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 |
|---|---|---|---|---|---|---|
IsPrimitiveRoot.pow_mul_pow_lcm | Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.lean | theorem pow_mul_pow_lcm {ζ' : M} {k' : ℕ} (hζ : IsPrimitiveRoot ζ k) (hζ' : IsPrimitiveRoot ζ' k')
(hk : k ≠ 0) (hk' : k' ≠ 0) :
IsPrimitiveRoot
(ζ ^ (k / Nat.factorizationLCMLeft k k') * ζ' ^ (k' / Nat.factorizationLCMRight k k'))
(Nat.lcm k k') | case h.e'_3.h.e'_3
M : Type u_1
inst✝ : CommMonoid M
k : ℕ
ζ ζ' : M
k' : ℕ
hζ : IsPrimitiveRoot ζ k
hζ' : IsPrimitiveRoot ζ' k'
hk : k ≠ 0
hk' : k' ≠ 0
⊢ ζ ^ (k / k.factorizationLCMLeft k') * ζ' ^ (k' / k.factorizationLCMRight k') =
ζ ^ (orderOf ζ / (orderOf ζ).factorizationLCMLeft (orderOf ζ')) *
ζ' ^ (order... | simp [hζ.eq_orderOf, hζ'.eq_orderOf] | no goals | d8f21129aa159a42 |
NumberField.house.lin_1 | Mathlib/NumberTheory/NumberField/House.lean | theorem lin_1 (l k r) : a k l * (newBasis K) r =
∑ u, (a' K a k l r u) * (newBasis K) u | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
l : β
k : α
r : K →+* ℂ
⊢ a k l * (NumberField.house.newBasis K) r =
∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l r u) * (NumberField.house.newBasis K) u | simp only [Basis.sum_repr (newBasis K) (a k l * (newBasis K) r), a', ← zsmul_eq_mul] | no goals | 0d554c02dfff4b86 |
List.length_splitLengths_getElem_eq | Mathlib/Data/List/SplitLengths.lean | theorem length_splitLengths_getElem_eq {i : ℕ} (hi : i < sz.length)
(h : (sz.take (i + 1)).sum ≤ l.length) :
((sz.splitLengths l)[i]'(by simpa)).length = sz[i] | α : Type u_1
l : List α
sz : List ℕ
i : ℕ
hi : i < sz.length
h : (take (i + 1) sz).sum ≤ l.length
⊢ i < (sz.splitLengths l).length | simpa | no goals | 66eb11c40c1e8446 |
CategoryTheory.Presieve.Arrows.pullbackCompatible_iff | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | theorem Arrows.pullbackCompatible_iff (x : (i : I) → P.obj (op (X i))) :
Compatible P π x ↔ PullbackCompatible P π x | C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (ofArrows X π).hasPullbacks
x : (i : I) → P.obj (op (X i))
⊢ Compatible P π x ↔ PullbackCompatible P π x | refine ⟨fun t i j ↦ ?_, fun t i j Z gi gj comm ↦ ?_⟩ | case refine_1
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
B : C
I : Type u_1
X : I → C
π : (i : I) → X i ⟶ B
inst✝ : (ofArrows X π).hasPullbacks
x : (i : I) → P.obj (op (X i))
t : Compatible P π x
i j : I
⊢ P.map (pullback.fst (π i) (π j)).op (x i) = P.map (pullback.snd (π i) (π j)).op (x j)
case refine_... | 2754281d88b44b36 |
Finset.shatterer_eq | Mathlib/Combinatorics/SetFamily/Shatter.lean | @[simp] lemma shatterer_eq : 𝒜.shatterer = 𝒜 ↔ IsLowerSet (𝒜 : Set (Finset α)) | case refine_2
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
h : IsLowerSet ↑𝒜
s : Finset α
hs : s ∈ 𝒜.shatterer
⊢ s ∈ 𝒜 | obtain ⟨t, ht, hst⟩ := (mem_shatterer.1 hs).exists_superset | case refine_2.intro.intro
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
h : IsLowerSet ↑𝒜
s : Finset α
hs : s ∈ 𝒜.shatterer
t : Finset α
ht : t ∈ 𝒜
hst : s ⊆ t
⊢ s ∈ 𝒜 | 46cb82815a11788b |
EReal.coe_ennreal_top_mul | Mathlib/Data/Real/EReal.lean | theorem coe_ennreal_top_mul (x : ℝ≥0) : ((⊤ * x : ℝ≥0∞) : EReal) = ⊤ * x | case inr
x : ℝ≥0
h0 : x ≠ 0
⊢ ↑⊤ = ⊤ * ↑↑x | exact Eq.symm <| if_pos <| NNReal.coe_pos.2 h0.bot_lt | no goals | 133d558b813487e6 |
WithTop.untopD_zero_mul | Mathlib/Algebra/Order/Ring/WithTop.lean | @[simp]
lemma untopD_zero_mul (a b : WithTop α) : (a * b).untopD 0 = a.untopD 0 * b.untopD 0 | case neg.coe.coe
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
a✝¹ : α
ha : ¬↑a✝¹ = 0
a✝ : α
hb : ¬↑a✝ = 0
⊢ untopD 0 (↑a✝¹ * ↑a✝) = untopD 0 ↑a✝¹ * untopD 0 ↑a✝ | rw [← coe_mul, untopD_coe, untopD_coe, untopD_coe] | no goals | ebaa5001124c917e |
Finsupp.mapDomain_mono | Mathlib/Data/Finsupp/Order.lean | lemma mapDomain_mono : Monotone (mapDomain f : (ι →₀ α) → (κ →₀ α)) | ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝ : OrderedAddCommMonoid α
f : ι → κ
g₁ g₂ : ι →₀ α
h : g₁ ≤ g₂
⊢ ∀ i ∈ g₁.support ∪ g₂.support, single (f i) 0 = 0 | simp | no goals | a0409aa672f4d3c0 |
AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal.prime | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | theorem carrier.asIdeal.prime : (carrier.asIdeal f_deg hm q).IsPrime :=
(carrier.asIdeal.homogeneous f_deg hm q).isPrime_of_homogeneous_mem_or_mem
(carrier.asIdeal.ne_top f_deg hm q) fun {x y} ⟨nx, hnx⟩ ⟨ny, hny⟩ hxy =>
show (∀ _, _ ∈ _) ∨ ∀ _, _ ∈ _ by
rw [← and_forall_ne nx, and_iff_left, ← and_forall... | case h.e'_5
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm ... | simp_rw [proj_apply] | case h.e'_5
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm ... | 5ba35f6f859f61e0 |
Std.Sat.AIG.mkGateCached.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedLemmas.lean | theorem mkGateCached.go_decl_eq (aig : AIG α) (input : GateInput aig) :
∀ (idx : Nat) (h1) (h2), (go aig input).aig.decls[idx]'h1 = aig.decls[idx]'h2 | case h_2.h_8
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
res : Entrypoint α
x✝³ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝³ : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input... | rw [← hres] | case h_2.h_8
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
res : Entrypoint α
x✝³ : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝³ : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input... | f550d988487b2b3c |
convexHull_eq | Mathlib/Analysis/Convex/Combination.lean | theorem convexHull_eq (s : Set E) : convexHull R s =
{ x : E | ∃ (ι : Type) (t : Finset ι) (w : ι → R) (z : ι → E), (∀ i ∈ t, 0 ≤ w i) ∧
∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x } | case right
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s : Set E
x : E
hx : x ∈ s
⊢ ({PUnit.unit}.centerMass (fun x => 1) fun x_1 => x) = x | simp only [Finset.centerMass, Finset.sum_singleton, inv_one, one_smul] | no goals | 84416efa323f8fb0 |
ProbabilityTheory.IndepFun.variance_sum | Mathlib/Probability/Variance.lean | theorem IndepFun.variance_sum [IsProbabilityMeasure μ] {ι : Type*} {X : ι → Ω → ℝ}
{s : Finset ι} (hs : ∀ i ∈ s, MemLp (X i) 2 μ)
(h : Set.Pairwise ↑s fun i j => IndepFun (X i) (X j) μ) :
variance (∑ i ∈ s, X i) μ = ∑ i ∈ s, variance (X i) μ | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
X : ι → Ω → ℝ
k : ι
s : Finset ι
ks : k ∉ s
IH :
(∀ i ∈ s, MemLp (X i) 2 μ) →
((↑s).Pairwise fun i j => IndepFun (X i) (X j) μ) → Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]
hs : ∀ i ∈ insert k s, MemLp (X i) 2 μ
h : (↑(... | rw [integral_add', integral_add', integral_add'] | case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
ι : Type u_2
X : ι → Ω → ℝ
k : ι
s : Finset ι
ks : k ∉ s
IH :
(∀ i ∈ s, MemLp (X i) 2 μ) →
((↑s).Pairwise fun i j => IndepFun (X i) (X j) μ) → Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]
hs : ∀ i ∈ insert k s, MemLp (X i) 2 μ... | 1d5529d2c6cdc1ee |
CategoryTheory.epi_iff_inl_eq_inr | Mathlib/CategoryTheory/Limits/EpiMono.lean | lemma epi_iff_inl_eq_inr (hc : IsColimit c) : Epi f ↔ c.inl = c.inr | case mpr
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
hf : c.inl = c.inr
⊢ Epi f | constructor | case mpr.left_cancellation
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
hf : c.inl = c.inr
⊢ ∀ {Z : C} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h | c90317b4ece5c50a |
List.flatten_splitLengths | Mathlib/Data/List/SplitLengths.lean | theorem flatten_splitLengths (h : l.length ≤ sz.sum) : (sz.splitLengths l).flatten = l | case cons
α : Type u_1
head✝ : ℕ
tail✝ : List ℕ
tail_ih✝ : ∀ (l : List α), l.length ≤ tail✝.sum → (tail✝.splitLengths l).flatten = l
l : List α
h : l.length ≤ (head✝ :: tail✝).sum
⊢ ((head✝ :: tail✝).splitLengths l).flatten = l | case cons head tail ih =>
simp only [splitLengths_cons, flatten_cons]
rw [ih, take_append_drop]
simpa [add_comm] using h | no goals | f83714d3d5fad5e5 |
Polynomial.le_natTrailingDegree | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | theorem le_natTrailingDegree (hp : p ≠ 0) (hn : ∀ m < n, p.coeff m = 0) :
n ≤ p.natTrailingDegree | R : Type u
n : ℕ
inst✝ : Semiring R
p : R[X]
hp : p ≠ 0
hn : ∀ m < n, p.coeff m = 0
⊢ n ≤ p.support.min' ⋯ | exact Finset.le_min' _ _ _ fun m hm => not_lt.1 fun hmn => mem_support_iff.1 hm <| hn _ hmn | no goals | e6daf4dd4d532f5b |
convex_hyperplane | Mathlib/Analysis/Convex/Basic.lean | theorem convex_hyperplane {f : E → β} (h : IsLinearMap 𝕜 f) (r : β) : Convex 𝕜 { w | f w = r } | 𝕜 : Type u_1
E : Type u_2
β : Type u_4
inst✝⁵ : OrderedSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : OrderedAddCommMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
f : E → β
h : IsLinearMap 𝕜 f
r : β
⊢ Convex 𝕜 {w | f w = r} | simp_rw [le_antisymm_iff] | 𝕜 : Type u_1
E : Type u_2
β : Type u_4
inst✝⁵ : OrderedSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : Module 𝕜 E
inst✝² : OrderedAddCommMonoid β
inst✝¹ : Module 𝕜 β
inst✝ : OrderedSMul 𝕜 β
f : E → β
h : IsLinearMap 𝕜 f
r : β
⊢ Convex 𝕜 {w | f w ≤ r ∧ r ≤ f w} | fffcb3ce2a4744e3 |
AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux | Mathlib/Analysis/Complex/OpenMapping.lean | theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux (hf : AnalyticAt ℂ f z₀) :
(∀ᶠ z in 𝓝 z₀, f z = f z₀) ∨ 𝓝 (f z₀) ≤ map f (𝓝 z₀) | case intro.intro.intro.intro.intro
f : ℂ → ℂ
z₀ : ℂ
hf : AnalyticAt ℂ f z₀
h : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
R : ℝ
hR : 0 < R
h1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀
h2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z
ρ : ℝ
hρ : ρ > 0
h4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀
h3 : DiffContOnCl ℂ f (ball z₀ ρ)
r : ℝ := ... | refine ⟨‖f x - f z₀‖ / 2, half_pos (norm_sub_pos_iff.mpr (h7 x hx)), ?_⟩ | case intro.intro.intro.intro.intro
f : ℂ → ℂ
z₀ : ℂ
hf : AnalyticAt ℂ f z₀
h : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀
R : ℝ
hR : 0 < R
h1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀
h2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z
ρ : ℝ
hρ : ρ > 0
h4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀
h3 : DiffContOnCl ℂ f (ball z₀ ρ)
r : ℝ := ... | 3b877a39b266ab74 |
tendstoUniformlyOn_tsum | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | theorem tendstoUniformlyOn_tsum {f : α → β → F} (hu : Summable u) {s : Set β}
(hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
TendstoUniformlyOn (fun t : Finset α => fun x => ∑ n ∈ t, f n x) (fun x => ∑' n, f n x) atTop
s | α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
s : Set β
hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n
ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (n : Finset α) in atTop, ∀ x ∈ s, dist (∑' (n : α), f n x) (∑ n ∈ n, f n x) < ε | filter_upwards [(tendsto_order.1 (tendsto_tsum_compl_atTop_zero u)).2 _ εpos] with t ht x hx | case h
α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
s : Set β
hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n
ε : ℝ
εpos : ε > 0
t : Finset α
ht : ∑' (a : { x // x ∉ t }), u ↑a < ε
x : β
hx : x ∈ s
⊢ dist (∑' (n : α), f n x) (∑ n ∈ t, f n x... | f6fd8bf2ef5cc1be |
List.any_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem _root_.List.any_toArray (p : α → Bool) (l : List α) : l.toArray.any p = l.any p | α : Type u_1
p : α → Bool
l : List α
⊢ l.toArray.any p = l.any p | rw [any_toList] | no goals | 635ba922e08b3188 |
minpoly.natDegree_pos | Mathlib/FieldTheory/Minpoly/Basic.lean | theorem natDegree_pos [Nontrivial B] (hx : IsIntegral A x) : 0 < natDegree (minpoly A x) | A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : Ring B
inst✝¹ : Algebra A B
x : B
inst✝ : Nontrivial B
hx : IsIntegral A x
ndeg_eq_zero : (minpoly A x).natDegree = 0
⊢ C ((minpoly A x).coeff 0) = 1 | convert C_1 (R := A) | case h.e'_2.h.e'_6
A : Type u_1
B : Type u_2
inst✝³ : CommRing A
inst✝² : Ring B
inst✝¹ : Algebra A B
x : B
inst✝ : Nontrivial B
hx : IsIntegral A x
ndeg_eq_zero : (minpoly A x).natDegree = 0
⊢ (minpoly A x).coeff 0 = 1 | 6a5f0c636a2a1747 |
Polynomial.natDegree_C_mul_X | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | theorem natDegree_C_mul_X (a : R) (ha : a ≠ 0) : natDegree (C a * X) = 1 | R : Type u
inst✝ : Semiring R
a : R
ha : a ≠ 0
⊢ (C a * X).natDegree = 1 | simpa only [pow_one] using natDegree_C_mul_X_pow 1 a ha | no goals | dad1e5babf68b162 |
ComputablePred.not | Mathlib/Computability/Halting.lean | theorem not {p : α → Prop} (hp : ComputablePred p) : ComputablePred fun a => ¬p a | α : Type u_1
inst✝ : Primcodable α
f : α → Bool
hf : Computable f
hp : ComputablePred fun a => f a = true
⊢ DecidablePred fun a => ¬(fun a => f a = true) a | infer_instance | no goals | 3653fc258816fcbb |
isAdjointPair_toLinearMap₂' | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | theorem isAdjointPair_toLinearMap₂' :
LinearMap.IsAdjointPair (Matrix.toLinearMap₂' R J) (Matrix.toLinearMap₂' R J')
(Matrix.toLin' A) (Matrix.toLin' A') ↔
Matrix.IsAdjointPair J J' A A' | R : Type u_1
n : Type u_11
n' : Type u_13
inst✝⁴ : CommRing R
inst✝³ : Fintype n
inst✝² : Fintype n'
J : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
⊢ B = B' ↔ (toMatrix₂' R) B = (toMatrix₂' R) B' | constructor <;> intro h | case mp
R : Type u_1
n : Type u_11
n' : Type u_13
inst✝⁴ : CommRing R
inst✝³ : Fintype n
inst✝² : Fintype n'
J : Matrix n n R
J' : Matrix n' n' R
A : Matrix n' n R
A' : Matrix n n' R
inst✝¹ : DecidableEq n
inst✝ : DecidableEq n'
B B' : (n → R) →ₗ[R] (n' → R) →ₗ[R] R
h : B = B'
⊢ (toMatrix₂' R) B = (toMatrix₂' R) B'
ca... | d3f9398d16094479 |
Compactum.isClosed_iff | Mathlib/Topology/Category/Compactum.lean | theorem isClosed_iff {X : Compactum} (S : Set X) :
IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S | case mp
X : Compactum
S : Set X.A
F : Ultrafilter X.A
h : S ∈ F
c : X.str F ∉ S
cond : Sᶜ ∈ F
⊢ False | rw [compl_mem_iff_not_mem] at cond | case mp
X : Compactum
S : Set X.A
F : Ultrafilter X.A
h : S ∈ F
c : X.str F ∉ S
cond : S ∉ F
⊢ False | 1b8ad681ff710a41 |
PiLp.basis_toMatrix_basisFun_mul | Mathlib/Analysis/Normed/Lp/PiLp.lean | theorem basis_toMatrix_basisFun_mul [Fintype ι]
{𝕜} [SeminormedCommRing 𝕜] (b : Basis ι 𝕜 (PiLp p fun _ : ι => 𝕜))
(A : Matrix ι ι 𝕜) :
b.toMatrix (PiLp.basisFun _ _ _) * A =
Matrix.of fun i j => b.repr ((WithLp.equiv _ _).symm (Aᵀ j)) i | p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
𝕜 : Type u_5
inst✝ : SeminormedCommRing 𝕜
b : Basis ι 𝕜 (PiLp p fun x => 𝕜)
A : Matrix ι ι 𝕜
⊢ b.toMatrix ⇑(basisFun p 𝕜 ι) * A = of fun i j => (b.repr ((WithLp.equiv p ((i : ι) → (fun x => 𝕜) i)).symm (Aᵀ j))) i | have := basis_toMatrix_basisFun_mul (b.map (WithLp.linearEquiv _ 𝕜 _)) A | p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
𝕜 : Type u_5
inst✝ : SeminormedCommRing 𝕜
b : Basis ι 𝕜 (PiLp p fun x => 𝕜)
A : Matrix ι ι 𝕜
this :
(b.map (WithLp.linearEquiv p 𝕜 ((i : ι) → (fun x => 𝕜) i))).toMatrix ⇑(Pi.basisFun 𝕜 ι) * A =
of fun i j => ((b.map (WithLp.linearEquiv p 𝕜 ((i : ι) → (fun x => 𝕜)... | adf6746c6a6c2baa |
Std.Sat.AIG.denote.go_eq_of_isPrefix | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/LawfulOperator.lean | theorem denote.go_eq_of_isPrefix (decls1 decls2 : Array (Decl α)) (start : Nat) {hdag1} {hdag2}
{hbounds1} {hbounds2} (hprefix : IsPrefix decls1 decls2) :
denote.go start decls2 assign hbounds2 hdag2
=
denote.go start decls1 assign hbounds1 hdag1 | α : Type
assign : α → Bool
decls1 decls2 : Array (Decl α)
start : Nat
hdag1 : IsDAG α decls1
hdag2 : IsDAG α decls2
hbounds1 : start < decls1.size
hbounds2 : start < decls2.size
hprefix : IsPrefix decls1 decls2
hidx1 : decls2[start] = decls1[start]
⊢ (match elem : decls2[start] with
| Decl.const b => b
| Decl.a... | split | case h_1
α : Type
assign : α → Bool
decls1 decls2 : Array (Decl α)
start : Nat
hdag1 : IsDAG α decls1
hdag2 : IsDAG α decls2
hbounds1 : start < decls1.size
hbounds2 : start < decls2.size
hprefix : IsPrefix decls1 decls2
hidx1 : decls2[start] = decls1[start]
b✝ : Bool
heq✝ : decls2[start] = Decl.const b✝
⊢ b✝ =
matc... | c2a2ed6f9996b875 |
Polynomial.smeval_neg_nat | Mathlib/Algebra/Polynomial/Smeval.lean | theorem smeval_neg_nat (S : Type*) [NonAssocRing S] [Pow S ℕ] [NatPowAssoc S] (q : ℕ[X])
(n : ℕ) : q.smeval (-(n : S)) = q.smeval (-n : ℤ) | S : Type u_3
inst✝² : NonAssocRing S
inst✝¹ : Pow S ℕ
inst✝ : NatPowAssoc S
q : ℕ[X]
n : ℕ
⊢ q.smeval (-↑n) = ↑(q.smeval (-↑n)) | rw [smeval_eq_sum, smeval_eq_sum] | S : Type u_3
inst✝² : NonAssocRing S
inst✝¹ : Pow S ℕ
inst✝ : NatPowAssoc S
q : ℕ[X]
n : ℕ
⊢ q.sum (smul_pow (-↑n)) = ↑(q.sum (smul_pow (-↑n))) | c242622875ddf7b1 |
splits_X_pow_sub_one_of_X_pow_sub_C | Mathlib/FieldTheory/AbelRuffini.lean | theorem splits_X_pow_sub_one_of_X_pow_sub_C {F : Type*} [Field F] {E : Type*} [Field E]
(i : F →+* E) (n : ℕ) {a : F} (ha : a ≠ 0) (h : (X ^ n - C a).Splits i) :
(X ^ n - 1 : F[X]).Splits i | case h
F : Type u_3
inst✝¹ : Field F
E : Type u_4
inst✝ : Field E
i : F →+* E
n : ℕ
a : F
ha : a ≠ 0
h : Splits i (X ^ n - C a)
ha' : i a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : (X ^ n - C a).degree ≠ 0
b : E
hb : b ^ n = i a
hb' : b ≠ 0
s : Multiset E := (Polynomial.map i (X ^ n - C a)).roots
hs : Polynomial.map i (X ^ n -... | dsimp only [Function.comp_apply] | case h
F : Type u_3
inst✝¹ : Field F
E : Type u_4
inst✝ : Field E
i : F →+* E
n : ℕ
a : F
ha : a ≠ 0
h : Splits i (X ^ n - C a)
ha' : i a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : (X ^ n - C a).degree ≠ 0
b : E
hb : b ^ n = i a
hb' : b ≠ 0
s : Multiset E := (Polynomial.map i (X ^ n - C a)).roots
hs : Polynomial.map i (X ^ n -... | 3246ca7f177f02a2 |
Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod | Mathlib/LinearAlgebra/FreeModule/Int.lean | /-- Given a submodule `N` in Smith normal form of a free `R`-module, its index as an additive
subgroup is an appropriate power of the cardinality of `R` multiplied by the product of the
indexes of the ideals generated by each basis vector. -/
lemma toAddSubgroup_index_eq_pow_mul_prod [Module R M] {N : Submodule R M}
... | case h.refine_2.h
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map ... | by_cases hj : ∃ j, f j = i | case pos
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivF... | 3d7a9cb7455a5c0f |
DFinsupp.mapRange.addMonoidHom_comp | Mathlib/Data/DFinsupp/Defs.lean | theorem mapRange.addMonoidHom_comp (f : ∀ i, β₁ i →+ β₂ i) (f₂ : ∀ i, β i →+ β₁ i) :
(mapRange.addMonoidHom fun i => (f i).comp (f₂ i)) =
(mapRange.addMonoidHom f).comp (mapRange.addMonoidHom f₂) | case refine_1
ι : Type u
β : ι → Type v
β₁ : ι → Type v₁
β₂ : ι → Type v₂
inst✝² : (i : ι) → AddZeroClass (β i)
inst✝¹ : (i : ι) → AddZeroClass (β₁ i)
inst✝ : (i : ι) → AddZeroClass (β₂ i)
f : (i : ι) → β₁ i →+ β₂ i
f₂ : (i : ι) → β i →+ β₁ i
i✝ : ι
⊢ (fun i x => (f i) x) i✝ 0 = 0 | apply map_zero | no goals | f5b74fc335cfe878 |
Complex.abs_arg_lt_pi_div_two_iff | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem abs_arg_lt_pi_div_two_iff {z : ℂ} : |arg z| < π / 2 ↔ 0 < re z ∨ z = 0 | z : ℂ
⊢ 0 < z.re ∨ 0 ≤ z.im ∧ (z.im < 0 ∨ z = 0) ↔ 0 < z.re ∨ z = 0 | rcases eq_or_ne z 0 with hz | hz | case inl
z : ℂ
hz : z = 0
⊢ 0 < z.re ∨ 0 ≤ z.im ∧ (z.im < 0 ∨ z = 0) ↔ 0 < z.re ∨ z = 0
case inr
z : ℂ
hz : z ≠ 0
⊢ 0 < z.re ∨ 0 ≤ z.im ∧ (z.im < 0 ∨ z = 0) ↔ 0 < z.re ∨ z = 0 | 86a383eb96fa5adb |
CategoryTheory.nerve.δ₁_mk₂_eq | Mathlib/AlgebraicTopology/SimplicialSet/Nerve.lean | theorem δ₁_mk₂_eq : (nerve C).δ 1 (ComposableArrows.mk₂ f g) = ComposableArrows.mk₁ (f ≫ g) :=
ComposableArrows.ext₁ rfl rfl (by simp; rfl)
| C : Type u_1
inst✝ : Category.{u_2, u_1} C
X₀ X₁ X₂ : C
f : X₀ ⟶ X₁
g : X₁ ⟶ X₂
⊢ ComposableArrows.hom (SimplicialObject.δ (nerve C) 1 (ComposableArrows.mk₂ f g)) =
eqToHom ⋯ ≫ (ComposableArrows.mk₁ (f ≫ g)).hom ≫ eqToHom ⋯ | simp | C : Type u_1
inst✝ : Category.{u_2, u_1} C
X₀ X₁ X₂ : C
f : X₀ ⟶ X₁
g : X₁ ⟶ X₂
⊢ ComposableArrows.hom (SimplicialObject.δ (nerve C) 1 ((ComposableArrows.mk₁ g).precomp f)) = f ≫ g | ce33b0f9b07aee0a |
bergelson' | Mathlib/MeasureTheory/Function/Intersectivity.lean | /-- **Bergelson Intersectivity Lemma**: In a finite measure space, a sequence of events that have
measure at least `r` has an infinite subset whose finite intersections all have positive volume.
TODO: The infinity of `t` should be strengthened to `t` having positive natural density. -/
lemma bergelson' {s : ℕ → Set α}... | α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
N : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator 1)
hN₀ : μ N = 0
hN₁ : ∀ (u :... | simp | no goals | 8eca484049265c21 |
Ordinal.not_bddAbove_principal | Mathlib/SetTheory/Ordinal/Principal.lean | theorem not_bddAbove_principal (op : Ordinal → Ordinal → Ordinal) :
¬ BddAbove { o | Principal op o } | case intro
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a : Ordinal.{u_1}
ha : a ∈ upperBounds {o | Principal op o}
⊢ False | exact ((le_nfp _ _).trans (ha (principal_nfp_iSup op (succ a)))).not_lt (lt_succ a) | no goals | b833ef87aba4a278 |
MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1 | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε) :
(∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) + 2 * ε * μ s | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εp... | rotate_left | case hn
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε ... | 10b49d97c45d9730 |
sdiff_sdiff_right | Mathlib/Order/BooleanAlgebra.lean | theorem sdiff_sdiff_right : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z | α : Type u
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊓ (z ⊓ x ⊔ x \ y) = x ⊓ y \ z ⊓ (z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y | rw [inf_sup_left] | no goals | 7b53ef024b0eda3c |
Array.back?_attachWith | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Attach.lean | theorem back?_attachWith {P : α → Prop} {xs : Array α}
{H : ∀ (a : α), a ∈ xs → P a} :
(xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back? h)⟩) | α : Type u_1
P : α → Prop
xs : Array α
H : ∀ (a : α), a ∈ xs → P a
⊢ (xs.attachWith P H).back? = xs.back?.pbind fun a h => some ⟨a, ⋯⟩ | cases xs | case mk
α : Type u_1
P : α → Prop
toList✝ : List α
H : ∀ (a : α), a ∈ { toList := toList✝ } → P a
⊢ ({ toList := toList✝ }.attachWith P H).back? = { toList := toList✝ }.back?.pbind fun a h => some ⟨a, ⋯⟩ | e31d0a4f1b583f14 |
Vector.findSome?_mkVector_of_pos | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Find.lean | theorem findSome?_mkVector_of_pos (h : 0 < n) : findSome? f (mkVector n a) = f a | n : Nat
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → Option α✝
a : α✝¹
h : 0 < n
⊢ findSome? f (mkVector n a) = f a | simp [findSome?_mkVector, Nat.ne_of_gt h] | no goals | 20745acc7b7056eb |
Polynomial.X_pow_sub_X_sub_one_irreducible_rat | Mathlib/RingTheory/Polynomial/Selmer.lean | theorem X_pow_sub_X_sub_one_irreducible_rat (hn1 : n ≠ 1) : Irreducible (X ^ n - X - 1 : ℚ[X]) | case neg
n : ℕ
hn1 : n ≠ 1
hn0 : ¬n = 0
⊢ Irreducible (X ^ n - X - 1) | have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 := by
simp only [trinomial, C_neg, C_1]; ring | case neg
n : ℕ
hn1 : n ≠ 1
hn0 : ¬n = 0
hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1
⊢ Irreducible (X ^ n - X - 1) | 45b2105ebcfc534a |
RCLike.wInner_one_const_right | Mathlib/Analysis/RCLike/Inner.lean | @[simp] lemma wInner_one_const_right (f : ι → 𝕜) (a : 𝕜) :
⟪f, const _ a⟫_[𝕜] = (∑ i, conj (f i)) * a | ι : Type u_1
𝕜 : Type u_3
inst✝¹ : Fintype ι
inst✝ : RCLike 𝕜
f : ι → 𝕜
a : 𝕜
⊢ wInner 1 f (const ι a) = (∑ i : ι, (starRingEnd 𝕜) (f i)) * a | simp [wInner_one_eq_sum, sum_mul] | no goals | a48fca30ffe68b37 |
List.monotone_filter_right | Mathlib/Data/List/Basic.lean | theorem monotone_filter_right (l : List α) ⦃p q : α → Bool⦄
(h : ∀ a, p a → q a) : l.filter p <+ l.filter q | α : Type u
l : List α
p q : α → Bool
h : ∀ (a : α), p a = true → q a = true
⊢ filter p l <+ filter q l | induction' l with hd tl IH | case nil
α : Type u
p q : α → Bool
h : ∀ (a : α), p a = true → q a = true
⊢ filter p [] <+ filter q []
case cons
α : Type u
p q : α → Bool
h : ∀ (a : α), p a = true → q a = true
hd : α
tl : List α
IH : filter p tl <+ filter q tl
⊢ filter p (hd :: tl) <+ filter q (hd :: tl) | ec84442b80b4abf0 |
Real.eulerMascheroniSeq'_six_lt_two_thirds | Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | lemma eulerMascheroniSeq'_six_lt_two_thirds : eulerMascheroniSeq' 6 < 2 / 3 | h1 : eulerMascheroniSeq' 6 = 49 / 20 - log 6
⊢ 0 < 6 | positivity | no goals | bd6c8b8313725a15 |
Ordinal.cof_iSup_le | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem cof_iSup_le {ι} {f : ι → Ordinal} (H : ∀ i, f i < iSup f) :
cof (iSup f) ≤ #ι | ι : Type u_1
f : ι → Ordinal.{u_1}
H : ∀ (i : ι), f i < iSup f
⊢ (iSup f).cof ≤ Cardinal.lift.{u_1, u_1} #ι | exact cof_iSup_le_lift H | no goals | e8ab74fcd5eb12b4 |
SimplexCategory.mkOfSucc_subinterval_eq | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | @[simp]
lemma mkOfSucc_subinterval_eq {n} (j l : ℕ) (hjl : j + l ≤ n) (i : Fin l) :
mkOfSucc i ≫ subinterval j l hjl =
mkOfSucc ⟨j + i.1, Nat.lt_of_lt_of_le (Nat.add_lt_add_left i.2 j) hjl⟩ | n j l : ℕ
hjl : j + l ≤ n
i✝ : Fin l
i : Fin (⦋1⦌.len + 1)
⊢ ↑(match 1 with
| 0 => i✝.castSucc
| 1 => i✝.succ) +
j =
↑(match 1 with
| 0 => ⟨j + ↑i✝, ⋯⟩
| 1 => ⟨j + ↑i✝ + 1, ⋯⟩) | rw [← Nat.add_comm j _] | n j l : ℕ
hjl : j + l ≤ n
i✝ : Fin l
i : Fin (⦋1⦌.len + 1)
⊢ j +
↑(match 1 with
| 0 => i✝.castSucc
| 1 => i✝.succ) =
↑(match 1 with
| 0 => ⟨j + ↑i✝, ⋯⟩
| 1 => ⟨j + ↑i✝ + 1, ⋯⟩) | 34ac17f755ced5f6 |
Set.Finite.exists_maximal_wrt' | Mathlib/Data/Set/Finite/Basic.lean | theorem Finite.exists_maximal_wrt' [PartialOrder β] (f : α → β) (s : Set α) (h : (f '' s).Finite)
(hs : s.Nonempty) : (∃ a ∈ s, ∀ (a' : α), a' ∈ s → f a ≤ f a' → f a = f a') | α : Type u
β : Type v
inst✝ : PartialOrder β
f : α → β
s : Set α
h : (f '' s).Finite
hs : s.Nonempty
⊢ ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a' | obtain ⟨_, ⟨a, ha, rfl⟩, hmax⟩ := Finite.exists_maximal_wrt id (f '' s) h (hs.image f) | case intro.intro.intro.intro
α : Type u
β : Type v
inst✝ : PartialOrder β
f : α → β
s : Set α
h : (f '' s).Finite
hs : s.Nonempty
a : α
ha : a ∈ s
hmax : ∀ a' ∈ f '' s, id (f a) ≤ id a' → id (f a) = id a'
⊢ ∃ a ∈ s, ∀ a' ∈ s, f a ≤ f a' → f a = f a' | 82760c068d5c9c48 |
Vector.getElem?_flatten | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem getElem?_flatten (l : Vector (Vector β m) n) (i : Nat) :
l.flatten[i]? =
if hi : i < n * m then
haveI : i / m < n | β : Type u_1
m n : Nat
l : Vector (Vector β m) n
i : Nat
⊢ l.flatten[i]? = if hi : i < n * m then some l[i / m][i % m] else none | simp [getElem?_def] | no goals | 67943409f27d03d5 |
HNNExtension.ReducedWord.exists_normalWord_prod_eq | Mathlib/GroupTheory/HNNExtension.lean | theorem exists_normalWord_prod_eq
(d : TransversalPair G A B) (w : ReducedWord G A B) :
∃ w' : NormalWord d, w'.prod φ = w.prod φ ∧
w'.toList.map Prod.fst = w.toList.map Prod.fst ∧
∀ u ∈ w.toList.head?.map Prod.fst,
w'.head⁻¹ * w.head ∈ toSubgroup A B (-u) | case pos
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
w : ReducedWord G A B
this :
∀ (w : ReducedWord G A B),
w.head = 1 →
∃ w',
ReducedWord.prod φ w'.toReducedWord = ReducedWord.prod φ w ∧
List.map Prod.fst w'.toList = List.map Prod.fst w.toList ∧
... | exact this w hw1 | no goals | 4623268a783c6740 |
List.getElem_insertIdx_of_lt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/InsertIdx.lean | theorem getElem_insertIdx_of_lt {l : List α} {x : α} {n k : Nat} (hn : k < n)
(hk : k < (insertIdx n x l).length) :
(insertIdx n x l)[k] = l[k]'(by simp [length_insertIdx] at hk; split at hk <;> omega) | case succ.cons.succ
α : Type u
x : α
n : Nat
ih : ∀ {l : List α} {k : Nat} (hn : k < n) (hk : k < (insertIdx n x l).length), (insertIdx n x l)[k] = l[k]
head✝ : α
tail✝ : List α
n✝ : Nat
hn : n✝ + 1 < n + 1
hk : n✝ + 1 < (insertIdx (n + 1) x (head✝ :: tail✝)).length
⊢ (insertIdx (n + 1) x (head✝ :: tail✝))[n✝ + 1] = (h... | rw [Nat.succ_lt_succ_iff] at hn | case succ.cons.succ
α : Type u
x : α
n : Nat
ih : ∀ {l : List α} {k : Nat} (hn : k < n) (hk : k < (insertIdx n x l).length), (insertIdx n x l)[k] = l[k]
head✝ : α
tail✝ : List α
n✝ : Nat
hn✝ : n✝ + 1 < n + 1
hn : n✝ < n
hk : n✝ + 1 < (insertIdx (n + 1) x (head✝ :: tail✝)).length
⊢ (insertIdx (n + 1) x (head✝ :: tail✝))... | c6321edf020f1bc8 |
List.prev_ne_cons_cons | Mathlib/Data/List/Cycle.lean | theorem prev_ne_cons_cons (y z : α) (h : x ∈ y :: z :: l) (hy : x ≠ y) (hz : x ≠ z) :
prev (y :: z :: l) x h = prev (z :: l) x (by simpa [hy] using h) | case cons
α : Type u_1
inst✝ : DecidableEq α
x y z : α
hy : x ≠ y
hz : x ≠ z
head✝ : α
tail✝ : List α
h : x ∈ y :: z :: head✝ :: tail✝
⊢ (y :: z :: head✝ :: tail✝).prev x h = (z :: head✝ :: tail✝).prev x ⋯ | rw [prev, dif_neg hy, if_neg hz] | no goals | f42b696839174de2 |
List.append_flatten_map_append | Mathlib/Data/List/Flatten.lean | theorem append_flatten_map_append (L : List (List α)) (x : List α) :
x ++ (L.map (· ++ x)).flatten = (L.map (x ++ ·)).flatten ++ x | α : Type u_1
L : List (List α)
x : List α
⊢ x ++ (map (fun x_1 => x_1 ++ x) L).flatten = (map (fun x_1 => x ++ x_1) L).flatten ++ x | induction L with
| nil => rw [map_nil, flatten, append_nil, map_nil, flatten, nil_append]
| cons _ _ ih =>
rw [map_cons, flatten, map_cons, flatten, append_assoc, ih, append_assoc, append_assoc] | no goals | 0ef7703076ce4e58 |
CategoryTheory.Limits.preservesBinaryBiproduct_of_epi_biprodComparison' | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | /-- If the (coproduct-like) biproduct comparison for `F`, `X` and `Y` is an epimorphism, then
`F` preserves the biproduct of `X` and `Y`. For the converse, see `mapBiprod`. -/
lemma preservesBinaryBiproduct_of_epi_biprodComparison' {X Y : C} [HasBinaryBiproduct X Y]
[HasBinaryBiproduct (F.obj X) (F.obj Y)] [Epi... | C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : Preadditive C
D : Type u'
inst✝⁵ : Category.{v', u'} D
inst✝⁴ : Preadditive D
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
X Y : C
inst✝² : HasBinaryBiproduct X Y
inst✝¹ : HasBinaryBiproduct (F.obj X) (F.obj Y)
inst✝ : Epi (F.biprodComparison' X Y)
this✝ : Epi (F.splitEpiBi... | apply preservesBinaryBiproduct_of_mono_biprodComparison | no goals | 52754a0e6125a449 |
isNoetherian_iff_fg_wellFounded | Mathlib/RingTheory/Noetherian/Defs.lean | theorem isNoetherian_iff_fg_wellFounded :
IsNoetherian R M ↔ WellFoundedGT { N : Submodule R M // N.FG } | case h.e'_6.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
α : Type (max 0 u_2) := { N // N.FG }
H : WellFoundedGT { N // N.FG }
N N₀ : Submodule R M
h₁ : N₀.FG
e : N₀ ≤ N
h₂ : ∀ x ∈ {N' | ↑N' ≤ N}, ¬x > ⟨N₀, h₁⟩
h₃ : ¬N ≤ N₀
x : M
hx₁ : x ∈ N
hx₂ : x ∉ N₀
⊢ x ∈ sp... | exact (le_sup_left : (R ∙ x) ≤ _) (Submodule.mem_span_singleton_self _) | no goals | 3acc6e2f9e0e2255 |
MonomialOrder.div | Mathlib/RingTheory/MvPolynomial/Groebner.lean | theorem div {ι : Type*} {b : ι → MvPolynomial σ R}
(hb : ∀ i, IsUnit (m.leadingCoeff (b i))) (f : MvPolynomial σ R) :
∃ (g : ι →₀ (MvPolynomial σ R)) (r : MvPolynomial σ R),
f = Finsupp.linearCombination _ b g + r ∧
(∀ i, m.degree (b i * (g i)) ≼[m] m.degree f) ∧
(∀ c ∈ r.support, ∀ i, ¬ (... | case pos
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
b✝ : ι → MvPolynomial σ R
hb : ∀ (i : ι), IsUnit (m.leadingCoeff (b✝ i))
f : MvPolynomial σ R
hb' : ∀ (i : ι), m.degree (b✝ i) ≠ 0
hf0 : f = 0
b : ι
⊢ 0 ≤ m.toSyn (m.degree f) | exact bot_le | no goals | f995acdac7645677 |
Complex.arg_eq_nhds_of_re_neg_of_im_neg | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) :
arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / ‖x‖) - π | x : ℂ
hx_re : x.re < 0
hx_im : x.im < 0
⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ y.im < 0 | refine IsOpen.eventually_mem ?_ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) | x : ℂ
hx_re : x.re < 0
hx_im : x.im < 0
⊢ IsOpen fun y => y.re < 0 ∧ y.im < 0 | f5169d04c761e55b |
Finset.erase_image_subset_image_erase | Mathlib/Data/Finset/Image.lean | theorem erase_image_subset_image_erase [DecidableEq α] (f : α → β) (s : Finset α) (a : α) :
(s.image f).erase (f a) ⊆ (s.erase a).image f | α : Type u_1
β : Type u_2
inst✝¹ : DecidableEq β
inst✝ : DecidableEq α
f : α → β
s : Finset α
a x : α
hx : x ∈ s
hb : f x ≠ f a
⊢ ∃ a_1, (a_1 ≠ a ∧ a_1 ∈ s) ∧ f a_1 = f x | exact ⟨_, ⟨ne_of_apply_ne f hb, hx⟩, rfl⟩ | no goals | 11d3ff2d879d689d |
Subbimodule.smul_mem' | Mathlib/Algebra/Module/Bimodule.lean | theorem smul_mem' (p : Submodule (A ⊗[R] B) M) (b : B) {m : M} (hm : m ∈ p) : b • m ∈ p | R : Type u_1
A : Type u_2
B : Type u_3
M : Type u_4
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : AddCommMonoid M
inst✝⁹ : Module R M
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Module A M
inst✝⁵ : Module B M
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
inst✝² : IsScalarTower R A M
inst✝¹ : IsScalarTower R B M
inst✝ : SMulCommC... | exact this.symm ▸ p.smul_mem _ hm | no goals | 56cdcdc8f9388211 |
ProbabilityTheory.mgf_pos' | Mathlib/Probability/Moments/Basic.lean | theorem mgf_pos' (hμ : μ ≠ 0) (h_int_X : Integrable (fun ω => exp (t * X ω)) μ) :
0 < mgf X μ t | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hμ : μ ≠ 0
h_int_X : Integrable (fun ω => rexp (t * X ω)) μ
this : ∫ (x : Ω), rexp (t * X x) ∂μ = ∫ (x : Ω) in Set.univ, rexp (t * X x) ∂μ
h_eq_univ : (Function.support fun x => rexp (t * X x)) = Set.univ
⊢ 0 < μ ((Function.support fun x => rexp (t * X x)... | rw [h_eq_univ, Set.inter_univ _] | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
hμ : μ ≠ 0
h_int_X : Integrable (fun ω => rexp (t * X ω)) μ
this : ∫ (x : Ω), rexp (t * X x) ∂μ = ∫ (x : Ω) in Set.univ, rexp (t * X x) ∂μ
h_eq_univ : (Function.support fun x => rexp (t * X x)) = Set.univ
⊢ 0 < μ Set.univ | b968b1589af35a47 |
MonoidHom.noncommCoprod_range | Mathlib/GroupTheory/NoncommCoprod.lean | lemma noncommCoprod_range {M N P : Type*} [Group M] [Group N] [Group P]
(f : M →* P) (g : N →* P) (comm : ∀ (m : M) (n : N), Commute (f m) (g n)) :
(noncommCoprod f g comm).range = f.range ⊔ g.range | case a.intro
M : Type u_4
N : Type u_5
P : Type u_6
inst✝² : Group M
inst✝¹ : Group N
inst✝ : Group P
f : M →* P
g : N →* P
comm : ∀ (m : M) (n : N), Commute (f m) (g n)
a : M × N
⊢ (f.noncommCoprod g comm) a ∈ f.range ⊔ g.range | exact mul_mem (mem_sup_left ⟨a.1, rfl⟩) (mem_sup_right ⟨a.2, rfl⟩) | no goals | a43981614b1c4850 |
AlgebraicGeometry.Scheme.IdealSheafData.ideal_le_ker_glueDataObjι | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma ideal_le_ker_glueDataObjι (U V : X.affineOpens) :
I.ideal V ≤ RingHom.ker (U.1.ι.app V.1 ≫ (I.glueDataObjι U).app _).hom | X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑Γ(X, ↑V)
hx : x ∈ I.ideal V
p : ↑↑(I.glueDataObj U).toPresheafedSpace
hp : p ∈ Opposite.unop (Opposite.op (I.glueDataObjι U ⁻¹ᵁ (↑U).ι ⁻¹ᵁ ↑V))
f : ↑Γ(X, ↑U)
g : ↑Γ(X, ↑V)
hfg : X.basicOpen f = X.basicOpen g
hf : ↑((ConcreteCategory.hom (I.glueDataObjι U).base) ... | rw [opensRange_glueDataObjMap] | no goals | 9da7a9bdc95919a0 |
MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure | Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean | theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure [NormedAddCommGroup E]
[NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [Nontrivial E] [IsAddHaarMeasure μ]
{L : AddSubgroup E} [Countable L] [DiscreteTopology L] (fund : IsAddFundamentalDomain L F μ)
(h_symm : ∀ x ∈ s, -x ∈ s) (h_... | E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomai... | rintro _ ⟨y, hy, rfl⟩ | case intro.intro
E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAd... | c80ef6cf21c9e752 |
Set.IsWF.min_union | Mathlib/Order/WellFoundedSet.lean | theorem IsWF.min_union (hs : s.IsWF) (hsn : s.Nonempty) (ht : t.IsWF) (htn : t.Nonempty) :
(hs.union ht).min (union_nonempty.2 (Or.intro_left _ hsn)) =
Min.min (hs.min hsn) (ht.min htn) | α : Type u_2
inst✝ : LinearOrder α
s t : Set α
hs : s.IsWF
hsn : s.Nonempty
ht : t.IsWF
htn : t.Nonempty
⊢ hs.min hsn ≤ ⋯.min ⋯ ∨ ht.min htn ≤ ⋯.min ⋯ | exact ((mem_union _ _ _).1 ((hs.union ht).min_mem (union_nonempty.2 (.inl hsn)))).imp
(hs.min_le _) (ht.min_le _) | no goals | 40ab8b3df5835adf |
Multiset.gcd_map_mul | Mathlib/Algebra/GCDMonoid/Multiset.lean | theorem gcd_map_mul (a : α) (s : Multiset α) : (s.map (a * ·)).gcd = normalize a * s.gcd | case refine_2
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
a : α
s✝ : Multiset α
b : α
s : Multiset α
ih : (map (fun x => a * x) s).gcd = normalize a * s.gcd
⊢ GCDMonoid.gcd (a * b) (map (fun x => a * x) s).gcd = GCDMonoid.gcd (a * b) (a * s.gcd) | rw [ih] | case refine_2
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
a : α
s✝ : Multiset α
b : α
s : Multiset α
ih : (map (fun x => a * x) s).gcd = normalize a * s.gcd
⊢ GCDMonoid.gcd (a * b) (normalize a * s.gcd) = GCDMonoid.gcd (a * b) (a * s.gcd) | d2ad33f97004f5bc |
stronglyMeasurable_const_smul_iff | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | theorem _root_.stronglyMeasurable_const_smul_iff {m : MeasurableSpace α} (c : G) :
(StronglyMeasurable fun x => c • f x) ↔ StronglyMeasurable f :=
⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
| α : Type u_1
β : Type u_2
f : α → β
G : Type u_6
inst✝³ : TopologicalSpace β
inst✝² : Group G
inst✝¹ : MulAction G β
inst✝ : ContinuousConstSMul G β
m : MeasurableSpace α
c : G
h : StronglyMeasurable fun x => c • f x
⊢ StronglyMeasurable f | simpa only [inv_smul_smul] using h.const_smul' c⁻¹ | no goals | 0e68e36ad43d1db1 |
CategoryTheory.Limits.braid_natural | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | theorem braid_natural [HasBinaryProducts C] {W X Y Z : C} (f : X ⟶ Y) (g : Z ⟶ W) :
prod.map f g ≫ (prod.braiding _ _).hom = (prod.braiding _ _).hom ≫ prod.map g f | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasBinaryProducts C
W X Y Z : C
f : X ⟶ Y
g : Z ⟶ W
⊢ prod.map f g ≫ (prod.braiding Y W).hom = (prod.braiding X Z).hom ≫ prod.map g f | simp | no goals | fe194938c0fbe68c |
AkraBazziRecurrence.isBigO_apply_r_sub_b | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma isBigO_apply_r_sub_b (q : ℝ → ℝ) (hq_diff : DifferentiableOn ℝ q (Set.Ioi 1))
(hq_poly : GrowsPolynomially fun x => ‖deriv q x‖) (i : α) :
(fun n => q (r i n) - q (b i * n)) =O[atTop] fun n => (deriv q n) * (r i n - b i * n) | case diff
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : b' ... | case diff =>
refine hq_diff.differentiableAt (Ioi_mem_nhds ?_)
calc 1 < b' * n := by exact h_bn
_ ≤ z := hz.1 | case mem_Icc
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
q : ℝ → ℝ
hq_diff : DifferentiableOn ℝ q (Set.Ioi 1)
hq_poly✝ : GrowsPolynomially fun x => ‖deriv q x‖
i : α
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
hb_lt_one : b' < 1
hb : ... | b25323efa1ebb7f3 |
BoundedContinuousFunction.exists_extension_forall_exists_le_ge_of_isClosedEmbedding | Mathlib/Topology/TietzeExtension.lean | theorem exists_extension_forall_exists_le_ge_of_isClosedEmbedding [Nonempty X] (f : X →ᵇ ℝ)
{e : X → Y} (he : IsClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f | X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : NormalSpace Y
inst✝ : Nonempty X
f : X →ᵇ ℝ
e : X → Y
he : IsClosedEmbedding e
inhabited_h : Inhabited X
a : ℝ
ha : IsGLB (range ⇑f) a
hb : IsLUB (range ⇑f) a
hmem : ∀ (x : X), f x ∈ Icc a a
hle : a ≤ a
⊢ ∀ (x : X), f x = a | simpa using hmem | no goals | c0f84dcf09f7662d |
TopCat.Presheaf.app_surjective_of_injective_of_locally_surjective | Mathlib/Topology/Sheaves/Stalks.lean | theorem app_surjective_of_injective_of_locally_surjective {F G : Sheaf C X} (f : F ⟶ G)
(U : Opens X) (hinj : ∀ x ∈ U, Function.Injective ((stalkFunctor C x).map f.1))
(hsurj : ∀ (t x) (_ : x ∈ U), ∃ (V : Opens X) (_ : x ∈ V) (iVU : V ⟶ U)
(s : ToType (F.1.obj (op V))), f.1.app (op V) s = G.1.map iVU.op t) ... | C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ : (forget C).Refle... | obtain ⟨s, s_spec, -⟩ := F.existsUnique_gluing' V U iVU V_cover sf this | case intro.intro
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasColimits C
X : TopCat
FC : C → C → Type u_1
CC : C → Type v
inst✝⁴ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
instCC : ConcreteCategory C FC
inst✝³ : PreservesFilteredColimits (forget C)
inst✝² : HasLimits C
inst✝¹ : PreservesLimits (forget C)
inst✝ :... | 8069452fab42619a |
PosNum.divMod_to_nat_aux | Mathlib/Data/Num/Lemmas.lean | theorem divMod_to_nat_aux {n d : PosNum} {q r : Num} (h₁ : (r : ℕ) + d * ((q : ℕ) + q) = n)
(h₂ : (r : ℕ) < 2 * d) :
((divModAux d q r).2 + d * (divModAux d q r).1 : ℕ) = ↑n ∧ ((divModAux d q r).2 : ℕ) < d | case some
n d : PosNum
q r : Num
h₁ : ↑r + ↑d * (↑q + ↑q) = ↑n
h₂ : ↑r < 2 * ↑d
this✝ : ∀ {r₂ : Num}, Num.ofZNum' (r.sub' (Num.pos d)) = some r₂ ↔ ↑r = ↑r₂ + ↑d
r₂ : Num
e : Num.ofZNum' (r.sub' (Num.pos d)) = some r₂
this : ↑r = ↑r₂ + ↑d
⊢ ↑r₂ + ↑d * (2 * ↑q + 1) = ↑n ∧ ↑r₂ < ↑d | constructor | case some.left
n d : PosNum
q r : Num
h₁ : ↑r + ↑d * (↑q + ↑q) = ↑n
h₂ : ↑r < 2 * ↑d
this✝ : ∀ {r₂ : Num}, Num.ofZNum' (r.sub' (Num.pos d)) = some r₂ ↔ ↑r = ↑r₂ + ↑d
r₂ : Num
e : Num.ofZNum' (r.sub' (Num.pos d)) = some r₂
this : ↑r = ↑r₂ + ↑d
⊢ ↑r₂ + ↑d * (2 * ↑q + 1) = ↑n
case some.right
n d : PosNum
q r : Num
h₁ : ↑... | 37662baae67c37c0 |
UniformOnFun.uniformSpace_eq_inf_precomp_of_cover | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | theorem uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} (φ₁ : δ₁ → α) (φ₂ : δ₂ → α)
(𝔗₁ : Set (Set δ₁)) (𝔗₂ : Set (Set δ₂))
(h_image₁ : MapsTo (φ₁ '' ·) 𝔗₁ 𝔖) (h_image₂ : MapsTo (φ₂ '' ·) 𝔗₂ 𝔖)
(h_preimage₁ : MapsTo (φ₁ ⁻¹' ·) 𝔖 𝔗₁) (h_preimage₂ : MapsTo (φ₂ ⁻¹' ·) 𝔖 𝔗₂)
(h_cover : ∀ S ∈ ... | case refine_1
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
δ₁ : Type u_5
δ₂ : Type u_6
φ₁ : δ₁ → α
φ₂ : δ₂ → α
𝔗₁ : Set (Set δ₁)
𝔗₂ : Set (Set δ₂)
h_image₁ : MapsTo (fun x => φ₁ '' x) 𝔗₁ 𝔖
h_image₂ : MapsTo (fun x => φ₂ '' x) 𝔗₂ 𝔖
h_preimage₁ : MapsTo (fun x => φ₁ ⁻¹' x) 𝔖 𝔗₁
h_preimage₂ : ... | exact UniformOnFun.precomp_uniformContinuous h_image₁ | no goals | 687e94ee2bcbcd2a |
Nat.pow_eq_zero | Mathlib/Data/Nat/Init.lean | @[simp, nolint simpNF] protected lemma pow_eq_zero {a : ℕ} : ∀ {n : ℕ}, a ^ n = 0 ↔ a = 0 ∧ n ≠ 0
| 0 => by simp
| n + 1 => by rw [Nat.pow_succ, mul_eq_zero, Nat.pow_eq_zero]; omega
| a n : ℕ
⊢ a ^ (n + 1) = 0 ↔ a = 0 ∧ n + 1 ≠ 0 | rw [Nat.pow_succ, mul_eq_zero, Nat.pow_eq_zero] | a n : ℕ
⊢ a = 0 ∧ n ≠ 0 ∨ a = 0 ↔ a = 0 ∧ n + 1 ≠ 0 | b909cb114af1c98e |
Dynamics.coverEntropy_image_of_comap | Mathlib/Dynamics/TopologicalEntropy/Semiconj.lean | theorem coverEntropy_image_of_comap (u : UniformSpace Y) {S : X → X} {T : Y → Y} {φ : X → Y}
(h : Semiconj φ S T) (F : Set X) :
coverEntropy T (φ '' F) = @coverEntropy X (comap φ u) S F | case a
X : Type u_1
Y : Type u_2
u : UniformSpace Y
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
⊢ coverEntropy S F ≤ coverEntropy T (φ '' F) | refine iSup₂_le fun U U_uni ↦ ?_ | case a
X : Type u_1
Y : Type u_2
u : UniformSpace Y
S : X → X
T : Y → Y
φ : X → Y
h : Semiconj φ S T
F : Set X
U : Set (X × X)
U_uni : U ∈ 𝓤 X
⊢ coverEntropyEntourage S F U ≤ coverEntropy T (φ '' F) | 78de7c310336c26f |
AkraBazziRecurrence.exists_eventually_const_mul_le_r | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma exists_eventually_const_mul_le_r :
∃ c ∈ Set.Ioo (0 : ℝ) 1, ∀ᶠ (n : ℕ) in atTop, ∀ i, c * n ≤ r i n | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
gt_zero : 0 < b (min_bi b)
⊢ 0 < b (min_bi b) / 2 | positivity | no goals | bc15a40b01d9e13e |
ZMod.pow_totient | Mathlib/FieldTheory/Finite/Basic.lean | theorem ZMod.pow_totient {n : ℕ} (x : (ZMod n)ˣ) : x ^ φ n = 1 | case zero
x : (ZMod 0)ˣ
⊢ x ^ φ 0 = 1 | rw [Nat.totient_zero, pow_zero] | no goals | 9bb1b1d29801a0f8 |
List.findIdx?_subtype | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem findIdx?_subtype {p : α → Prop} {l : List { x // p x }}
{f : { x // p x } → Bool} {g : α → Bool} (hf : ∀ x h, f ⟨x, h⟩ = g x) :
l.findIdx? f = l.unattach.findIdx? g | α : Type u_1
p : α → Prop
l : List { x // p x }
f : { x // p x } → Bool
g : α → Bool
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
⊢ findIdx? f l = findIdx? g l.unattach | unfold unattach | α : Type u_1
p : α → Prop
l : List { x // p x }
f : { x // p x } → Bool
g : α → Bool
hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x
⊢ findIdx? f l = findIdx? g (map (fun x => x.val) l) | 8dd711f0d5a822d1 |
Nat.Subtype.ofNat_surjective | Mathlib/Logic/Denumerable.lean | theorem ofNat_surjective : Surjective (ofNat s)
| ⟨x, hx⟩ => by
set t : List s :=
((List.range x).filter fun y => y ∈ s).pmap
(fun (y : ℕ) (hy : y ∈ s) => ⟨y, hy⟩)
(by intros a ha; simpa using (List.mem_filter.mp ha).2) with ht
have hmt : ∀ {y : s}, y ∈ t ↔ y < ⟨x, hx⟩ | s : Set ℕ
inst✝¹ : Infinite ↑s
inst✝ : DecidablePred fun x => x ∈ s
x : ℕ
hx : x ∈ s
a : ℕ
ha : a ∈ List.filter (fun y => decide (y ∈ s)) (List.range x)
⊢ a ∈ s | simpa using (List.mem_filter.mp ha).2 | no goals | 83d4af376ec3e51f |
Polynomial.degree_derivative_eq | Mathlib/Algebra/Polynomial/Derivative.lean | theorem degree_derivative_eq [NoZeroSMulDivisors ℕ R] (p : R[X]) (hp : 0 < natDegree p) :
degree (derivative p) = (natDegree p - 1 : ℕ) | R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroSMulDivisors ℕ R
p : R[X]
hp : 0 < p.natDegree
n : ℕ
hn : n ∈ p.support
⊢ n ≤ p.natDegree | apply le_natDegree_of_mem_supp _ hn | no goals | 3fd314c621a50c27 |
SimpleGraph.ComponentCompl.exists_adj_boundary_pair | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | theorem exists_adj_boundary_pair (Gc : G.Preconnected) (hK : K.Nonempty) :
∀ C : G.ComponentCompl K, ∃ ck : V × V, ck.1 ∈ C ∧ ck.2 ∈ K ∧ G.Adj ck.1 ck.2 | V : Type u
G : SimpleGraph V
K : Set V
Gc : G.Preconnected
hK : K.Nonempty
v : V
vnK : v ∉ K
C : G.ComponentCompl K := G.componentComplMk vnK
dis : K ∩ ↑C ⊆ ∅ := Set.disjoint_iff.mp (ComponentCompl.disjoint_right C)
h : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.2
⊢ False | suffices Set.univ = (C : Set V) by exact dis ⟨hK.choose_spec, this ▸ Set.mem_univ hK.some⟩ | V : Type u
G : SimpleGraph V
K : Set V
Gc : G.Preconnected
hK : K.Nonempty
v : V
vnK : v ∉ K
C : G.ComponentCompl K := G.componentComplMk vnK
dis : K ∩ ↑C ⊆ ∅ := Set.disjoint_iff.mp (ComponentCompl.disjoint_right C)
h : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.2
⊢ Set.univ = ↑C | 391e6e06597f6fa6 |
CauSeq.limZero_sub_rev | Mathlib/Algebra/Order/CauSeq/Basic.lean | theorem limZero_sub_rev {f g : CauSeq β abv} (hfg : LimZero (f - g)) : LimZero (g - f) | α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hfg : (f - g).LimZero
⊢ (g - f).LimZero | simpa using neg_limZero hfg | no goals | c479b6d55cafa2ae |
AlgebraicGeometry.Scheme.PartialMap.equiv_of_fromSpecStalkOfMem_eq | Mathlib/AlgebraicGeometry/RationalMap.lean | lemma equiv_of_fromSpecStalkOfMem_eq [IrreducibleSpace X]
{x : X} [X.IsGermInjectiveAt x] (f g : X.PartialMap Y)
(hxf : x ∈ f.domain) (hxg : x ∈ g.domain)
(H : f.fromSpecStalkOfMem hxf = g.fromSpecStalkOfMem hxg) : f.equiv g | case refine_1
X Y : Scheme
inst✝¹ : IrreducibleSpace ↑↑X.toPresheafedSpace
x : ↑↑X.toPresheafedSpace
inst✝ : X.IsGermInjectiveAt x
f g : X.PartialMap Y
hxf : x ∈ f.domain
hxg : x ∈ g.domain
H : f.fromSpecStalkOfMem hxf = g.fromSpecStalkOfMem hxg
hdense : Dense (↑f.domain ⊓ ↑g.domain)
this : (↑(f.domain ⊓ g.domain)).IsG... | rw [← f.fromSpecStalkOfMem_restrict hdense inf_le_left ⟨hxf, hxg⟩,
← g.fromSpecStalkOfMem_restrict hdense inf_le_right ⟨hxf, hxg⟩] at H | case refine_1
X Y : Scheme
inst✝¹ : IrreducibleSpace ↑↑X.toPresheafedSpace
x : ↑↑X.toPresheafedSpace
inst✝ : X.IsGermInjectiveAt x
f g : X.PartialMap Y
hxf : x ∈ f.domain
hxg : x ∈ g.domain
hdense : Dense (↑f.domain ⊓ ↑g.domain)
H :
(f.restrict (f.domain ⊓ g.domain) hdense ⋯).fromSpecStalkOfMem ⋯ =
(g.restrict (f... | 72b5375ab01aa1fb |
CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_iff | Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean | lemma LeftHomologyMapData.quasiIso_iff {φ : S₁ ⟶ S₂} {h₁ : S₁.LeftHomologyData}
{h₂ : S₂.LeftHomologyData} (γ : LeftHomologyMapData φ h₁ h₂) :
QuasiIso φ ↔ IsIso γ.φH | C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ QuasiIso φ ↔ IsIso γ.φH | rw [ShortComplex.quasiIso_iff, γ.homologyMap_eq] | C : Type u_2
inst✝³ : Category.{u_1, u_2} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) ↔ IsIso γ.φH | 659f2b93e716db47 |
Complex.cos_sq_add_sin_sq | Mathlib/Data/Complex/Trigonometric.lean | theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 | x : ℂ
⊢ cos x ^ 2 + sin x ^ 2 = 1 | rw [add_comm, sin_sq_add_cos_sq] | no goals | 4c68ad511a8b3d4d |
Submodule.mem_one | Mathlib/Algebra/Algebra/Operations.lean | theorem mem_one {x : A} : x ∈ (1 : Submodule R A) ↔ ∃ y, algebraMap R A y = x | R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Algebra R A
x : A
⊢ x ∈ 1 ↔ ∃ y, (algebraMap R A) y = x | simp [one_eq_range] | no goals | 5b7b21512e8300a3 |
NumberField.house.ξ_mulVec_eq_0 | Mathlib/NumberTheory/NumberField/House.lean | theorem ξ_mulVec_eq_0 : a *ᵥ ξ K x = 0 | case e_f.h.e_f
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
inst✝ : Fintype β
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r... | ext1 r | case e_f.h.e_f.h
K : Type u_1
inst✝² : Field K
inst✝¹ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
x : β × (K →+* ℂ) → ℤ
hxl : x ≠ 0
inst✝ : Fintype β
hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0
k : α
lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l,... | dbdfd50ef1433d12 |
Stream'.zip_inits_tails | Mathlib/Data/Stream/Init.lean | theorem zip_inits_tails (s : Stream' α) : zip appendStream' (inits s) (tails s) = const s | case a
α : Type u
s : Stream' α
n : ℕ
⊢ (zip appendStream' s.inits s.tails).get n = (const s).get n | rw [get_zip, get_inits, get_tails, get_const, take_succ, cons_append_stream, append_take_drop,
Stream'.eta] | no goals | 112f93d37b2ad7d1 |
MonoidHom.injective_noncommPiCoprod_of_iSupIndep | Mathlib/GroupTheory/NoncommPiCoprod.lean | theorem injective_noncommPiCoprod_of_iSupIndep [Fintype ι]
{hcomm : Pairwise fun i j : ι => ∀ (x : H i) (y : H j), Commute (ϕ i x) (ϕ j y)}
(hind : iSupIndep fun i => (ϕ i).range)
(hinj : ∀ i, Function.Injective (ϕ i)) : Function.Injective (noncommPiCoprod ϕ hcomm) | case h.a
G : Type u_1
inst✝² : Group G
ι : Type u_2
H : ι → Type u_3
inst✝¹ : (i : ι) → Group (H i)
ϕ : (i : ι) → H i →* G
inst✝ : Fintype ι
hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)
hind : iSupIndep fun i => (ϕ i).range
hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)
f : (i : ι) → H i
... | simp [this i (Finset.mem_univ i)] | no goals | 4f895bd19c338ae6 |
exists_dist_le_le | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem exists_dist_le_le (hδ : 0 ≤ δ) (hε : 0 ≤ ε) (h : dist x z ≤ ε + δ) :
∃ y, dist x y ≤ δ ∧ dist y z ≤ ε | case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ ε
h : dist x z ≤ ε + δ
hε' : 0 < ε
hεδ : 0 < ε + δ
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x y ≤ δ ∧ dist y z ≤ ε | rw [hy.1, hy.2, div_mul_comm, div_mul_comm ε] | case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 ≤ δ
hε : 0 ≤ ε
h : dist x z ≤ ε + δ
hε' : 0 < ε
hεδ : 0 < ε + δ
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x z / (ε + δ) * δ ≤ δ ∧ dist x z / (ε + δ) * ε ≤ ε | 59238fa715097bae |
Nat.Ico_filter_coprime_le | Mathlib/Data/Nat/Totient.lean | theorem Ico_filter_coprime_le {a : ℕ} (k n : ℕ) (a_pos : 0 < a) :
#{x ∈ Ico k (k + n) | a.Coprime x} ≤ totient a * (n / a + 1) | case zero.a.h.h.bc
a k n : ℕ
a_pos : 0 < a
⊢ n % a ≤ a | exact le_of_lt (mod_lt n a_pos) | no goals | 392fd5c3c0759474 |
dist_prod_prod_le_of_le | Mathlib/Analysis/Normed/Group/Basic.lean | theorem dist_prod_prod_le_of_le (s : Finset ι) {f a : ι → E} {d : ι → ℝ}
(h : ∀ b ∈ s, dist (f b) (a b) ≤ d b) :
dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, d b | ι : Type u_3
E : Type u_5
inst✝ : SeminormedCommGroup E
s : Finset ι
f a : ι → E
d : ι → ℝ
h : ∀ b ∈ s, dist (f b) (a b) ≤ d b
⊢ dist (∏ b ∈ s, f b) (∏ b ∈ s, a b) ≤ ∑ b ∈ s, d b | simp only [dist_eq_norm_div, ← Finset.prod_div_distrib] at * | ι : Type u_3
E : Type u_5
inst✝ : SeminormedCommGroup E
s : Finset ι
f a : ι → E
d : ι → ℝ
h : ∀ b ∈ s, ‖f b / a b‖ ≤ d b
⊢ ‖∏ x ∈ s, f x / a x‖ ≤ ∑ b ∈ s, d b | be10c7f3d5cfcefc |
exists_locallyFinite_subset_iUnion_ball_radius_lt | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | theorem exists_locallyFinite_subset_iUnion_ball_radius_lt (hs : IsClosed s) {R : α → ℝ}
(hR : ∀ x ∈ s, 0 < R x) :
∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ),
(∀ i, c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
(LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) | α : Type u
inst✝¹ : MetricSpace α
inst✝ : ProperSpace α
s : Set α
hs : IsClosed s
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
⊢ ∃ ι c r r',
(∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
(LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) | have : ∀ x ∈ s, (𝓝 x).HasBasis (fun r : ℝ => 0 < r ∧ r < R x) fun r => ball x r := fun x hx =>
nhds_basis_uniformity (uniformity_basis_dist_lt (hR x hx)) | α : Type u
inst✝¹ : MetricSpace α
inst✝ : ProperSpace α
s : Set α
hs : IsClosed s
R : α → ℝ
hR : ∀ x ∈ s, 0 < R x
this : ∀ x ∈ s, (𝓝 x).HasBasis (fun r => 0 < r ∧ r < R x) fun r => ball x r
⊢ ∃ ι c r r',
(∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
(LocallyFinite fun i => ball (c i) (r' i))... | 15507c30e9a4d0ca |
Matrix.det_kroneckerTMul | Mathlib/Data/Matrix/Kronecker.lean | theorem det_kroneckerTMul [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n]
(A : Matrix m m α) (B : Matrix n n β) :
det (A ⊗ₖₜ[R] B) = (det A ^ Fintype.card n) ⊗ₜ[R] (det B ^ Fintype.card m) | R : Type u_1
α : Type u_2
β : Type u_4
m : Type u_9
n : Type u_10
inst✝⁸ : CommRing R
inst✝⁷ : CommRing α
inst✝⁶ : CommRing β
inst✝⁵ : Algebra R α
inst✝⁴ : Algebra R β
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
A : Matrix m m α
B : Matrix n n β
⊢ (A.map fun a => includeLeft a).de... | simp only [← AlgHom.mapMatrix_apply, ← AlgHom.map_det] | R : Type u_1
α : Type u_2
β : Type u_4
m : Type u_9
n : Type u_10
inst✝⁸ : CommRing R
inst✝⁷ : CommRing α
inst✝⁶ : CommRing β
inst✝⁵ : Algebra R α
inst✝⁴ : Algebra R β
inst✝³ : Fintype m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
A : Matrix m m α
B : Matrix n n β
⊢ includeLeft A.det ^ Fintype.card ... | d4ecfa797464d2f8 |
IsOpenMap.continuousOn_range_of_leftInverse | Mathlib/Topology/ContinuousOn.lean | theorem IsOpenMap.continuousOn_range_of_leftInverse {f : α → β} (hf : IsOpenMap f) {finv : β → α}
(hleft : Function.LeftInverse finv f) : ContinuousOn finv (range f) | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
hf : IsOpenMap f
finv : β → α
hleft : LeftInverse finv f
⊢ ContinuousOn finv (f '' univ) | exact (hf.restrict isOpen_univ).continuousOn_image_of_leftInvOn fun x _ => hleft x | no goals | c74e8c3ce6cccd25 |
Field.nonempty_algEquiv_of_range_minpoly_eq | Mathlib/FieldTheory/Isaacs.lean | theorem nonempty_algEquiv_of_range_minpoly_eq
(h : Set.range (@minpoly F E _ _ _) = Set.range (@minpoly F K _ _ _)) :
Nonempty (E ≃ₐ[F] K) :=
have ⟨σ⟩ := nonempty_algHom_of_range_minpoly_subset h.le
have : Algebra.IsAlgebraic F K := ⟨fun y ↦ IsIntegral.isAlgebraic <| by
by_contra hy
have ⟨x, hx⟩ := ... | F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
alg : Algebra.IsAlgebraic F E
h : Set.range (minpoly F) = Set.range (minpoly F)
σ : E →ₐ[F] K
y : K
⊢ IsIntegral F y | by_contra hy | F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
alg : Algebra.IsAlgebraic F E
h : Set.range (minpoly F) = Set.range (minpoly F)
σ : E →ₐ[F] K
y : K
hy : ¬IsIntegral F y
⊢ False | a978256364842fdb |
InnerProductGeometry.norm_add_eq_norm_sub_iff_angle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | theorem norm_add_eq_norm_sub_iff_angle_eq_pi_div_two (x y : V) :
‖x + y‖ = ‖x - y‖ ↔ angle x y = π / 2 | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
⊢ ‖x‖ ^ 2 + 2 * inner x y + ‖y‖ ^ 2 = ‖x‖ ^ 2 - 2 * inner x y + ‖y‖ ^ 2 ↔ inner x y = 0 | constructor <;> intro h <;> linarith | no goals | 2b6e97cab0fd5a34 |
ProbabilityTheory.MemLp.uniformIntegrable_of_identDistrib_aux | Mathlib/Probability/IdentDistrib.lean | theorem MemLp.uniformIntegrable_of_identDistrib_aux {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : MemLp (f j) p μ) (hfmeas : ∀ i, StronglyMeasurable (f i))
(hf : ∀ i, IdentDistrib (f i) (f j) μ μ) : UniformIntegrable f p μ | α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i : ι), ... | ext x | case h
α : Type u_1
inst✝⁴ : MeasurableSpace α
E : Type u_5
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace E
μ : Measure α
inst✝ : IsFiniteMeasure μ
ι : Type u_6
f : ι → α → E
j : ι
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
hℒp : MemLp (f j) p μ
hfmeas : ∀ (i : ι), StronglyMeasurable (f i)
hf : ∀ (i... | e0093e34b057b36a |
cfcₙAux_mem_range_inr | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean | lemma cfcₙAux_mem_range_inr (f : C(σₙ 𝕜 a, 𝕜)₀) :
cfcₙAux hp₁ a ha f ∈ NonUnitalStarAlgHom.range (Unitization.inrNonUnitalStarAlgHom 𝕜 A) | case h.e'_3
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NonUnitalNormedRing A
inst✝⁶ : StarRing A
inst✝⁵ : NormedSpace 𝕜 A
inst✝⁴ : IsScalarTower 𝕜 A A
inst✝³ : SMulCommClass 𝕜 A A
inst✝² : StarModule 𝕜 A
p : A → Prop
p₁ : Unitization 𝕜 A → Prop
hp₁ : ∀ {x : A}, p₁ ↑x ↔ p x
a : A
ha : p a
inst✝¹ : Conti... | aesop | no goals | 694a4ae2d7053273 |
Polynomial.exists_partition_polynomial_aux | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | theorem exists_partition_polynomial_aux (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : Fq[X]} (hb : b ≠ 0)
(A : Fin n → Fq[X]) : ∃ t : Fin n → Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊),
∀ i₀ i₁ : Fin n, t i₀ = t i₁ ↔
(cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε | Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
ε : ℝ
hε : 0 < ε
b : Fq[X]
hb : b ≠ 0
hbε : 0 < cardPowDegree b • ε
n : ℕ
ih :
∀ (A : Fin n → Fq[X]),
∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε
A : Fin (n + 1) → Fq[X]
anti_archim' :
∀ {i j k : Fin (n + 1... | rw [AbsoluteValue.map_sub] | Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
ε : ℝ
hε : 0 < ε
b : Fq[X]
hb : b ≠ 0
hbε : 0 < cardPowDegree b • ε
n : ℕ
ih :
∀ (A : Fin n → Fq[X]),
∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε
A : Fin (n + 1) → Fq[X]
anti_archim' :
∀ {i j k : Fin (n + 1... | d529c9a5494d9218 |
Computation.lift_eq_iff_equiv | Mathlib/Data/Seq/Computation.lean | theorem lift_eq_iff_equiv (c₁ c₂ : Computation α) : LiftRel (· = ·) c₁ c₂ ↔ c₁ ~ c₂ :=
⟨fun ⟨h1, h2⟩ a =>
⟨fun a1 => by let ⟨b, b2, ab⟩ := h1 a1; rwa [ab],
fun a2 => by let ⟨b, b1, ab⟩ := h2 a2; rwa [← ab]⟩,
fun e => ⟨fun {a} a1 => ⟨a, (e _).1 a1, rfl⟩, fun {a} a2 => ⟨a, (e _).2 a2, rfl⟩⟩⟩
| α : Type u
c₁ c₂ : Computation α
x✝ : LiftRel (fun x1 x2 => x1 = x2) c₁ c₂
a : α
h1 : ∀ {a : α}, a ∈ c₁ → ∃ b, b ∈ c₂ ∧ (fun x1 x2 => x1 = x2) a b
h2 : ∀ {b : α}, b ∈ c₂ → ∃ a, a ∈ c₁ ∧ (fun x1 x2 => x1 = x2) a b
a1 : a ∈ c₁
b : α
b2 : b ∈ c₂
ab : (fun x1 x2 => x1 = x2) a b
⊢ a ∈ c₂ | rwa [ab] | no goals | 04f27608cd88ce15 |
MeasureTheory.isPiSystem_squareCylinders | Mathlib/MeasureTheory/Constructions/Cylinders.lean | theorem isPiSystem_squareCylinders {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsPiSystem (C i))
(hC_univ : ∀ i, univ ∈ C i) :
IsPiSystem (squareCylinders C) | case neg
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set (α i)
h₁ : t₁ ∈ univ.pi C
s₂ : Finset ι
t₂ : (i : ι) → Set (α i)
h₂ : t₂ ∈ univ.pi C
t₁' : (i : ι) → Set (α i) := s₁.piecewise t₁ fun i => univ
t₂' : (i ... | rw [h2' i hi₂] | case neg
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsPiSystem (C i)
hC_univ : ∀ (i : ι), univ ∈ C i
s₁ : Finset ι
t₁ : (i : ι) → Set (α i)
h₁ : t₁ ∈ univ.pi C
s₂ : Finset ι
t₂ : (i : ι) → Set (α i)
h₂ : t₂ ∈ univ.pi C
t₁' : (i : ι) → Set (α i) := s₁.piecewise t₁ fun i => univ
t₂' : (i ... | 93f3d5a8b0403be2 |
Set.definable_iff_finitely_definable | Mathlib/ModelTheory/Definability.lean | theorem definable_iff_finitely_definable :
A.Definable L s ↔ ∃ (A0 : Finset M), (A0 : Set M) ⊆ A ∧
(A0 : Set M).Definable L s | case mpr
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
s : Set (α → M)
⊢ (∃ A0, ↑A0 ⊆ A ∧ (↑A0).Definable L s) → A.Definable L s | rintro ⟨A0, hA0, hd⟩ | case mpr.intro.intro
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
s : Set (α → M)
A0 : Finset M
hA0 : ↑A0 ⊆ A
hd : (↑A0).Definable L s
⊢ A.Definable L s | 86029d1ceb82f9e6 |
AlgebraicGeometry.Scheme.Pullback.t'_snd_fst_fst | Mathlib/AlgebraicGeometry/Pullbacks.lean | theorem t'_snd_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.snd _ _ ≫ pullback.fst _ _ ≫ pullback.fst _ _ =
pullback.fst _ _ ≫ pullback.snd _ _ | X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
i j k : 𝒰.J
⊢ t' 𝒰 f g i j k ≫
pullback.snd (fV 𝒰 f g j k) (fV 𝒰 f g j i) ≫
pullback.fst (pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) ≫ pullback.fst (𝒰.map j ≫ f) g =
pullback.fst (fV �... | simp only [t', Category.assoc, pullbackSymmetry_hom_comp_snd_assoc,
pullbackRightPullbackFstIso_inv_fst_assoc, pullback.lift_fst_assoc, t_fst_fst,
pullbackRightPullbackFstIso_hom_fst_assoc] | no goals | eb11413885f2269d |
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