : R
hL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]
n : ℕ
⊢ (1 + -(x * y)) * L - 1 ≡ 0 [SMOD I ^ n • ⊤]
|
b6dffcb57fa8856c
|
IsOpen.exists_msmooth_support_eq
|
Mathlib/Geometry/Manifold/PartitionOfUnity.lean
|
theorem IsOpen.exists_msmooth_support_eq {s : Set M} (hs : IsOpen s) :
∃ f : M → ℝ, f.support = s ∧ ContMDiff I 𝓘(ℝ) ∞ f ∧ ∀ x, 0 ≤ f x
|
case intro.refine_1.refine_2.h
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x => (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) ∞ (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∉ s
⊢ ∀ (x_1 : M), (f x_1) x * g x_1 (↑(chartAt H x_1) x) = 0
|
intro c
|
case intro.refine_1.refine_2.h
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
s : Set M
hs : IsOpen s
f : SmoothPartitionOfUnity M I M
hf : f.IsSubordinate fun x => (chartAt H x).source
g : M → H → ℝ
g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s
g_diff : ∀ (c : M), ContMDiff I 𝓘(ℝ, ℝ) ∞ (g c)
hg : ∀ (c : M), range (g c) ⊆ Icc 0 1
h'g : ∀ (c : M) (x : H), 0 ≤ g c x
h''g : ∀ (c x : M), 0 ≤ (f c) x * g c (↑(chartAt H c) x)
x : M
hx : x ∉ s
c : M
⊢ (f c) x * g c (↑(chartAt H c) x) = 0
|
1c8b08bf6bae7592
|
ProbabilityTheory.Kernel.snd_prod
|
Mathlib/Probability/Kernel/Composition/Prod.lean
|
@[simp] lemma snd_prod (κ : Kernel α β) [IsMarkovKernel κ] (η : Kernel α γ) [IsSFiniteKernel η] :
snd (κ ×ₖ η) = η
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsMarkovKernel κ
η : Kernel α γ
inst✝ : IsSFiniteKernel η
⊢ (κ ×ₖ η).snd = η
|
ext x
|
case h.h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsMarkovKernel κ
η : Kernel α γ
inst✝ : IsSFiniteKernel η
x : α
s✝ : Set γ
a✝ : MeasurableSet s✝
⊢ ((κ ×ₖ η).snd x) s✝ = (η x) s✝
|
2321a1081df35ddf
|
Int.nneg_mul_add_sq_of_abs_le_one
|
Mathlib/Algebra/Order/Ring/Cast.lean
|
lemma nneg_mul_add_sq_of_abs_le_one (n : ℤ) (hx : |x| ≤ 1) : (0 : R) ≤ n * x + n * n
|
R : Type u_1
inst✝ : LinearOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hnx : 0 < n → 0 ≤ x + ↑n
hnx' : n < 0 → x + ↑n ≤ 0
⊢ 0 ≤ ↑n * x + ↑n * ↑n
|
rw [← mul_add, mul_nonneg_iff]
|
R : Type u_1
inst✝ : LinearOrderedRing R
x : R
n : ℤ
hx : |x| ≤ 1
hnx : 0 < n → 0 ≤ x + ↑n
hnx' : n < 0 → x + ↑n ≤ 0
⊢ 0 ≤ ↑n ∧ 0 ≤ x + ↑n ∨ ↑n ≤ 0 ∧ x + ↑n ≤ 0
|
bd5028d11b4586e9
|
RingSeminorm.exists_index_pow_le
|
Mathlib/Analysis/Normed/Ring/Seminorm.lean
|
theorem exists_index_pow_le (hna : IsNonarchimedean p) (x y : R) (n : ℕ) :
∃ (m : ℕ), m < n + 1 ∧ p ((x + y) ^ (n : ℕ)) ^ (1 / (n : ℝ)) ≤
(p (x ^ m) * p (y ^ (n - m : ℕ))) ^ (1 / (n : ℝ))
|
R : Type u_1
inst✝ : CommRing R
p : RingSeminorm R
hna : IsNonarchimedean ⇑p
x y : R
n : ℕ
⊢ ∃ m < n + 1, p ((x + y) ^ n) ^ (1 / ↑n) ≤ (p (x ^ m) * p (y ^ (n - m))) ^ (1 / ↑n)
|
obtain ⟨m, hm_lt, hm⟩ := IsNonarchimedean.add_pow_le hna n x y
|
case intro.intro
R : Type u_1
inst✝ : CommRing R
p : RingSeminorm R
hna : IsNonarchimedean ⇑p
x y : R
n m : ℕ
hm_lt : m < n + 1
hm : p ((x + y) ^ n) ≤ p (x ^ m) * p (y ^ (n - m))
⊢ ∃ m < n + 1, p ((x + y) ^ n) ^ (1 / ↑n) ≤ (p (x ^ m) * p (y ^ (n - m))) ^ (1 / ↑n)
|
af6022ef7d300c81
|
t2Space_quotient_mulAction_of_properSMul
|
Mathlib/Topology/Algebra/ProperAction/Basic.lean
|
theorem t2Space_quotient_mulAction_of_properSMul [ProperSMul G X] :
T2Space (Quotient (MulAction.orbitRel G X))
|
G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : ProperSMul G X
R : Setoid X := MulAction.orbitRel G X
π : X → Quotient R := Quotient.mk'
this : IsOpenQuotientMap (Prod.map π π)
⊢ IsClosed (diagonal (Quotient R))
|
rw [← this.isQuotientMap.isClosed_preimage]
|
G : Type u_1
X : Type u_2
inst✝⁴ : Group G
inst✝³ : MulAction G X
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalSpace X
inst✝ : ProperSMul G X
R : Setoid X := MulAction.orbitRel G X
π : X → Quotient R := Quotient.mk'
this : IsOpenQuotientMap (Prod.map π π)
⊢ IsClosed (Prod.map π π ⁻¹' diagonal (Quotient R))
|
55cc0959d3fdc2ab
|
MeasureTheory.tilted_zero'
|
Mathlib/MeasureTheory/Measure/Tilted.lean
|
@[simp]
lemma tilted_zero' (μ : Measure α) : μ.tilted 0 = (μ Set.univ)⁻¹ • μ
|
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
⊢ μ.tilted 0 = (μ Set.univ)⁻¹ • μ
|
change μ.tilted (fun _ ↦ 0) = (μ Set.univ)⁻¹ • μ
|
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
⊢ (μ.tilted fun x => 0) = (μ Set.univ)⁻¹ • μ
|
1a8638fac4774a18
|
EuclideanSpace.single_apply
|
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
theorem EuclideanSpace.single_apply (i : ι) (a : 𝕜) (j : ι) :
(EuclideanSpace.single i a) j = ite (j = i) a 0
|
ι : Type u_1
𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq ι
i : ι
a : 𝕜
j : ι
⊢ single i a j = if j = i then a else 0
|
rw [EuclideanSpace.single, WithLp.equiv_symm_pi_apply, ← Pi.single_apply i a j]
|
no goals
|
db204a4603e31523
|
frontier_inter_open_inter
|
Mathlib/Topology/Constructions.lean
|
theorem frontier_inter_open_inter {s t : Set X} (ht : IsOpen t) :
frontier (s ∩ t) ∩ t = frontier s ∩ t
|
X : Type u
inst✝ : TopologicalSpace X
s t : Set X
ht : IsOpen t
⊢ frontier (s ∩ t) ∩ t = frontier s ∩ t
|
simp only [Set.inter_comm _ t, ← Subtype.preimage_coe_eq_preimage_coe_iff,
ht.isOpenMap_subtype_val.preimage_frontier_eq_frontier_preimage continuous_subtype_val,
Subtype.preimage_coe_self_inter]
|
no goals
|
55361fcb299f8882
|
Submodule.nonempty_basis_of_pid
|
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
(N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N)
|
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop ⊤ ⋯).symm
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
∃ bN,
∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
∃ (hnm : n' + 1 ≤ m' + 1),
∃ bM,
∀ (as : Fin n' → R),
(∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
n' : ℕ
bN' : Basis (Fin n') R ↥N'
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
|
obtain ⟨bN, _hbN⟩ := h' n' bN'
|
case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_2
inst✝⁵ : CommRing R
M : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : IsDomain R
inst✝¹ : IsPrincipalIdealRing R
ι : Type u_4
inst✝ : Finite ι
b : Basis ι R M
this : DecidableEq M
val✝ : Fintype ι
N : Submodule R M
ih : ∀ N' ≤ N, ∀ x ∈ N, (∀ (c : R), ∀ y ∈ N', c • x + y = 0 → c = 0) → ∃ n, Nonempty (Basis (Fin n) R ↥N')
b' : Basis (Fin (Fintype.card ι)) R ↥⊤ := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop ⊤ ⋯).symm
N_bot : ¬N = ⊥
y : M
a : R
hay : a • y ∈ N
M' N' : Submodule R M
N'_le_N : N' ≤ N
ay_ortho : ∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0
h' :
∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'),
∃ bN,
∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
∃ (hnm : n' + 1 ≤ m' + 1),
∃ bM,
∀ (as : Fin n' → R),
(∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
n' : ℕ
bN' : Basis (Fin n') R ↥N'
bN : Basis (Fin (n' + 1)) R ↥N
_hbN :
∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'),
∃ (hnm : n' + 1 ≤ m' + 1),
∃ bM,
∀ (as : Fin n' → R),
(∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) →
∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))
⊢ ∃ n, Nonempty (Basis (Fin n) R ↥N)
|
73a8c31f0fc25373
|
max_assoc
|
Mathlib/Order/Defs/LinearOrder.lean
|
lemma max_assoc (a b c : α) : max (max a b) c = max a (max b c)
|
case h₁
α : Type u_1
inst✝ : LinearOrder α
a b c : α
⊢ a ≤ max (max a b) c
|
apply le_trans (le_max_left a b) (le_max_left ..)
|
no goals
|
749ba1aebfbd6b8b
|
ae_eq_restrict_iff_indicator_ae_eq
|
Mathlib/MeasureTheory/Measure/Restrict.lean
|
theorem ae_eq_restrict_iff_indicator_ae_eq {g : α → β} (hs : MeasurableSet s) :
f =ᵐ[μ.restrict s] g ↔ s.indicator f =ᵐ[μ] s.indicator g
|
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
g : α → β
hs : MeasurableSet s
⊢ f =ᶠ[ae (μ.restrict s)] g ↔ s.indicator f =ᶠ[ae μ] s.indicator g
|
rw [Filter.EventuallyEq, ae_restrict_iff' hs]
|
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
μ : Measure α
s : Set α
f : α → β
inst✝ : Zero β
g : α → β
hs : MeasurableSet s
⊢ (∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x) ↔ s.indicator f =ᶠ[ae μ] s.indicator g
|
2261d3e8339b53b4
|
CliffordAlgebra.ι_mul_ι_add_swap_of_isOrtho
|
Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean
|
theorem ι_mul_ι_add_swap_of_isOrtho {a b : M} (h : Q.IsOrtho a b) :
ι Q a * ι Q b + ι Q b * ι Q a = 0
|
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
a b : M
h : QuadraticMap.IsOrtho Q a b
⊢ (ι Q) a * (ι Q) b + (ι Q) b * (ι Q) a = 0
|
rw [ι_mul_ι_add_swap, h.polar_eq_zero]
|
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
a b : M
h : QuadraticMap.IsOrtho Q a b
⊢ (algebraMap R (CliffordAlgebra Q)) 0 = 0
|
235177264bd47bd1
|
Mathlib.Tactic.Module.NF.zero_sub_eq_eval
|
Mathlib/Tactic/Module.lean
|
theorem zero_sub_eq_eval [AddCommGroup M] [Ring R] [Module R M] (l : NF R M) :
0 - l.eval = (-l).eval
|
R : Type u_2
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Ring R
inst✝ : Module R M
l : NF R M
⊢ 0 - l.eval = (-l).eval
|
simp [eval_neg]
|
no goals
|
f20da5fb84464c9f
|
LinearRecurrence.sol_eq_of_eq_init
|
Mathlib/Algebra/LinearRecurrence.lean
|
theorem sol_eq_of_eq_init (u v : ℕ → α) (hu : E.IsSolution u) (hv : E.IsSolution v) :
u = v ↔ Set.EqOn u v ↑(range E.order)
|
α : Type u_1
inst✝ : CommSemiring α
E : LinearRecurrence α
u v : ℕ → α
hu : E.IsSolution u
hv : E.IsSolution v
h : Set.EqOn u v ↑(range E.order)
u' : ↥E.solSpace := ⟨u, hu⟩
v' : ↥E.solSpace := ⟨v, hv⟩
⊢ E.toInit u' = E.toInit v'
|
ext x
|
case h
α : Type u_1
inst✝ : CommSemiring α
E : LinearRecurrence α
u v : ℕ → α
hu : E.IsSolution u
hv : E.IsSolution v
h : Set.EqOn u v ↑(range E.order)
u' : ↥E.solSpace := ⟨u, hu⟩
v' : ↥E.solSpace := ⟨v, hv⟩
x : Fin E.order
⊢ E.toInit u' x = E.toInit v' x
|
5fb080a45c739da9
|
Ordinal.not_bddAbove_isInitial
|
Mathlib/SetTheory/Cardinal/Aleph.lean
|
theorem not_bddAbove_isInitial : ¬ BddAbove {x | IsInitial x}
|
case intro
a : Ordinal.{u_1}
ha : a ∈ upperBounds {x | x.IsInitial}
this : (succ a.card).ord ≤ a
⊢ False
|
rw [ord_le] at this
|
case intro
a : Ordinal.{u_1}
ha : a ∈ upperBounds {x | x.IsInitial}
this : succ a.card ≤ a.card
⊢ False
|
35664785dffb95ea
|
FirstOrder.Language.dlo_isExtensionPair
|
Mathlib/ModelTheory/Order.lean
|
lemma dlo_isExtensionPair
(M : Type w) [Language.order.Structure M] [M ⊨ Language.order.linearOrderTheory]
(N : Type w') [Language.order.Structure N] [N ⊨ Language.order.dlo] [Nonempty N] :
Language.order.IsExtensionPair M N
|
M : Type w
inst✝⁴ : Language.order.Structure M
inst✝³ : M ⊨ Language.order.linearOrderTheory
N : Type w'
inst✝² : Language.order.Structure N
inst✝¹ : N ⊨ Language.order.dlo
inst✝ : Nonempty N
⊢ Language.order.IsExtensionPair M N
|
classical
rw [isExtensionPair_iff_exists_embedding_closure_singleton_sup]
intro S S_fg f m
letI := Language.order.linearOrderOfModels M
letI := Language.order.linearOrderOfModels N
have := Language.order.denselyOrdered_of_dlo N
have := Language.order.noBotOrder_of_dlo N
have := Language.order.noTopOrder_of_dlo N
have := NoBotOrder.to_noMinOrder N
have := NoTopOrder.to_noMaxOrder N
have hS : Set.Finite (S : Set M) := (S.fg_iff_structure_fg.1 S_fg).finite
obtain ⟨g, hg⟩ := Order.exists_orderEmbedding_insert hS.toFinset
((OrderIso.setCongr hS.toFinset (S : Set M) hS.coe_toFinset).toOrderEmbedding.trans
(OrderEmbedding.ofStrictMono f (HomClass.strictMono f))) m
let g' :
((Substructure.closure Language.order).toFun {m} ⊔ S : Language.order.Substructure M) ↪o N :=
((OrderIso.setCongr _ _ (by
convert LowerAdjoint.closure_eq_self_of_mem_closed _
(Substructure.mem_closed_of_isRelational Language.order
((insert m hS.toFinset : Finset M) : Set M))
simp only [Finset.coe_insert, Set.Finite.coe_toFinset, Substructure.closure_insert,
Substructure.closure_eq])).toOrderEmbedding.trans g)
use StrongHomClass.toEmbedding g'
ext ⟨x, xS⟩
refine congr_fun hg.symm ⟨x, (?_ : x ∈ hS.toFinset)⟩
simp only [Set.Finite.mem_toFinset, SetLike.mem_coe, xS]
|
no goals
|
688ee7e1f57a4d8d
|
Ergodic.ae_empty_or_univ_of_preimage_ae_le'
|
Mathlib/Dynamics/Ergodic/Ergodic.lean
|
theorem ae_empty_or_univ_of_preimage_ae_le' (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
(hs' : f ⁻¹' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ
|
α : Type u_1
m : MeasurableSpace α
s : Set α
f : α → α
μ : Measure α
hf : Ergodic f μ
hs : NullMeasurableSet s μ
hs' : f ⁻¹' s ≤ᶠ[ae μ] s
h_fin : μ s ≠ ⊤
⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ
|
refine hf.quasiErgodic.ae_empty_or_univ₀ hs ?_
|
α : Type u_1
m : MeasurableSpace α
s : Set α
f : α → α
μ : Measure α
hf : Ergodic f μ
hs : NullMeasurableSet s μ
hs' : f ⁻¹' s ≤ᶠ[ae μ] s
h_fin : μ s ≠ ⊤
⊢ f ⁻¹' s =ᶠ[ae μ] s
|
bb8c7ce97a4e44bf
|
Multiset.mem_sub_of_nodup
|
Mathlib/Data/Multiset/Nodup.lean
|
theorem mem_sub_of_nodup [DecidableEq α] {a : α} {s t : Multiset α} (d : Nodup s) :
a ∈ s - t ↔ a ∈ s ∧ a ∉ t :=
⟨fun h =>
⟨mem_of_le (Multiset.sub_le_self ..) h, fun h' => by
refine count_eq_zero.1 ?_ h
rw [count_sub a s t, Nat.sub_eq_zero_iff_le]
exact le_trans (nodup_iff_count_le_one.1 d _) (count_pos.2 h')⟩,
fun ⟨h₁, h₂⟩ => Or.resolve_right (mem_add.1 <| mem_of_le Multiset.le_sub_add h₁) h₂⟩
|
α : Type u_1
inst✝ : DecidableEq α
a : α
s t : Multiset α
d : s.Nodup
h : a ∈ s - t
h' : a ∈ t
⊢ count a s ≤ count a t
|
exact le_trans (nodup_iff_count_le_one.1 d _) (count_pos.2 h')
|
no goals
|
731241905e736d7a
|
MeasureTheory.analyticSet_empty
|
Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
|
theorem analyticSet_empty : AnalyticSet (∅ : Set α)
|
α : Type u_1
inst✝ : TopologicalSpace α
⊢ ∅ = ∅ ∨ ∃ f, Continuous f ∧ range f = ∅
|
exact Or.inl rfl
|
no goals
|
96d6590387ca3fc0
|
convexHull_range_eq_exists_affineCombination
|
Mathlib/Analysis/Convex/Combination.lean
|
theorem convexHull_range_eq_exists_affineCombination (v : ι → E) : convexHull R (range v) =
{ x | ∃ (s : Finset ι) (w : ι → R), (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧
s.affineCombination R v w = x }
|
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i => (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
⊢ (s ∪ s').sum W = 1
|
rw [sum_add_distrib, ← sum_subset subset_union_left,
← sum_subset subset_union_right, sum_ite_of_true,
sum_ite_of_true, ← mul_sum, ← mul_sum, hw₁, hw₁', ← add_mul, hab,
mul_one] <;> intros <;> simp_all
|
no goals
|
c246f547060b42da
|
Set.toFinset_setOf
|
Mathlib/Data/Fintype/Sets.lean
|
theorem toFinset_setOf [Fintype α] (p : α → Prop) [DecidablePred p] [Fintype { x | p x }] :
Set.toFinset {x | p x} = Finset.univ.filter p
|
α : Type u_1
inst✝² : Fintype α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : Fintype ↑{x | p x}
⊢ {x | p x}.toFinset = filter p Finset.univ
|
ext
|
case h
α : Type u_1
inst✝² : Fintype α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : Fintype ↑{x | p x}
a✝ : α
⊢ a✝ ∈ {x | p x}.toFinset ↔ a✝ ∈ filter p Finset.univ
|
1860bcb393ffcedc
|
OrthogonalFamily.orthonormal_sigma_orthonormal
|
Mathlib/Analysis/InnerProductSpace/Subspace.lean
|
theorem OrthogonalFamily.orthonormal_sigma_orthonormal {α : ι → Type*} {v_family : ∀ i, α i → G i}
(hv_family : ∀ i, Orthonormal 𝕜 (v_family i)) :
Orthonormal 𝕜 fun a : Σi, α i => V a.1 (v_family a.1 a.2)
|
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
α : ι → Type u_6
v_family : (i : ι) → α i → G i
hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i)
i : ι
v : α i
hvw : ⟨i, v⟩ ≠ ⟨i, v⟩
⊢ False
|
exact hvw rfl
|
no goals
|
d3fff3fd19f98338
|
Set.ruzsa_covering_mul
|
Mathlib/Combinatorics/Additive/RuzsaCovering.lean
|
/-- **Ruzsa's covering lemma** for sets. See also `Finset.ruzsa_covering_mul`. -/
@[to_additive "**Ruzsa's covering lemma** for sets. See also `Finset.ruzsa_covering_add`."]
lemma ruzsa_covering_mul (hA : A.Finite) (hB : B.Finite) (hB₀ : B.Nonempty)
(hK : Nat.card (A * B) ≤ K * Nat.card B) :
∃ F ⊆ A, Nat.card F ≤ K ∧ A ⊆ F * (B / B) ∧ F.Finite
|
case intro.intro
G : Type u_1
inst✝ : Group G
K : ℝ
A B : Finset G
hB₀ : (↑B).Nonempty
hK : ↑(Nat.card ↑(↑A * ↑B)) ≤ K * ↑(Nat.card ↑↑B)
⊢ ∃ F ⊆ ↑A, ↑(Nat.card ↑F) ≤ K ∧ ↑A ⊆ F * (↑B / ↑B) ∧ F.Finite
|
obtain ⟨F, hFA, hF, hAF⟩ := Finset.ruzsa_covering_mul hB₀ (by simpa [← Finset.coe_mul] using hK)
|
case intro.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
K : ℝ
A B : Finset G
hB₀ : (↑B).Nonempty
hK : ↑(Nat.card ↑(↑A * ↑B)) ≤ K * ↑(Nat.card ↑↑B)
F : Finset G
hFA : F ⊆ A
hF : ↑F.card ≤ K
hAF : A ⊆ F * (B / B)
⊢ ∃ F ⊆ ↑A, ↑(Nat.card ↑F) ≤ K ∧ ↑A ⊆ F * (↑B / ↑B) ∧ F.Finite
|
ad264201fa97dd6b
|
SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₁
|
Mathlib/AlgebraicTopology/SimplicialSet/Coskeletal.lean
|
lemma fac_aux₁ {n : ℕ}
(s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X))
(x : s.pt) (i : ℕ) (hi : i < n) :
X.map (mkOfSucc ⟨i, hi⟩).op (lift s x) =
s.π.app (strArrowMk₂ (mkOfSucc ⟨i, hi⟩)) x
|
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hi : i < n
⊢ X.map (mkOfSucc ⟨i, hi⟩).op (lift s x) = s.π.app (strArrowMk₂ (mkOfSucc ⟨i, hi⟩) ⋯) x
|
dsimp [lift]
|
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hi : i < n
⊢ X.map (mkOfSucc ⟨i, hi⟩).op
(spineToSimplex
{
vertex := fun i =>
s.π.app
(StructuredArrow.mk
(((Truncated.inclusion 2).obj { obj := ⦋0⦌, property := lift.proof_1 }).const ⦋n⦌ i).op)
x,
arrow := fun i => s.π.app (StructuredArrow.mk (mkOfLe i.castSucc i.succ ⋯).op) x, arrow_src := ⋯,
arrow_tgt := ⋯ }) =
s.π.app (strArrowMk₂ (mkOfSucc ⟨i, hi⟩) ⋯) x
|
97963738e37dfef3
|
MeasureTheory.condExpL2_indicator_nonneg
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
|
theorem condExpL2_indicator_nonneg (hm : m ≤ m0) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
[SigmaFinite (μ.trim hm)] : (0 : α → ℝ) ≤ᵐ[μ]
condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1)
|
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
t : Set α
ht : MeasurableSet t
hμt : (μ.trim hm) t < ⊤
h_ae :
∀ᵐ (x : α) ∂μ,
x ∈ t →
AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h x =
↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) x
⊢ 0 ≤ ∫ (x : α) in t, AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h x ∂μ
|
rw [setIntegral_congr_ae (hm t ht) h_ae,
setIntegral_condExpL2_indicator ht hs ((le_trim hm).trans_lt hμt).ne hμs]
|
case refine_2
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
t : Set α
ht : MeasurableSet t
hμt : (μ.trim hm) t < ⊤
h_ae :
∀ᵐ (x : α) ∂μ,
x ∈ t →
AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h x =
↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) x
⊢ 0 ≤ (μ (s ∩ t)).toReal
|
d88e3b2d7a38a88c
|
MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
|
Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
|
theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
(f : ℝⁿ⁺¹ → Eⁿ⁺¹)
(f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
(s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
(Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
(Hi : IntegrableOn (∑ i, f' · (e i) i) (Box.Icc I)) :
(∫ x in Box.Icc I, ∑ i, f' x (e i) i) =
∑ i : Fin (n + 1),
((∫ x in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
∫ x in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)
|
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : Tendsto (Box.lower ∘ ⇑J) atTop (𝓝 I.lower)
hJu : Tendsto (Box.upper ∘ ⇑J) atTop (𝓝 I.upper)
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
∀ (k : ℕ),
∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i : Fin (n + 1), (f' x) (e i) i =
∑ i : Fin (n + 1),
((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
Hi : Integrable (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (volume.restrict (Box.Ioo I))
⊢ Tendsto (fun k => ∫ (x : Fin (n + 1) → ℝ) in Box.Ioo (J k), ∑ i : Fin (n + 1), (f' x) (e i) i) atTop
(𝓝 (∫ (x : Fin (n + 1) → ℝ) in Box.Ioo I, ∑ i : Fin (n + 1), (f' x) (e i) i))
|
rw [← Box.iUnion_Ioo_of_tendsto J.monotone hJl hJu] at Hi ⊢
|
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : Tendsto (Box.lower ∘ ⇑J) atTop (𝓝 I.lower)
hJu : Tendsto (Box.upper ∘ ⇑J) atTop (𝓝 I.upper)
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
∀ (k : ℕ),
∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i : Fin (n + 1), (f' x) (e i) i =
∑ i : Fin (n + 1),
((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
Hi : Integrable (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (volume.restrict (⋃ n_1, Box.Ioo (J n_1)))
⊢ Tendsto (fun k => ∫ (x : Fin (n + 1) → ℝ) in Box.Ioo (J k), ∑ i : Fin (n + 1), (f' x) (e i) i) atTop
(𝓝 (∫ (x : Fin (n + 1) → ℝ) in ⋃ n_1, Box.Ioo (J n_1), ∑ i : Fin (n + 1), (f' x) (e i) i))
|
d25a0c4e5a28a1f3
|
pow_padicValNat_dvd
|
Mathlib/NumberTheory/Padics/PadicVal/Basic.lean
|
theorem pow_padicValNat_dvd {n : ℕ} : p ^ padicValNat p n ∣ n
|
case inr
p n : ℕ
hn : n > 0
⊢ p ^ padicValNat p n ∣ n
|
rcases eq_or_ne p 1 with (rfl | hp)
|
case inr.inl
n : ℕ
hn : n > 0
⊢ 1 ^ padicValNat 1 n ∣ n
case inr.inr
p n : ℕ
hn : n > 0
hp : p ≠ 1
⊢ p ^ padicValNat p n ∣ n
|
b80d28e2b2224789
|
AlgebraicGeometry.HasAffineProperty.diagonal_of_diagonal_of_isPullback
|
Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
|
theorem HasAffineProperty.diagonal_of_diagonal_of_isPullback
(P) {Q} [HasAffineProperty P Q]
{X Y U V : Scheme.{u}} {f : X ⟶ Y} {g : U ⟶ Y}
[IsAffine U] [IsOpenImmersion g]
{iV : V ⟶ X} {f' : V ⟶ U} (h : IsPullback iV f' f g) (H : P.diagonal f) :
Q.diagonal f'
|
case h.e'_3.h
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝² : HasAffineProperty P Q
X Y U✝ V✝ : Scheme
f : X ⟶ Y
g : U✝ ⟶ Y
inst✝¹ : IsAffine U✝
inst✝ : IsOpenImmersion g
iV : V✝ ⟶ X
f' : V✝ ⟶ U✝
h : IsPullback iV f' f g
H : P.diagonal f
this : Q.IsLocal := isLocal_affineProperty P
U V : Scheme
f₁ : U ⟶ pullback f g
f₂ : V ⟶ pullback f g
hU : IsAffine U
hV : IsAffine V
hf₁ : IsOpenImmersion f₁
hf₂ : IsOpenImmersion f₂
e_1✝ :
pullback (pullback.diagonal f)
(pullback.map (f₁ ≫ pullback.snd f g) (f₂ ≫ pullback.snd f g) f f (f₁ ≫ pullback.fst f g) (f₂ ≫ pullback.fst f g)
g ⋯ ⋯) =
pullback (pullback.diagonal f)
(pullback.map (f₁ ≫ pullback.snd f g) (f₂ ≫ pullback.snd f g) f f (f₁ ≫ pullback.fst f g) (f₂ ≫ pullback.fst f g)
g ⋯ ⋯)
e_2✝ :
pullback (f₁ ≫ pullback.snd f g) (f₂ ≫ pullback.snd f g) = pullback (f₁ ≫ pullback.snd f g) (f₂ ≫ pullback.snd f g)
⊢ (pullbackDiagonalMapIso f g f₁ f₂).hom ≫ pullback.mapDesc f₁ f₂ (pullback.snd f g) =
pullback.snd (pullback.diagonal f)
(pullback.map (f₁ ≫ pullback.snd f g) (f₂ ≫ pullback.snd f g) f f (f₁ ≫ pullback.fst f g) (f₂ ≫ pullback.fst f g)
g ⋯ ⋯)
|
apply pullback.hom_ext <;> simp
|
no goals
|
6247e87fffb2135c
|
Finset.prod_ite_mem
|
Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
theorem prod_ite_mem [DecidableEq α] (s t : Finset α) (f : α → β) :
∏ i ∈ s, (if i ∈ t then f i else 1) = ∏ i ∈ s ∩ t, f i
|
α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s t : Finset α
f : α → β
⊢ (∏ i ∈ s, if i ∈ t then f i else 1) = ∏ i ∈ s ∩ t, f i
|
rw [← Finset.prod_filter, Finset.filter_mem_eq_inter]
|
no goals
|
4415ba4b31d04aed
|
card_classGroup_eq_one_iff
|
Mathlib/RingTheory/ClassGroup.lean
|
theorem card_classGroup_eq_one_iff [IsDedekindDomain R] [Fintype (ClassGroup R)] :
Fintype.card (ClassGroup R) = 1 ↔ IsPrincipalIdealRing R
|
case mp
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsDedekindDomain R
inst✝ : Fintype (ClassGroup R)
⊢ Fintype.card (ClassGroup R) = 1 → IsPrincipalIdealRing R
case mpr
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsDedekindDomain R
inst✝ : Fintype (ClassGroup R)
⊢ IsPrincipalIdealRing R → Fintype.card (ClassGroup R) = 1
|
swap
|
case mpr
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsDedekindDomain R
inst✝ : Fintype (ClassGroup R)
⊢ IsPrincipalIdealRing R → Fintype.card (ClassGroup R) = 1
case mp
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : IsDedekindDomain R
inst✝ : Fintype (ClassGroup R)
⊢ Fintype.card (ClassGroup R) = 1 → IsPrincipalIdealRing R
|
6bf7e5af8c06407b
|
MeasureTheory.Measure.count_apply_infinite
|
Mathlib/MeasureTheory/Measure/Count.lean
|
theorem count_apply_infinite (hs : s.Infinite) : count s = ∞
|
α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
hs : s.Infinite
n : ℕ
⊢ ↑n ≤ count s
|
rcases hs.exists_subset_card_eq n with ⟨t, ht, rfl⟩
|
case intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : Set α
hs : s.Infinite
t : Finset α
ht : ↑t ⊆ s
⊢ ↑t.card ≤ count s
|
ed96a5bdd1a2bdb2
|
List.flatten_reverse
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem flatten_reverse (L : List (List α)) :
L.reverse.flatten = (L.map reverse).flatten.reverse
|
α : Type u_1
L : List (List α)
⊢ L.reverse.flatten = (map reverse L).flatten.reverse
|
induction L <;> simp_all
|
no goals
|
30e1614dac48f0c6
|
CochainComplex.mappingCone.map_δ
|
Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean
|
lemma map_δ :
(G.mapHomologicalComplex (ComplexShape.up ℤ)).map (triangle φ).mor₃ ≫
NatTrans.app ((Functor.mapHomologicalComplex G (ComplexShape.up ℤ)).commShiftIso 1).hom K =
(mapHomologicalComplexIso φ G).hom ≫
(triangle ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).mor₃
|
case h
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
n : ℤ
⊢ G.map ((triangle φ).mor₃.f n) ≫ 𝟙 (G.obj (K.X (n + 1))) =
(G.map ((↑(fst φ)).v n (n + 1) ⋯) ≫ (inl ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 1) n ⋯ +
G.map ((snd φ).v n n ⋯) ≫ (inr ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) ≫
(triangle ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).mor₃.f n
|
simp only [Functor.mapHomologicalComplex_obj_X, add_comp, assoc, inl_v_triangle_mor₃_f,
shiftFunctor_obj_X, shiftFunctorObjXIso, HomologicalComplex.XIsoOfEq_rfl, Iso.refl_inv,
comp_neg, comp_id, inr_f_triangle_mor₃_f, comp_zero, add_zero]
|
case h
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
inst✝⁴ : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
n : ℤ
⊢ G.map ((triangle φ).mor₃.f n) = -G.map ((↑(fst φ)).v n (n + 1) ⋯)
|
31723edde10b5c70
|
deriv.lhopital_zero_nhds_right
|
Mathlib/Analysis/Calculus/LHopital.lean
|
theorem lhopital_zero_nhds_right (hdf : ∀ᶠ x in 𝓝[>] a, DifferentiableAt ℝ f x)
(hg' : ∀ᶠ x in 𝓝[>] a, deriv g x ≠ 0) (hfa : Tendsto f (𝓝[>] a) (𝓝 0))
(hga : Tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l
|
a : ℝ
l : Filter ℝ
f g : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in 𝓝[>] a, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in 𝓝[>] a, deriv g x ≠ 0
hfa : Tendsto f (𝓝[>] a) (𝓝 0)
hga : Tendsto g (𝓝[>] a) (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[>] a) l
hdg : ∀ᶠ (x : ℝ) in 𝓝[>] a, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in 𝓝[>] a, HasDerivAt f (deriv f x) x
hdg' : ∀ᶠ (x : ℝ) in 𝓝[>] a, HasDerivAt g (deriv g x) x
⊢ Tendsto (fun x => f x / g x) (𝓝[>] a) l
|
exact HasDerivAt.lhopital_zero_nhds_right hdf' hdg' hg' hfa hga hdiv
|
no goals
|
cd7c4c838ec90e49
|
AddCircle.ergodic_nsmul_add
|
Mathlib/Dynamics/Ergodic/AddCircle.lean
|
theorem ergodic_nsmul_add (x : AddCircle T) {n : ℕ} (h : 1 < n) : Ergodic fun y => n • y + x :=
ergodic_zsmul_add x (by simp [h] : 1 < |(n : ℤ)|)
|
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
n : ℕ
h : 1 < n
⊢ 1 < |↑n|
|
simp [h]
|
no goals
|
9e072f1a9fcf95e4
|
CategoryTheory.Iso.eHomCongr_comp
|
Mathlib/CategoryTheory/Enriched/HomCongr.lean
|
/-- `eHomCongr` respects composition of morphisms. Recall that for any
composable pair of arrows `f : X ⟶ Y` and `g : Y ⟶ Z` in `C`, the composite
`f ≫ g` in `C` defines a morphism `𝟙_ V ⟶ (X ⟶[V] Z)` in `V`. Composing with
the isomorphism `eHomCongr V α γ` yields a morphism in `V` that can be factored
through the enriched composition map as shown:
`𝟙_ V ⟶ 𝟙_ V ⊗ 𝟙_ V ⟶ (X₁ ⟶[V] Y₁) ⊗ (Y₁ ⟶[V] Z₁) ⟶ (X₁ ⟶[V] Z₁)`. -/
@[reassoc]
lemma eHomCongr_comp {X Y Z X₁ Y₁ Z₁ : C} (α : X ≅ X₁) (β : Y ≅ Y₁) (γ : Z ≅ Z₁)
(f : X ⟶ Y) (g : Y ⟶ Z) :
eHomEquiv V (f ≫ g) ≫ (eHomCongr V α γ).hom =
(λ_ _).inv ≫ (eHomEquiv V f ≫ (eHomCongr V α β).hom) ▷ _ ≫
_ ◁ (eHomEquiv V g ≫ (eHomCongr V β γ).hom) ≫ eComp V X₁ Y₁ Z₁
|
V : Type u'
inst✝³ : Category.{v', u'} V
inst✝² : MonoidalCategory V
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : EnrichedOrdinaryCategory V C
X Y Z X₁ Y₁ Z₁ : C
α : X ≅ X₁
β : Y ≅ Y₁
γ : Z ≅ Z₁
f : X ⟶ Y
g : Y ⟶ Z
⊢ (eHomEquiv V) (f ≫ g) ≫ (eHomCongr V α γ).hom =
(λ_ (𝟙_ V)).inv ≫
((eHomEquiv V) f ≫ (eHomCongr V α β).hom) ▷ 𝟙_ V ≫
EnrichedCategory.Hom X₁ Y₁ ◁ ((eHomEquiv V) g ≫ (eHomCongr V β γ).hom) ≫ eComp V X₁ Y₁ Z₁
|
simp only [eHomCongr, MonoidalCategory.whiskerRight_id, assoc,
MonoidalCategory.whiskerLeft_comp]
|
V : Type u'
inst✝³ : Category.{v', u'} V
inst✝² : MonoidalCategory V
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : EnrichedOrdinaryCategory V C
X Y Z X₁ Y₁ Z₁ : C
α : X ≅ X₁
β : Y ≅ Y₁
γ : Z ≅ Z₁
f : X ⟶ Y
g : Y ⟶ Z
⊢ (eHomEquiv V) (f ≫ g) ≫ eHomWhiskerRight V α.inv Z ≫ eHomWhiskerLeft V X₁ γ.hom =
(λ_ (𝟙_ V)).inv ≫
(ρ_ (𝟙_ V)).hom ≫
(eHomEquiv V) f ≫
eHomWhiskerRight V α.inv Y ≫
eHomWhiskerLeft V X₁ β.hom ≫
(ρ_ (EnrichedCategory.Hom X₁ Y₁)).inv ≫
EnrichedCategory.Hom X₁ Y₁ ◁ (eHomEquiv V) g ≫
EnrichedCategory.Hom X₁ Y₁ ◁ eHomWhiskerRight V β.inv Z ≫
EnrichedCategory.Hom X₁ Y₁ ◁ eHomWhiskerLeft V Y₁ γ.hom ≫ eComp V X₁ Y₁ Z₁
|
9ba22ae25e903236
|
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three
|
Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean
|
private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E'
|
case h.a₆
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
hu6 : u ^ 6 = E.a₆ / E'.a₆
hu0 : u ≠ 0
⊢ (E.variableChange { u := Units.mk0 u hu0, r := 0, s := 0, t := 0 }).a₆ = E'.a₆
|
simp_rw [variableChange_a₆, a₁_of_isShortNF, a₂_of_isShortNF, a₃_of_isShortNF,
Units.val_inv_eq_inv_val, Units.val_mk0, inv_pow, inv_mul_eq_div, hu6]
|
case h.a₆
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
hu6 : u ^ 6 = E.a₆ / E'.a₆
hu0 : u ≠ 0
⊢ (E.a₆ + 0 * E.a₄ + 0 ^ 2 * 0 + 0 ^ 3 - 0 * 0 - 0 ^ 2 - 0 * 0 * 0) / (E.a₆ / E'.a₆) = E'.a₆
|
5599f8482dc6841e
|
IsFractionRing.integerNormalization_eq_zero_iff
|
Mathlib/RingTheory/Localization/Integral.lean
|
theorem integerNormalization_eq_zero_iff {p : K[X]} :
integerNormalization (nonZeroDivisors A) p = 0 ↔ p = 0
|
A : Type u_3
K : Type u_4
inst✝⁴ : CommRing A
inst✝³ : IsDomain A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : K[X]
⊢ (∀ (n : ℕ), p.coeff n = coeff 0 n) ↔ ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
|
obtain ⟨⟨b, nonzero⟩, hb⟩ := integerNormalization_spec (nonZeroDivisors A) p
|
case intro.mk
A : Type u_3
K : Type u_4
inst✝⁴ : CommRing A
inst✝³ : IsDomain A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : K[X]
b : A
nonzero : b ∈ nonZeroDivisors A
hb : ∀ (i : ℕ), (algebraMap A K) ((integerNormalization (nonZeroDivisors A) p).coeff i) = ↑⟨b, nonzero⟩ • p.coeff i
⊢ (∀ (n : ℕ), p.coeff n = coeff 0 n) ↔ ∀ (n : ℕ), (integerNormalization (nonZeroDivisors A) p).coeff n = coeff 0 n
|
4085e90fc2cd39dd
|
LaurentSeries.inducing_coe
|
Mathlib/RingTheory/LaurentSeries.lean
|
theorem inducing_coe : IsUniformInducing ((↑) : RatFunc K → K⸨X⸩)
|
case h.mpr.intro.intro
K : Type u_2
inst✝ : Field K
S : Set (RatFunc K × RatFunc K)
w✝ : Set (RatFunc K)
hT : w✝ ∈ nhds 0
pre_T : (fun x => x.2 - x.1) ⁻¹' w✝ ⊆ S
⊢ ∃ t, (∃ t_1 ∈ nhds 0, (fun x => x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧ (fun x => (↑x.1, ↑x.2)) ⁻¹' t ⊆ S
|
obtain ⟨d, hd⟩ := Valued.mem_nhds.mp hT
|
case h.mpr.intro.intro.intro
K : Type u_2
inst✝ : Field K
S : Set (RatFunc K × RatFunc K)
w✝ : Set (RatFunc K)
hT : w✝ ∈ nhds 0
pre_T : (fun x => x.2 - x.1) ⁻¹' w✝ ⊆ S
d : (WithZero (Multiplicative ℤ))ˣ
hd : {y | Valued.v (y - 0) < ↑d} ⊆ w✝
⊢ ∃ t, (∃ t_1 ∈ nhds 0, (fun x => x.2 - x.1) ⁻¹' t_1 ⊆ t) ∧ (fun x => (↑x.1, ↑x.2)) ⁻¹' t ⊆ S
|
b5e123cc0ae5a26a
|
Polynomial.taylor_coeff
|
Mathlib/Algebra/Polynomial/Taylor.lean
|
theorem taylor_coeff (n : ℕ) : (taylor r f).coeff n = (hasseDeriv n f).eval r :=
show (lcoeff R n).comp (taylor r) f = (leval r).comp (hasseDeriv n) f by
congr 1; clear! f; ext i
simp only [leval_apply, mul_one, one_mul, eval_monomial, LinearMap.comp_apply, coeff_C_mul,
hasseDeriv_monomial, taylor_apply, monomial_comp, C_1, (commute_X (C r)).add_pow i,
map_sum]
simp only [lcoeff_apply, ← C_eq_natCast, mul_assoc, ← C_pow, ← C_mul, coeff_mul_C,
(Nat.cast_commute _ _).eq, coeff_X_pow, boole_mul, Finset.sum_ite_eq, Finset.mem_range]
split_ifs with h; · rfl
push_neg at h; rw [Nat.choose_eq_zero_of_lt h, Nat.cast_zero, mul_zero]
|
case e_a.h.h
R : Type u_1
inst✝ : Semiring R
r : R
n i : ℕ
⊢ ((lcoeff R n ∘ₗ taylor r) ∘ₗ monomial i) 1 = ((leval r ∘ₗ hasseDeriv n) ∘ₗ monomial i) 1
|
simp only [leval_apply, mul_one, one_mul, eval_monomial, LinearMap.comp_apply, coeff_C_mul,
hasseDeriv_monomial, taylor_apply, monomial_comp, C_1, (commute_X (C r)).add_pow i,
map_sum]
|
case e_a.h.h
R : Type u_1
inst✝ : Semiring R
r : R
n i : ℕ
⊢ ∑ x ∈ Finset.range (i + 1), (lcoeff R n) (X ^ x * C r ^ (i - x) * ↑(i.choose x)) = ↑(i.choose n) * r ^ (i - n)
|
5fdac828253feeff
|
Set.offDiag_insert
|
Mathlib/Data/Set/Prod.lean
|
theorem offDiag_insert (ha : a ∉ s) : (insert a s).offDiag = s.offDiag ∪ {a} ×ˢ s ∪ s ×ˢ {a}
|
α : Type u_1
s : Set α
b : α
hb : b ∈ s
ha : b ∉ s
⊢ False
|
exact ha hb
|
no goals
|
c20b88b8124221df
|
MeasurableEquiv.map_measurableEquiv_injective
|
Mathlib/MeasureTheory/Measure/Map.lean
|
theorem map_measurableEquiv_injective (e : α ≃ᵐ β) : Injective (Measure.map e)
|
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
⊢ Injective (map ⇑e)
|
intro μ₁ μ₂ hμ
|
α : Type u_1
β : Type u_2
x✝ : MeasurableSpace α
inst✝ : MeasurableSpace β
e : α ≃ᵐ β
μ₁ μ₂ : Measure α
hμ : map (⇑e) μ₁ = map (⇑e) μ₂
⊢ μ₁ = μ₂
|
72954bef4fbcbcc6
|
isClosedMap_swap
|
Mathlib/Topology/Constructions.lean
|
lemma isClosedMap_swap : IsClosedMap (Prod.swap : X × Y → Y × X) := fun s hs ↦ by
rw [image_swap_eq_preimage_swap]
exact hs.preimage continuous_swap
|
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set (X × Y)
hs : IsClosed s
⊢ IsClosed (Prod.swap '' s)
|
rw [image_swap_eq_preimage_swap]
|
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set (X × Y)
hs : IsClosed s
⊢ IsClosed (Prod.swap ⁻¹' s)
|
b5e19039f16567ae
|
ClassGroup.mk_eq_mk_of_coe_ideal
|
Mathlib/RingTheory/ClassGroup.lean
|
theorem ClassGroup.mk_eq_mk_of_coe_ideal {I J : (FractionalIdeal R⁰ <| FractionRing R)ˣ}
{I' J' : Ideal R} (hI : (I : FractionalIdeal R⁰ <| FractionRing R) = I')
(hJ : (J : FractionalIdeal R⁰ <| FractionRing R) = J') :
ClassGroup.mk I = ClassGroup.mk J ↔
∃ x y : R, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J'
|
case mp.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' J' : Ideal R
hI : ↑I = ↑I'
x : (FractionRing R)ˣ
hJ : ↑(I * (toPrincipalIdeal R (FractionRing R)) x) = ↑J'
⊢ ∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J'
|
rw [Units.val_mul, hI, coe_toPrincipalIdeal, mul_comm,
spanSingleton_mul_coeIdeal_eq_coeIdeal] at hJ
|
case mp.intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
I : (FractionalIdeal R⁰ (FractionRing R))ˣ
I' J' : Ideal R
hI : ↑I = ↑I'
x : (FractionRing R)ˣ
hJ : Ideal.span {(sec R⁰ ↑x).1} * I' = Ideal.span {↑(sec R⁰ ↑x).2} * J'
⊢ ∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * J'
|
dd1e2a77bf98cf5c
|
Real.tanh_eq_sinh_div_cosh
|
Mathlib/Data/Complex/Trigonometric.lean
|
theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=
ofReal_inj.1 <| by simp [tanh_eq_sinh_div_cosh]
|
x : ℝ
⊢ ↑(tanh x) = ↑(sinh x / cosh x)
|
simp [tanh_eq_sinh_div_cosh]
|
no goals
|
29b98f20bbd1dac7
|
quadraticChar_card_card
|
Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean
|
theorem quadraticChar_card_card [DecidableEq F] (hF : ringChar F ≠ 2) {F' : Type*} [Field F']
[Fintype F'] [DecidableEq F'] (hF' : ringChar F' ≠ 2) (h : ringChar F' ≠ ringChar F) :
quadraticChar F (Fintype.card F') =
quadraticChar F' (quadraticChar F (-1) * Fintype.card F)
|
F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
⊢ χ ≠ 1
|
obtain ⟨a, ha⟩ := quadraticChar_exists_neg_one' hF
|
case intro
F : Type u_1
inst✝⁵ : Field F
inst✝⁴ : Fintype F
inst✝³ : DecidableEq F
hF : ringChar F ≠ 2
F' : Type u_2
inst✝² : Field F'
inst✝¹ : Fintype F'
inst✝ : DecidableEq F'
hF' : ringChar F' ≠ 2
h : ringChar F' ≠ ringChar F
χ : MulChar F F' := (quadraticChar F).ringHomComp (algebraMap ℤ F')
a : Fˣ
ha : (quadraticChar F) ↑a = -1
⊢ χ ≠ 1
|
0521617d4359a853
|
Nat.mod_le
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Div/Basic.lean
|
theorem mod_le (x y : Nat) : x % y ≤ x
|
x y : Nat
h₁ : x < y
⊢ x ≤ x
|
apply Nat.le_refl
|
no goals
|
804d0bc58bde878c
|
IsPrimitiveRoot.card_primitiveRoots
|
Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.lean
|
theorem card_primitiveRoots {ζ : R} {k : ℕ} (h : IsPrimitiveRoot ζ k) :
#(primitiveRoots k R) = φ k
|
case neg.refine_3
R : Type u_4
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
⊢ ∀ b ∈ primitiveRoots k R, ∃ a, (a < k ∧ k.Coprime a) ∧ ζ ^ a = b
|
intro ξ hξ
|
case neg.refine_3
R : Type u_4
inst✝¹ : CommRing R
inst✝ : IsDomain R
ζ : R
k : ℕ
h : IsPrimitiveRoot ζ k
h0 : ¬k = 0
this : NeZero k
ξ : R
hξ : ξ ∈ primitiveRoots k R
⊢ ∃ a, (a < k ∧ k.Coprime a) ∧ ζ ^ a = ξ
|
a39b2aaac60d8b86
|
Differentiable.eq_const_of_tendsto_cocompact
|
Mathlib/Analysis/Complex/Liouville.lean
|
theorem eq_const_of_tendsto_cocompact [Nontrivial E] {f : E → F} (hf : Differentiable ℂ f) {c : F}
(hb : Tendsto f (cocompact E) (𝓝 c)) : f = Function.const E c
|
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
h_bdd : Bornology.IsBounded (range f)
⊢ f = const E c
|
obtain ⟨c', hc'⟩ := hf.exists_eq_const_of_bounded h_bdd
|
case intro
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : Nontrivial E
f : E → F
hf : Differentiable ℂ f
c : F
hb : Tendsto f (cocompact E) (𝓝 c)
h_bdd : Bornology.IsBounded (range f)
c' : F
hc' : f = const E c'
⊢ f = const E c
|
e9c3924ef5c1ad0a
|
SimpleGraph.coe_finsetWalkLength_eq
|
Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkCounting.lean
|
theorem coe_finsetWalkLength_eq (n : ℕ) (u v : V) :
(G.finsetWalkLength n u v : Set (G.Walk u v)) = {p : G.Walk u v | p.length = n}
|
V : Type u
G : SimpleGraph V
inst✝¹ : DecidableEq V
inst✝ : G.LocallyFinite
n : ℕ
u v : V
⊢ ↑(G.finsetWalkLength n u v) = {p | p.length = n}
|
induction n generalizing u v with
| zero =>
obtain rfl | huv := eq_or_ne u v <;> simp [finsetWalkLength, set_walk_length_zero_eq_of_ne, *]
| succ n ih =>
simp only [finsetWalkLength, set_walk_length_succ_eq, Finset.coe_biUnion, Finset.mem_coe,
Finset.mem_univ, Set.iUnion_true]
ext p
simp only [mem_neighborSet, Finset.coe_map, Embedding.coeFn_mk, Set.iUnion_coe_set,
Set.mem_iUnion, Set.mem_image, Finset.mem_coe, Set.mem_setOf_eq]
congr!
rename_i w _ q
have := Set.ext_iff.mp (ih w v) q
simp only [Finset.mem_coe, Set.mem_setOf_eq] at this
rw [← this]
|
no goals
|
bd2484cc02b0ec2f
|
RingHom.Flat.isStableUnderBaseChange
|
Mathlib/RingTheory/RingHom/Flat.lean
|
lemma isStableUnderBaseChange : IsStableUnderBaseChange Flat
|
R S T : Type u_4
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Module.Flat R T
this : Module.Flat S (S ⊗[R] T)
⊢ Module.Flat S (S ⊗[R] T)
|
convert this
|
case h.e'_5.h
R S T : Type u_4
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Module.Flat R T
this : Module.Flat S (S ⊗[R] T)
e_4✝ : NonUnitalNonAssocSemiring.toAddCommMonoid = addCommMonoid
⊢ Algebra.toModule = leftModule
|
de2a47f537ad5f13
|
List.mapFinIdx_eq_append_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean
|
theorem mapFinIdx_eq_append_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
l.mapFinIdx f = l₁ ++ l₂ ↔
∃ (l₁' : List α) (l₂' : List α) (w : l = l₁' ++ l₂'),
l₁'.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₁ ∧
l₂'.mapFinIdx (fun i a h => f (i + l₁'.length) a (by simp [w]; omega)) = l₂
|
case mp.right.h
α : Type u_1
β : Type u_2
l₁ l₂ : List β
l : List α
f : (i : Nat) → α → i < l.length → β
h✝ : (l₁ ++ l₂).length = l.length
w : ∀ (i : Nat) (h : i < l.length), (l₁ ++ l₂)[i] = f i l[i] h
h : l₁.length + l₂.length = l.length
i : Nat
hi₁ : i < ((drop l₁.length l).mapFinIdx fun i a h => f (i + (take l₁.length l).length) a ⋯).length
hi₂ : i < l₂.length
this : l₁.length ≤ l.length
⊢ f (i + min l₁.length l.length) l[l₁.length + i] ⋯ = l₂[i]
|
simp only [Nat.min_eq_left this, Nat.add_comm]
|
case mp.right.h
α : Type u_1
β : Type u_2
l₁ l₂ : List β
l : List α
f : (i : Nat) → α → i < l.length → β
h✝ : (l₁ ++ l₂).length = l.length
w : ∀ (i : Nat) (h : i < l.length), (l₁ ++ l₂)[i] = f i l[i] h
h : l₁.length + l₂.length = l.length
i : Nat
hi₁ : i < ((drop l₁.length l).mapFinIdx fun i a h => f (i + (take l₁.length l).length) a ⋯).length
hi₂ : i < l₂.length
this : l₁.length ≤ l.length
⊢ f (i + l₁.length) l[i + l₁.length] ⋯ = l₂[i]
|
6ee35de95da05087
|
FractionalIdeal.isPrincipal.of_finite_maximals_of_inv
|
Mathlib/RingTheory/DedekindDomain/PID.lean
|
theorem FractionalIdeal.isPrincipal.of_finite_maximals_of_inv {A : Type*} [CommRing A]
[Algebra R A] {S : Submonoid R} [IsLocalization S A] (hS : S ≤ R⁰)
(hf : {I : Ideal R | I.IsMaximal}.Finite) (I I' : FractionalIdeal S A) (hinv : I * I' = 1) :
Submodule.IsPrincipal (I : Submodule R A)
|
R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M
a : Ideal R → A
ha : ∀ M ∈ s, a M ∈ I
b : Ideal R → A
hb : ∀ M ∈ s, b M ∈ I'
hm : ∀ M ∈ s, a M * b M ∉ IsLocalization.coeSubmodule A M
u : Ideal R → R
hu : ∀ M ∈ s, u M ∈ ⨅ M' ∈ s.erase M, M'
hum : ∀ M ∈ s, u M ∉ M
v : A := ∑ M ∈ s, u M • b M
⊢ (↑I).IsPrincipal
|
have hv : v ∈ I' := Submodule.sum_mem _ fun M hM => Submodule.smul_mem _ _ <| hb M hM
|
R : Type u_1
inst✝³ : CommRing R
A : Type u_2
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
hS : S ≤ R⁰
hf : {I | I.IsMaximal}.Finite
I I' : FractionalIdeal S A
hinv : ↑I * ↑I' = ↑1
hinv' : I * I' = 1
s : Finset (Ideal R) := hf.toFinset
this : DecidableEq (Ideal R)
coprime : ∀ M ∈ s, ∀ M' ∈ s.erase M, M ⊔ M' = ⊤
nle : ∀ M ∈ s, ¬⨅ M' ∈ s.erase M, M' ≤ M
a : Ideal R → A
ha : ∀ M ∈ s, a M ∈ I
b : Ideal R → A
hb : ∀ M ∈ s, b M ∈ I'
hm : ∀ M ∈ s, a M * b M ∉ IsLocalization.coeSubmodule A M
u : Ideal R → R
hu : ∀ M ∈ s, u M ∈ ⨅ M' ∈ s.erase M, M'
hum : ∀ M ∈ s, u M ∉ M
v : A := ∑ M ∈ s, u M • b M
hv : v ∈ I'
⊢ (↑I).IsPrincipal
|
739ab9d4c311b5e5
|
LSeries.convolution_congr
|
Mathlib/NumberTheory/LSeries/Convolution.lean
|
lemma LSeries.convolution_congr {R : Type*} [Semiring R] {f f' g g' : ℕ → R}
(hf : ∀ {n}, n ≠ 0 → f n = f' n) (hg : ∀ {n}, n ≠ 0 → g n = g' n) :
f ⍟ g = f' ⍟ g'
|
R : Type u_1
inst✝ : Semiring R
f f' g g' : ℕ → R
hf : ∀ {n : ℕ}, n ≠ 0 → f n = f' n
hg : ∀ {n : ℕ}, n ≠ 0 → g n = g' n
⊢ f ⍟ g = f' ⍟ g'
|
simp [convolution, toArithmeticFunction_congr hf, toArithmeticFunction_congr hg]
|
no goals
|
8790498b364f4457
|
MeasureTheory.rnDeriv_tilted_right
|
Mathlib/MeasureTheory/Measure/Tilted.lean
|
lemma rnDeriv_tilted_right (μ ν : Measure α) [SigmaFinite μ] [SigmaFinite ν]
(hf : Integrable (fun x ↦ exp (f x)) ν) :
μ.rnDeriv (ν.tilted f)
=ᵐ[ν] fun x ↦ ENNReal.ofReal (exp (- f x) * ∫ x, exp (f x) ∂ν) * μ.rnDeriv ν x
|
case inr.refine_2.h
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
h0 : NeZero ν
⊢ ∀ (a : α), ENNReal.ofReal (rexp (f a) / ∫ (x : α), rexp (f x) ∂ν) ≠ 0
|
simp only [ne_eq, ENNReal.ofReal_eq_zero, not_le]
|
case inr.refine_2.h
α : Type u_1
mα : MeasurableSpace α
f : α → ℝ
μ ν : Measure α
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hf : Integrable (fun x => rexp (f x)) ν
h0 : NeZero ν
⊢ ∀ (a : α), 0 < rexp (f a) / ∫ (x : α), rexp (f x) ∂ν
|
4634d04eea4db0da
|
BoxIntegral.integralSum_disjUnion
|
Mathlib/Analysis/BoxIntegral/Basic.lean
|
theorem integralSum_disjUnion (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) {π₁ π₂ : TaggedPrepartition I}
(h : Disjoint π₁.iUnion π₂.iUnion) :
integralSum f vol (π₁.disjUnion π₂ h) = integralSum f vol π₁ + integralSum f vol π₂
|
case refine_1
ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I : Box ι
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π₁ π₂ : TaggedPrepartition I
h : Disjoint π₁.iUnion π₂.iUnion
J : Box ι
hJ : J ∈ π₁.boxes
⊢ (vol J) (f ((π₁.disjUnion π₂ h).tag J)) = (vol J) (f (π₁.tag J))
|
rw [disjUnion_tag_of_mem_left _ hJ]
|
no goals
|
39c8d39fe49c69e6
|
PrincipalSeg.irrefl
|
Mathlib/Order/InitialSeg.lean
|
theorem irrefl {r : α → α → Prop} [IsWellOrder α r] (f : r ≺i r) : False
|
α : Type u_1
r : α → α → Prop
inst✝ : IsWellOrder α r
f : r ≺i r
h : r f.top f.top
⊢ False
|
exact _root_.irrefl _ h
|
no goals
|
f181ad2164d3fedb
|
Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul
|
Mathlib/RingTheory/Finiteness/Nakayama.lean
|
theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M)
|
case intro.intro.refine_2.intro.intro.intro.intro.intro.refine_2.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : s✝.Finite
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : s.Finite
ih :
(∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
z : M
hz : z ∈ I • span R s
d : R
left✝ : d ∈ I
hyz : d • i + z = r • n
⊢ (c * r) • n ∈ I • span R s
|
rw [mul_smul, ← hyz, smul_add, smul_smul, mul_comm, mul_smul]
|
case intro.intro.refine_2.intro.intro.intro.intro.intro.refine_2.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
s✝ : Set M
hfs : s✝.Finite
i : M
s : Set M
x✝¹ : i ∉ s
x✝ : s.Finite
ih :
(∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0
r : R
hr1 : r - 1 ∈ I
hs : i ∈ ↑N ∧ s ⊆ ↑N
c : R
hc1 : c - 1 ∈ I
hci : c • i ∈ I • span R s
n : M
hn : n ∈ N
z : M
hz : z ∈ I • span R s
d : R
left✝ : d ∈ I
hyz : d • i + z = r • n
⊢ d • c • i + c • z ∈ I • span R s
|
ddde3c87abdd1812
|
MeasureTheory.maximal_ineq
|
Mathlib/Probability/Martingale/OptionalStopping.lean
|
theorem maximal_ineq [IsFiniteMeasure μ] (hsub : Submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0}
(n : ℕ) : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω} ≤
ENNReal.ofReal (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω},
f n ω ∂μ)
|
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} +
ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}, f n ω ∂μ) ≤
ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ ENNReal.ofReal (∫ (x : Ω) in Set.univ, f n x ∂μ) =
ENNReal.ofReal
(∫ (x : Ω) in
{ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} ∪ {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}, f n x ∂μ)
|
convert rfl
|
case h.e'_3.h.e'_1.h.e'_6.h.e'_4
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
this :
ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} +
ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}, f n ω ∂μ) ≤
ENNReal.ofReal (∫ (x : Ω), f n x ∂μ)
⊢ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} ∪ {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε} = Set.univ
|
7389c69afd10c321
|
HasFPowerSeriesAt.eq_pow_order_mul_iterate_dslope
|
Mathlib/Analysis/Analytic/IsolatedZeros.lean
|
theorem eq_pow_order_mul_iterate_dslope (hp : HasFPowerSeriesAt f p z₀) :
∀ᶠ z in 𝓝 z₀, f z = (z - z₀) ^ p.order • (swap dslope z₀)^[p.order] f z
|
case h.e'_3.h.e'_6
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hp : HasFPowerSeriesAt f p z₀
hq : ∀ᶠ (z : 𝕜) in 𝓝 z₀, HasSum (fun n => (z - z₀) ^ n • (fslope^[p.order] p).coeff n) ((swap dslope z₀)^[p.order] f z)
x : 𝕜
hx2 : HasSum (fun n => (x - z₀) ^ n • p.coeff n) (f x)
this : ∀ k < p.order, p.coeff k = 0
s : E
hs1 : (x - z₀) ^ p.order • s = f x
hs2 : HasSum (fun m => (x - z₀) ^ m • p.coeff (m + p.order)) s
hx1 : HasSum (fun n => (x - z₀) ^ n • p.coeff (n + p.order)) ((swap dslope z₀)^[p.order] f x)
⊢ (swap dslope z₀)^[p.order] f x = s
|
exact hx1.unique hs2
|
no goals
|
d47ea1e5dc2b340e
|
MeasureTheory.measurableSet_exists_tendsto
|
Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
|
theorem measurableSet_exists_tendsto [TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ]
[hγ : OpensMeasurableSpace γ] [Countable ι] {l : Filter ι}
[l.IsCountablyGenerated] {f : ι → β → γ} (hf : ∀ i, Measurable (f i)) :
MeasurableSet { x | ∃ c, Tendsto (fun n => f n x) l (𝓝 c) }
|
case inr.intro
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁵ : MeasurableSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : PolishSpace γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
u : ℕ → Set ι
hu : l.HasAntitoneBasis u
this :
∀ (x : β),
(map (fun x_1 => f x_1 x) l ×ˢ map (fun x_1 => f x_1 x) l).HasAntitoneBasis fun n =>
((fun x_1 => f x_1 x) '' u n) ×ˢ ((fun x_1 => f x_1 x) '' u n)
K : ℕ
x✝ : K ∈ fun i => True
N : ℕ
⊢ MeasurableSet {x | ((fun n => f n x) '' u N) ×ˢ ((fun n => f n x) '' u N) ⊆ {p | dist p.1 p.2 < 1 / (↑K + 1)}}
|
simp_rw [prod_image_image_eq, image_subset_iff, prod_subset_iff, Set.setOf_forall]
|
case inr.intro
ι : Type u_2
γ : Type u_3
β : Type u_5
inst✝⁵ : MeasurableSpace β
inst✝⁴ : TopologicalSpace γ
inst✝³ : PolishSpace γ
inst✝² : MeasurableSpace γ
hγ : OpensMeasurableSpace γ
inst✝¹ : Countable ι
l : Filter ι
inst✝ : l.IsCountablyGenerated
f : ι → β → γ
hf : ∀ (i : ι), Measurable (f i)
hl : l.NeBot
this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ
u : ℕ → Set ι
hu : l.HasAntitoneBasis u
this :
∀ (x : β),
(map (fun x_1 => f x_1 x) l ×ˢ map (fun x_1 => f x_1 x) l).HasAntitoneBasis fun n =>
((fun x_1 => f x_1 x) '' u n) ×ˢ ((fun x_1 => f x_1 x) '' u n)
K : ℕ
x✝ : K ∈ fun i => True
N : ℕ
⊢ MeasurableSet
(⋂ i ∈ u N, ⋂ i_1 ∈ u N, {x | (i, i_1) ∈ (fun p => (f p.1 x, f p.2 x)) ⁻¹' {p | dist p.1 p.2 < 1 / (↑K + 1)}})
|
7e9dea099a7e1607
|
Nat.mul_mod_mul_left
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Div/Basic.lean
|
theorem mul_mod_mul_left (z x y : Nat) : (z * x) % (z * y) = z * (x % y) :=
if y0 : y = 0 then by
rw [y0, Nat.mul_zero, mod_zero, mod_zero]
else if z0 : z = 0 then by
rw [z0, Nat.zero_mul, Nat.zero_mul, Nat.zero_mul, mod_zero]
else by
induction x using Nat.strongRecOn with
| _ n IH =>
have y0 : y > 0 := Nat.pos_of_ne_zero y0
have z0 : z > 0 := Nat.pos_of_ne_zero z0
cases Nat.lt_or_ge n y with
| inl yn => rw [mod_eq_of_lt yn, mod_eq_of_lt (Nat.mul_lt_mul_of_pos_left yn z0)]
| inr yn =>
rw [mod_eq_sub_mod yn, mod_eq_sub_mod (Nat.mul_le_mul_left z yn),
← Nat.mul_sub_left_distrib]
exact IH _ (sub_lt (Nat.lt_of_lt_of_le y0 yn) y0)
|
case ind
z y : Nat
y0✝ : ¬y = 0
z0✝ : ¬z = 0
n : Nat
IH : ∀ (m : Nat), m < n → z * m % (z * y) = z * (m % y)
y0 : y > 0
z0 : z > 0
⊢ z * n % (z * y) = z * (n % y)
|
cases Nat.lt_or_ge n y with
| inl yn => rw [mod_eq_of_lt yn, mod_eq_of_lt (Nat.mul_lt_mul_of_pos_left yn z0)]
| inr yn =>
rw [mod_eq_sub_mod yn, mod_eq_sub_mod (Nat.mul_le_mul_left z yn),
← Nat.mul_sub_left_distrib]
exact IH _ (sub_lt (Nat.lt_of_lt_of_le y0 yn) y0)
|
no goals
|
e8c2ffc687851db8
|
ContinuousLinearMap.integral_comp_comm'
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
theorem integral_comp_comm' (L : E →L[𝕜] F) {K} (hL : AntilipschitzWith K L) (φ : X → E) :
∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ)
|
X : Type u_1
E : Type u_3
F : Type u_4
inst✝⁹ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
L : E →L[𝕜] F
K : ℝ≥0
hL : AntilipschitzWith K ⇑L
φ : X → E
h : ¬Integrable φ μ
⊢ ¬Integrable (fun x => L (φ x)) μ
|
rwa [← Function.comp_def,
LipschitzWith.integrable_comp_iff_of_antilipschitz L.lipschitz hL L.map_zero]
|
no goals
|
9748060e06158f2a
|
Nat.psp_from_prime_psp
|
Mathlib/NumberTheory/FermatPsp.lean
|
theorem psp_from_prime_psp {b : ℕ} (b_ge_two : 2 ≤ b) {p : ℕ} (p_prime : p.Prime)
(p_gt_two : 2 < p) (not_dvd : ¬p ∣ b * (b ^ 2 - 1)) : FermatPsp (psp_from_prime b p) b
|
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (b ^ 2 - 1)
q₂ : p * (b ^ 2 - 1) ∣ b ^ (p - 1) - 1
q₃ : p * (b ^ 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b)
⊢ 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
|
have q₄ : p * (b ^ 2 - 1) * 2 ∣ b * ((b ^ (p - 1) - 1) * (b ^ p + b)) :=
dvd_mul_of_dvd_right q₃ _
|
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
ha₂ : 2 ∣ b ^ p + b
ha₃ : p ∣ b ^ (p - 1) - 1
ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1
q₁ : p.Coprime (b ^ 2 - 1)
q₂ : p * (b ^ 2 - 1) ∣ b ^ (p - 1) - 1
q₃ : p * (b ^ 2 - 1) * 2 ∣ (b ^ (p - 1) - 1) * (b ^ p + b)
q₄ : p * (b ^ 2 - 1) * 2 ∣ b * ((b ^ (p - 1) - 1) * (b ^ p + b))
⊢ 2 * p * (b ^ 2 - 1) ∣ b * (b ^ (p - 1) - 1) * (b ^ p + b)
|
6f5ab4acf8539f9d
|
MulSemiringAction.eval_charpoly
|
Mathlib/RingTheory/Invariant.lean
|
theorem eval_charpoly (b : B) : (charpoly G b).eval b = 0
|
B : Type u_2
G : Type u_3
inst✝³ : CommRing B
inst✝² : Group G
inst✝¹ : MulSemiringAction G B
inst✝ : Fintype G
b : B
⊢ ∏ j : G, eval b (X - C (j • b)) = 0
|
apply Finset.prod_eq_zero (Finset.mem_univ (1 : G))
|
B : Type u_2
G : Type u_3
inst✝³ : CommRing B
inst✝² : Group G
inst✝¹ : MulSemiringAction G B
inst✝ : Fintype G
b : B
⊢ eval b (X - C (1 • b)) = 0
|
c8dfec67acb3bbef
|
TensorProduct.piScalarRightInv_single
|
Mathlib/LinearAlgebra/TensorProduct/Pi.lean
|
@[simp]
private lemma piScalarRightInv_single (x : N) (i : ι) :
piScalarRightInv R S N ι (Pi.single i x) = x ⊗ₜ Pi.single i 1
|
R : Type u_1
inst✝⁸ : CommSemiring R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Type u_3
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
inst✝³ : Module S N
inst✝² : IsScalarTower R S N
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
x : N
i : ι
⊢ (TensorProduct.piScalarRightInv R S N ι) (Pi.single i x) = x ⊗ₜ[R] Pi.single i 1
|
simp [piScalarRightInv, Pi.single_apply, TensorProduct.ite_tmul]
|
no goals
|
b922b7eb6c89cf47
|
CategoryTheory.PreGaloisCategory.initial_iff_fiber_empty
|
Mathlib/CategoryTheory/Galois/Basic.lean
|
/-- An object is initial if and only if its fiber is empty. -/
lemma initial_iff_fiber_empty (X : C) : Nonempty (IsInitial X) ↔ IsEmpty (F.obj X)
|
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : PreGaloisCategory C
inst✝ : FiberFunctor F
X : C
this : PreservesFiniteColimits (forget FintypeCat)
⊢ ReflectsColimit (empty FintypeCat) (forget FintypeCat)
|
show ReflectsColimit (Functor.empty.{0} _) FintypeCat.incl
|
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : PreGaloisCategory C
inst✝ : FiberFunctor F
X : C
this : PreservesFiniteColimits (forget FintypeCat)
⊢ ReflectsColimit (empty FintypeCat) FintypeCat.incl
|
15a0099a3b27cb76
|
AlgebraicTopology.map_alternatingFaceMapComplex
|
Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
|
theorem map_alternatingFaceMapComplex {D : Type*} [Category D] [Preadditive D] (F : C ⥤ D)
[F.Additive] :
alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex _ =
(SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D
|
case h_obj
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
X : SimplicialObject C
⊢ (alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex (ComplexShape.down ℕ)).obj X =
((SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D).obj X
|
apply HomologicalComplex.ext
|
case h_obj.h_d
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
X : SimplicialObject C
⊢ ∀ (i j : ℕ),
(ComplexShape.down ℕ).Rel i j →
((alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex (ComplexShape.down ℕ)).obj X).d i j ≫ eqToHom ⋯ =
eqToHom ⋯ ≫ (((SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D).obj X).d i j
case h_obj.h_X
C : Type u_1
inst✝⁴ : Category.{u_4, u_1} C
inst✝³ : Preadditive C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
X : SimplicialObject C
⊢ ((alternatingFaceMapComplex C ⋙ F.mapHomologicalComplex (ComplexShape.down ℕ)).obj X).X =
(((SimplicialObject.whiskering C D).obj F ⋙ alternatingFaceMapComplex D).obj X).X
|
b8fade3875a479e6
|
CategoryTheory.Limits.pullbackDiagonalMapIdIso_inv_snd_fst
|
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
theorem pullbackDiagonalMapIdIso_inv_snd_fst :
(pullbackDiagonalMapIdIso f g i).inv ≫ pullback.snd _ _ ≫ pullback.fst _ _ =
pullback.fst _ _
|
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
X Y : C
inst✝⁴ : HasPullbacks C
S T : C
f : X ⟶ T
g : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f g
inst✝¹ : HasPullback (f ≫ i) (g ≫ i)
inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)
⊢ (pullbackDiagonalMapIdIso f g i).inv ≫
snd (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯) ≫ fst (f ≫ i) (g ≫ i) =
fst f g
|
rw [Iso.inv_comp_eq]
|
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
X Y : C
inst✝⁴ : HasPullbacks C
S T : C
f : X ⟶ T
g : Y ⟶ T
i : T ⟶ S
inst✝³ : HasPullback i i
inst✝² : HasPullback f g
inst✝¹ : HasPullback (f ≫ i) (g ≫ i)
inst✝ : HasPullback (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯)
⊢ snd (diagonal i) (map (f ≫ i) (g ≫ i) i i f g (𝟙 S) ⋯ ⋯) ≫ fst (f ≫ i) (g ≫ i) =
(pullbackDiagonalMapIdIso f g i).hom ≫ fst f g
|
b31e9cfbf33e9257
|
Algebra.toMatrix_lmul'
|
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
theorem toMatrix_lmul' (x : S) (i j) :
LinearMap.toMatrix b b (lmul R S x) i j = b.repr (x * b j) i
|
R : Type u_1
S : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
m : Type u_3
inst✝¹ : Fintype m
inst✝ : DecidableEq m
b : Basis m R S
x : S
i j : m
⊢ (toMatrix b b) ((lmul R S) x) i j = (b.repr (x * b j)) i
|
simp only [LinearMap.toMatrix_apply', coe_lmul_eq_mul, LinearMap.mul_apply']
|
no goals
|
602839d874935104
|
AddMonoidAlgebra.mapDomainAlgHom_id
|
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
@[simp]
lemma mapDomainAlgHom_id (k A) [CommSemiring k] [Semiring A] [Algebra k A] [AddMonoid G] :
mapDomainAlgHom k A (AddMonoidHom.id G) = AlgHom.id k (AddMonoidAlgebra A G)
|
case H.H
G : Type u₂
k : Type u_3
A : Type u_4
inst✝³ : CommSemiring k
inst✝² : Semiring A
inst✝¹ : Algebra k A
inst✝ : AddMonoid G
x✝¹ : A[G]
x✝ : G
⊢ ((mapDomainAlgHom k A (AddMonoidHom.id G)) x✝¹) x✝ = ((AlgHom.id k A[G]) x✝¹) x✝
|
simp [AddMonoidHom.id, ← Function.id_def]
|
no goals
|
f65a43974ba03501
|
Pell.matiyasevic
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
theorem matiyasevic {a k x y} :
(∃ a1 : 1 < a, xn a1 k = x ∧ yn a1 k = y) ↔
1 < a ∧ k ≤ y ∧ (x = 1 ∧ y = 0 ∨
∃ u v s t b : ℕ,
x * x - (a * a - 1) * y * y = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧
s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * y] ∧
b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]) :=
⟨fun ⟨a1, hx, hy⟩ => by
rw [← hx, ← hy]
refine ⟨a1,
(Nat.eq_zero_or_pos k).elim (fun k0 => by rw [k0]; exact ⟨le_rfl, Or.inl ⟨rfl, rfl⟩⟩)
fun kpos => ?_⟩
exact
let x := xn a1 k
let y := yn a1 k
let m := 2 * (k * y)
let u := xn a1 m
let v := yn a1 m
have ky : k ≤ y := yn_ge_n a1 k
have yv : y * y ∣ v := (ysq_dvd_yy a1 k).trans <| (y_dvd_iff _ _ _).2 <| dvd_mul_left _ _
have uco : Nat.Coprime u (4 * y) :=
have : 2 ∣ v :=
modEq_zero_iff_dvd.1 <| (yn_modEq_two _ _).trans (dvd_mul_right _ _).modEq_zero_nat
have : Nat.Coprime u 2 := (xy_coprime a1 m).coprime_dvd_right this
(this.mul_right this).mul_right <|
(xy_coprime _ _).coprime_dvd_right (dvd_of_mul_left_dvd yv)
let ⟨b, ba, bm1⟩ := chineseRemainder uco a 1
have m1 : 1 < m :=
have : 0 < k * y := mul_pos kpos (strictMono_y a1 kpos)
Nat.mul_le_mul_left 2 this
have vp : 0 < v := strictMono_y a1 (lt_trans zero_lt_one m1)
have b1 : 1 < b :=
have : xn a1 1 < u := strictMono_x a1 m1
have : a < u
|
a k x y : ℕ
x✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y
a1 : 1 < a
hx : xn a1 k = x
hy : yn a1 k = y
⊢ 1 < a ∧
k ≤ y ∧
(x = 1 ∧ y = 0 ∨
∃ u v s t b,
x * x - (a * a - 1) * y * y = 1 ∧
u * u - (a * a - 1) * v * v = 1 ∧
s * s - (b * b - 1) * t * t = 1 ∧
1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])
|
rw [← hx, ← hy]
|
a k x y : ℕ
x✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y
a1 : 1 < a
hx : xn a1 k = x
hy : yn a1 k = y
⊢ 1 < a ∧
k ≤ yn a1 k ∧
(xn a1 k = 1 ∧ yn a1 k = 0 ∨
∃ u v s t b,
xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 ∧
u * u - (a * a - 1) * v * v = 1 ∧
s * s - (b * b - 1) * t * t = 1 ∧
1 < b ∧
b ≡ 1 [MOD 4 * yn a1 k] ∧
b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 k * yn a1 k ∣ v ∧ s ≡ xn a1 k [MOD u] ∧ t ≡ k [MOD 4 * yn a1 k])
|
86da2519f71702c3
|
Subgroup.exists_finsupp_of_mem_closure_range
|
Mathlib/Algebra/Group/Subgroup/Finsupp.lean
|
theorem exists_finsupp_of_mem_closure_range (hx : x ∈ closure (Set.range f)) :
∃ a : ι →₀ ℤ, x = a.prod (f · ^ ·)
|
case mem
M : Type u_1
inst✝ : CommGroup M
ι : Type u_2
f : ι → M
x✝ x : M
h : x ∈ Set.range f
⊢ ∃ a, x = a.prod fun x1 x2 => f x1 ^ x2
|
obtain ⟨i, rfl⟩ := h
|
case mem.intro
M : Type u_1
inst✝ : CommGroup M
ι : Type u_2
f : ι → M
x : M
i : ι
⊢ ∃ a, f i = a.prod fun x1 x2 => f x1 ^ x2
|
ee4be18dd2e022fe
|
CochainComplex.isStrictlyLE_of_le
|
Mathlib/Algebra/Homology/Embedding/CochainComplex.lean
|
lemma isStrictlyLE_of_le (p q : ℤ) (hpq : p ≤ q) [K.IsStrictlyLE p] :
K.IsStrictlyLE q
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsStrictlyLE p
⊢ K.IsStrictlyLE q
|
rw [isStrictlyLE_iff]
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsStrictlyLE p
⊢ ∀ (i : ℤ), q < i → IsZero (K.X i)
|
9e62042f615bf74a
|
hasFDerivAt_exp_smul_const_of_mem_ball'
|
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
theorem hasFDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕊)
(htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
HasFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
(((1 : 𝕊 →L[𝕂] 𝕊).smulRight x).smulRight (exp 𝕂 (t • x))) t
|
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ HasFDerivAt (fun u => exp 𝕂 (u • x)) ((ContinuousLinearMap.smulRight 1 x).smulRight (exp 𝕂 (t • x))) t
|
convert hasFDerivAt_exp_smul_const_of_mem_ball 𝕂 _ _ htx using 1
|
case h.e'_12.h.h
𝕂 : Type u_1
𝕊 : Type u_2
𝔸 : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕂
inst✝⁸ : CharZero 𝕂
inst✝⁷ : NormedCommRing 𝕊
inst✝⁶ : NormedRing 𝔸
inst✝⁵ : NormedSpace 𝕂 𝕊
inst✝⁴ : NormedAlgebra 𝕂 𝔸
inst✝³ : Algebra 𝕊 𝔸
inst✝² : ContinuousSMul 𝕊 𝔸
inst✝¹ : IsScalarTower 𝕂 𝕊 𝔸
inst✝ : CompleteSpace 𝔸
x : 𝔸
t : 𝕊
htx : t • x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
e_8✝ : NormedAddCommGroup.toAddCommGroup = SeminormedAddCommGroup.toAddCommGroup
⊢ (ContinuousLinearMap.smulRight 1 x).smulRight (exp 𝕂 (t • x)) = exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x
|
a03bb16c07893e27
|
UniformSpace.Completion.mem_uniformity_dist
|
Mathlib/Topology/MetricSpace/Completion.lean
|
theorem mem_uniformity_dist (s : Set (Completion α × Completion α)) :
s ∈ 𝓤 (Completion α) ↔ ∃ ε > 0, ∀ {a b}, dist a b < ε → (a, b) ∈ s
|
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ ε ≤ dist x y ∨ (x, y) ∈ t
|
refine induction_on₂ x y ?_ ?_
|
case refine_1
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ IsClosed {x | ε ≤ dist x.1 x.2 ∨ (x.1, x.2) ∈ t}
case refine_2
α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
hs : s ∈ 𝓤 (Completion α)
t : Set (Completion α × Completion α)
ht : t ∈ 𝓤 (Completion α)
tclosed : IsClosed t
ts : t ⊆ s
A : {x | (↑x.1, ↑x.2) ∈ t} ∈ 𝓤 α
ε : ℝ
εpos : ε > 0
hε : ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ {x | (↑x.1, ↑x.2) ∈ t}
x y : Completion α
hxy : dist x y < ε
⊢ ∀ (a b : α), ε ≤ dist ↑a ↑b ∨ (↑a, ↑b) ∈ t
|
c4dfc92c04080ea5
|
Profinite.NobelingProof.C0_projOrd
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem C0_projOrd {x : I → Bool} (hx : x ∈ C0 C ho) : Proj (ord I · < o) x = x
|
case h.inr
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
x : I → Bool
hx : x ∈ C0 C ho
i : I
hi : o = ord I i
⊢ false = x i
|
simp only [C0, Set.mem_inter_iff, Set.mem_setOf_eq] at hx
|
case h.inr
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x1 x2 => x1 < x2
x : I → Bool
i : I
hi : o = ord I i
hx : x ∈ C ∧ x (term I ho) = false
⊢ false = x i
|
93a5fd18327066a4
|
Rel.gc_leftDual_rightDual
|
Mathlib/Order/Rel/GaloisConnection.lean
|
theorem gc_leftDual_rightDual : GaloisConnection (toDual ∘ R.leftDual) (R.rightDual ∘ ofDual) :=
fun _ _ ↦ ⟨fun h _ ha _ hb ↦ h (by simpa) ha, fun h _ hb _ ha ↦ h (by simpa) hb⟩
|
α : Type u_1
β : Type u_2
R : Rel α β
x✝³ : Set α
x✝² : (Set β)ᵒᵈ
h : (⇑toDual ∘ R.leftDual) x✝³ ≤ x✝²
x✝¹ : α
ha : x✝¹ ∈ x✝³
x✝ : β
hb : x✝ ∈ ofDual x✝²
⊢ x✝ ∈ x✝²
|
simpa
|
no goals
|
c0c4bbebfd7bb08d
|
intervalIntegral.continuousOn_primitive
|
Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
|
theorem continuousOn_primitive (h_int : IntegrableOn f (Icc a b) μ) :
ContinuousOn (fun x => ∫ t in Ioc a x, f t ∂μ) (Icc a b)
|
case pos
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NoAtoms μ
h_int : IntegrableOn f (Icc a b) μ
h : a ≤ b
this : ∀ x ∈ Icc a b, ∫ (t : ℝ) in Ioc a x, f t ∂μ = ∫ (t : ℝ) in a..x, f t ∂μ
x₀ : ℝ
a✝ : x₀ ∈ Icc a b
⊢ IntegrableOn f (Ioc a b) μ
|
exact h_int.mono Ioc_subset_Icc_self le_rfl
|
no goals
|
4480467b3d286247
|
Padic.exi_rat_seq_conv_cauchy
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
let ⟨N2, hN2⟩ := f.cauchy₂ hε3
exists max N N2
intro j hj
suffices
padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
ring_nf at this ⊢
rw [← padicNormE.eq_padic_norm']
exact mod_cast this
apply lt_of_le_of_lt
· apply padicNormE.add_le
· rw [← add_thirds ε]
apply _root_.add_lt_add
· suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
simpa only [sub_add_sub_cancel]
apply lt_of_le_of_lt
· apply padicNormE.add_le
· apply _root_.add_lt_add
· rw [padicNormE.map_sub]
apply mod_cast hN j
exact le_of_max_le_left hj
· exact hN2 _ (le_of_max_le_right hj) _ (le_max_right _ _)
· apply mod_cast hN (max N N2)
apply le_max_left
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ N ⊔ N2
this : padicNormE (↑(limSeq f j) - ↑f (N ⊔ N2) + (↑f (N ⊔ N2) - ↑(limSeq f (N ⊔ N2)))) < ε
⊢ padicNorm p (limSeq f j - limSeq f (N ⊔ N2)) < ε
|
ring_nf at this ⊢
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ N ⊔ N2
this : padicNormE (↑(limSeq f j) - ↑(limSeq f (N ⊔ N2))) < ε
⊢ padicNorm p (limSeq f j - limSeq f (N ⊔ N2)) < ε
|
5532e11826254be6
|
Zsqrtd.not_divides_sq
|
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
theorem not_divides_sq (x y) : (x + 1) * (x + 1) ≠ d * (y + 1) * (y + 1) := fun e => by
have t := (divides_sq_eq_zero e).left
contradiction
|
d : ℕ
dnsq : Nonsquare d
x y : ℕ
e : (x + 1) * (x + 1) = d * (y + 1) * (y + 1)
t : x + 1 = 0
⊢ False
|
contradiction
|
no goals
|
9529b3c349df2e6e
|
Equiv.Perm.mclosure_swap_castSucc_succ
|
Mathlib/GroupTheory/Perm/Sign.lean
|
theorem mclosure_swap_castSucc_succ (n : ℕ) :
Submonoid.closure (Set.range fun i : Fin n ↦ swap i.castSucc i.succ) = ⊤
|
n : ℕ
i j : Fin (n + 1)
ne : i ≠ j
lt : i < j
⊢ swap i j ∈ ↑(Submonoid.closure (Set.range fun i => swap i.castSucc i.succ))
|
induction' j using Fin.induction with j ih
|
case zero
n : ℕ
i : Fin (n + 1)
ne : i ≠ 0
lt : i < 0
⊢ swap i 0 ∈ ↑(Submonoid.closure (Set.range fun i => swap i.castSucc i.succ))
case succ
n : ℕ
i : Fin (n + 1)
j : Fin n
ih :
i ≠ j.castSucc → i < j.castSucc → swap i j.castSucc ∈ ↑(Submonoid.closure (Set.range fun i => swap i.castSucc i.succ))
ne : i ≠ j.succ
lt : i < j.succ
⊢ swap i j.succ ∈ ↑(Submonoid.closure (Set.range fun i => swap i.castSucc i.succ))
|
6a1796e117466965
|
Function.Surjective.iSup_comp
|
Mathlib/Order/CompleteLattice.lean
|
theorem Function.Surjective.iSup_comp {f : ι → ι'} (hf : Surjective f) (g : ι' → α) :
⨆ x, g (f x) = ⨆ y, g y
|
α : Type u_1
ι : Sort u_4
ι' : Sort u_5
inst✝ : SupSet α
f : ι → ι'
hf : Surjective f
g : ι' → α
⊢ ⨆ x, g (f x) = ⨆ y, g y
|
simp only [iSup.eq_1]
|
α : Type u_1
ι : Sort u_4
ι' : Sort u_5
inst✝ : SupSet α
f : ι → ι'
hf : Surjective f
g : ι' → α
⊢ sSup (range fun x => g (f x)) = sSup (range fun y => g y)
|
bb4449b14580f170
|
Polynomial.coeff_map
|
Mathlib/Algebra/Polynomial/Eval/Coeff.lean
|
theorem coeff_map (n : ℕ) : coeff (p.map f) n = f (coeff p n)
|
R : Type u
S : Type v
inst✝¹ : Semiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
n : ℕ
⊢ ∑ n_1 ∈ p.support, ((C.comp f) (p.coeff n_1) * X ^ n_1).coeff n = ∑ x ∈ p.support, f ((C (p.coeff x) * X ^ x).coeff n)
|
refine Finset.sum_congr rfl fun x _hx => ?_
|
R : Type u
S : Type v
inst✝¹ : Semiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
n x : ℕ
_hx : x ∈ p.support
⊢ ((C.comp f) (p.coeff x) * X ^ x).coeff n = f ((C (p.coeff x) * X ^ x).coeff n)
|
c6b3aab6eefd374b
|
PowerBasis.exists_eq_aeval'
|
Mathlib/RingTheory/PowerBasis.lean
|
theorem exists_eq_aeval' (pb : PowerBasis R S) (y : S) : ∃ f : R[X], y = aeval pb.gen f
|
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
pb : PowerBasis R S
y : S
a✝ : Nontrivial S
⊢ ∃ f, y = (aeval pb.gen) f
|
obtain ⟨f, _, hf⟩ := exists_eq_aeval pb y
|
case intro.intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
pb : PowerBasis R S
y : S
a✝ : Nontrivial S
f : R[X]
left✝ : f.natDegree < pb.dim
hf : y = (aeval pb.gen) f
⊢ ∃ f, y = (aeval pb.gen) f
|
ddcc9e5e2f91f77d
|
intervalIntegral.intervalIntegrable_deriv_of_nonneg
|
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
theorem intervalIntegrable_deriv_of_nonneg (hcont : ContinuousOn g (uIcc a b))
(hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x)
(hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x) : IntervalIntegrable g' volume a b
|
case inr
g' g : ℝ → ℝ
a b : ℝ
hab : b ≤ a
hcont : ContinuousOn g (Icc b a)
hderiv : ∀ x ∈ Ioo b a, HasDerivAt g (g' x) x
hpos : ∀ x ∈ Ioo b a, 0 ≤ g' x
⊢ IntegrableOn g' (Ioc b a) volume
|
exact integrableOn_deriv_of_nonneg hcont hderiv hpos
|
no goals
|
1cdf5ec3b7a39a4e
|
Int.natAbs_add_le
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean
|
theorem natAbs_add_le (a b : Int) : natAbs (a + b) ≤ natAbs a + natAbs b
|
a b : Int
this : ∀ (a b : Nat), (subNatNat a b.succ).natAbs ≤ (a + b).succ
⊢ (a + b).natAbs ≤ a.natAbs + b.natAbs
|
match a, b with
| (a:Nat), (b:Nat) => rw [ofNat_add_ofNat, natAbs_ofNat]; apply Nat.le_refl
| (a:Nat), -[b+1] => rw [natAbs_ofNat, natAbs_negSucc]; apply this
| -[a+1], (b:Nat) =>
rw [natAbs_negSucc, natAbs_ofNat, Nat.succ_add, Nat.add_comm a b]; apply this
| -[a+1], -[b+1] => rw [natAbs_negSucc, succ_add]; apply Nat.le_refl
|
no goals
|
2617f5dcc449a695
|
ConvexOn.lipschitzOnWith_of_abs_le
|
Mathlib/Analysis/Convex/Continuous.lean
|
lemma ConvexOn.lipschitzOnWith_of_abs_le (hf : ConvexOn ℝ (ball x₀ r) f) (hε : 0 < ε)
(hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ball x₀ r
hy' : y ∈ ball x₀ r
z : E := x + (ε / ‖x - y‖) • (x - y)
hxy : 0 < ‖x - y‖
hz : z ∈ ball x₀ r
a : ℝ := ε / (ε + ‖x - y‖)
b : ℝ := ‖x - y‖ / (ε + ‖x - y‖)
hab : a + b = 1
hxyz : x = a • y + b • z
⊢ (ε + ‖x - y‖) * f x ≤ ‖x - y‖ * f z + ε * f y
|
have h := hf.2 hy' hz (by positivity) (by positivity) hab
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
ε r M : ℝ
hf : ConvexOn ℝ (ball x₀ r) f
hε : 0 < ε
hM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M
K : ℝ := 2 * M / ε
hK : K = 2 * M / ε
x y : E
hx : x ∈ ball x₀ (r - ε)
hy : y ∈ ball x₀ (r - ε)
hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r
hx' : x ∈ ball x₀ r
hy' : y ∈ ball x₀ r
z : E := x + (ε / ‖x - y‖) • (x - y)
hxy : 0 < ‖x - y‖
hz : z ∈ ball x₀ r
a : ℝ := ε / (ε + ‖x - y‖)
b : ℝ := ‖x - y‖ / (ε + ‖x - y‖)
hab : a + b = 1
hxyz : x = a • y + b • z
h : f (a • y + b • z) ≤ a • f y + b • f z
⊢ (ε + ‖x - y‖) * f x ≤ ‖x - y‖ * f z + ε * f y
|
f794e80453a6298e
|
Matrix.Pivot.exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec
|
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
theorem exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec
(M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜) :
∃ L L' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜),
IsTwoBlockDiagonal ((L.map toMatrix).prod * M * (L'.map toMatrix).prod)
|
case inl.unit
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) = 0
H : ∀ (i : Fin r), M (inl i) (inr ()) = 0 ∧ M (inr ()) (inl i) = 0
i : Fin r
⊢ of (fun i j => M (inl i) (inr j)) i PUnit.unit = 0 i PUnit.unit
|
exact (H i).1
|
no goals
|
ef47f5e7783abe91
|
DeltaGeneratedSpace.continuous_iff
|
Mathlib/Topology/Compactness/DeltaGeneratedSpace.lean
|
/-- A map out of a delta-generated space is continuous iff it preserves continuity of maps
from ℝⁿ into X. -/
lemma DeltaGeneratedSpace.continuous_iff [DeltaGeneratedSpace X] {f : X → Y} :
Continuous f ↔ ∀ (n : ℕ) (p : C(((Fin n) → ℝ), X)), Continuous (f ∘ p)
|
case mp
X : Type u_1
Y : Type u_2
tX : TopologicalSpace X
tY : TopologicalSpace Y
inst✝ : DeltaGeneratedSpace X
f : X → Y
h : ∀ (i : (n : ℕ) × C(Fin n → ℝ, X)), coinduced (f ∘ ⇑i.snd) inferInstance ≤ tY
n : ℕ
p : C(Fin n → ℝ, X)
⊢ coinduced (f ∘ ⇑p) Pi.topologicalSpace ≤ tY
|
apply h ⟨n, p⟩
|
no goals
|
722db935464c7d28
|
ProbabilityTheory.integrable_rpow_abs_of_integrable_exp_mul
|
Mathlib/Probability/Moments/IntegrableExpMul.lean
|
/-- If `ω ↦ exp (t * X ω)` is integrable at `t` and `-t` for `t ≠ 0`, then `ω ↦ |X ω| ^ p` is
integrable for all nonnegative `p : ℝ`. -/
lemma integrable_rpow_abs_of_integrable_exp_mul (ht : t ≠ 0)
(ht_int_pos : Integrable (fun ω ↦ exp (t * X ω)) μ)
(ht_int_neg : Integrable (fun ω ↦ exp (- t * X ω)) μ) {p : ℝ} (hp : 0 ≤ p) :
Integrable (fun ω ↦ |X ω| ^ p) μ
|
case refine_3
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
ht : t ≠ 0
ht_int_pos : Integrable (fun ω => rexp (t * X ω)) μ
ht_int_neg : Integrable (fun ω => rexp (-t * X ω)) μ
p : ℝ
hp : 0 ≤ p
h : Integrable (fun ω => |X ω| ^ p * rexp (0 * X ω)) μ
⊢ Integrable (fun ω => |X ω| ^ p) μ
|
simpa using h
|
no goals
|
9251f8aeb0444eb7
|
isBounded_pow
|
Mathlib/Topology/Bornology/BoundedOperation.lean
|
lemma isBounded_pow {R : Type*} [Bornology R] [Monoid R] [BoundedMul R] {s : Set R}
(s_bdd : Bornology.IsBounded s) (n : ℕ) :
Bornology.IsBounded ((fun x ↦ x ^ n) '' s)
|
case neg
R : Type u_2
inst✝² : Bornology R
inst✝¹ : Monoid R
inst✝ : BoundedMul R
s : Set R
s_bdd : Bornology.IsBounded s
s_empty : ¬s = ∅
⊢ Bornology.IsBounded ((fun x => x ^ 0) '' s)
|
simp_rw [← nonempty_iff_ne_empty] at s_empty
|
case neg
R : Type u_2
inst✝² : Bornology R
inst✝¹ : Monoid R
inst✝ : BoundedMul R
s : Set R
s_bdd : Bornology.IsBounded s
s_empty : s.Nonempty
⊢ Bornology.IsBounded ((fun x => x ^ 0) '' s)
|
72761af82af91aa5
|
smul_top_inf_eq_smul_of_isSMulRegular_on_quot
|
Mathlib/RingTheory/Regular/IsSMulRegular.lean
|
lemma smul_top_inf_eq_smul_of_isSMulRegular_on_quot :
IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N
|
R : Type u_1
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
r : R
⊢ IsSMulRegular (M ⧸ N) r → r • ⊤ ⊓ N ≤ r • N
|
convert map_mono ∘ (isSMulRegular_on_quot_iff_lsmul_comap_le N r).mp using 2
|
case h'.h.e'_3
R : Type u_1
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
N : Submodule R M
r : R
a✝ : IsSMulRegular (M ⧸ N) r
⊢ r • ⊤ ⊓ N = map (DistribMulAction.toLinearMap R M r) (comap ((LinearMap.lsmul R M) r) N)
|
501e1cac1d97bfca
|
Matrix.IsTotallyUnimodular.fromRows_unitlike
|
Mathlib/LinearAlgebra/Matrix/Determinant/TotallyUnimodular.lean
|
/-- If `A` is totally unimodular and each row of `B` is all zeros except for at most a single `1` or
a single `-1` then `fromRows A B` is totally unimodular. -/
lemma IsTotallyUnimodular.fromRows_unitlike [DecidableEq n] {A : Matrix m n R} {B : Matrix m' n R}
(hA : A.IsTotallyUnimodular)
(hB : Nonempty n → ∀ i : m', ∃ j : n, ∃ s : SignType, B i = Pi.single j s.cast) :
(fromRows A B).IsTotallyUnimodular
|
m : Type u_1
m' : Type u_2
n : Type u_3
R : Type u_5
inst✝¹ : CommRing R
inst✝ : DecidableEq n
A : Matrix m n R
B : Matrix m' n R
hA : A.IsTotallyUnimodular
k : ℕ
ih :
∀ (f : Fin k → m ⊕ m') (g : Fin k → n),
Function.Injective f → Function.Injective g → ((A.fromRows B).submatrix f g).det ∈ Set.range SignType.cast
f : Fin (k + 1) → m ⊕ m'
g : Fin (k + 1) → n
hf : Function.Injective f
hg : Function.Injective g
hB : ∀ (i : m'), ∃ j s, B i = Pi.single j ↑s
i : Fin (k + 1)
j : m'
hfi : f i = Sum.inr j
s : SignType
x : Fin (k + 1)
hj' : B j = Pi.single (g x) ↑s
hAB :
((A.fromRows B).submatrix f g).det =
(-1) ^ (↑i + ↑x) * ↑s * (((A.fromRows B).submatrix f g).submatrix i.succAbove x.succAbove).det
⊢ ↑SignType.castHom (-1) = -1
|
simp
|
no goals
|
9c8e0262dc5799a2
|
linearIndependent_of_top_le_span_of_card_eq_finrank
|
Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
|
theorem linearIndependent_of_top_le_span_of_card_eq_finrank {ι : Type*} [Fintype ι] {b : ι → V}
(spans : ⊤ ≤ span K (Set.range b)) (card_eq : Fintype.card ι = finrank K V) :
LinearIndependent K b :=
linearIndependent_iff'.mpr fun s g dependent i i_mem_s => by
classical
by_contra gx_ne_zero
-- We'll derive a contradiction by showing `b '' (univ \ {i})` of cardinality `n - 1`
-- spans a vector space of dimension `n`.
refine not_le_of_gt (span_lt_top_of_card_lt_finrank
(show (b '' (Set.univ \ {i})).toFinset.card < finrank K V from ?_)) ?_
· calc
(b '' (Set.univ \ {i})).toFinset.card = ((Set.univ \ {i}).toFinset.image b).card
|
case pos.intro
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
ι : Type u_2
inst✝ : Fintype ι
b : ι → V
spans : ⊤ ≤ span K (range b)
card_eq : Fintype.card ι = finrank K V
s : Finset ι
g : ι → K
dependent : ∑ i ∈ s, g i • b i = 0
i : ι
i_mem_s : i ∈ s
gx_ne_zero : ¬g i = 0
j : ι
j_eq : j = i
k : ι
hk : k ∈ s.erase i
k_ne_i : k ≠ i
right✝ : k ∈ s
⊢ g k • b k ∈ span K (b '' (Set.univ \ {i}))
|
refine smul_mem _ _ (subset_span ⟨k, ?_, rfl⟩)
|
case pos.intro
K : Type u
V : Type v
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
ι : Type u_2
inst✝ : Fintype ι
b : ι → V
spans : ⊤ ≤ span K (range b)
card_eq : Fintype.card ι = finrank K V
s : Finset ι
g : ι → K
dependent : ∑ i ∈ s, g i • b i = 0
i : ι
i_mem_s : i ∈ s
gx_ne_zero : ¬g i = 0
j : ι
j_eq : j = i
k : ι
hk : k ∈ s.erase i
k_ne_i : k ≠ i
right✝ : k ∈ s
⊢ k ∈ Set.univ \ {i}
|
db71af46429798d0
|
PseudoMetricSpace.le_two_mul_dist_ofPreNNDist
|
Mathlib/Topology/Metrizable/Uniformity.lean
|
theorem le_two_mul_dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x, d x x = 0)
(dist_comm : ∀ x y, d x y = d y x)
(hd : ∀ x₁ x₂ x₃ x₄, d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))) (x y : X) :
↑(d x y) ≤ 2 * @dist X
(@PseudoMetricSpace.toDist X (PseudoMetricSpace.ofPreNNDist d dist_self dist_comm)) x y
|
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * (d x₁ x₂ ⊔ (d x₂ x₃ ⊔ d x₃ x₄))
hd₀_trans : Transitive fun x y => d x y = 0
this : IsTrans X fun x y => d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.length = m → d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
⊢ ∃ z z', d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
|
have hsne : s.Nonempty := ⟨0, hs₀⟩
|
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * (d x₁ x₂ ⊔ (d x₂ x₃ ⊔ d x₃ x₄))
hd₀_trans : Transitive fun x y => d x y = 0
this : IsTrans X fun x y => d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.length = m → d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
⊢ ∃ z z', d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
|
a217eb6776c388b0
|
IntermediateField.sup_toSubfield
|
Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.lean
|
theorem sup_toSubfield (S T : IntermediateField F E) :
(S ⊔ T).toSubfield = S.toSubfield ⊔ T.toSubfield
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S T : IntermediateField F E
⊢ Subfield.closure (Set.range ⇑(algebraMap F E) ∪ (↑S ∪ ↑T)) = Subfield.closure (↑S ∪ ↑T)
|
congr 1
|
case e_s
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S T : IntermediateField F E
⊢ Set.range ⇑(algebraMap F E) ∪ (↑S ∪ ↑T) = ↑S ∪ ↑T
|
b50505e8feceb97d
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.