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
|
|---|---|---|---|---|---|---|
Nat.log_div_mul_self
|
Mathlib/Data/Nat/Log.lean
|
theorem log_div_mul_self (b n : ℕ) : log b (n / b * b) = log b n
|
b n : ℕ
⊢ log b (n / b * b) = log b n
|
rcases le_or_lt b 1 with hb | hb
|
case inl
b n : ℕ
hb : b ≤ 1
⊢ log b (n / b * b) = log b n
case inr
b n : ℕ
hb : 1 < b
⊢ log b (n / b * b) = log b n
|
e0df15fc98715957
|
MeasureTheory.Lp.simpleFunc.toSimpleFunc_toLp
|
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
theorem toSimpleFunc_toLp (f : α →ₛ E) (hfi : MemLp f p μ) : toSimpleFunc (toLp f hfi) =ᵐ[μ] f
|
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ ⇑(toSimpleFunc (toLp f hfi)) =ᶠ[ae μ] ⇑f
|
rw [← AEEqFun.mk_eq_mk]
|
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEEqFun.mk ⇑(toSimpleFunc (toLp f hfi)) ?m.234584 = AEEqFun.mk ⇑f ?m.234585
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑(toSimpleFunc (toLp f hfi))) μ
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑f) μ
|
833193308ce49be3
|
hasFDerivAt_update
|
Mathlib/Analysis/Calculus/FDeriv/Pi.lean
|
theorem hasFDerivAt_update (x : ∀ i, E i) {i : ι} (y : E i) :
HasFDerivAt (Function.update x i) (.pi (Pi.single i (.id 𝕜 (E i)))) y
|
𝕜 : Type u_1
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : (i : ι) → E i
i : ι
y : E i
l : E i →L[𝕜] (i : ι) → E i := ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))
update_eq : Function.update x i = (fun x_1 => x) + ⇑l ∘ fun x_1 => x_1 - x i
⊢ HasFDerivAt ((fun x_1 => x) + ⇑l ∘ fun x_1 => x_1 - x i) l y
|
convert (hasFDerivAt_const _ _).add (l.hasFDerivAt.comp y (hasFDerivAt_sub_const (x i)))
|
case h.e'_12.h.h
𝕜 : Type u_1
ι : Type u_2
inst✝⁴ : DecidableEq ι
inst✝³ : Fintype ι
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
x : (i : ι) → E i
i : ι
y : E i
l : E i →L[𝕜] (i : ι) → E i := ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))
update_eq : Function.update x i = (fun x_1 => x) + ⇑l ∘ fun x_1 => x_1 - x i
e_8✝ : Pi.addCommGroup = SeminormedAddCommGroup.toAddCommGroup
he✝ : Pi.module ι E 𝕜 = NormedSpace.toModule
e_10✝ : Pi.topologicalSpace = UniformSpace.toTopologicalSpace
⊢ l = 0 + l.comp (ContinuousLinearMap.id 𝕜 (E i))
|
1a0c7205e23fe373
|
IsCauSeq.series_ratio_test
|
Mathlib/Algebra/Order/CauSeq/BigOperators.lean
|
lemma series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m ↦ ∑ n ∈ range m, f n
|
case inr.succ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrderedField α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : |r| < 1
hr : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)) →
∀ (m : ℕ), n.succ ≤ m → m = k + n.succ → abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
m : ℕ
hmn : n.succ ≤ m
hk : m = k + 1 + n.succ
kn : k + n.succ ≥ n.succ
⊢ abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
|
rw [hk, Nat.succ_add, pow_succ r, ← mul_assoc]
|
case inr.succ
α : Type u_1
β : Type u_2
inst✝³ : LinearOrderedField α
inst✝² : Ring β
abv : β → α
inst✝¹ : IsAbsoluteValue abv
inst✝ : Archimedean α
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : |r| < 1
hr : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)) →
∀ (m : ℕ), n.succ ≤ m → m = k + n.succ → abv (f m) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ m
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)
m : ℕ
hmn : n.succ ≤ m
hk : m = k + 1 + n.succ
kn : k + n.succ ≥ n.succ
⊢ abv (f (k + n.succ).succ) ≤ abv (f n.succ) * r⁻¹ ^ n.succ * r ^ (k + n.succ) * r
|
1eb29c24e9473c8d
|
fixed_of_fixed1_aux1
|
Mathlib/RingTheory/Invariant.lean
|
theorem fixed_of_fixed1_aux1 [DecidableEq (Ideal B)] :
∃ a b : B, (∀ g : G, g • a = a) ∧ a ∉ Q ∧
∀ g : G, algebraMap B (B ⧸ Q) (g • b) = algebraMap B (B ⧸ Q) (if g • Q = Q then a else 0)
|
case intro.intro.intro.intro.intro
B : Type u_2
inst✝⁵ : CommRing B
G : Type u_3
inst✝⁴ : Group G
inst✝³ : Finite G
inst✝² : MulSemiringAction G B
Q : Ideal B
inst✝¹ : Q.IsPrime
inst✝ : DecidableEq (Ideal B)
val✝ : Fintype G
P : Ideal B := (Finset.filter (fun g => g • Q ≠ Q) Finset.univ).inf fun g => g • Q
h1 : ¬P ≤ Q
b : B
hbQ : b ∉ Q
hbP : ∀ (g : G), g • Q ≠ Q → b ∈ g • Q
f : B[X] := MulSemiringAction.charpoly G b
q : (B ⧸ Q)[X]
hq : map (algebraMap B (B ⧸ Q)) f = X ^ rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f) * q
hq0 : ¬X ∣ q
j : ℕ := rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f)
k : ℕ := q.natDegree
r : B[X] := ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
hr✝ : map (algebraMap B (B ⧸ Q)) r = q
hf : eval b f = 0
hr : eval b r ∈ Q
⊢ ∃ a b,
(∀ (g : G), g • a = a) ∧
a ∉ Q ∧ ∀ (g : G), (algebraMap B (B ⧸ Q)) (g • b) = (algebraMap B (B ⧸ Q)) (if g • Q = Q then a else 0)
|
let a := f.coeff j
|
case intro.intro.intro.intro.intro
B : Type u_2
inst✝⁵ : CommRing B
G : Type u_3
inst✝⁴ : Group G
inst✝³ : Finite G
inst✝² : MulSemiringAction G B
Q : Ideal B
inst✝¹ : Q.IsPrime
inst✝ : DecidableEq (Ideal B)
val✝ : Fintype G
P : Ideal B := (Finset.filter (fun g => g • Q ≠ Q) Finset.univ).inf fun g => g • Q
h1 : ¬P ≤ Q
b : B
hbQ : b ∉ Q
hbP : ∀ (g : G), g • Q ≠ Q → b ∈ g • Q
f : B[X] := MulSemiringAction.charpoly G b
q : (B ⧸ Q)[X]
hq : map (algebraMap B (B ⧸ Q)) f = X ^ rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f) * q
hq0 : ¬X ∣ q
j : ℕ := rootMultiplicity 0 (map (algebraMap B (B ⧸ Q)) f)
k : ℕ := q.natDegree
r : B[X] := ∑ i ∈ Finset.range (k + 1), (monomial i) (f.coeff (i + j))
hr✝ : map (algebraMap B (B ⧸ Q)) r = q
hf : eval b f = 0
hr : eval b r ∈ Q
a : B := f.coeff j
⊢ ∃ a b,
(∀ (g : G), g • a = a) ∧
a ∉ Q ∧ ∀ (g : G), (algebraMap B (B ⧸ Q)) (g • b) = (algebraMap B (B ⧸ Q)) (if g • Q = Q then a else 0)
|
807fb7b15a430402
|
Complex.isTheta_cpow_rpow
|
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
theorem isTheta_cpow_rpow (hl_im : IsBoundedUnder (· ≤ ·) l fun x => |(g x).im|)
(hl : ∀ᶠ x in l, f x = 0 → re (g x) = 0 → g x = 0) :
(fun x => f x ^ g x) =Θ[l] fun x => ‖f x‖ ^ (g x).re :=
calc
(fun x => f x ^ g x) =Θ[l]
(fun x => ‖f x‖ ^ (g x).re / Real.exp (arg (f x) * im (g x))) :=
.of_norm_eventuallyEq <| hl.mono fun _ => norm_cpow_of_imp
_ =Θ[l] fun x => ‖f x‖ ^ (g x).re / (1 : ℝ) :=
(isTheta_refl _ _).div (isTheta_exp_arg_mul_im hl_im)
_ =ᶠ[l] (fun x => ‖f x‖ ^ (g x).re)
|
α : Type u_1
l : Filter α
f g : α → ℂ
hl_im : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l fun x => |(g x).im|
hl : ∀ᶠ (x : α) in l, f x = 0 → (g x).re = 0 → g x = 0
⊢ (fun x => ‖f x‖ ^ (g x).re / 1) =ᶠ[l] fun x => ‖f x‖ ^ (g x).re
|
simp only [ofReal_one, div_one, EventuallyEq.rfl]
|
no goals
|
1d341a53588709b8
|
Ordnode.all_node4R
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem all_node4R {P l x m y r} :
@All α P (node4R l x m y r) ↔ All P l ∧ P x ∧ All P m ∧ P y ∧ All P r
|
α : Type u_1
P : α → Prop
l : Ordnode α
x : α
m : Ordnode α
y : α
r : Ordnode α
⊢ All P (l.node4R x m y r) ↔ All P l ∧ P x ∧ All P m ∧ P y ∧ All P r
|
cases m <;> simp [node4R, all_node', All, all_node3R, and_assoc]
|
no goals
|
d6ed2f27bea4da55
|
gauge_lt_one_of_mem_of_isOpen
|
Mathlib/Analysis/Convex/Gauge.lean
|
theorem gauge_lt_one_of_mem_of_isOpen (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
gauge s x < 1 :=
interior_subset_gauge_lt_one s <| by rwa [hs₂.interior_eq]
|
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs₂ : IsOpen s
x : E
hx : x ∈ s
⊢ x ∈ interior s
|
rwa [hs₂.interior_eq]
|
no goals
|
e4ebe6789d2a641b
|
RingHom.finitePresentation_ofLocalizationSpanTarget
|
Mathlib/RingTheory/RingHom/FinitePresentation.lean
|
theorem finitePresentation_ofLocalizationSpanTarget :
OfLocalizationSpanTarget @FinitePresentation
|
case mk
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this✝ : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
hfI : ∀ a ∈ I, f a = 0
f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI
hf' : Function.Surjective ⇑f'
t : Finset A := Finset.image (fun g => (Ideal.Quotient.mk I) (g' g)) Finset.univ
ht : Ideal.span ↑t = ⊤
this : Algebra.FinitePresentation R A
g : A
hg : g ∈ t
⊢ ∃ a, ∃ (hb : a ∈ s), (Ideal.Quotient.mk I) (g' ⟨a, hb⟩) = ↑⟨g, hg⟩
|
convert hg
|
case a
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f✝ : R →+* S
s : Finset S
hs : Ideal.span ↑s = ⊤
this✝ : Algebra R S := f✝.toAlgebra
H : ∀ (r : { x // x ∈ s }), Algebra.FinitePresentation R (Localization.Away ↑r)
hfintype : Algebra.FiniteType R S
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] S
hf : Function.Surjective ⇑f
l : ↑↑s →₀ S
hl : (Finsupp.linearCombination S Subtype.val) l = 1
g' : { x // x ∈ s } → MvPolynomial (Fin n) R
hg' : ∀ (g : { x // x ∈ s }), f (g' g) = ↑g
h' : { x // x ∈ s } → MvPolynomial (Fin n) R
hh' : ∀ (g : { x // x ∈ s }), f (h' g) = l g
I : Ideal (MvPolynomial (Fin n) R) := Ideal.span {∑ g : { x // x ∈ s }, g' g * h' g - 1}
A : Type u_1 := MvPolynomial (Fin n) R ⧸ I
hfI : ∀ a ∈ I, f a = 0
f' : A →ₐ[R] S := Ideal.Quotient.liftₐ I f hfI
hf' : Function.Surjective ⇑f'
t : Finset A := Finset.image (fun g => (Ideal.Quotient.mk I) (g' g)) Finset.univ
ht : Ideal.span ↑t = ⊤
this : Algebra.FinitePresentation R A
g : A
hg : g ∈ t
⊢ (∃ a, ∃ (hb : a ∈ s), (Ideal.Quotient.mk I) (g' ⟨a, hb⟩) = ↑⟨g, hg⟩) ↔ g ∈ t
|
1e270402a60647b4
|
Submodule.linearMap_eq_iff_of_span_eq_top
|
Mathlib/LinearAlgebra/Span/Basic.lean
|
lemma linearMap_eq_iff_of_span_eq_top (f g : M →ₗ[R] N)
{S : Set M} (hM : span R S = ⊤) :
f = g ↔ ∀ (s : S), f s = g s
|
case h.e'_1.a.mpr
R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_9
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
f g : M →ₗ[R] N
S : Set M
hM : span R S = ⊤
h : f ∘ₗ ⊤.subtype = g ∘ₗ ⊤.subtype
⊢ f = g
|
ext x
|
case h.e'_1.a.mpr.h
R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_9
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
f g : M →ₗ[R] N
S : Set M
hM : span R S = ⊤
h : f ∘ₗ ⊤.subtype = g ∘ₗ ⊤.subtype
x : M
⊢ f x = g x
|
0643c3f955b1db39
|
Algebra.IsAlgebraic.exists_integral_multiples
|
Mathlib/RingTheory/Algebraic/Integral.lean
|
theorem _root_.Algebra.IsAlgebraic.exists_integral_multiples [NoZeroDivisors R]
[alg : Algebra.IsAlgebraic R A] (s : Finset A) :
∃ y ≠ (0 : R), ∀ z ∈ s, IsIntegral R (y • z)
|
R : Type u_1
A : Type u_3
inst✝³ : CommRing R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : NoZeroDivisors R
alg : Algebra.IsAlgebraic R A
s : Finset A
this : Nontrivial R
r : A → R
hr : ∀ (x : A), r x ≠ 0
int : ∀ (x : A), IsIntegral R (r x • x)
x✝ : A
h : x✝ ∈ s
⊢ IsIntegral R ((∏ x ∈ s, r x) • x✝)
|
classical rw [← Finset.prod_erase_mul _ _ h, mul_smul]
|
R : Type u_1
A : Type u_3
inst✝³ : CommRing R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : NoZeroDivisors R
alg : Algebra.IsAlgebraic R A
s : Finset A
this : Nontrivial R
r : A → R
hr : ∀ (x : A), r x ≠ 0
int : ∀ (x : A), IsIntegral R (r x • x)
x✝ : A
h : x✝ ∈ s
⊢ IsIntegral R ((∏ x ∈ s.erase x✝, r x) • r x✝ • x✝)
|
918ed071598a1fa2
|
MvPolynomial.support_mul_X
|
Mathlib/Algebra/MvPolynomial/Basic.lean
|
theorem support_mul_X (s : σ) (p : MvPolynomial σ R) :
(p * X s).support = p.support.map (addRightEmbedding (Finsupp.single s 1)) :=
AddMonoidAlgebra.support_mul_single p _ (by simp) _
|
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
s : σ
p : MvPolynomial σ R
⊢ ∀ (y : R), y * 1 = 0 ↔ y = 0
|
simp
|
no goals
|
ac13dbd48e946e2f
|
Algebra.basicOpen_subset_unramifiedLocus_iff
|
Mathlib/RingTheory/Unramified/Locus.lean
|
lemma basicOpen_subset_unramifiedLocus_iff {f : A} :
↑(PrimeSpectrum.basicOpen f) ⊆ unramifiedLocus R A ↔
Algebra.FormallyUnramified R (Localization.Away f)
|
R A : Type u
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
⊢ ↑(PrimeSpectrum.basicOpen f) ⊆ unramifiedLocus R A ↔ FormallyUnramified R (Localization.Away f)
|
rw [unramifiedLocus_eq_compl_support, Set.subset_compl_comm,
PrimeSpectrum.basicOpen_eq_zeroLocus_compl, compl_compl,
← LocalizedModule.subsingleton_iff_support_subset, Algebra.formallyUnramified_iff]
|
R A : Type u
inst✝² : CommRing R
inst✝¹ : CommRing A
inst✝ : Algebra R A
f : A
⊢ Subsingleton (LocalizedModule (Submonoid.powers f) (Ω[A⁄R])) ↔ Subsingleton (Ω[Localization.Away f⁄R])
|
897aba214e911642
|
Module.End.eigenspace_restrict_eq_bot
|
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
theorem eigenspace_restrict_eq_bot {f : End R M} {p : Submodule R M} (hfp : ∀ x ∈ p, f x ∈ p)
{μ : R} (hμp : Disjoint (f.eigenspace μ) p) : eigenspace (f.restrict hfp) μ = ⊥
|
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
p : Submodule R M
hfp : ∀ x ∈ p, f x ∈ p
μ : R
hμp : Disjoint (f.eigenspace μ) p
⊢ eigenspace (LinearMap.restrict f hfp) μ ≤ ⊥
|
intro x hx
|
R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
p : Submodule R M
hfp : ∀ x ∈ p, f x ∈ p
μ : R
hμp : Disjoint (f.eigenspace μ) p
x : ↥p
hx : x ∈ eigenspace (LinearMap.restrict f hfp) μ
⊢ x ∈ ⊥
|
234c8da254101d84
|
EMetric.NonemptyCompacts.isClosed_in_closeds
|
Mathlib/Topology/MetricSpace/Closeds.lean
|
theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
IsClosed (range <| @NonemptyCompacts.toCloseds α _ _)
|
case refine_1.intro.intro
α : Type u
inst✝¹ : EMetricSpace α
inst✝ : CompleteSpace α
this : range NonemptyCompacts.toCloseds = {s | (↑s).Nonempty ∧ IsCompact ↑s}
s : Closeds α
hs : s ∈ closure {s | (↑s).Nonempty ∧ IsCompact ↑s}
t : Closeds α
ht : t ∈ {s | (↑s).Nonempty ∧ IsCompact ↑s}
Dst : edist t s < ⊤
⊢ (↑s).Nonempty
|
exact nonempty_of_hausdorffEdist_ne_top ht.1 (ne_of_lt Dst)
|
no goals
|
47173bb814bf33e2
|
List.max?_mem
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MinMax.lean
|
theorem max?_mem [Max α] (min_eq_or : ∀ a b : α, max a b = a ∨ max a b = b) :
{xs : List α} → xs.max? = some a → a ∈ xs
| nil => by simp
| cons x xs => by
rw [max?]; rintro ⟨⟩
induction xs generalizing x with simp at *
| cons y xs ih =>
rcases ih (max x y) with h | h <;> simp [h]
simp [← or_assoc, min_eq_or x y]
|
α : Type u_1
a : α
inst✝ : Max α
min_eq_or : ∀ (a b : α), max a b = a ∨ max a b = b
⊢ [].max? = some a → a ∈ []
|
simp
|
no goals
|
7317c10ebce96da2
|
Batteries.LTCmp.eq_compareOfLessAndEq
|
Mathlib/.lake/packages/batteries/Batteries/Classes/Order.lean
|
theorem LTCmp.eq_compareOfLessAndEq
[LT α] [DecidableEq α] [BEq α] [LawfulBEq α] [BEqCmp cmp] [LTCmp cmp]
(x y : α) [Decidable (x < y)] : cmp x y = compareOfLessAndEq x y
|
case isFalse.isFalse.eq
α : Type u_1
cmp : α → α → Ordering
inst✝⁶ : LT α
inst✝⁵ : DecidableEq α
inst✝⁴ : BEq α
inst✝³ : LawfulBEq α
inst✝² : BEqCmp cmp
inst✝¹ : LTCmp cmp
x y : α
inst✝ : Decidable (x < y)
h1 : ¬x < y
h2 : ¬x = y
e : cmp x y = Ordering.eq
⊢ Ordering.eq = Ordering.gt
|
cases h2 (BEqCmp.cmp_iff_eq.1 e)
|
no goals
|
f00bdc3f2d775dc4
|
Directed.strictConvex_iUnion
|
Mathlib/Analysis/Convex/Strict.lean
|
theorem Directed.strictConvex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)) : StrictConvex 𝕜 (⋃ i, s i)
|
case intro
𝕜 : Type u_1
E : Type u_3
inst✝³ : OrderedSemiring 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
ι : Sort u_6
s : ι → Set E
hdir : Directed (fun x1 x2 => x1 ⊆ x2) s
hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)
x y : E
hy : ∃ i, y ∈ s i
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
i : ι
hx : x ∈ s i
⊢ a • x + b • y ∈ interior (⋃ i, s i)
|
obtain ⟨j, hy⟩ := hy
|
case intro.intro
𝕜 : Type u_1
E : Type u_3
inst✝³ : OrderedSemiring 𝕜
inst✝² : TopologicalSpace E
inst✝¹ : AddCommMonoid E
inst✝ : SMul 𝕜 E
ι : Sort u_6
s : ι → Set E
hdir : Directed (fun x1 x2 => x1 ⊆ x2) s
hs : ∀ ⦃i : ι⦄, StrictConvex 𝕜 (s i)
x y : E
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
i : ι
hx : x ∈ s i
j : ι
hy : y ∈ s j
⊢ a • x + b • y ∈ interior (⋃ i, s i)
|
5a828b9512ea5146
|
AddCommGroup.DirectLimit.map_id
|
Mathlib/Algebra/Colimit/Module.lean
|
@[simp] lemma map_id :
map (fun _ ↦ AddMonoidHom.id _) (fun _ _ _ ↦ rfl) = AddMonoidHom.id (DirectLimit G f) :=
DFunLike.ext _ _ <| by
rintro ⟨x⟩; refine x.induction_on (by simp) (fun _ ↦ map_apply_of _ _) (by simp +contextual)
|
ι : Type u_2
inst✝² : Preorder ι
G : ι → Type u_3
inst✝¹ : (i : ι) → AddCommMonoid (G i)
f : (i j : ι) → i ≤ j → G i →+ G j
inst✝ : DecidableEq ι
x✝ : DirectLimit G f
x : DirectSum ι G
⊢ (map (fun x => AddMonoidHom.id (G x)) ⋯)
(Quot.mk (⇑(addConGen (Module.DirectLimit.Eqv fun i j hij => (f i j hij).toNatLinearMap)).toSetoid) 0) =
(AddMonoidHom.id (DirectLimit G f))
(Quot.mk (⇑(addConGen (Module.DirectLimit.Eqv fun i j hij => (f i j hij).toNatLinearMap)).toSetoid) 0)
|
simp
|
no goals
|
909e489a38ff0289
|
Tuple.sort_eq_refl_iff_monotone
|
Mathlib/Data/Fin/Tuple/Sort.lean
|
theorem sort_eq_refl_iff_monotone : sort f = Equiv.refl _ ↔ Monotone f
|
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
⊢ sort f = Equiv.refl (Fin n) ↔ Monotone f
|
rw [eq_comm, eq_sort_iff, Equiv.coe_refl, Function.comp_id]
|
n : ℕ
α : Type u_1
inst✝ : LinearOrder α
f : Fin n → α
⊢ (Monotone f ∧ ∀ (i j : Fin n), i < j → f (id i) = f (id j) → id i < id j) ↔ Monotone f
|
1b1478e08b64e551
|
Metric.infDist_le_dist_of_mem
|
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
theorem infDist_le_dist_of_mem (h : y ∈ s) : infDist x s ≤ dist x y
|
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
x y : α
h : y ∈ s
⊢ (infEdist x s).toReal ≤ (edist x y).toReal
|
exact ENNReal.toReal_mono (edist_ne_top _ _) (infEdist_le_edist_of_mem h)
|
no goals
|
998d62bdb3720f39
|
CategoryTheory.Sieve.generate_le_iff
|
Mathlib/CategoryTheory/Sites/Sieves.lean
|
theorem generate_le_iff (R : Presieve X) (S : Sieve X) : generate R ≤ S ↔ R ≤ S :=
⟨fun H _ _ hg => H _ ⟨_, 𝟙 _, _, hg, id_comp _⟩, fun ss Y f => by
rintro ⟨Z, f, g, hg, rfl⟩
exact S.downward_closed (ss Z hg) f⟩
|
C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
R : Presieve X
S : Sieve X
ss : R ≤ S.arrows
Y : C
f : Y ⟶ X
⊢ (generate R).arrows f → S.arrows f
|
rintro ⟨Z, f, g, hg, rfl⟩
|
case intro.intro.intro.intro
C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
R : Presieve X
S : Sieve X
ss : R ≤ S.arrows
Y Z : C
f : Y ⟶ Z
g : Z ⟶ X
hg : R g
⊢ S.arrows (f ≫ g)
|
b906d353ddc8fcba
|
Zlattice.FG
|
Mathlib/Algebra/Module/ZLattice/Basic.lean
|
theorem Zlattice.FG [hs : IsZLattice K L] : L.FG
|
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
⊢ s ⊆ Set.range Subtype.val
|
simp only [Subtype.range_coe_subtype, Set.setOf_mem_eq, subset_rfl]
|
no goals
|
68cb8b310e25c5ae
|
LieHom.surjective_rangeRestrict
|
Mathlib/Algebra/Lie/Subalgebra.lean
|
theorem surjective_rangeRestrict : Function.Surjective f.rangeRestrict
|
case mk
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
y : L₂
hy : y ∈ f.range
⊢ ∃ a, f.rangeRestrict a = ⟨y, hy⟩
|
rw [mem_range] at hy
|
case mk
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
y : L₂
hy✝ : y ∈ f.range
hy : ∃ y_1, f y_1 = y
⊢ ∃ a, f.rangeRestrict a = ⟨y, hy✝⟩
|
31a7c87f261059cd
|
AddMonoidAlgebra.supDegree_sub_lt_of_leadingCoeff_eq
|
Mathlib/Algebra/MonoidAlgebra/Degree.lean
|
lemma supDegree_sub_lt_of_leadingCoeff_eq (hD : D.Injective) {R} [CommRing R] {p q : R[A]}
(hd : p.supDegree D = q.supDegree D) (hc : p.leadingCoeff D = q.leadingCoeff D) :
(p - q).supDegree D < p.supDegree D ∨ p = q
|
case refine_2
A : Type u_3
B : Type u_5
inst✝³ : LinearOrder B
inst✝² : OrderBot B
D : A → B
inst✝¹ : AddZeroClass A
hD : Function.Injective D
R : Type u_8
inst✝ : CommRing R
p q : R[A]
hd : supDegree D p = supDegree D q
hc : leadingCoeff D p = leadingCoeff D q
he : ¬(p - q) (Function.invFun D (supDegree D (p - q))) = 0
⊢ supDegree D (p - q) ≠ supDegree D p
|
refine fun h => he ?_
|
case refine_2
A : Type u_3
B : Type u_5
inst✝³ : LinearOrder B
inst✝² : OrderBot B
D : A → B
inst✝¹ : AddZeroClass A
hD : Function.Injective D
R : Type u_8
inst✝ : CommRing R
p q : R[A]
hd : supDegree D p = supDegree D q
hc : leadingCoeff D p = leadingCoeff D q
he : ¬(p - q) (Function.invFun D (supDegree D (p - q))) = 0
h : supDegree D (p - q) = supDegree D p
⊢ (p - q) (Function.invFun D (supDegree D (p - q))) = 0
|
8a45d765e6091c8f
|
OrthogonalFamily.summable_iff_norm_sq_summable
|
Mathlib/Analysis/InnerProductSpace/Subspace.lean
|
theorem OrthogonalFamily.summable_iff_norm_sq_summable [CompleteSpace E] (f : ∀ i, G i) :
(Summable fun i => V i (f i)) ↔ Summable fun i => ‖f i‖ ^ 2
|
case mpr
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
⊢ ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε
|
obtain ⟨a, H⟩ := hf _ hε'
|
case mpr.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : CompleteSpace E
f : (i : ι) → G i
hf : ∀ ε > 0, ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε
ε : ℝ
hε : ε > 0
hε' : 0 < ε ^ 2 / 2
a : Finset ι
H : ∀ (m : Finset ι), a ≤ m → ∀ (n : Finset ι), a ≤ n → |∑ i ∈ m, ‖f i‖ ^ 2 - ∑ i ∈ n, ‖f i‖ ^ 2| < ε ^ 2 / 2
⊢ ∃ N, ∀ (m : Finset ι), N ≤ m → ∀ (n : Finset ι), N ≤ n → ‖∑ i ∈ m, (V i) (f i) - ∑ i ∈ n, (V i) (f i)‖ < ε
|
ef6c5d8e8db28ff0
|
round_sub_int
|
Mathlib/Algebra/Order/Round.lean
|
theorem round_sub_int (x : α) (y : ℤ) : round (x - y) = round x - y
|
α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
y : ℤ
⊢ round (x + -↑y) = round x - y
|
norm_cast
|
α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
x : α
y : ℤ
⊢ round (x + ↑(-y)) = round x - y
|
bbc77d223e07e350
|
Submodule.prod_sup_prod
|
Mathlib/LinearAlgebra/Span/Basic.lean
|
theorem prod_sup_prod : prod p q₁ ⊔ prod p' q₁' = prod (p ⊔ p') (q₁ ⊔ q₁')
|
R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
p p' : Submodule R M
M' : Type u_9
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
q₁ q₁' : Submodule R M'
⊢ ∀ (a : M) (b : M'), a ∈ p ⊔ p' → b ∈ q₁ ⊔ q₁' → (a, b) ∈ p.prod q₁ ⊔ p'.prod q₁'
|
intro xx yy hxx hyy
|
R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
p p' : Submodule R M
M' : Type u_9
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
q₁ q₁' : Submodule R M'
xx : M
yy : M'
hxx : xx ∈ p ⊔ p'
hyy : yy ∈ q₁ ⊔ q₁'
⊢ (xx, yy) ∈ p.prod q₁ ⊔ p'.prod q₁'
|
6bc60ca9bdd6f33f
|
ExtremallyDisconnected.homeoCompactToT2_injective
|
Mathlib/Topology/ExtremallyDisconnected.lean
|
private lemma ExtremallyDisconnected.homeoCompactToT2_injective [ExtremallyDisconnected A]
[T2Space A] [T2Space E] [CompactSpace E] {ρ : E → A} (ρ_cont : Continuous ρ)
(ρ_surj : ρ.Surjective) (zorn_subset : ∀ E₀ : Set E, E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ) :
ρ.Injective
|
case intro.intro.intro.intro.intro.intro
A E : Type u
inst✝⁵ : TopologicalSpace A
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
⊢ False
|
have G₂_comp : IsCompact G₂ᶜ := IsClosed.isCompact G₂_open.isClosed_compl
|
case intro.intro.intro.intro.intro.intro
A E : Type u
inst✝⁵ : TopologicalSpace A
inst✝⁴ : TopologicalSpace E
inst✝³ : ExtremallyDisconnected A
inst✝² : T2Space A
inst✝¹ : T2Space E
inst✝ : CompactSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
x₁ x₂ : E
hρx : ρ x₁ = ρ x₂
hx : ¬x₁ = x₂
G₁ G₂ : Set E
G₁_open : IsOpen G₁
G₂_open : IsOpen G₂
hx₁ : x₁ ∈ G₁
hx₂ : x₂ ∈ G₂
disj : Disjoint G₁ G₂
G₁_comp : IsCompact G₁ᶜ
G₂_comp : IsCompact G₂ᶜ
⊢ False
|
76a8066ab0a4e7d4
|
MvPolynomial.C_mem_coeffsIn
|
Mathlib/Algebra/MvPolynomial/Basic.lean
|
@[simp]
lemma C_mem_coeffsIn : C x ∈ coeffsIn σ M ↔ x ∈ M
|
R : Type u_2
S : Type u_3
σ : Type u_4
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Module R S
M : Submodule R S
x : S
⊢ C x ∈ coeffsIn σ M ↔ x ∈ M
|
simpa using monomial_mem_coeffsIn (i := 0)
|
no goals
|
169ffb43e59394e5
|
CategoryTheory.Presieve.extend_restrict
|
Mathlib/CategoryTheory/Sites/IsSheafFor.lean
|
theorem extend_restrict {x : FamilyOfElements P (generate R).arrows} (t : x.Compatible) :
(x.restrict (le_generate R)).sieveExtend = x
|
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
X : C
R : Presieve X
x : FamilyOfElements P (generate R).arrows
t : x.SieveCompatible
⊢ (FamilyOfElements.restrict ⋯ x).sieveExtend = x
|
funext _ _ h
|
case h.h.h
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
X : C
R : Presieve X
x : FamilyOfElements P (generate R).arrows
t : x.SieveCompatible
x✝¹ : C
x✝ : x✝¹ ⟶ X
h : (generate R).arrows x✝
⊢ (FamilyOfElements.restrict ⋯ x).sieveExtend x✝ h = x x✝ h
|
ee85a8bbbc52acdf
|
ProbabilityTheory.iteratedDeriv_two_cgf
|
Mathlib/Probability/Moments/MGFAnalytic.lean
|
lemma iteratedDeriv_two_cgf (h : v ∈ interior (integrableExpSet X μ)) :
iteratedDeriv 2 (cgf X μ) v
= μ[fun ω ↦ (X ω)^2 * exp (v * X ω)] / mgf X μ v - deriv (cgf X μ) v ^ 2
|
case pos
Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
v : ℝ
h : v ∈ interior (integrableExpSet X μ)
hμ : μ = 0
this : deriv 0 = 0
⊢ deriv (deriv (cgf X μ)) v = (∫ (x : Ω), (fun ω => X ω ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v - deriv (cgf X μ) v ^ 2
|
simp [hμ, this]
|
no goals
|
24002d4e2edb89d3
|
SimpleGraph.Walk.snd_takeUntil
|
Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkDecomp.lean
|
lemma snd_takeUntil (hsu : w ≠ u) (p : G.Walk u v) (h : w ∈ p.support) :
(p.takeUntil w h).snd = p.snd
|
V : Type u
G : SimpleGraph V
v w u : V
inst✝ : DecidableEq V
hsu : w ≠ u
p : G.Walk u v
h : w ∈ p.support
⊢ 1 ≤ (p.takeUntil w h).length
|
by_contra! hc
|
V : Type u
G : SimpleGraph V
v w u : V
inst✝ : DecidableEq V
hsu : w ≠ u
p : G.Walk u v
h : w ∈ p.support
hc : (p.takeUntil w h).length < 1
⊢ False
|
0b75184ebe6222be
|
intervalIntegral.integral_eq_zero_iff_of_le_of_nonneg_ae
|
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
theorem integral_eq_zero_iff_of_le_of_nonneg_ae (hab : a ≤ b) (hf : 0 ≤ᵐ[μ.restrict (Ioc a b)] f)
(hfi : IntervalIntegrable f μ a b) :
∫ x in a..b, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict (Ioc a b)] 0
|
f : ℝ → ℝ
a b : ℝ
μ : Measure ℝ
hab : a ≤ b
hf : 0 ≤ᶠ[ae (μ.restrict (Ioc a b))] f
hfi : IntervalIntegrable f μ a b
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = 0 ↔ f =ᶠ[ae (μ.restrict (Ioc a b))] 0
|
rw [integral_of_le hab, integral_eq_zero_iff_of_nonneg_ae hf hfi.1]
|
no goals
|
63ad7e2fd29d7d88
|
Lean.Omega.IntList.dot_sdiv_left
|
Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean
|
theorem dot_sdiv_left (xs ys : IntList) {d : Int} (h : d ∣ xs.gcd) :
dot (xs.sdiv d) ys = (dot xs ys) / d
|
case cons.cons
d x : Int
xs : List Int
ih : ∀ (ys : IntList), d ∣ ↑(gcd xs) → (sdiv xs d).dot ys = dot xs ys / d
h : d ∣ ↑(gcd (x :: xs))
y : Int
ys : List Int
wx : d ∣ x
⊢ (sdiv (x :: xs) d).dot (y :: ys) = dot (x :: xs) (y :: ys) / d
|
have wxy : d ∣ x * y := Int.dvd_trans wx (Int.dvd_mul_right x y)
|
case cons.cons
d x : Int
xs : List Int
ih : ∀ (ys : IntList), d ∣ ↑(gcd xs) → (sdiv xs d).dot ys = dot xs ys / d
h : d ∣ ↑(gcd (x :: xs))
y : Int
ys : List Int
wx : d ∣ x
wxy : d ∣ x * y
⊢ (sdiv (x :: xs) d).dot (y :: ys) = dot (x :: xs) (y :: ys) / d
|
77e481063fc36653
|
ae_eq_const_or_exists_average_ne_compl
|
Mathlib/Analysis/Convex/Integral.lean
|
theorem ae_eq_const_or_exists_average_ne_compl [IsFiniteMeasure μ] (hfi : Integrable f μ) :
f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨
∃ t, MeasurableSet t ∧ μ t ≠ 0 ∧ μ tᶜ ≠ 0 ∧ (⨍ x in t, f x ∂μ) ≠ ⨍ x in tᶜ, f x ∂μ
|
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ
|
by_cases h₀' : μ tᶜ = 0
|
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : μ tᶜ = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ
case neg
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : ¬μ t = 0
h₀' : ¬μ tᶜ = 0
⊢ ∫ (x : α) in t, f x ∂μ = (μ t).toReal • ⨍ (x : α), f x ∂μ
|
fdfdf49480067dc7
|
List.not_lex_antisymm
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/BasicAux.lean
|
theorem not_lex_antisymm [DecidableEq α] {r : α → α → Prop} [DecidableRel r]
(antisymm : ∀ x y : α, ¬ r x y → ¬ r y x → x = y)
{as bs : List α} (h₁ : ¬ Lex r bs as) (h₂ : ¬ Lex r as bs) : as = bs :=
match as, bs with
| [], [] => rfl
| [], _::_ => False.elim <| h₂ (List.Lex.nil ..)
| _::_, [] => False.elim <| h₁ (List.Lex.nil ..)
| a::as, b::bs => by
by_cases hab : r a b
· exact False.elim <| h₂ (List.Lex.rel hab)
· by_cases eq : a = b
· subst eq
have h₁ : ¬ Lex r bs as := fun h => h₁ (List.Lex.cons h)
have h₂ : ¬ Lex r as bs := fun h => h₂ (List.Lex.cons h)
simp [not_lex_antisymm antisymm h₁ h₂]
· exfalso
by_cases hba : r b a
· exact h₁ (Lex.rel hba)
· exact eq (antisymm _ _ hab hba)
|
case pos
α : Type u_1
inst✝¹ : DecidableEq α
r : α → α → Prop
inst✝ : DecidableRel r
antisymm : ∀ (x y : α), ¬r x y → ¬r y x → x = y
as✝ bs✝ : List α
a : α
as bs : List α
h₁✝ : ¬Lex r (a :: bs) (a :: as)
h₂ : ¬Lex r (a :: as) (a :: bs)
hab : ¬r a a
h₁ : ¬Lex r bs as
⊢ a :: as = a :: bs
|
have h₂ : ¬ Lex r as bs := fun h => h₂ (List.Lex.cons h)
|
case pos
α : Type u_1
inst✝¹ : DecidableEq α
r : α → α → Prop
inst✝ : DecidableRel r
antisymm : ∀ (x y : α), ¬r x y → ¬r y x → x = y
as✝ bs✝ : List α
a : α
as bs : List α
h₁✝ : ¬Lex r (a :: bs) (a :: as)
h₂✝ : ¬Lex r (a :: as) (a :: bs)
hab : ¬r a a
h₁ : ¬Lex r bs as
h₂ : ¬Lex r as bs
⊢ a :: as = a :: bs
|
bc20d604ba52bc59
|
Complex.HadamardThreeLines.sSupNormIm_scale_right
|
Mathlib/Analysis/Complex/Hadamard.lean
|
/-- The supremum of the norm of `scale f l u` on the line `z.re = 1` is the same as
the supremum of `f` on the line `z.re = u`. -/
lemma sSupNormIm_scale_right (f : ℂ → E) {l u : ℝ} (hul : l < u) :
sSupNormIm (scale f l u) 1 = sSupNormIm f u
|
case h.right
E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℂ → E
l u : ℝ
hul : l < u
e : E
z : ℂ
hz₁ : z.re = u
hz₂ : f z = e
⊢ f (↑l + 1 * (z - ↑l)) = e
|
simp only [one_mul, add_sub_cancel, hz₂]
|
no goals
|
3565defd141fb614
|
List.rel_perm_imp
|
Mathlib/Data/List/Perm/Basic.lean
|
theorem rel_perm_imp (hr : RightUnique r) : (Forall₂ r ⇒ Forall₂ r ⇒ (· → ·)) Perm Perm :=
fun a b h₁ c d h₂ h =>
have : (flip (Forall₂ r) ∘r Perm ∘r Forall₂ r) b d := ⟨a, h₁, c, h, h₂⟩
have : ((flip (Forall₂ r) ∘r Forall₂ r) ∘r Perm) b d
|
α : Type u_1
β : Type u_2
r : α → β → Prop
hr : RightUnique r
a : List α
b : List β
h₁ : Forall₂ r a b
c : List α
d : List β
h₂ : Forall₂ r c d
h : a ~ c
this : (flip (Forall₂ r) ∘r Perm ∘r Forall₂ r) b d
⊢ ((flip (Forall₂ r) ∘r Forall₂ r) ∘r Perm) b d
|
rwa [← forall₂_comp_perm_eq_perm_comp_forall₂, ← Relation.comp_assoc] at this
|
no goals
|
b9753c3618954b7a
|
ZetaAsymptotics.term_nonneg
|
Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean
|
lemma term_nonneg (n : ℕ) (s : ℝ) : 0 ≤ term n s
|
n : ℕ
s : ℝ
⊢ 0 ≤ term n s
|
rw [term, intervalIntegral.integral_of_le (by simp)]
|
n : ℕ
s : ℝ
⊢ 0 ≤ ∫ (x : ℝ) in Ioc (↑n) (↑n + 1), (x - ↑n) / x ^ (s + 1) ∂volume
|
53bd7b8921be03b5
|
ContinuousLinearMap.closed_complemented_range_of_isCompl_of_ker_eq_bot
|
Mathlib/Analysis/Normed/Operator/Banach.lean
|
theorem closed_complemented_range_of_isCompl_of_ker_eq_bot {F : Type*} [NormedAddCommGroup F]
[NormedSpace 𝕜 F] [CompleteSpace F] (f : E →L[𝕜] F) (G : Submodule 𝕜 F)
(h : IsCompl (LinearMap.range f) G) (hG : IsClosed (G : Set F)) (hker : ker f = ⊥) :
IsClosed (LinearMap.range f : Set F)
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (LinearMap.range f) G
hG : IsClosed ↑G
hker : LinearMap.ker f = ⊥
⊢ IsClosed ↑(LinearMap.range f)
|
haveI : CompleteSpace G := hG.completeSpace_coe
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (LinearMap.range f) G
hG : IsClosed ↑G
hker : LinearMap.ker f = ⊥
this : CompleteSpace ↥G
⊢ IsClosed ↑(LinearMap.range f)
|
3ffb2bcc15aa85d2
|
SetTheory.PGame.zero_lt_neg_iff
|
Mathlib/SetTheory/Game/PGame.lean
|
theorem zero_lt_neg_iff {x : PGame} : 0 < -x ↔ x < 0
|
x : PGame
⊢ 0 < -x ↔ x < 0
|
rw [lt_neg_iff, neg_zero]
|
no goals
|
1fca89bf4fe60a8a
|
Ideal.exists_ideal_over_prime_of_isIntegral_of_isPrime
|
Mathlib/RingTheory/Ideal/GoingUp.lean
|
theorem exists_ideal_over_prime_of_isIntegral_of_isPrime
[Algebra.IsIntegral R S] (P : Ideal R) [IsPrime P]
(I : Ideal S) [IsPrime I] (hIP : I.comap (algebraMap R S) ≤ P) :
∃ Q ≥ I, IsPrime Q ∧ Q.comap (algebraMap R S) = P
|
R : Type u_1
inst✝⁵ : CommRing R
S : Type u_2
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Algebra.IsIntegral R S
P : Ideal R
inst✝¹ : P.IsPrime
I : Ideal S
inst✝ : I.IsPrime
hIP : comap (algebraMap R S) I ≤ P
⊢ RingHom.ker (Quotient.mk (comap (algebraMap R S) I)) ≤ P
|
simp [hIP]
|
no goals
|
8b33395f9cef74a0
|
AddCircle.isAddFundamentalDomain_of_ae_ball
|
Mathlib/MeasureTheory/Group/AddCircle.lean
|
theorem isAddFundamentalDomain_of_ae_ball (I : Set <| AddCircle T) (u x : AddCircle T)
(hu : IsOfFinAddOrder u) (hI : I =ᵐ[volume] ball x (T / (2 * addOrderOf u))) :
IsAddFundamentalDomain (AddSubgroup.zmultiples u) I
|
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
⊢ 1 ≤ ↑n
|
norm_cast
|
T : ℝ
hT : Fact (0 < T)
I : Set (AddCircle T)
u x : AddCircle T
hu : IsOfFinAddOrder u
G : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u
n : ℕ := addOrderOf u
B : Set (AddCircle T) := ball x (T / (2 * ↑n))
hI : I =ᶠ[ae volume] B
⊢ 1 ≤ n
|
3261dbc8487460b6
|
PowerSeries.hasSum_of_monomials_self
|
Mathlib/RingTheory/PowerSeries/PiTopology.lean
|
theorem hasSum_of_monomials_self (f : PowerSeries R) :
HasSum (fun d : ℕ => monomial R d (coeff R d f)) f
|
case h.e'_5.h.h.e_5.h.e_n
R : Type u_1
inst✝¹ : Semiring R
inst✝ : TopologicalSpace R
f : R⟦X⟧
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () (x✝ PUnit.unit) = x✝
case h.e'_5.h.h.e_6.h.e_a.e_n
R : Type u_1
inst✝¹ : Semiring R
inst✝ : TopologicalSpace R
f : R⟦X⟧
x✝ : Unit →₀ ℕ
⊢ Finsupp.single () (x✝ PUnit.unit) = x✝
|
all_goals { ext; simp }
|
no goals
|
413402dfd03be0a0
|
CategoryTheory.GrothendieckTopology.W_inverseImage_whiskeringLeft
|
Mathlib/CategoryTheory/Sites/Equivalence.lean
|
lemma W_inverseImage_whiskeringLeft :
K.W.inverseImage ((whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op) = J.W
|
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
⊢ K.W = LeftBousfield.W fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj
|
rw [W_eq_W_range_sheafToPresheaf_obj, ← LeftBousfield.W_isoClosure]
|
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁴ : Category.{v₃, u₃} A
inst✝³ : G.IsCoverDense J
inst✝² : G.Full
inst✝¹ : G.IsContinuous K J
inst✝ : (G.sheafPushforwardContinuous A K J).EssSurj
P Q : Cᵒᵖ ⥤ A
f : P ⟶ Q
⊢ LeftBousfield.W (ObjectProperty.isoClosure fun x => x ∈ Set.range (sheafToPresheaf K A).obj) =
LeftBousfield.W fun x => x ∈ Set.range (sheafToPresheaf J A ⋙ (whiskeringLeft Dᵒᵖ Cᵒᵖ A).obj G.op).obj
|
5d43f5abfc41ecac
|
Stream'.WSeq.liftRel_flatten
|
Mathlib/Data/Seq/WSeq.lean
|
theorem liftRel_flatten {R : α → β → Prop} {c1 : Computation (WSeq α)} {c2 : Computation (WSeq β)}
(h : c1.LiftRel (LiftRel R) c2) : LiftRel R (flatten c1) (flatten c2) :=
let S s t := ∃ c1 c2, s = flatten c1 ∧ t = flatten c2 ∧ Computation.LiftRel (LiftRel R) c1 c2
⟨S, ⟨c1, c2, rfl, rfl, h⟩, fun {s t} h =>
match s, t, h with
| _, _, ⟨c1, c2, rfl, rfl, h⟩ => by
simp only [destruct_flatten]; apply liftRel_bind _ _ h
intro a b ab; apply Computation.LiftRel.imp _ _ _ (liftRel_destruct ab)
intro a b; apply LiftRelO.imp_right
intro s t h; refine ⟨Computation.pure s, Computation.pure t, ?_, ?_, ?_⟩ <;>
-- Porting note: These 2 theorems should be excluded.
simp [h, -liftRel_pure_left, -liftRel_pure_right]⟩
|
α : Type u
β : Type v
R : α → β → Prop
c1✝ : Computation (WSeq α)
c2✝ : Computation (WSeq β)
h✝¹ : Computation.LiftRel (LiftRel R) c1✝ c2✝
S : WSeq α → WSeq β → Prop :=
fun s t => ∃ c1 c2, s = flatten c1 ∧ t = flatten c2 ∧ Computation.LiftRel (LiftRel R) c1 c2
s : WSeq α
t : WSeq β
h✝ : S s t
c1 : Computation (WSeq α)
c2 : Computation (WSeq β)
h : Computation.LiftRel (LiftRel R) c1 c2
⊢ ∀ {a : WSeq α} {b : WSeq β}, LiftRel R a b → Computation.LiftRel (LiftRelO R S) a.destruct b.destruct
|
intro a b ab
|
α : Type u
β : Type v
R : α → β → Prop
c1✝ : Computation (WSeq α)
c2✝ : Computation (WSeq β)
h✝¹ : Computation.LiftRel (LiftRel R) c1✝ c2✝
S : WSeq α → WSeq β → Prop :=
fun s t => ∃ c1 c2, s = flatten c1 ∧ t = flatten c2 ∧ Computation.LiftRel (LiftRel R) c1 c2
s : WSeq α
t : WSeq β
h✝ : S s t
c1 : Computation (WSeq α)
c2 : Computation (WSeq β)
h : Computation.LiftRel (LiftRel R) c1 c2
a : WSeq α
b : WSeq β
ab : LiftRel R a b
⊢ Computation.LiftRel (LiftRelO R S) a.destruct b.destruct
|
e2278a6885cf1b11
|
Option.get_attach
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Attach.lean
|
theorem get_attach {o : Option α} (h : o.attach.isSome = true) :
o.attach.get h = ⟨o.get (by simpa using h), by simp⟩
|
case some
α : Type u_1
val✝ : α
h : (some val✝).attach.isSome = true
⊢ (some val✝).attach.get h = ⟨(some val✝).get ⋯, ⋯⟩
|
simp [get_some]
|
no goals
|
4c6e8d0a413cc9e7
|
continuous_algebraMap_iff_smul
|
Mathlib/Topology/Algebra/Algebra.lean
|
theorem continuous_algebraMap_iff_smul [IsTopologicalSemiring A] :
Continuous (algebraMap R A) ↔ Continuous fun p : R × A => p.1 • p.2
|
R : Type u_1
A : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : TopologicalSpace R
inst✝¹ : TopologicalSpace A
inst✝ : IsTopologicalSemiring A
h : Continuous ⇑(algebraMap R A)
⊢ Continuous fun p => p.1 • p.2
|
simp only [Algebra.smul_def]
|
R : Type u_1
A : Type u
inst✝⁵ : CommSemiring R
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
inst✝² : TopologicalSpace R
inst✝¹ : TopologicalSpace A
inst✝ : IsTopologicalSemiring A
h : Continuous ⇑(algebraMap R A)
⊢ Continuous fun p => (algebraMap R A) p.1 * p.2
|
cf5084d686cf2d34
|
ContinuousMapZero.induction_on
|
Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
|
/-- An induction principle for `C(s, 𝕜)₀`. -/
@[elab_as_elim]
lemma ContinuousMapZero.induction_on {s : Set 𝕜} [Zero s] (h0 : ((0 : s) : 𝕜) = 0)
{p : C(s, 𝕜)₀ → Prop} (zero : p 0) (id : p (.id h0)) (star_id : p (star (.id h0)))
(add : ∀ f g, p f → p g → p (f + g)) (mul : ∀ f g, p f → p g → p (f * g))
(smul : ∀ (r : 𝕜) f, p f → p (r • f))
(closure : (∀ f ∈ adjoin 𝕜 {(.id h0 : C(s, 𝕜)₀)}, p f) → ∀ f, p f) (f : C(s, 𝕜)₀) :
p f
|
case mem.inl
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Zero ↑s
h0 : ↑0 = 0
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id h0)
star_id : p (star (ContinuousMapZero.id h0))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id h0}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f✝ f : C(↑s, 𝕜)₀
⊢ p (ContinuousMapZero.id h0)
case mem.inr
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
s : Set 𝕜
inst✝ : Zero ↑s
h0 : ↑0 = 0
p : C(↑s, 𝕜)₀ → Prop
zero : p 0
id : p (ContinuousMapZero.id h0)
star_id : p (star (ContinuousMapZero.id h0))
add : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)
mul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)
smul : ∀ (r : 𝕜) (f : C(↑s, 𝕜)₀), p f → p (r • f)
closure : (∀ f ∈ adjoin 𝕜 {ContinuousMapZero.id h0}, p f) → ∀ (f : C(↑s, 𝕜)₀), p f
f✝ f : C(↑s, 𝕜)₀
⊢ p (star (ContinuousMapZero.id h0))
|
all_goals assumption
|
no goals
|
0762fdd5f4173a39
|
List.reduceOption_map
|
Mathlib/Data/List/ReduceOption.lean
|
theorem reduceOption_map {l : List (Option α)} {f : α → β} :
reduceOption (map (Option.map f) l) = map f (reduceOption l)
|
case nil
α : Type u_1
β : Type u_2
f : α → β
⊢ (map (Option.map f) []).reduceOption = map f [].reduceOption
|
simp only [reduceOption_nil, map_nil]
|
no goals
|
cc6651bb2af464b1
|
Monoid.CoprodI.NeWord.inv_prod
|
Mathlib/GroupTheory/CoprodI.lean
|
theorem inv_prod {i j} (w : NeWord G i j) : w.inv.prod = w.prod⁻¹
|
ι : Type u_1
G : ι → Type u_4
inst✝ : (i : ι) → Group (G i)
i j : ι
w : NeWord G i j
⊢ w.inv.prod = w.prod⁻¹
|
induction w <;> simp [inv, *]
|
no goals
|
ac1d569673d1fc27
|
Orientation.oangle_sign_neg_right
|
Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
|
theorem oangle_sign_neg_right (x y : V) : (o.oangle x (-y)).sign = -(o.oangle x y).sign
|
case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
⊢ (o.oangle x (-y)).sign = -(o.oangle x y).sign
|
by_cases hy : y = 0
|
case pos
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
hy : y = 0
⊢ (o.oangle x (-y)).sign = -(o.oangle x y).sign
case neg
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : ¬x = 0
hy : ¬y = 0
⊢ (o.oangle x (-y)).sign = -(o.oangle x y).sign
|
5c2729a61cddd080
|
UnitAddTorus.mFourierSubalgebra_separatesPoints
|
Mathlib/Analysis/Fourier/AddCircleMulti.lean
|
theorem mFourierSubalgebra_separatesPoints : (mFourierSubalgebra d).SeparatesPoints
|
case intro
d : Type u_1
inst✝ : Fintype d
x y : UnitAddTorus d
i : d
hi : ¬x i = y i
⊢ ∃ f ∈ (fun f => ⇑f) '' ↑(mFourierSubalgebra d).toSubalgebra, f x ≠ f y
|
refine ⟨_, ⟨mFourier (Pi.single i 1), subset_adjoin ⟨Pi.single i 1, rfl⟩, rfl⟩, ?_⟩
|
case intro
d : Type u_1
inst✝ : Fintype d
x y : UnitAddTorus d
i : d
hi : ¬x i = y i
⊢ (fun f => ⇑f) (mFourier (Pi.single i 1)) x ≠ (fun f => ⇑f) (mFourier (Pi.single i 1)) y
|
62716958135bef29
|
List.mapFinIdx_eq_cons_iff'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean
|
theorem mapFinIdx_eq_cons_iff' {l : List α} {b : β} {f : (i : Nat) → α → (h : i < l.length) → β} :
l.mapFinIdx f = b :: l₂ ↔
l.head?.pbind (fun x m => (f 0 x (by cases l <;> simp_all))) = some b ∧
l.tail?.attach.map (fun ⟨t, m⟩ => t.mapFinIdx fun i a h => f (i + 1) a (by cases l <;> simp_all)) = some l₂
|
α : Type u_1
β : Type u_2
l₂ : List β
l : List α
b : β
f : (i : Nat) → α → i < l.length → β
⊢ l.mapFinIdx f = b :: l₂ ↔
(l.head?.pbind fun x m => some (f 0 x ⋯)) = some b ∧
Option.map
(fun x =>
match x with
| ⟨t, m⟩ => t.mapFinIdx fun i a h => f (i + 1) a ⋯)
l.tail?.attach =
some l₂
|
cases l <;> simp
|
no goals
|
2970fe8e7c7f7a5b
|
ONote.scale_opowAux
|
Mathlib/SetTheory/Ordinal/Notation.lean
|
theorem scale_opowAux (e a0 a : ONote) [NF e] [NF a0] [NF a] :
∀ k m, repr (opowAux e a0 a k m) = ω ^ repr e * repr (opowAux 0 a0 a k m)
| 0, m => by cases m <;> simp [opowAux]
| k + 1, m => by
by_cases h : m = 0
· simp [h, opowAux, mul_add, opow_add, mul_assoc, scale_opowAux _ _ _ k]
· -- Porting note: rewrote proof
rw [opowAux]; swap
· assumption
rw [opowAux]; swap
· assumption
rw [repr_add, repr_scale, scale_opowAux _ _ _ k]
simp only [repr_add, repr_scale, opow_add, mul_assoc, zero_add, mul_add]
|
case neg
e a0 a : ONote
inst✝² : e.NF
inst✝¹ : a0.NF
inst✝ : a.NF
k m : ℕ
h : ¬m = 0
⊢ ((e + a0.mulNat k).scale a + e.opowAux a0 a k m).repr = ω ^ e.repr * (opowAux 0 a0 a (k + 1) m).repr
|
rw [opowAux]
|
case neg
e a0 a : ONote
inst✝² : e.NF
inst✝¹ : a0.NF
inst✝ : a.NF
k m : ℕ
h : ¬m = 0
⊢ ((e + a0.mulNat k).scale a + e.opowAux a0 a k m).repr =
ω ^ e.repr * ((0 + a0.mulNat k).scale a + opowAux 0 a0 a k m).repr
case neg.x_2
e a0 a : ONote
inst✝² : e.NF
inst✝¹ : a0.NF
inst✝ : a.NF
k m : ℕ
h : ¬m = 0
⊢ m = 0 → False
|
b7f13d71a1db69ef
|
mul_le_mul_iff_of_ge
|
Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean
|
@[to_additive] lemma mul_le_mul_iff_of_ge [MulLeftStrictMono α]
[MulRightStrictMono α] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) :
a₂ * b₂ ≤ a₁ * b₁ ↔ a₁ = a₂ ∧ b₁ = b₂
|
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a₁ a₂ b₁ b₂ : α
ha : a₁ ≤ a₂
hb : b₁ ≤ b₂
this✝ : MulLeftMono α
this : MulRightMono α
⊢ a₁ = a₂ ∧ b₁ = b₂ → a₂ * b₂ ≤ a₁ * b₁
|
rintro ⟨rfl, rfl⟩
|
case intro
α : Type u_1
inst✝³ : Mul α
inst✝² : PartialOrder α
inst✝¹ : MulLeftStrictMono α
inst✝ : MulRightStrictMono α
a₁ b₁ : α
this✝ : MulLeftMono α
this : MulRightMono α
ha : a₁ ≤ a₁
hb : b₁ ≤ b₁
⊢ a₁ * b₁ ≤ a₁ * b₁
|
9c6e86cfc88a99bc
|
SeminormFamily.basisSets_smul_left
|
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
theorem basisSets_smul_left (x : 𝕜) (U : Set E) (hU : U ∈ p.basisSets) :
∃ V ∈ p.addGroupFilterBasis.sets, V ⊆ (fun y : E => x • y) ⁻¹' U
|
case pos
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : Nonempty ι
x : 𝕜
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : x ≠ 0
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun y => x • y) ⁻¹' (s.sup p).ball 0 r
|
rw [(s.sup p).smul_ball_preimage 0 r x h, smul_zero]
|
case pos
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : Nonempty ι
x : 𝕜
U : Set E
hU✝ : U ∈ p.basisSets
s : Finset ι
r : ℝ
hr : 0 < r
hU : U = (s.sup p).ball 0 r
h : x ≠ 0
⊢ ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (s.sup p).ball 0 (r / ‖x‖)
|
746096ac124c9ebb
|
solvableByRad.isSolvable'
|
Mathlib/FieldTheory/AbelRuffini.lean
|
theorem isSolvable' {α : E} {q : F[X]} (q_irred : Irreducible q) (q_aeval : aeval α q = 0)
(hα : IsSolvableByRad F α) : IsSolvable q.Gal
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α : E
q : F[X]
q_irred : Irreducible q
q_aeval : (aeval α) q = 0
hα : IsSolvableByRad F α
this : IsSolvable (q * C q.leadingCoeff⁻¹).Gal
⊢ ¬q = 0 ∧ ¬q.leadingCoeff = 0
|
exact ⟨q_irred.ne_zero, leadingCoeff_ne_zero.mpr q_irred.ne_zero⟩
|
no goals
|
d267ed0ecc2c4dbe
|
MeasureTheory.volume_sum_rpow_le
|
Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
|
theorem MeasureTheory.volume_sum_rpow_le [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : ℝ) :
volume {x : ι → ℝ | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} = (.ofReal r) ^ card ι *
.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1))
|
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
⊢ volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) ≤ r} =
ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
|
have eq_zero := fun x : ι → ℝ => norm_eq_zero (E := PiLp (.ofReal p) (fun _ : ι => ℝ)) (a := x)
|
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : Nonempty ι
p : ℝ
hp : 1 ≤ p
r : ℝ
h₁ : 0 < p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
nm_zero : ‖0‖ = 0
eq_zero : ∀ (x : ι → ℝ), ‖x‖ = 0 ↔ x = 0
⊢ volume {x | (∑ i : ι, |x i| ^ p) ^ (1 / p) ≤ r} =
ENNReal.ofReal r ^ card ι * ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))
|
b43945fead980666
|
Real.pi_gt_d2
|
Mathlib/Data/Real/Pi/Bounds.lean
|
theorem pi_gt_d2 : 3.14 < π
|
⊢ 3.14 < π
|
pi_lower_bound [338 / 239, 704 / 381, 1940 / 989, 1447 / 727]
|
no goals
|
77800e38bc315fa9
|
Zsqrtd.le_arch
|
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
theorem le_arch (a : ℤ√d) : ∃ n : ℕ, a ≤ n
|
case intro.intro
d : ℕ
a : ℤ√↑d
x y : ℕ
h : a ≤ { re := ↑x, im := ↑y }
⊢ { re := ↑x, im := ↑y } ≤ ↑(x + d * y)
|
change Nonneg ⟨↑x + d * y - ↑x, 0 - ↑y⟩
|
case intro.intro
d : ℕ
a : ℤ√↑d
x y : ℕ
h : a ≤ { re := ↑x, im := ↑y }
⊢ { re := ↑x + ↑d * ↑y - ↑x, im := 0 - ↑y }.Nonneg
|
8d26cc96bf8abedc
|
Bimod.whiskerRight_comp_bimod
|
Mathlib/CategoryTheory/Monoidal/Bimod.lean
|
theorem whiskerRight_comp_bimod {W X Y Z : Mon_ C} {M M' : Bimod W X} (f : M ⟶ M') (N : Bimod X Y)
(P : Bimod Y Z) :
whiskerRight f (N.tensorBimod P) =
(associatorBimod M N P).inv ≫
whiskerRight (whiskerRight f N) P ≫ (associatorBimod M' N P).hom
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ whiskerRight f (N.tensorBimod P) =
(M.associatorBimod N P).inv ≫ whiskerRight (whiskerRight f N) P ≫ (M'.associatorBimod N P).hom
|
dsimp [tensorHom, tensorBimod, associatorBimod]
|
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
⊢ whiskerRight f
{ X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } =
(isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).inv ≫
whiskerRight (whiskerRight f N) P ≫
(isoOfIso
{ hom := AssociatorBimod.hom M' N P, inv := AssociatorBimod.inv M' N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯
⋯).hom
|
2725ee4aa075b079
|
doublyStochastic_sum_perm_aux
|
Mathlib/Analysis/Convex/Birkhoff.lean
|
/--
If M is a scalar multiple of a doubly stochastic matrix, then it is a conical combination of
permutation matrices. This is most useful when M is a doubly stochastic matrix, in which case
the combination is convex.
This particular formulation is chosen to make the inductive step easier: we no longer need to
rescale each time a permutation matrix is subtracted.
-/
private lemma doublyStochastic_sum_perm_aux (M : Matrix n n R)
(s : R) (hs : 0 ≤ s)
(hM : ∃ M' ∈ doublyStochastic R n, M = s • M') :
∃ w : Equiv.Perm n → R, (∀ σ, 0 ≤ w σ) ∧ ∑ σ, w σ • σ.permMatrix R = M
|
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : LinearOrderedField R
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : ∃ M' ∈ doublyStochastic R n, M = s • M'
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ : Equiv.Perm n, w σ • Equiv.Perm.permMatrix R σ = M
|
rcases isEmpty_or_nonempty n
|
case inl
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : LinearOrderedField R
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : ∃ M' ∈ doublyStochastic R n, M = s • M'
h✝ : IsEmpty n
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ : Equiv.Perm n, w σ • Equiv.Perm.permMatrix R σ = M
case inr
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : LinearOrderedField R
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : ∃ M' ∈ doublyStochastic R n, M = s • M'
h✝ : Nonempty n
⊢ ∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ : Equiv.Perm n, w σ • Equiv.Perm.permMatrix R σ = M
|
fe82d3c50ee8248f
|
Batteries.UnionFind.root_link
|
Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean
|
theorem root_link {self : UnionFind} {x y : Fin self.size}
(xroot : self.parent x = x) (yroot : self.parent y = y) :
∃ r, (r = x ∨ r = y) ∧ ∀ i,
(link self x y yroot).rootD i =
if self.rootD i = x ∨ self.rootD i = y then r.1 else self.rootD i
|
self : UnionFind
x y : Fin self.size
xroot : self.parent ↑x = ↑x
yroot : self.parent ↑y = ↑y
h : ¬↑x = ↑y
this :
∀ {x y : Fin self.size},
self.parent ↑x = ↑x →
self.parent ↑y = ↑y →
∀ {m : UnionFind},
(∀ (i : Nat), m.parent i = if ↑y = i then ↑x else self.parent i) →
∃ r,
(r = x ∨ r = y) ∧
∀ (i : Nat), m.rootD i = if self.rootD i = ↑x ∨ self.rootD i = ↑y then ↑r else self.rootD i
hr : self.rank ↑y < self.rank ↑x
⊢ ∃ r,
(r = x ∨ r = y) ∧
∀ (i : Nat), (self.link x y yroot).rootD i = if self.rootD i = ↑x ∨ self.rootD i = ↑y then ↑r else self.rootD i
|
exact this xroot yroot fun i => by simp [parent_link, h, hr]
|
no goals
|
d17902f7f8b46530
|
MeasureTheory.Measure.addHaar_unitClosedBall_eq_addHaar_unitBall
|
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
theorem addHaar_unitClosedBall_eq_addHaar_unitBall :
μ (closedBall (0 : E) 1) = μ (ball 0 1)
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A :
Tendsto (fun r => ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1)) (𝓝[<] 1)
(𝓝 (ENNReal.ofReal (1 ^ finrank ℝ E) * μ (closedBall 0 1)))
⊢ μ (closedBall 0 1) ≤ μ (ball 0 1)
|
simp only [one_pow, one_mul, ENNReal.ofReal_one] at A
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : Tendsto (fun r => ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1)) (𝓝[<] 1) (𝓝 (μ (closedBall 0 1)))
⊢ μ (closedBall 0 1) ≤ μ (ball 0 1)
|
fec794d9d0962e21
|
Matrix.toLin_one
|
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
theorem Matrix.toLin_one : Matrix.toLin v₁ v₁ 1 = LinearMap.id
|
R : Type u_1
inst✝⁴ : CommSemiring R
n : Type u_4
inst✝³ : Fintype n
inst✝² : DecidableEq n
M₁ : Type u_5
inst✝¹ : AddCommMonoid M₁
inst✝ : Module R M₁
v₁ : Basis n R M₁
⊢ (toLin v₁ v₁) 1 = LinearMap.id
|
rw [← LinearMap.toMatrix_id v₁, Matrix.toLin_toMatrix]
|
no goals
|
0fc717586a9e23ee
|
Option.get_ite'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean
|
theorem get_ite' {p : Prop} [Decidable p] (h) :
(if p then none else some b).get h = b
|
α✝ : Type u_1
b : α✝
p : Prop
inst✝ : Decidable p
h : (if p then none else some b).isSome = true
⊢ (if h : p then none else some ((fun x => b) h)).isSome = true
|
simpa using h
|
no goals
|
9191a3197025dfa5
|
Ideal.inertiaDeg_comap_eq
|
Mathlib/NumberTheory/RamificationInertia/Basic.lean
|
lemma inertiaDeg_comap_eq (e : S ≃ₐ[R] S₁) (P : Ideal S₁) [p.IsMaximal] :
inertiaDeg p (P.comap e) = inertiaDeg p P
|
case pos
R : Type u
inst✝⁵ : CommRing R
S : Type v
inst✝⁴ : CommRing S
p : Ideal R
S₁ : Type u_1
inst✝³ : CommRing S₁
inst✝² : Algebra R S₁
inst✝¹ : Algebra R S
e : S ≃ₐ[R] S₁
P : Ideal S₁
inst✝ : p.IsMaximal
he : comap f (comap e P) = p ↔ comap (algebraMap R S₁) P = p
h : P.LiesOver p
⊢ finrank (R ⧸ p) (S ⧸ comap e P) = finrank (R ⧸ p) (S₁ ⧸ P)
|
exact (Quotient.algEquivOfEqComap p e rfl).toLinearEquiv.finrank_eq
|
no goals
|
808161996d92d945
|
RelSeries.nonempty_of_finiteDimensional
|
Mathlib/Order/RelSeries.lean
|
lemma nonempty_of_finiteDimensional [r.FiniteDimensional] : Nonempty α
|
case intro
α : Type u_1
r : Rel α α
inst✝ : r.FiniteDimensional
p : RelSeries r
h✝ : ∀ (y : RelSeries r), y.length ≤ p.length
⊢ Nonempty α
|
exact ⟨p 0⟩
|
no goals
|
a04bbe4eba35b6fb
|
ClassGroup.exists_mem_finsetApprox
|
Mathlib/NumberTheory/ClassNumber/Finite.lean
|
theorem exists_mem_finsetApprox (a : S) {b} (hb : b ≠ (0 : R)) :
∃ q : S,
∃ r ∈ finsetApprox bS adm, abv (Algebra.norm R (r • a - b • q)) <
abv (Algebra.norm R (algebraMap R S b))
|
R : Type u_1
S : Type u_2
inst✝⁷ : EuclideanDomain R
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : Algebra R S
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
bS : Basis ι R S
adm : abv.IsAdmissible
inst✝¹ : Infinite R
inst✝ : DecidableEq R
a : S
b : R
hb : b ≠ 0
dim_pos : 0 < Fintype.card ι
ε : ℝ := ↑(normBound abv bS) ^ (-1 / ↑(Fintype.card ι))
ε_eq : ε = ↑(normBound abv bS) ^ (-1 / ↑(Fintype.card ι))
hε : 0 < ε
ε_le : ↑(normBound abv bS) * (abv b • ε) ^ ↑(Fintype.card ι) ≤ ↑(abv b) ^ ↑(Fintype.card ι)
μ : Fin (cardM bS adm).succ ↪ R := distinctElems bS adm
hμ : μ = distinctElems bS adm
s : ι →₀ R := bS.repr a
s_eq : ∀ (i : ι), s i = (bS.repr a) i
qs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i / b
⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - b • q)) < abv ((Algebra.norm R) ((algebraMap R S) b))
|
let rs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i % b
|
R : Type u_1
S : Type u_2
inst✝⁷ : EuclideanDomain R
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : Algebra R S
abv : AbsoluteValue R ℤ
ι : Type u_5
inst✝³ : DecidableEq ι
inst✝² : Fintype ι
bS : Basis ι R S
adm : abv.IsAdmissible
inst✝¹ : Infinite R
inst✝ : DecidableEq R
a : S
b : R
hb : b ≠ 0
dim_pos : 0 < Fintype.card ι
ε : ℝ := ↑(normBound abv bS) ^ (-1 / ↑(Fintype.card ι))
ε_eq : ε = ↑(normBound abv bS) ^ (-1 / ↑(Fintype.card ι))
hε : 0 < ε
ε_le : ↑(normBound abv bS) * (abv b • ε) ^ ↑(Fintype.card ι) ≤ ↑(abv b) ^ ↑(Fintype.card ι)
μ : Fin (cardM bS adm).succ ↪ R := distinctElems bS adm
hμ : μ = distinctElems bS adm
s : ι →₀ R := bS.repr a
s_eq : ∀ (i : ι), s i = (bS.repr a) i
qs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i / b
rs : Fin (cardM bS adm).succ → ι → R := fun j i => μ j * s i % b
⊢ ∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - b • q)) < abv ((Algebra.norm R) ((algebraMap R S) b))
|
57a52f480b46c47f
|
List.ext_getElem
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem ext_getElem {l₁ l₂ : List α} (hl : length l₁ = length l₂)
(h : ∀ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length), l₁[i]'h₁ = l₂[i]'h₂) : l₁ = l₂ :=
ext_getElem? fun n =>
if h₁ : n < length l₁ then by
simp_all [getElem?_eq_getElem]
else by
have h₁ := Nat.le_of_not_lt h₁
rw [getElem?_eq_none h₁, getElem?_eq_none]; rwa [← hl]
|
α : Type u_1
l₁ l₂ : List α
hl : l₁.length = l₂.length
h : ∀ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length), l₁[i] = l₂[i]
n : Nat
h₁ : ¬n < l₁.length
⊢ l₁[n]? = l₂[n]?
|
have h₁ := Nat.le_of_not_lt h₁
|
α : Type u_1
l₁ l₂ : List α
hl : l₁.length = l₂.length
h : ∀ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length), l₁[i] = l₂[i]
n : Nat
h₁✝ : ¬n < l₁.length
h₁ : l₁.length ≤ n
⊢ l₁[n]? = l₂[n]?
|
6ddc71b98482ae57
|
AlgebraicGeometry.HasRingHomProperty.isLocal_ringHomProperty_of_isLocalAtSource_of_isLocalAtTarget
|
Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
|
lemma isLocal_ringHomProperty_of_isLocalAtSource_of_isLocalAtTarget
[IsLocalAtTarget P] [IsLocalAtSource P] :
RingHom.PropertyIsLocal fun f ↦ P (Spec.map (CommRingCat.ofHom f))
|
case ofLocalizationSpanTarget
P : MorphismProperty Scheme
inst✝¹ : IsLocalAtTarget P
inst✝ : IsLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
⊢ RingHom.OfLocalizationSpanTarget fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
|
intros R S _ _ f s hs H
|
case ofLocalizationSpanTarget
P : MorphismProperty Scheme
inst✝³ : IsLocalAtTarget P
inst✝² : IsLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f))
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
H :
∀ (r : ↑s),
(fun {R S} [CommRing R] [CommRing S] f => P (Spec.map (CommRingCat.ofHom f)))
((algebraMap S (Localization.Away ↑r)).comp f)
⊢ P (Spec.map (CommRingCat.ofHom f))
|
b9bae3dd3914b2b4
|
intervalIntegral.intervalIntegrable_log'
|
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
theorem intervalIntegrable_log' : IntervalIntegrable log volume a b
|
case h₂f.hab.h.hpos
a b x : ℝ
hx : 0 < x
s : ℝ
hs₁ : 0 < s
hs₂ : s < 1
⊢ log s ≤ 0
|
exact (log_nonpos_iff hs₁.le).mpr hs₂.le
|
no goals
|
d485782e2df5cd2b
|
EReal.nhdsWithin_top
|
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
lemma nhdsWithin_top : 𝓝[≠] (⊤ : EReal) = (atTop).map Real.toEReal
|
case pos
x : EReal
hx : x < ⊤
hx_bot : x = ⊥
⊢ ∃ i', Ico ↑i' ⊤ ⊆ Ici x ∩ {⊤}ᶜ
|
simp [hx_bot]
|
no goals
|
44a55101f1792898
|
Std.DHashMap.Internal.Raw₀.get!_eq_default
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean
|
theorem get!_eq_default [LawfulBEq α] (h : m.1.WF) {a : α} [Inhabited (β a)] :
m.contains a = false → m.get! a = default
|
α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : LawfulBEq α
h : m.val.WF
a : α
inst✝ : Inhabited (β a)
⊢ m.contains a = false → m.get! a = default
|
simp_to_model using List.getValueCast!_eq_default
|
no goals
|
00227ee183807c1b
|
AlgebraicGeometry.isPullback_opens_inf
|
Mathlib/AlgebraicGeometry/Restrict.lean
|
lemma isPullback_opens_inf {X : Scheme} (U V : X.Opens) :
IsPullback (X.homOfLE inf_le_left) (X.homOfLE inf_le_right) U.ι V.ι :=
(isPullback_morphismRestrict V.ι U).of_iso (V.ι.isoImage _ ≪≫ X.isoOfEq
(V.functor_map_eq_inf U)) (Iso.refl _) (Iso.refl _) (Iso.refl _) (by simp [← cancel_mono U.ι])
(by simp [← cancel_mono V.ι]) (by simp) (by simp)
|
X : Scheme
U V : X.Opens
⊢ V.ι ≫ (Iso.refl X).hom = (Iso.refl ↑V).hom ≫ V.ι
|
simp
|
no goals
|
d85d3594e7244cad
|
Option.attachWith_eq_none_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Attach.lean
|
theorem attachWith_eq_none_iff {p : α → Prop} {o : Option α} (H : ∀ a ∈ o, p a) :
o.attachWith p H = none ↔ o = none
|
α : Type u_1
p : α → Prop
o : Option α
H : ∀ (a : α), a ∈ o → p a
⊢ o.attachWith p H = none ↔ o = none
|
cases o <;> simp
|
no goals
|
fe17c1b9bea5c567
|
Vitali.exists_disjoint_covering_ae
|
Mathlib/MeasureTheory/Covering/Vitali.lean
|
theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ C * μ (B a))
(ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a))
(hf : ∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ t, r a ≤ ε ∧ c a = x) :
∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
|
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ a ∈ t', r a ≤ 1
A' : ∀ a ∈ t', (B a).Nonempty
a : ι
ha : a ∈ t'
⊢ 0 ≤ r a
|
exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1))
|
no goals
|
4466d2fc3e80cb3f
|
Lagrange.interpolate_eq_sum_interpolate_insert_sdiff
|
Mathlib/LinearAlgebra/Lagrange.lean
|
theorem interpolate_eq_sum_interpolate_insert_sdiff (hvt : Set.InjOn v t) (hs : s.Nonempty)
(hst : s ⊆ t) :
interpolate t v r = ∑ i ∈ s, interpolate (insert i (t \ s)) v r * Lagrange.basis s v i
|
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s t : Finset ι
v r : ι → F
hvt : Set.InjOn v ↑t
hs✝ : s.Nonempty
hst : s ⊆ t
i : ι
hi : i ∈ s
hs : 1 ≤ #s
hst' : #s ≤ #t
⊢ #t = 1 + (#t - #s) + (#s - 1)
|
rw [add_assoc, tsub_add_tsub_cancel hst' hs, ← add_tsub_assoc_of_le (hs.trans hst'),
Nat.succ_add_sub_one, zero_add]
|
no goals
|
071129da26397277
|
toIcoMod_apply_left
|
Mathlib/Algebra/Order/ToIntervalMod.lean
|
theorem toIcoMod_apply_left (a : α) : toIcoMod hp a a = a
|
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a : α
⊢ a = a + 0 • p
|
simp
|
no goals
|
543de7a2df64cb87
|
EuclideanGeometry.Sphere.oangle_center_eq_two_zsmul_oangle
|
Mathlib/Geometry/Euclidean/Angle/Sphere.lean
|
theorem oangle_center_eq_two_zsmul_oangle {s : Sphere P} {p₁ p₂ p₃ : P} (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₃ : p₂ ≠ p₃) :
∡ p₁ s.center p₃ = (2 : ℤ) • ∡ p₁ p₂ p₃
|
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
s : Sphere P
p₁ p₂ p₃ : P
hp₁ : ‖p₁ -ᵥ s.center‖ = s.radius
hp₂ : ‖p₂ -ᵥ s.center‖ = s.radius
hp₃ : ‖p₃ -ᵥ s.center‖ = s.radius
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
⊢ ∡ p₁ s.center p₃ = 2 • ∡ p₁ p₂ p₃
|
rw [oangle, oangle, o.oangle_eq_two_zsmul_oangle_sub_of_norm_eq_real _ _ hp₂ hp₁ hp₃] <;>
simp [hp₂p₁, hp₂p₃]
|
no goals
|
528367b8f3bfb27f
|
CategoryTheory.Limits.inr_inr_pushoutRightPushoutInlIso_hom
|
Mathlib/CategoryTheory/Limits/Shapes/Pullback/Pasting.lean
|
theorem inr_inr_pushoutRightPushoutInlIso_hom :
pushout.inr _ _ ≫ pushout.inr _ _ ≫ (pushoutRightPushoutInlIso f g f').hom =
pushout.inr _ _
|
C : Type u
inst✝² : Category.{v, u} C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
f' : Y ⟶ W
inst✝¹ : HasPushout f g
inst✝ : HasPushout f' (pushout.inl f g)
⊢ (pushout.inr f g ≫ pushout.inr f' (pushout.inl f g)) ≫ (pushoutRightPushoutInlIso f g f').hom = pushout.inr (f ≫ f') g
|
apply IsColimit.comp_coconePointUniqueUpToIso_hom (pasteVertIsPushout rfl _ _) _ WalkingSpan.right
|
no goals
|
3bb021acc1f9a6f6
|
Nat.Prime.pow_inj
|
Mathlib/Data/Nat/Prime/Int.lean
|
/-- Two prime powers with positive exponents are equal only when the primes and the
exponents are equal. -/
lemma Prime.pow_inj {p q m n : ℕ} (hp : p.Prime) (hq : q.Prime)
(h : p ^ (m + 1) = q ^ (n + 1)) : p = q ∧ m = n
|
p q m n : ℕ
hp : Prime p
hq : Prime q
h : p ^ (m + 1) = q ^ (n + 1)
H : p = q
⊢ p = q ∧ m = n
|
exact ⟨H, succ_inj'.mp <| Nat.pow_right_injective hq.two_le (H ▸ h)⟩
|
no goals
|
1ef68e89a4d2ad2f
|
CategoryTheory.SmallObject.succStruct_prop_le_propArrow
|
Mathlib/CategoryTheory/SmallObject/IsCardinalForSmallObjectArgument.lean
|
lemma succStruct_prop_le_propArrow :
(succStruct I κ).prop ≤ (propArrow.{w} I).functorCategory (Arrow C)
|
case mk
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.toType
inst✝ : I.IsCardinalForSmallObjectArgument κ
this✝² : LocallySmall.{w, v, u} C
this✝¹ : IsSmall.{w, v, u} I
this✝ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this : HasPushouts C
X✝ Y✝ : Arrow C ⥤ Arrow C
f✝ : X✝ ⟶ Y✝
F : Arrow C ⥤ Arrow C
f : Arrow C
j : FunctorObjIndex I.homFamily (F.obj f).hom
⊢ ofHoms I.homFamily ((Discrete.natTrans fun X => functorObjLeftFamily I.homFamily (F.obj f).hom X.as).app { as := j })
|
constructor
|
no goals
|
c7ca3759aa559e4f
|
mem_generatePiSystem_iUnion_elim'
|
Mathlib/MeasureTheory/PiSystem.lean
|
theorem mem_generatePiSystem_iUnion_elim' {α β} {g : β → Set (Set α)} {s : Set β}
(h_pi : ∀ b ∈ s, IsPiSystem (g b)) (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b)) :
∃ (T : Finset β) (f : β → Set α), ↑T ⊆ s ∧ (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b
|
case intro.intro.intro.refine_1.h.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
⊢ a ∈ ⋂ b ∈ Finset.image (fun x => ↑x) T, Function.extend (fun x => ↑x) f (fun x => ∅) b → a ∈ ⋂ b ∈ T, f b
|
simp (config := { proj := false }) only
[Set.mem_iInter, Subtype.forall, Finset.set_biInter_finset_image]
|
case intro.intro.intro.refine_1.h.mpr
α : Type u_3
β : Type u_4
g : β → Set (Set α)
s : Set β
h_pi : ∀ b ∈ s, IsPiSystem (g b)
T : Finset (Subtype s)
f : Subtype s → Set α
h_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b
h_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)
this : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, (g ∘ Subtype.val) b)
a : α
⊢ (∀ (a_1 : β) (b : a_1 ∈ s), ⟨a_1, b⟩ ∈ T → a ∈ Function.extend (fun x => ↑x) f (fun x => ∅) ↑⟨a_1, b⟩) →
∀ (a_2 : β) (b : s a_2), ⟨a_2, b⟩ ∈ T → a ∈ f ⟨a_2, b⟩
|
61b8f425d6f37a71
|
Cardinal.derivFamily_lt_ord_lift
|
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
theorem derivFamily_lt_ord_lift {ι : Type u} {f : ι → Ordinal → Ordinal} {c} (hc : IsRegular c)
(hι : lift.{v} #ι < c) (hc' : c ≠ ℵ₀) (hf : ∀ i, ∀ b < c.ord, f i b < c.ord) {a} :
a < c.ord → derivFamily f a < c.ord
|
ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
hω : ℵ₀ < c.ord.cof
⊢ lift.{?u.136517, u} #ι < c.ord.cof
|
rwa [hc.cof_eq]
|
no goals
|
a8993783bfefcfe4
|
Nat.sub_add_lt_sub
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean
|
theorem sub_add_lt_sub (h₁ : m + k ≤ n) (h₂ : 0 < k) : n - (m + k) < n - m
|
m k n : Nat
h₁ : m + k ≤ n
h₂ : 0 < k
⊢ n - (m + k) < n - m
|
rw [← Nat.sub_sub]
|
m k n : Nat
h₁ : m + k ≤ n
h₂ : 0 < k
⊢ n - m - k < n - m
|
7010e8c373a07d29
|
Matrix.vec2_add
|
Mathlib/Data/Matrix/Notation.lean
|
theorem vec2_add [Add α] (a₀ a₁ b₀ b₁ : α) : ![a₀, a₁] + ![b₀, b₁] = ![a₀ + b₀, a₁ + b₁]
|
α : Type u
inst✝ : Add α
a₀ a₁ b₀ b₁ : α
⊢ ![a₀, a₁] + ![b₀, b₁] = ![a₀ + b₀, a₁ + b₁]
|
rw [cons_add_cons, cons_add_cons, empty_add_empty]
|
no goals
|
1843967cad1c041b
|
PreTilt.isDomain
|
Mathlib/RingTheory/Perfection.lean
|
theorem isDomain : IsDomain (PreTilt O p)
|
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
hp : Nat.Prime p
this : Nontrivial (PreTilt O p)
⊢ IsDomain (PreTilt O p)
|
haveI : NoZeroDivisors (PreTilt O p) :=
⟨fun hfg => by
simp_rw [← map_eq_zero hv] at hfg ⊢; contrapose! hfg; rw [Valuation.map_mul]
exact mul_ne_zero hfg.1 hfg.2⟩
|
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
hp : Nat.Prime p
this✝ : Nontrivial (PreTilt O p)
this : NoZeroDivisors (PreTilt O p)
⊢ IsDomain (PreTilt O p)
|
2dffd41f28def1f7
|
Directed.rel_sequence
|
Mathlib/Logic/Encodable/Basic.lean
|
theorem rel_sequence {r : β → β → Prop} {f : α → β} (hf : Directed r f) (a : α) :
r (f a) (f (hf.sequence f (encode a + 1)))
|
α : Type u_1
β : Type u_2
inst✝¹ : Encodable α
inst✝ : Inhabited α
r : β → β → Prop
f : α → β
hf : Directed r f
a : α
⊢ r (f a) (f (Directed.sequence f hf (encode a + 1)))
|
simp only [Directed.sequence, add_eq, Nat.add_zero, encodek, and_self]
|
α : Type u_1
β : Type u_2
inst✝¹ : Encodable α
inst✝ : Inhabited α
r : β → β → Prop
f : α → β
hf : Directed r f
a : α
⊢ r (f a) (f (Classical.choose ⋯))
|
b03bb4bc3e2c3e3e
|
Complex.ofReal_zsmul
|
Mathlib/Data/Complex/Basic.lean
|
@[norm_cast] lemma ofReal_zsmul (n : ℤ) (r : ℝ) : ↑(n • r) = n • (r : ℂ)
|
n : ℤ
r : ℝ
⊢ ↑(n • r) = n • ↑r
|
simp
|
no goals
|
47613677c80f8fa6
|
MeasureTheory.L1.setToL1_add_left
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem setToL1_add_left (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ T' C') (f : α →₁[μ] E) :
setToL1 (hT.add hT') f = setToL1 hT f + setToL1 hT' f
|
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
f : ↥(Lp E 1 μ)
this : setToL1 ⋯ = setToL1 hT + setToL1 hT'
⊢ (setToL1 ⋯) f = (setToL1 hT) f + (setToL1 hT') f
|
rw [this, ContinuousLinearMap.add_apply]
|
no goals
|
1f9caebaa466b1da
|
Function.Semiconj.mapsTo_preimage
|
Mathlib/Data/Set/Function.lean
|
theorem mapsTo_preimage (h : Semiconj f fa fb) {s t : Set β} (hb : MapsTo fb s t) :
MapsTo fa (f ⁻¹' s) (f ⁻¹' t) := fun x hx => by simp only [mem_preimage, h x, hb hx]
|
α : Type u_1
β : Type u_2
fa : α → α
fb : β → β
f : α → β
h : Semiconj f fa fb
s t : Set β
hb : MapsTo fb s t
x : α
hx : x ∈ f ⁻¹' s
⊢ fa x ∈ f ⁻¹' t
|
simp only [mem_preimage, h x, hb hx]
|
no goals
|
44955fef6b6923dd
|
Nat.Partrec.Code.evaln_mono
|
Mathlib/Computability/PartrecCode.lean
|
theorem evaln_mono : ∀ {k₁ k₂ c n x}, k₁ ≤ k₂ → x ∈ evaln k₁ c n → x ∈ evaln k₂ c n
| 0, k₂, c, n, x, _, h => by simp [evaln] at h
| k + 1, k₂ + 1, c, n, x, hl, h => by
have hl' := Nat.le_of_succ_le_succ hl
have :
∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ → (x ∈ o₁ → x ∈ o₂) →
x ∈ do { guard (n ≤ k); o₁ } → x ∈ do { guard (n ≤ k₂); o₂ }
|
k k₂ : ℕ
c : Code
n x : ℕ
hl : k + 1 ≤ k₂ + 1
h : x ∈ evaln (k + 1) c n
hl' : k ≤ k₂
⊢ ∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ},
k ≤ k₂ →
(x ∈ o₁ → x ∈ o₂) →
(x ∈ do
guard (n ≤ k)
o₁) →
x ∈ do
guard (n ≤ k₂)
o₂
|
simp only [Option.mem_def, bind, Option.bind_eq_some, Option.guard_eq_some', exists_and_left,
exists_const, and_imp]
|
k k₂ : ℕ
c : Code
n x : ℕ
hl : k + 1 ≤ k₂ + 1
h : x ∈ evaln (k + 1) c n
hl' : k ≤ k₂
⊢ ∀ {k k₂ n x : ℕ} {o₁ o₂ : Option ℕ}, k ≤ k₂ → (o₁ = some x → o₂ = some x) → n ≤ k → o₁ = some x → n ≤ k₂ ∧ o₂ = some x
|
231a1123011d0046
|
approxOrderOf.image_pow_subset
|
Mathlib/NumberTheory/WellApproximable.lean
|
theorem image_pow_subset (n : ℕ) (hm : 0 < m) :
(fun (y : A) => y ^ m) '' approxOrderOf A (n * m) δ ⊆ approxOrderOf A n (m * δ)
|
case intro.intro
A : Type u_1
inst✝ : SeminormedCommGroup A
m : ℕ
δ : ℝ
n : ℕ
hm : 0 < m
a : A
ha : a ∈ approxOrderOf A (n * m) δ
⊢ (fun y => y ^ m) a ∈ approxOrderOf A n (↑m * δ)
|
obtain ⟨b, hb : orderOf b = n * m, hab : a ∈ ball b δ⟩ := mem_approxOrderOf_iff.mp ha
|
case intro.intro.intro.intro
A : Type u_1
inst✝ : SeminormedCommGroup A
m : ℕ
δ : ℝ
n : ℕ
hm : 0 < m
a : A
ha : a ∈ approxOrderOf A (n * m) δ
b : A
hb : orderOf b = n * m
hab : a ∈ ball b δ
⊢ (fun y => y ^ m) a ∈ approxOrderOf A n (↑m * δ)
|
b8cbbccee333ecb1
|
ascPochhammer_eval_neg_eq_descPochhammer
|
Mathlib/RingTheory/Polynomial/Pochhammer.lean
|
theorem ascPochhammer_eval_neg_eq_descPochhammer (r : R) : ∀ (k : ℕ),
(ascPochhammer R k).eval (-r) = (-1)^k * (descPochhammer R k).eval r
| 0 => by
rw [ascPochhammer_zero, descPochhammer_zero]
simp only [eval_one, pow_zero, mul_one]
| (k+1) => by
rw [ascPochhammer_succ_right, mul_add, eval_add, eval_mul_X, ← Nat.cast_comm, eval_natCast_mul,
Nat.cast_comm, ← mul_add, ascPochhammer_eval_neg_eq_descPochhammer r k, mul_assoc,
descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, ← Nat.cast_comm, eval_natCast_mul,
pow_add, pow_one, mul_assoc ((-1)^k) (-1), mul_sub, neg_one_mul, neg_mul_eq_mul_neg,
Nat.cast_comm, sub_eq_add_neg, neg_one_mul, neg_neg, ← mul_add]
|
R : Type u
inst✝ : Ring R
r : R
k : ℕ
⊢ eval (-r) (ascPochhammer R (k + 1)) = (-1) ^ (k + 1) * eval r (descPochhammer R (k + 1))
|
rw [ascPochhammer_succ_right, mul_add, eval_add, eval_mul_X, ← Nat.cast_comm, eval_natCast_mul,
Nat.cast_comm, ← mul_add, ascPochhammer_eval_neg_eq_descPochhammer r k, mul_assoc,
descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, ← Nat.cast_comm, eval_natCast_mul,
pow_add, pow_one, mul_assoc ((-1)^k) (-1), mul_sub, neg_one_mul, neg_mul_eq_mul_neg,
Nat.cast_comm, sub_eq_add_neg, neg_one_mul, neg_neg, ← mul_add]
|
no goals
|
91e878a9eb30688b
|
Submodule.mem_sSup_iff_exists_finset
|
Mathlib/LinearAlgebra/Finsupp/Span.lean
|
theorem Submodule.mem_sSup_iff_exists_finset {S : Set (Submodule R M)} {m : M} :
m ∈ sSup S ↔ ∃ s : Finset (Submodule R M), ↑s ⊆ S ∧ m ∈ ⨆ i ∈ s, i
|
case refine_2
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set (Submodule R M)
m : M
x✝ : ∃ s, m ∈ ⨆ i ∈ s, ↑i
s : Finset (Subtype (Membership.mem S))
hs : m ∈ ⨆ i ∈ s, ↑i
⊢ m ∈ ⨆ i, ⨆ (hi : i ∈ S), ⨆ (_ : ⟨i, hi⟩ ∈ s), i
|
rwa [iSup_subtype']
|
no goals
|
4a53431e2aac6cd1
|
Polynomial.isUnitTrinomial_iff
|
Mathlib/Algebra/Polynomial/UnitTrinomial.lean
|
theorem isUnitTrinomial_iff :
p.IsUnitTrinomial ↔ #p.support = 3 ∧ ∀ k ∈ p.support, IsUnit (p.coeff k)
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
k m n : ℕ
hkm : k < m
hmn : m < n
x y z : ℤ
hp : #{k, m, n} = 3 ∧ ∀ k_1 ∈ {k, m, n}, IsUnit ((C x * X ^ k + C y * X ^ m + C z * X ^ n).coeff k_1)
hx : IsUnit ((x + y * if k = m then 1 else 0) + z * if k = n then 1 else 0)
hy : IsUnit ((x * if m = k then 1 else 0) + y + z * if m = n then 1 else 0)
hz : IsUnit (((x * if n = k then 1 else 0) + y * if n = m then 1 else 0) + z)
⊢ (C x * X ^ k + C y * X ^ m + C z * X ^ n).IsUnitTrinomial
|
rw [if_neg hkm.ne, if_neg (hkm.trans hmn).ne] at hx
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
k m n : ℕ
hkm : k < m
hmn : m < n
x y z : ℤ
hp : #{k, m, n} = 3 ∧ ∀ k_1 ∈ {k, m, n}, IsUnit ((C x * X ^ k + C y * X ^ m + C z * X ^ n).coeff k_1)
hx : IsUnit (x + y * 0 + z * 0)
hy : IsUnit ((x * if m = k then 1 else 0) + y + z * if m = n then 1 else 0)
hz : IsUnit (((x * if n = k then 1 else 0) + y * if n = m then 1 else 0) + z)
⊢ (C x * X ^ k + C y * X ^ m + C z * X ^ n).IsUnitTrinomial
|
649352ed7b796161
|
Nat.le_div_iff_mul_le
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Div/Basic.lean
|
theorem le_div_iff_mul_le (k0 : 0 < k) : x ≤ y / k ↔ x * k ≤ y
|
case ind.zero
y k : Nat
h : 0 < k ∧ k ≤ y
IH : ∀ {x : Nat}, 0 < k → (x ≤ (y - k) / k ↔ x * k ≤ y - k)
k0 : 0 < k
⊢ 0 ≤ (y - k) / k + 1 ↔ 0 * k ≤ y
|
simp [zero_le]
|
no goals
|
50e411e07a3e340f
|
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