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 |
|---|---|---|---|---|---|---|
Bornology.IsVonNBounded.image_multilinear' | Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean | theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBounded 𝕜 s)
(f : ContinuousMultilinearMap 𝕜 E F) : IsVonNBounded 𝕜 (f '' s) := fun V hV ↦ by
classical
if h₁ : ∀ c : 𝕜, ‖c‖ ≤ 1 then
exact absorbs_iff_norm.2 ⟨2, fun c hc ↦ by linarith [h₁ c]⟩
else
let _ : NontriviallyNormed... | ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : ∀ (... | have := (hs i).tendsto_smallSets_nhds.eventually (mem_lift' (ht₀ i)) | ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : ∀ (... | b4505873b047f890 |
PartitionOfUnity.coe_finsupport | Mathlib/Topology/PartitionOfUnity.lean | theorem coe_finsupport (x₀ : X) :
(ρ.finsupport x₀ : Set ι) = support fun i ↦ ρ i x₀ | ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
⊢ ↑(ρ.finsupport x₀) = support fun i => (ρ i) x₀ | ext | case h
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
x✝ : ι
⊢ x✝ ∈ ↑(ρ.finsupport x₀) ↔ x✝ ∈ support fun i => (ρ i) x₀ | 81c20c813d4e57e2 |
CategoryTheory.Subpresheaf.equalizer.condition | Mathlib/CategoryTheory/Subpresheaf/Equalizer.lean | @[reassoc]
lemma equalizer.condition : equalizer.ι f g ≫ f = equalizer.ι f g ≫ g | C : Type u
inst✝ : Category.{v, u} C
F₁ F₂ : Cᵒᵖ ⥤ Type w
A : Subpresheaf F₁
f g : A.toPresheaf ⟶ F₂
⊢ ι f g ≫ f = ι f g ≫ g | simp [← range_le_equalizer_iff] | no goals | dfe45ce9a8693387 |
Nat.psp_from_prime_gt_p | Mathlib/NumberTheory/FermatPsp.lean | theorem psp_from_prime_gt_p {b : ℕ} (b_ge_two : 2 ≤ b) {p : ℕ} (p_prime : p.Prime)
(p_gt_two : 2 < p) : p < psp_from_prime b p | b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
AB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
⊢ p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2 | suffices h : p < (b ^ 2) ^ (p - 1) by gcongr | b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
AB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
⊢ p < (b ^ 2) ^ (p - 1) | 260daf6c75f986dc |
MeasureTheory.lintegral_eq_zero_of_isMulLeftInvariant | Mathlib/MeasureTheory/Group/LIntegral.lean | theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
(hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 | G : Type u_1
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : BorelSpace G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : μ.Regular
inst✝ : NeZero μ
f : G → ℝ≥0∞
hf : Continuous f
⊢ ∫⁻ (x : G), f x ∂μ = 0 ↔ f = 0 | rw [lintegral_eq_zero_iff hf.measurable, hf.ae_eq_iff_eq μ continuous_zero] | no goals | d692fa342adf93d5 |
exists_dist_lt_lt | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem exists_dist_lt_lt (hδ : 0 < δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
∃ y, dist x y < δ ∧ dist y z < ε | 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) < 1
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 | exact ⟨mul_lt_of_lt_one_left hδ h, mul_lt_of_lt_one_left hε h⟩ | no goals | 3c2b87a2394cdde0 |
CategoryTheory.Arrow.finite_iff | Mathlib/CategoryTheory/Comma/CardinalArrow.lean | lemma Arrow.finite_iff (C : Type u) [SmallCategory C] :
Finite (Arrow C) ↔ Nonempty (FinCategory C) | case mp.refine_2
C : Type u
inst✝ : SmallCategory C
a✝ : Finite (Arrow C)
a b : C
⊢ Fintype (a ⟶ b) | have := Finite.of_injective (fun (f : a ⟶ b) ↦ Arrow.mk f)
(fun f g h ↦ by
change (Arrow.mk f).hom = (Arrow.mk g).hom
congr) | case mp.refine_2
C : Type u
inst✝ : SmallCategory C
a✝ : Finite (Arrow C)
a b : C
this : Finite (a ⟶ b)
⊢ Fintype (a ⟶ b) | 98b324129af2b9de |
Vector.map_eq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem map_eq_iff {f : α → β} {l : Vector α n} {l' : Vector β n} :
map f l = l' ↔ ∀ i (h : i < n), l'[i] = f l[i] | α : Type u_1
β : Type u_2
n : Nat
f : α → β
l : Vector α n
l' : Vector β n
⊢ map f l = l' ↔ ∀ (i : Nat) (h : i < n), l'[i] = f l[i] | rcases l with ⟨l, rfl⟩ | case mk
α : Type u_1
β : Type u_2
f : α → β
l : Array α
l' : Vector β l.size
⊢ map f { toArray := l, size_toArray := ⋯ } = l' ↔
∀ (i : Nat) (h : i < l.size), l'[i] = f { toArray := l, size_toArray := ⋯ }[i] | 9e0ae22d1d915b68 |
CategoryTheory.projective_iff_llp_epimorphisms_of_isZero | Mathlib/CategoryTheory/Preadditive/Projective/LiftingProperties.lean | lemma projective_iff_llp_epimorphisms_of_isZero
[HasZeroMorphisms C] {P Z : C} (i : Z ⟶ P) (hZ : IsZero Z) :
Projective P ↔ (MorphismProperty.epimorphisms C).llp i | case mp
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
P Z : C
hZ : IsZero Z
⊢ Projective P → (MorphismProperty.epimorphisms C).llp 0 | intro _ X Y p (_ : Epi p) | case mp
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
P Z : C
hZ : IsZero Z
a✝ : Projective P
X Y : C
p : X ⟶ Y
x✝ : Epi p
⊢ HasLiftingProperty 0 p | f555dbbf0d0eaaca |
Filter.HasBasis.mem_lift_iff | Mathlib/Order/Filter/Lift.lean | theorem HasBasis.mem_lift_iff {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α}
(hf : f.HasBasis p s) {β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ}
{g : Set α → Filter γ} (hg : ∀ i, (g <| s i).HasBasis (pg i) (sg i)) (gm : Monotone g)
{s : Set γ} : s ∈ f.lift g ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg... | case refine_2
α : Type u_1
γ : Type u_3
ι : Sort u_6
p : ι → Prop
s✝ : ι → Set α
f : Filter α
hf : f.HasBasis p s✝
β : ι → Type u_5
pg : (i : ι) → β i → Prop
sg : (i : ι) → β i → Set γ
g : Set α → Filter γ
hg : ∀ (i : ι), (g (s✝ i)).HasBasis (pg i) (sg i)
gm : Monotone g
s : Set γ
⊢ (∃ i ∈ f.sets, s ∈ g i) ↔ ∃ i, p i ∧... | simp only [← (hg _).mem_iff] | case refine_2
α : Type u_1
γ : Type u_3
ι : Sort u_6
p : ι → Prop
s✝ : ι → Set α
f : Filter α
hf : f.HasBasis p s✝
β : ι → Type u_5
pg : (i : ι) → β i → Prop
sg : (i : ι) → β i → Set γ
g : Set α → Filter γ
hg : ∀ (i : ι), (g (s✝ i)).HasBasis (pg i) (sg i)
gm : Monotone g
s : Set γ
⊢ (∃ i ∈ f.sets, s ∈ g i) ↔ ∃ i, p i ∧... | 766e18f5697f1c3d |
Fin.tail_update_succ | Mathlib/Data/Fin/Tuple/Basic.lean | theorem tail_update_succ : tail (update q i.succ y) = update (tail q) i y | case pos
n : ℕ
α : Fin (n + 1) → Sort u
q : (i : Fin (n + 1)) → α i
i : Fin n
y : α i.succ
j : Fin n
h : j = i
⊢ tail (update q i.succ y) j = update (tail q) i y j | rw [h] | case pos
n : ℕ
α : Fin (n + 1) → Sort u
q : (i : Fin (n + 1)) → α i
i : Fin n
y : α i.succ
j : Fin n
h : j = i
⊢ tail (update q i.succ y) i = update (tail q) i y i | 344646d579394232 |
oneLePart_leOnePart_injective | Mathlib/Algebra/Order/Group/PosPart.lean | @[to_additive]
lemma oneLePart_leOnePart_injective : Injective fun a : α ↦ (a⁺ᵐ, a⁻ᵐ) | α : Type u_1
inst✝² : Lattice α
inst✝¹ : Group α
inst✝ : MulLeftMono α
a b : α
hpos : a⁺ᵐ = b⁺ᵐ
hneg : a⁻ᵐ = b⁻ᵐ
⊢ a = b | rw [← oneLePart_div_leOnePart a, ← oneLePart_div_leOnePart b, hpos, hneg] | no goals | f9ab77083dd7a53c |
Cubic.disc_ne_zero_iff_roots_nodup | Mathlib/Algebra/CubicDiscriminant.lean | theorem disc_ne_zero_iff_roots_nodup (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
P.disc ≠ 0 ↔ (map φ P).roots.Nodup | F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ {x, y, z}.Nodup | change _ ↔ (x ::ₘ y ::ₘ {z}).Nodup | F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ x ≠ y ∧ x ≠ z ∧ y ≠ z ↔ (x ::ₘ y ::ₘ {z}).Nodup | 2cef01e8fb0b75da |
Stream'.Seq.append_assoc | Mathlib/Data/Seq/Seq.lean | theorem append_assoc (s t u : Seq α) : append (append s t) u = append s (append t u) | α : Type u
s t u : Seq α
⊢ (s.append t).append u = s.append (t.append u) | apply eq_of_bisim fun s1 s2 => ∃ s t u, s1 = append (append s t) u ∧ s2 = append s (append t u) | case bisim
α : Type u
s t u : Seq α
⊢ IsBisimulation fun s1 s2 => ∃ s t u, s1 = (s.append t).append u ∧ s2 = s.append (t.append u)
case r
α : Type u
s t u : Seq α
⊢ ∃ s_1 t_1 u_1,
(s.append t).append u = (s_1.append t_1).append u_1 ∧ s.append (t.append u) = s_1.append (t_1.append u_1) | ece5c217f4d14d24 |
MeasureTheory.measurableSet_generateFrom_singleton_iff | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurableSet_generateFrom_singleton_iff {s t : Set α} :
MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ | α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
⊢ x = {True} | ext p | case h
α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
p : Prop
⊢ p ∈ x ↔ p ∈ {True} | 39127bdfcef76ebf |
Real.not_bddBelow_coe | Mathlib/Data/Real/Archimedean.lean | theorem not_bddBelow_coe : ¬ (BddBelow <| range (fun (x : ℚ) ↦ (x : ℝ))) | ⊢ ¬{x | ∀ ⦃a : ℝ⦄, (a ∈ range fun x => ↑x) → x ≤ a}.Nonempty | rw [Set.not_nonempty_iff_eq_empty] | ⊢ {x | ∀ ⦃a : ℝ⦄, (a ∈ range fun x => ↑x) → x ≤ a} = ∅ | 8b7b3266eceded65 |
isLocallyInjective_iff_isOpen_diagonal | Mathlib/Topology/SeparatedMap.lean | theorem isLocallyInjective_iff_isOpen_diagonal {f : X → Y} :
IsLocallyInjective f ↔ IsOpen f.pullbackDiagonal | case refine_2.intro.intro.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝ : TopologicalSpace X
f : X → Y
h :
∀ (a b : X) (b_1 : f a = f b),
⟨(a, b), b_1⟩ ∈ Function.pullbackDiagonal f → ∃ t ∈ 𝓝 a ×ˢ 𝓝 b, Subtype.val ⁻¹' t ⊆ Function.pullbackDiagonal f
x : X
t : Set (X × X)
ht : t ∈ 𝓝 x ×ˢ 𝓝 x
t_sub : ... | exact ⟨t₁ ∩ t₂, Filter.inter_mem h₁ h₂,
fun x₁ h₁ x₂ h₂ he ↦ @t_sub ⟨(x₁, x₂), he⟩ (prod_sub ⟨h₁.1, h₂.2⟩)⟩ | no goals | 81b1cbf2345d2091 |
isBigO_norm_restrict_cocompact | Mathlib/Analysis/Fourier/PoissonSummation.lean | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : f =O[cocompact ℝ] fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
(fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) =O[cocompact ℝ] (|·| ^ (-b)) | E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x : ℝ
⊢ ∀ (x_1 : ↑↑K),
‖(ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))) x_1‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖ | rintro ⟨y, hy⟩ | case mk
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x y : ℝ
hy : y ∈ ↑K
⊢ ‖(ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))) ⟨y, hy⟩‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖ | 6854796ce291b958 |
Algebra.FinitePresentation.of_restrict_scalars_finitePresentation | Mathlib/RingTheory/FinitePresentation.lean | theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
[FinitePresentation.{w₁, w₃} R B] [FiniteType R A] :
FinitePresentation.{w₂, w₃} A B | case refine_1.inl.intro.intro.hx
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf ... | exact Set.mem_range_self _ | no goals | 765dbb818bc3202d |
Complex.HadamardThreeLines.F_BddAbove | Mathlib/Analysis/Complex/Hadamard.lean | /-- When the function `f` is bounded above on a vertical strip, then so is `F`. -/
lemma F_BddAbove (f : ℂ → E) (ε : ℝ) (hε : ε > 0)
(hB : BddAbove ((norm ∘ f) '' verticalClosedStrip 0 1)) :
BddAbove ((norm ∘ (F f ε)) '' verticalClosedStrip 0 1) | case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
B : ℝ
hB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B
z : ℂ
hset : z ∈ verticalClosedStrip 0 1
⊢ ‖((↑ε + ↑(sSupNormIm f 0)) ^ (z - 1) * (↑ε + ↑(sSupNormIm f 1)) ^ (-z)) • f z‖ ≤
(1 ⊔ (ε + sSupNormIm f 0) ^ (-1... | specialize hB (‖f z‖) (by simpa [image_congr, mem_image, comp_apply] using ⟨z, hset, rfl⟩) | case h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
ε : ℝ
hε : ε > 0
B : ℝ
z : ℂ
hset : z ∈ verticalClosedStrip 0 1
hB : ‖f z‖ ≤ B
⊢ ‖((↑ε + ↑(sSupNormIm f 0)) ^ (z - 1) * (↑ε + ↑(sSupNormIm f 1)) ^ (-z)) • f z‖ ≤
(1 ⊔ (ε + sSupNormIm f 0) ^ (-1)) * (1 ⊔ (ε + sSupNormIm f 1) ^ (-1)) ... | 73d4e799cf93cedd |
Nat.minFac_eq_one_iff | Mathlib/Data/Nat/Prime/Defs.lean | theorem minFac_eq_one_iff {n : ℕ} : minFac n = 1 ↔ n = 1 | case mp
n : ℕ
h : n.minFac = 1
hn : ¬n = 1
this : Prime n.minFac
⊢ False | rw [h] at this | case mp
n : ℕ
h : n.minFac = 1
hn : ¬n = 1
this : Prime 1
⊢ False | cd9f6d188cf129cd |
pow_dvd_of_le_emultiplicity | Mathlib/RingTheory/Multiplicity.lean | theorem pow_dvd_of_le_emultiplicity {k : ℕ} (hk : k ≤ emultiplicity a b) :
a ^ k ∣ b | case succ.isTrue
α : Type u_1
inst✝ : Monoid α
a b : α
n✝ : ℕ
h✝ : FiniteMultiplicity a b
hk : n✝ + 1 ≤ Nat.find h✝
⊢ a ^ (n✝ + 1) ∣ b | simpa using (Nat.find_min _ (lt_of_succ_le hk)) | no goals | b583f365000b6572 |
Affine.Simplex.finrank_direction_altitude | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem finrank_direction_altitude {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2)) :
finrank ℝ (s.altitude i).direction = 1 | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 1)
i : Fin (n + 2)
h :
finrank ℝ ↥(vectorSpan ℝ (s.points '' ↑(univ.erase i))) +
finrank ℝ ↥((vectorSpan ℝ (s.points '' ↑(univ.erase i)))ᗮ ⊓ vector... | have hc : #(univ.erase i) = n + 1 := by rw [card_erase_of_mem (mem_univ _)]; simp | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 1)
i : Fin (n + 2)
h :
finrank ℝ ↥(vectorSpan ℝ (s.points '' ↑(univ.erase i))) +
finrank ℝ ↥((vectorSpan ℝ (s.points '' ↑(univ.erase i)))ᗮ ⊓ vector... | 8cce054b5cd905d5 |
CategoryTheory.IsPushout.isVanKampen_iff | Mathlib/CategoryTheory/Adhesive.lean | theorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :
H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w) | case mp
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H : IsPushout f g h i
⊢ H.IsVanKampen → IsVanKampenColimit (PushoutCocone.mk h i ⋯) | intro H F' c' α fα eα hα | case mp
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : H✝.IsVanKampen
F' : WalkingSpan ⥤ C
c' : Cocone F'
α : F' ⟶ span f g
fα : c'.pt ⟶ (PushoutCocone.mk h i ⋯).pt
eα : α ≫ (PushoutCocone.mk h i ⋯).ι = c'.ι ≫ (Functor.const WalkingSpan).map fα
hα : N... | 03b3318bb2e454c2 |
Real.deriv_Gamma_nat | Mathlib/NumberTheory/Harmonic/GammaDeriv.lean | /-- Explicit formula for the derivative of the Gamma function at positive integers, in terms of
harmonic numbers and the Euler-Mascheroni constant `γ`. -/
lemma deriv_Gamma_nat (n : ℕ) :
deriv Gamma (n + 1) = n ! * (-γ + harmonic n) | n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
⊢ -deriv f 1 = γ | have derivLB (n : ℕ) (hn : 0 < n) : log n ≤ deriv f (n + 1) := by
refine (le_of_eq ?_).trans <| hc.slope_le_deriv (mem_Ioi.mpr <| Nat.cast_pos.mpr hn)
(by positivity : _ < (_ : ℝ)) (by linarith) (hder <| by positivity)
rw [slope_def_field, show n + 1 - n = (1 : ℝ) by ring, div_one, h_rec n (by positivity),
... | n : ℕ
f : ℝ → ℝ := log ∘ Gamma
hc : ConvexOn ℝ (Ioi 0) f
h_rec : ∀ (x : ℝ), 0 < x → f (x + 1) = f x + log x
hder : ∀ {x : ℝ}, 0 < x → DifferentiableAt ℝ f x
hder_rec : ∀ (x : ℝ), 0 < x → deriv f (x + 1) = deriv f x + 1 / x
hder_nat : ∀ (n : ℕ), deriv f (↑n + 1) = deriv f 1 + ↑(harmonic n)
derivLB : ∀ (n : ℕ), 0 < n → l... | 0b4525a3b23b428f |
MeasureTheory.lintegral_const_mul' | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_const_mul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), r⁻¹ * (r * f a) ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ | simp only [(mul_assoc _ _ _).symm, rinv', one_mul] at this | α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hr : r ≠ ⊤
h : ¬r = 0
rinv : r * r⁻¹ = 1
rinv' : r⁻¹ * r = 1
this : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), f a ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ | 8660a935dda25562 |
Batteries.HashMap.Imp.Buckets.WF.mk' | Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean | theorem WF.mk' [BEq α] [Hashable α] (h) : (Buckets.mk n h : Buckets α β).WF | case refine_1
α : Type u_1
β : Type u_2
n : Nat
inst✝³ : BEq α
inst✝² : Hashable α
h✝ : 0 < n
inst✝¹ : LawfulHashable α
inst✝ : PartialEquivBEq α
x✝ : AssocList α β
h : ¬n = 0 ∧ x✝ = AssocList.nil
⊢ List.Pairwise (fun a b => ¬(a.fst == b.fst) = true) x✝.toList | simp [h, List.Pairwise.nil] | no goals | 44af9bb762dccc7e |
μ_nonempty | Mathlib/Analysis/Normed/Ring/SmoothingSeminorm.lean | theorem μ_nonempty {s : ℕ → ℕ} (hs_le : ∀ n : ℕ, s n ≤ n) {x : R} (ψ : ℕ → ℕ) :
{a : ℝ | ∀ᶠ n : ℝ in map (fun n : ℕ => μ x ^ (↑(s (ψ n)) * (1 / (ψ n : ℝ)))) atTop,
n ≤ a}.Nonempty | case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hμx : ¬μ x < 1
b : ℕ
a✝ : 0 ≤ b
⊢ μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ μ x ^ 1 | apply rpow_le_rpow_of_exponent_le (not_lt.mp hμx) | case h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hμx : ¬μ x < 1
b : ℕ
a✝ : 0 ≤ b
⊢ ↑(s (ψ b)) * (1 / ↑(ψ b)) ≤ 1 | f6e82807a41ed32a |
le_rieszMeasure_of_isCompact_tsupport_subset | Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean | lemma le_rieszMeasure_of_isCompact_tsupport_subset {f : C_c(X, ℝ≥0)} (hf : ∀ x, f x ≤ 1)
{K : Set X} (hK : IsCompact K) (h : tsupport f ⊆ K) : .ofNNReal (Λ f) ≤ rieszMeasure Λ K | case a
X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
K : Set X
hK : IsCompact K
h : tsupport ⇑f ⊆ K
ε : ℝ≥0
hε : 0 < ε
g : X →C_c ℝ≥0
hg :
(∀ x ∈ { carrier :... | intro x | case a
X : Type u_1
inst✝⁴ : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
f : X →C_c ℝ≥0
hf : ∀ (x : X), f x ≤ 1
K : Set X
hK : IsCompact K
h : tsupport ⇑f ⊆ K
ε : ℝ≥0
hε : 0 < ε
g : X →C_c ℝ≥0
hg :
(∀ x ∈ { carrier :... | c932490d6d76d2d6 |
List.sigma_eq_sigmaTR | Mathlib/.lake/packages/batteries/Batteries/Data/List/Basic.lean | theorem sigma_eq_sigmaTR : @List.sigma = @sigmaTR | case h.h.h.h
α : Type u_2
β : α → Type u_1
l₁ : List α
l₂ : (a : α) → List (β a)
⊢ l₁.sigma l₂ = l₁.sigmaTR l₂ | simp [List.sigma, sigmaTR] | case h.h.h.h
α : Type u_2
β : α → Type u_1
l₁ : List α
l₂ : (a : α) → List (β a)
⊢ flatMap (fun a => map (Sigma.mk a) (l₂ a)) l₁ =
(foldl (fun acc a => foldl (fun acc b => acc.push ⟨a, b⟩) acc (l₂ a)) #[] l₁).toList | 743475a23b138a94 |
LinearOrder.strictConvexOn_of_lt | Mathlib/Analysis/Convex/Function.lean | theorem LinearOrder.strictConvexOn_of_lt (hs : Convex 𝕜 s)
(hf : ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 →
f (a • x + b • y) < a • f x + b • f y) :
StrictConvexOn 𝕜 s f | case inr
𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁵ : OrderedSemiring 𝕜
inst✝⁴ : AddCommMonoid E
inst✝³ : OrderedAddCommMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : LinearOrder E
s : Set E
f : E → β
hs : Convex 𝕜 s
hf :
∀ ⦃x : E⦄,
x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x < y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b ... | exact this hs hf y hy x hx hxy.symm b a hb ha hab (hxy.lt_or_lt.resolve_left h) | no goals | 9cfac4d99efa429e |
Finsupp.mem_splitSupport_iff_nonzero | Mathlib/Data/Finsupp/Basic.lean | theorem mem_splitSupport_iff_nonzero (i : ι) : i ∈ splitSupport l ↔ split l i ≠ 0 | ι : Type u_4
M : Type u_5
αs : ι → Type u_13
inst✝ : Zero M
l : (i : ι) × αs i →₀ M
i : ι
⊢ i ∈ l.splitSupport ↔ l.split i ≠ 0 | rw [splitSupport, @mem_image _ _ (Classical.decEq _), Ne, ← support_eq_empty, ← Ne, ←
Finset.nonempty_iff_ne_empty, split, comapDomain, Finset.Nonempty] | ι : Type u_4
M : Type u_5
αs : ι → Type u_13
inst✝ : Zero M
l : (i : ι) × αs i →₀ M
i : ι
⊢ (∃ a ∈ l.support, a.fst = i) ↔
∃ x,
x ∈ { support := l.support.preimage (Sigma.mk i) ⋯, toFun := fun a => l ⟨i, a⟩, mem_support_toFun := ⋯ }.support | 67fb310cc0e05126 |
frobeniusNumber_pair | Mathlib/NumberTheory/FrobeniusNumber.lean | theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
FrobeniusNumber (m * n - m - n) {m, n} | case intro
m n : ℕ
cop : m.Coprime n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
k : ℕ
hk : m * n - m - n < k
x : { k_1 // k_1 ≡ 0 [MOD m] ∧ k_1 ≡ k [MOD n] } := chineseRemainder cop 0 k
hx : ↑x < m * n
key : ↑x ≤ k
a : ℕ
ha : ↑x = m * a
⊢ ∃ m_1 n_1, m_1 * m + n_1 * n = k | obtain ⟨b, hb⟩ := (modEq_iff_dvd' key).mp x.2.2 | case intro.intro
m n : ℕ
cop : m.Coprime n
hm : 1 < m
hn : 1 < n
hmn : m + n ≤ m * n
k : ℕ
hk : m * n - m - n < k
x : { k_1 // k_1 ≡ 0 [MOD m] ∧ k_1 ≡ k [MOD n] } := chineseRemainder cop 0 k
hx : ↑x < m * n
key : ↑x ≤ k
a : ℕ
ha : ↑x = m * a
b : ℕ
hb : k - ↑x = n * b
⊢ ∃ m_1 n_1, m_1 * m + n_1 * n = k | c8305ce6da9adabe |
Ideal.iSupIndep.linearIndependent' | Mathlib/Algebra/Module/Torsion.lean | theorem iSupIndep.linearIndependent' {ι R M : Type*} {v : ι → M} [Ring R]
[AddCommGroup M] [Module R M] (hv : iSupIndep fun i => R ∙ v i)
(h_ne_zero : ∀ i, Ideal.torsionOf R M (v i) = ⊥) : LinearIndependent R v | ι : Type u_3
R : Type u_4
M : Type u_5
v : ι → M
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h_ne_zero : ∀ (i : ι), torsionOf R M (v i) = ⊥
i : ι
r : R
hi : r • v i ∈ Submodule.span R (v '' (Set.univ \ {i}))
hv : Disjoint (Submodule.span R {v i}) (⨆ j, ⨆ (_ : j ≠ i), Submodule.span R {v j})
⊢ r = 0 | simp only [iSup_subtype', ← Submodule.span_range_eq_iSup (ι := Subtype _), disjoint_iff] at hv | ι : Type u_3
R : Type u_4
M : Type u_5
v : ι → M
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
h_ne_zero : ∀ (i : ι), torsionOf R M (v i) = ⊥
i : ι
r : R
hi : r • v i ∈ Submodule.span R (v '' (Set.univ \ {i}))
hv : Submodule.span R {v i} ⊓ Submodule.span R (Set.range fun i_1 => v ↑i_1) = ⊥
⊢ r = 0 | 7ccdd7b59f3a0647 |
EMetric.mem_ball_comm | Mathlib/Topology/EMetricSpace/Defs.lean | theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε | α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
ε : ℝ≥0∞
⊢ x ∈ ball y ε ↔ y ∈ ball x ε | rw [mem_ball', mem_ball] | no goals | 52a7ccc0b7a7d0d4 |
Std.DHashMap.Internal.Raw₀.containsThenInsertIfNew_eq_insertIfNewₘ | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Model.lean | theorem containsThenInsertIfNew_eq_insertIfNewₘ [BEq α] [Hashable α] (m : Raw₀ α β) (a : α)
(b : β a) : (m.containsThenInsertIfNew a b).2 = m.insertIfNewₘ a b | case isFalse
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw₀ α β
a : α
b : β a
h✝ : ¬AssocList.contains a m.1.buckets[(mkIdx m.1.buckets.size ⋯ (hash a)).val.toNat] = true
⊢ (false,
expandIfNecessary
⟨{ size := m.1.size + 1,
buckets :=
m.1.buckets.set... | rfl | no goals | 1cdd28730cc0764f |
EReal.coe_ennreal_toReal | Mathlib/Data/Real/EReal.lean | lemma coe_ennreal_toReal {x : ℝ≥0∞} (hx : x ≠ ∞) : (x.toReal : EReal) = x | x : ℝ≥0∞
hx : x ≠ ⊤
⊢ ↑x.toReal = ↑x | lift x to ℝ≥0 using hx | case intro
x : ℝ≥0
⊢ ↑(↑x).toReal = ↑↑x | 456df54b7f760cef |
NonUnitalAlgebra.span_eq_toSubmodule | Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | @[simp]
lemma span_eq_toSubmodule (s : NonUnitalSubalgebra R A) :
Submodule.span R (s : Set A) = s.toSubmodule | R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : NonUnitalNonAssocSemiring A
inst✝ : Module R A
s : NonUnitalSubalgebra R A
⊢ Submodule.span R ↑s = s.toSubmodule | simp [SetLike.ext'_iff, Submodule.coe_span_eq_self] | no goals | a12868adf6a84332 |
PiNat.isTopologicalBasis_cylinders | Mathlib/Topology/MetricSpace/PiNat.lean | theorem isTopologicalBasis_cylinders :
IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n) (n : ℕ), s = cylinder x n } | E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
⊢ IsTopologicalBasis {s | ∃ x n, s = cylinder x n} | apply isTopologicalBasis_of_isOpen_of_nhds | case h_open
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
⊢ ∀ u ∈ {s | ∃ x n, s = cylinder x n}, IsOpen u
case h_nhds
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
⊢ ∀ (a : (n : ℕ) → E n) (u : Set ((n : ℕ) → E... | 93a4ba35a83995ab |
Nat.primeFactorsPiBij_inj | Mathlib/Algebra/Order/Antidiag/Nat.lean | theorem primeFactorsPiBij_inj (d n : ℕ)
(f : (p : ℕ) → p ∈ n.primeFactors → Fin d) (hf : f ∈ pi n.primeFactors fun _ => univ)
(g : (p : ℕ) → p ∈ n.primeFactors → Fin d) (hg : g ∈ pi n.primeFactors fun _ => univ) :
Nat.primeFactorsPiBij d n f hf = Nat.primeFactorsPiBij d n g hg → f = g | case h.a
d n : ℕ
f : (p : ℕ) → p ∈ n.primeFactors → Fin d
hf : f ∈ n.primeFactors.pi fun x => univ
g : (p : ℕ) → p ∈ n.primeFactors → Fin d
hg : g ∈ n.primeFactors.pi fun x => univ
p : ℕ
hp✝ : p ∈ n.primeFactors
hp : Prime p ∧ p ∣ n ∧ n ≠ 0
hfg : f p hp✝ ≠ g p hp✝
q : { x // x ∈ n.primeFactors }
hq : q ∈ filter (fun p_... | rw [Nat.prime_dvd_prime_iff_eq hp.1 (Nat.prime_of_mem_primeFactorsList
<| List.mem_toFinset.mp q.2)] | case h.a
d n : ℕ
f : (p : ℕ) → p ∈ n.primeFactors → Fin d
hf : f ∈ n.primeFactors.pi fun x => univ
g : (p : ℕ) → p ∈ n.primeFactors → Fin d
hg : g ∈ n.primeFactors.pi fun x => univ
p : ℕ
hp✝ : p ∈ n.primeFactors
hp : Prime p ∧ p ∣ n ∧ n ≠ 0
hfg : f p hp✝ ≠ g p hp✝
q : { x // x ∈ n.primeFactors }
hq : q ∈ filter (fun p_... | 6425189256a15307 |
Stream'.Seq.join_cons | Mathlib/Data/Seq/Seq.lean | theorem join_cons (a : α) (s S) : join (cons (a, s) S) = cons a (append s (join S)) | α : Type u
a : α
s : Seq α
S : Seq (Seq1 α)
⊢ (cons (a, s) S).join = cons a (s.append S.join) | apply
eq_of_bisim
(fun s1 s2 => s1 = s2 ∨ ∃ a s S, s1 = join (cons (a, s) S) ∧ s2 = cons a (append s (join S)))
_ (Or.inr ⟨a, s, S, rfl, rfl⟩) | α : Type u
a : α
s : Seq α
S : Seq (Seq1 α)
⊢ IsBisimulation fun s1 s2 => s1 = s2 ∨ ∃ a s S, s1 = (cons (a, s) S).join ∧ s2 = cons a (s.append S.join) | ab6735a8c4d12fe1 |
WittVector.nth_mul_coeff' | Mathlib/RingTheory/WittVector/MulCoeff.lean | theorem nth_mul_coeff' (n : ℕ) :
∃ f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k,
∀ x y : 𝕎 k, f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
(x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) -
x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) | case mk
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : (↑(Membership.mem (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
x y : Trun... | simp_rw [product_val, this, range_val, Multiset.range_succ] at ha | case mk
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : (↑(Membership.mem (univ ×ˢ range (n + 1)).val) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
this : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b
x y : Trun... | e8f35526e8a3693d |
Function.Injective.exists_ne | Mathlib/Logic/Nontrivial/Basic.lean | theorem Function.Injective.exists_ne [Nontrivial α] {f : α → β}
(hf : Function.Injective f) (y : β) : ∃ x, f x ≠ y | case pos
α : Type u_1
β : Type u_2
inst✝ : Nontrivial α
f : α → β
hf : Injective f
y : β
x₁ x₂ : α
hx : x₁ ≠ x₂
h : f x₂ = y
⊢ ∃ x, f x ≠ y | exact ⟨x₁, (hf.ne_iff' h).2 hx⟩ | no goals | 88e66b4be33dae0c |
CochainComplex.mappingCone.liftCochain_v_fst_v | Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | @[reassoc (attr := simp)]
lemma liftCochain_v_fst_v (p₁ p₂ p₃ : ℤ) (h₁₂ : p₁ + n = p₂) (h₂₃ : p₂ + 1 = p₃) :
(liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (fst φ).1.v p₂ p₃ h₂₃ = α.v p₁ p₃ (by omega) | C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + 1 = p₃
⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (↑(fst φ)).v p₂ p₃ h₂... | simpa only [Cochain.comp_v _ _ h p₁ p₂ p₃ h₁₂ h₂₃]
using Cochain.congr_v (liftCochain_fst φ α β h) p₁ p₃ (by omega) | no goals | 27fc166adba6aa1c |
List.eraseIdx_eq_take_drop_succ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean | theorem eraseIdx_eq_take_drop_succ :
∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
| nil, _ => by simp
| a::l, 0 => by simp
| a::l, i + 1 => by simp [eraseIdx_eq_take_drop_succ l i]
| α : Type u_1
x✝ : Nat
⊢ [].eraseIdx x✝ = take x✝ [] ++ drop (x✝ + 1) [] | simp | no goals | 393992788c6150bf |
ProbabilityTheory.lintegral_paretoPDF_eq_one | Mathlib/Probability/Distributions/Pareto.lean | /-- The pdf of the Pareto distribution integrates to `1`. -/
@[simp]
lemma lintegral_paretoPDF_eq_one (ht : 0 < t) (hr : 0 < r) :
∫⁻ x, paretoPDF t r x = 1 | t r : ℝ
ht : 0 < t
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0
rightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))
⊢ ∫⁻ (x : ℝ), paretoPDF t r x = 1 | rw [← ENNReal.toReal_eq_one_iff, ← lintegral_add_compl _ measurableSet_Ici, compl_Ici,
leftSide, rightSide, add_zero, ← integral_eq_lintegral_of_nonneg_ae] | t r : ℝ
ht : 0 < t
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0
rightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))
⊢ ∫ (a : ℝ) in Ici t, r * t ^ r * a ^ (-(r + 1)) = 1
case hf
t r : ℝ
ht : 0 < t
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio t... | e4acb7fa15218c7c |
Nat.cast_eq_zero | Mathlib/Algebra/CharZero/Defs.lean | theorem cast_eq_zero {n : ℕ} : (n : R) = 0 ↔ n = 0 | R : Type u_1
inst✝¹ : AddMonoidWithOne R
inst✝ : CharZero R
n : ℕ
⊢ ↑n = 0 ↔ n = 0 | rw [← cast_zero, cast_inj] | no goals | efc0521b24619bfb |
CStarAlgebra.span_nonneg | Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart.lean | /-- A C⋆-algebra is spanned by its nonnegative elements. -/
lemma CStarAlgebra.span_nonneg : Submodule.span ℂ {a : A | 0 ≤ a} = ⊤ | A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℂ A
inst✝⁷ : SMulCommClass ℂ A A
inst✝⁶ : IsScalarTower ℂ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
x✝ : x ∈ ⊤
⊢ ... | apply_rules [sub_mem, Submodule.smul_mem, add_mem] | case a.hx
A : Type u_1
inst✝⁹ : NonUnitalRing A
inst✝⁸ : Module ℂ A
inst✝⁷ : SMulCommClass ℂ A A
inst✝⁶ : IsScalarTower ℂ A A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
x✝ ... | 9efd68269dbda90e |
Matrix.fromBlocks_mul_fromRows | Mathlib/Data/Matrix/ColumnRowPartitioned.lean | /-- A block matrix multiplied by a row partitioned matrix gives a row partitioned matrix. -/
lemma fromBlocks_mul_fromRows [Fintype n₁] [Fintype n₂] (A₁ : Matrix n₁ n R) (A₂ : Matrix n₂ n R)
(B₁₁ : Matrix m₁ n₁ R) (B₁₂ : Matrix m₁ n₂ R) (B₂₁ : Matrix m₂ n₁ R) (B₂₂ : Matrix m₂ n₂ R) :
fromBlocks B₁₁ B₁₂ B₂₁ B₂₂ ... | R : Type u_1
m₁ : Type u_3
m₂ : Type u_4
n : Type u_5
n₁ : Type u_6
n₂ : Type u_7
inst✝² : Semiring R
inst✝¹ : Fintype n₁
inst✝ : Fintype n₂
A₁ : Matrix n₁ n R
A₂ : Matrix n₂ n R
B₁₁ : Matrix m₁ n₁ R
B₁₂ : Matrix m₁ n₂ R
B₂₁ : Matrix m₂ n₁ R
B₂₂ : Matrix m₂ n₂ R
⊢ fromBlocks B₁₁ B₁₂ B₂₁ B₂₂ * A₁.fromRows A₂ = (B₁₁ * A₁... | ext (_ | _) _ <;> simp [mul_apply] | no goals | 532900959aca78c4 |
AlgebraicGeometry.isNoetherianRing_of_away | Mathlib/AlgebraicGeometry/Noetherian.lean | theorem isNoetherianRing_of_away : IsNoetherianRing R | R : Type u
inst✝ : CommRing R
S : Finset R
hS : Ideal.span ↑S = ⊤
hN : ∀ (s : { x // x ∈ S }), IsNoetherianRing (Localization.Away ↑s)
I : ℕ →o Submodule R R
floc : (s : R) → R →+* Localization.Away s := fun s => algebraMap R (Localization.Away s)
suitableN : R → Set ℕ := fun s => {n | ∀ (m : ℕ), n ≤ m → Ideal.map (flo... | have hSuit : ∀ s : S, minN s ∈ suitableN s := by
intro s
apply Nat.sInf_mem
let f : ℕ →o Ideal (Away (M := R) s) :=
⟨fun n ↦ Ideal.map (floc s) (I n), fun _ _ h ↦ Ideal.map_mono (I.monotone h)⟩
exact monotone_stabilizes_iff_noetherian.mpr (hN s) f | R : Type u
inst✝ : CommRing R
S : Finset R
hS : Ideal.span ↑S = ⊤
hN : ∀ (s : { x // x ∈ S }), IsNoetherianRing (Localization.Away ↑s)
I : ℕ →o Submodule R R
floc : (s : R) → R →+* Localization.Away s := fun s => algebraMap R (Localization.Away s)
suitableN : R → Set ℕ := fun s => {n | ∀ (m : ℕ), n ≤ m → Ideal.map (flo... | e9967a599c4e7dd7 |
Pell.y_mul_dvd | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem y_mul_dvd (n) : ∀ k, yn a1 n ∣ yn a1 (n * k)
| 0 => dvd_zero _
| k + 1 => by
rw [Nat.mul_succ, yn_add]; exact dvd_add (dvd_mul_left _ _) ((y_mul_dvd _ k).mul_right _)
| a : ℕ
a1 : 1 < a
n k : ℕ
⊢ yn a1 n ∣ yn a1 (n * (k + 1)) | rw [Nat.mul_succ, yn_add] | a : ℕ
a1 : 1 < a
n k : ℕ
⊢ yn a1 n ∣ xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n | fe09739c7a819004 |
Function.Periodic.intervalIntegrable | Mathlib/MeasureTheory/Integral/Periodic.lean | theorem intervalIntegrable {t : ℝ} (h₁f : Function.Periodic f T) (hT : 0 < T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) (a₁ a₂ : ℝ) :
IntervalIntegrable f MeasureTheory.volume a₁ a₂ | E : Type u_1
inst✝ : NormedAddCommGroup E
f : ℝ → E
T t : ℝ
h₁f : Periodic f T
hT : 0 < T
h₂f : IntervalIntegrable f volume t (t + T)
a₁ a₂ : ℝ
n₁ : ℕ
hn₁ : (t - a₁ ⊓ a₂) / T ≤ ↑n₁
n₂ : ℕ
hn₂ : (a₁ ⊔ a₂ - t) / T ≤ ↑n₂
⊢ t + ↑n₂ * T ≤ (t - ↑n₁ * T) ⊔ (t + ↑n₂ * T) | apply le_max_right | no goals | d88139d71939b23e |
IsClosed.smul_left_of_isCompact | Mathlib/Topology/Algebra/Group/Basic.lean | theorem IsClosed.smul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
IsClosed (s • t) | case intro.intro
α : Type u
β : Type v
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : Group α
inst✝² : MulAction α β
inst✝¹ : ContinuousInv α
inst✝ : ContinuousSMul α β
s : Set α
t : Set β
ht : IsClosed t
hs : IsCompact s
f : β → α
hf : ∀ x ∈ s • t, f x ∈ s ∧ (f x)⁻¹ • x ∈ t
x : β
hx : x ∈ closure (s ... | have : Ultrafilter.map f u ≤ 𝓟 s :=
calc Ultrafilter.map f u ≤ map f (𝓟 (s • t)) := map_mono (le_principal_iff.mpr hust)
_ = 𝓟 (f '' (s • t)) := map_principal
_ ≤ 𝓟 s := principal_mono.mpr (image_subset_iff.mpr (fun x hx ↦ (hf x hx).1)) | case intro.intro
α : Type u
β : Type v
inst✝⁵ : TopologicalSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : Group α
inst✝² : MulAction α β
inst✝¹ : ContinuousInv α
inst✝ : ContinuousSMul α β
s : Set α
t : Set β
ht : IsClosed t
hs : IsCompact s
f : β → α
hf : ∀ x ∈ s • t, f x ∈ s ∧ (f x)⁻¹ • x ∈ t
x : β
hx : x ∈ closure (s ... | 22da55322c3f5f93 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRupUnits_of_assignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem assignmentsInvariant_insertRupUnits_of_assignmentsInvariant {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRupUnits f units).1 | case pos
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsize... | rw [hb'] | case pos
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsize
hsize... | 7537b9cac0fe9952 |
HomologicalComplex₂.total.mapAux.d₁_mapMap | Mathlib/Algebra/Homology/TotalComplex.lean | @[reassoc (attr := simp)]
lemma d₁_mapMap (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂) :
K.d₁ c₁₂ i₁ i₂ i₁₂ ≫ GradedObject.mapMap (toGradedObjectMap φ) _ i₁₂ =
(φ.f i₁).f i₂ ≫ L.d₁ c₁₂ i₁ i₂ i₁₂ | case pos
C : Type u_1
inst✝⁵ : Category.{u_5, u_1} C
inst✝⁴ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L : HomologicalComplex₂ C c₁ c₂
φ : K ⟶ L
c₁₂ : ComplexShape I₁₂
inst✝³ : TotalComplexShape c₁ c₂ c₁₂
inst✝² : DecidableEq I₁₂
inst✝¹ : K.HasTotal c₁₂
inst✝ ... | simp [totalAux.d₁_eq' _ c₁₂ h] | no goals | 96dc7301489c7462 |
List.prod_map_ite_eq | Mathlib/Algebra/BigOperators/Group/List/Basic.lean | @[to_additive]
lemma prod_map_ite_eq {A : Type*} [DecidableEq A] (l : List A) (f g : A → G) (a : A) :
(l.map fun x => if x = a then f x else g x).prod
= (f a / g a) ^ (l.count a) * (l.map g).prod | case cons
G : Type u_7
inst✝¹ : CommGroup G
A : Type u_8
inst✝ : DecidableEq A
f g : A → G
a x : A
xs : List A
ih : (map (fun x => if x = a then f x else g x) xs).prod = (f a / g a) ^ count a xs * (map g xs).prod
⊢ (map (fun x => if x = a then f x else g x) (x :: xs)).prod = (f a / g a) ^ count a (x :: xs) * (map g (x ... | simp only [map_cons, prod_cons, nodup_cons, ne_eq, mem_cons, count_cons] at ih ⊢ | case cons
G : Type u_7
inst✝¹ : CommGroup G
A : Type u_8
inst✝ : DecidableEq A
f g : A → G
a x : A
xs : List A
ih : (map (fun x => if x = a then f x else g x) xs).prod = (f a / g a) ^ count a xs * (map g xs).prod
⊢ (if x = a then f x else g x) * (map (fun x => if x = a then f x else g x) xs).prod =
(f a / g a) ^ (c... | cf149ec2fcf674c8 |
MeasureTheory.SimpleFunc.induction | Mathlib/MeasureTheory/Function/SimpleFunc.lean | theorem induction {α γ} [MeasurableSpace α] [AddMonoid γ] {P : SimpleFunc α γ → Prop}
(h_ind :
∀ (c) {s} (hs : MeasurableSet s),
P (SimpleFunc.piecewise s hs (SimpleFunc.const _ c) (SimpleFunc.const _ 0)))
(h_add : ∀ ⦃f g : SimpleFunc α γ⦄, Disjoint (support f) (support g) → P f → P g → P (f + g))... | α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → P f → P g → P (f + g)
f : α →ₛ γ
⊢ P f | generalize h : f.range \ {0} = s | α : Type u_5
γ : Type u_6
inst✝¹ : MeasurableSpace α
inst✝ : AddMonoid γ
P : (α →ₛ γ) → Prop
h_ind : ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), P (piecewise s hs (const α c) (const α 0))
h_add : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → P f → P g → P (f + g)
f : α →ₛ γ
s : Finset γ
h : f.range \ {0} = s... | 01b1b313375647ab |
CategoryTheory.MonoidalClosed.FunctorCategory.homEquiv_naturality_three | Mathlib/CategoryTheory/Closed/FunctorCategory/Basic.lean | lemma homEquiv_naturality_three [∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂]
(f : F₁ ⊗ F₂ ⟶ F₃) (f₃ : F₃ ⟶ F₃') :
homEquiv (f ≫ f₃) = homEquiv f ≫ (ρ_ _).inv ≫ _ ◁ functorHomEquiv _ f₃ ≫
functorEnrichedComp C F₁ F₃ F₃' | C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : MonoidalCategory C
inst✝³ : MonoidalClosed C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
inst✝¹ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₃ F₃' : J ⥤ C
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂
f : F₁ ⊗ F₂ ⟶ F₃
f₃ : F₃ ⟶ F₃'
⊢ { app := fun j => end_... | ext j | case w.h
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : MonoidalCategory C
inst✝³ : MonoidalClosed C
J : Type u₂
inst✝² : Category.{v₂, u₂} J
inst✝¹ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₃ F₃' : J ⥤ C
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasEnrichedHom C F₁ F₂
f : F₁ ⊗ F₂ ⟶ F₃
f₃ : F₃ ⟶ F₃'
j : J
⊢ { app :... | 69e327ebe38fc471 |
Int.testBit_bit_succ | Mathlib/Data/Int/Bitwise.lean | theorem testBit_bit_succ (m b) : ∀ n, testBit (bit b n) (Nat.succ m) = testBit n m
| (n : ℕ) => by rw [bit_coe_nat]; apply Nat.testBit_bit_succ
| -[n+1] => by
dsimp only [testBit]
simp only [bit_negSucc]
cases b <;> simp only [Bool.not_false, Bool.not_true, Nat.testBit_bit_succ]
| m : ℕ
b : Bool
n : ℕ
⊢ (match bit b -[n+1], m.succ with
| ofNat m, n => m.testBit n
| -[m+1], n => !m.testBit n) =
!n.testBit m | simp only [bit_negSucc] | m : ℕ
b : Bool
n : ℕ
⊢ (!(Nat.bit (!b) n).testBit m.succ) = !n.testBit m | 3efbbc734e12bbc3 |
Monotone.leftLim_le | Mathlib/Topology/Order/LeftRightLim.lean | theorem leftLim_le (h : x ≤ y) : leftLim f x ≤ f y | case inr.refine_1
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] x ≠ ⊥
A : (𝓝[<] x).NeBot
⊢ (... | exact (forall_mem_nonempty_iff_neBot.2 A) _ self_mem_nhdsWithin | no goals | 0d90cf806eab6a36 |
SetTheory.PGame.Domineering.moveRight_card | Mathlib/SetTheory/Game/Domineering.lean | theorem moveRight_card {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) :
Finset.card (moveRight b m) + 2 = Finset.card b | b : Board
m : ℤ × ℤ
h : m ∈ right b
⊢ Finset.card b - 1 - 1 + 2 = Finset.card b | exact tsub_add_cancel_of_le (card_of_mem_right h) | no goals | 9e440b8eb7222f22 |
LieAlgebra.isSolvable_tensorProduct_iff | Mathlib/Algebra/Lie/Solvable.lean | theorem isSolvable_tensorProduct_iff : IsSolvable (A ⊗[R] L) ↔ IsSolvable L | case h
R : Type u
L : Type v
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
A : Type u_1
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Module.FaithfullyFlat R A
k : ℕ
h : derivedSeries A (A ⊗[R] L) k = ⊥
⊢ derivedSeries R L k = ⊥ | rw [eq_bot_iff] at h ⊢ | case h
R : Type u
L : Type v
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
A : Type u_1
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Module.FaithfullyFlat R A
k : ℕ
h : derivedSeries A (A ⊗[R] L) k ≤ ⊥
⊢ derivedSeries R L k ≤ ⊥ | 9c5c949fa63473f6 |
IsPrimitiveRoot.norm_sub_one_two | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem norm_sub_one_two {k : ℕ} (hζ : IsPrimitiveRoot ζ (2 ^ k)) (hk : 2 ≤ k)
[H : IsCyclotomicExtension {(2 : ℕ+) ^ k} K L] (hirr : Irreducible (cyclotomic (2 ^ k) K)) :
norm K (ζ - 1) = 2 | K : Type u
L : Type v
inst✝² : Field L
ζ : L
inst✝¹ : Field K
inst✝ : Algebra K L
k : ℕ
hζ : IsPrimitiveRoot ζ (2 ^ k)
hk : 2 ≤ k
H : IsCyclotomicExtension {2 ^ k} K L
hirr : Irreducible (cyclotomic (2 ^ k) K)
⊢ (Algebra.norm K) (ζ - 1) = 2 | have : 2 < (2 : ℕ+) ^ k := by
simp only [← coe_lt_coe, one_coe, pow_coe]
nth_rw 1 [← pow_one 2]
exact Nat.pow_lt_pow_right one_lt_two (lt_of_lt_of_le one_lt_two hk) | K : Type u
L : Type v
inst✝² : Field L
ζ : L
inst✝¹ : Field K
inst✝ : Algebra K L
k : ℕ
hζ : IsPrimitiveRoot ζ (2 ^ k)
hk : 2 ≤ k
H : IsCyclotomicExtension {2 ^ k} K L
hirr : Irreducible (cyclotomic (2 ^ k) K)
this : 2 < 2 ^ k
⊢ (Algebra.norm K) (ζ - 1) = 2 | 06a533966d791d57 |
Submodule.fg_iff_exists_finite_generating_family | Mathlib/RingTheory/Finiteness/Defs.lean | lemma fg_iff_exists_finite_generating_family {A : Type u} [Semiring A] {M : Type v}
[AddCommMonoid M] [Module A M] {N : Submodule A M} :
N.FG ↔ ∃ (G : Type w) (_ : Finite G) (g : G → M), Submodule.span A (Set.range g) = N | A : Type u
inst✝² : Semiring A
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
N : Submodule A M
⊢ N.FG ↔ ∃ G, ∃ (_ : Finite G), ∃ g, span A (range g) = N | constructor | case mp
A : Type u
inst✝² : Semiring A
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
N : Submodule A M
⊢ N.FG → ∃ G, ∃ (_ : Finite G), ∃ g, span A (range g) = N
case mpr
A : Type u
inst✝² : Semiring A
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module A M
N : Submodule A M
⊢ (∃ G, ∃ (_ : Finite G), ∃ g, span ... | af32e5a937d3676d |
CategoryTheory.Presieve.isSheaf_coverage | Mathlib/CategoryTheory/Sites/Coverage.lean | theorem isSheaf_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ Type*) :
Presieve.IsSheaf (toGrothendieck _ K) P ↔
(∀ {X : C} (R : Presieve X), R ∈ K X → Presieve.IsSheafFor P R) | case mp
C : Type u_2
inst✝ : Category.{u_3, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
H : IsSheaf (toGrothendieck C K) P
X : C
R : Presieve X
hR : R ∈ K.covering X
⊢ IsSheafFor P R | rw [Presieve.isSheafFor_iff_generate] | case mp
C : Type u_2
inst✝ : Category.{u_3, u_2} C
K : Coverage C
P : Cᵒᵖ ⥤ Type u_1
H : IsSheaf (toGrothendieck C K) P
X : C
R : Presieve X
hR : R ∈ K.covering X
⊢ IsSheafFor P (Sieve.generate R).arrows | b0ee5f44dbe2a8c4 |
MeasureTheory.submartingale_of_condExp_sub_nonneg | Mathlib/Probability/Martingale/Basic.lean | theorem submartingale_of_condExp_sub_nonneg [IsFiniteMeasure μ] {f : ι → Ω → ℝ} (hadp : Adapted ℱ f)
(hint : ∀ i, Integrable (f i) μ) (hf : ∀ i j, i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i|ℱ i]) :
Submartingale f ℱ μ | Ω : Type u_1
ι : Type u_3
inst✝¹ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ι m0
inst✝ : IsFiniteMeasure μ
f : ι → Ω → ℝ
hadp : Adapted ℱ f
hint : ∀ (i : ι), Integrable (f i) μ
hf : ∀ (i j : ι), i ≤ j → 0 ≤ᶠ[ae μ] μ[f j - f i|↑ℱ i]
i j : ι
hij : i ≤ j
⊢ f i ≤ᶠ[ae μ] μ[f j|↑ℱ i] | rw [← condExp_of_stronglyMeasurable (ℱ.le _) (hadp _) (hint _), ← eventually_sub_nonneg] | Ω : Type u_1
ι : Type u_3
inst✝¹ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ι m0
inst✝ : IsFiniteMeasure μ
f : ι → Ω → ℝ
hadp : Adapted ℱ f
hint : ∀ (i : ι), Integrable (f i) μ
hf : ∀ (i j : ι), i ≤ j → 0 ≤ᶠ[ae μ] μ[f j - f i|↑ℱ i]
i j : ι
hij : i ≤ j
⊢ 0 ≤ᶠ[ae μ] μ[f j|↑ℱ i] - μ[f i|↑ℱ i] | b169bd0b4667d46b |
Lean.Order.chain_apply | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Basic.lean | theorem chain_apply [∀ x, PartialOrder (β x)] {c : (∀ x, β x) → Prop} (hc : chain c) (x : α) :
chain (fun y => ∃ f, c f ∧ f x = y) | α : Sort u
β : α → Sort v
inst✝ : (x : α) → PartialOrder (β x)
c : ((x : α) → β x) → Prop
hc : chain c
x : α
⊢ chain fun y => ∃ f, c f ∧ f x = y | intro _ _ ⟨f, hf, hfeq⟩ ⟨g, hg, hgeq⟩ | α : Sort u
β : α → Sort v
inst✝ : (x : α) → PartialOrder (β x)
c : ((x : α) → β x) → Prop
hc : chain c
x : α
x✝ y✝ : β x
f : (x : α) → β x
hf : c f
hfeq : f x = x✝
g : (x : α) → β x
hg : c g
hgeq : g x = y✝
⊢ x✝ ⊑ y✝ ∨ y✝ ⊑ x✝ | 0b2f42980302a68d |
CategoryTheory.Presieve.compatible_iff_sieveCompatible | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | theorem compatible_iff_sieveCompatible (x : FamilyOfElements P (S : Presieve X)) :
x.Compatible ↔ x.SieveCompatible | C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
x : FamilyOfElements P S.arrows
⊢ x.Compatible ↔ x.SieveCompatible | constructor | case mp
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
x : FamilyOfElements P S.arrows
⊢ x.Compatible → x.SieveCompatible
case mpr
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type w
X : C
S : Sieve X
x : FamilyOfElements P S.arrows
⊢ x.SieveCompatible → x.Compatible | c9cba2214cb59bf0 |
Set.sdiff_singleton_wcovBy | Mathlib/Order/Cover.lean | @[simp] lemma sdiff_singleton_wcovBy (s : Set α) (a : α) : s \ {a} ⩿ s | case h.e'_4.h
α : Type u_1
s : Set α
a : α
ha : a ∈ s
x✝ : α
⊢ x✝ ∈ s ↔ x✝ ∈ insert a (s \ {a}) | simp [ha] | no goals | 655eed34228b7954 |
CategoryTheory.ShortComplex.RightHomologyData.homologyIso_rightHomologyData | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | @[simp]
lemma RightHomologyData.homologyIso_rightHomologyData [S.HasHomology] :
S.rightHomologyData.homologyIso = S.rightHomologyIso.symm | case w
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ S.rightHomologyData.homologyIso.hom = S.rightHomologyIso.symm.hom | dsimp [homologyIso, rightHomologyIso] | case w
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ S.rightHomologyIso.inv ≫ rightHomologyMap' (𝟙 S) S.rightHomologyData S.rightHomologyData = S.rightHomologyIso.inv | f91fa8e7af752a65 |
AEMeasurable.isLUB | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | theorem AEMeasurable.isLUB {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α}
(hf : ∀ i, AEMeasurable (f i) μ) (hg : ∀ᵐ b ∂μ, IsLUB { a | ∃ i, f i b = a } (g b)) :
AEMeasurable g μ | case h
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
g : δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
... | simp_rw [Set.mem_setOf_eq, aeSeq.mk_eq_fun_of_mem_aeSeqSet hf h] | no goals | 22b5a03f49a80425 |
IsConformalMap.is_complex_or_conj_linear | Mathlib/Analysis/Complex/Conformal.lean | theorem IsConformalMap.is_complex_or_conj_linear (h : IsConformalMap g) :
(∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g) ∨
∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g ∘L ↑conjCLE | case intro.intro.intro.intro.intro.inl
c : ℝ
a : Circle
⊢ restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ) = c • (rotation a).toLinearIsometry.toContinuousLinearMap | ext1 | case intro.intro.intro.intro.intro.inl.h
c : ℝ
a : Circle
x✝ : ℂ
⊢ (restrictScalars ℝ (c • ↑a • ContinuousLinearMap.id ℂ ℂ)) x✝ =
(c • (rotation a).toLinearIsometry.toContinuousLinearMap) x✝ | 03ba553ab0d0d9ad |
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
a : Ordinal.{max u v}
⊢ a < c.ord → derivFamily f a < c.ord | have hω : ℵ₀ < c.ord.cof := by
rw [hc.cof_eq]
exact lt_of_le_of_ne hc.1 hc'.symm | 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
a : Ordinal.{max u v}
hω : ℵ₀ < c.ord.cof
⊢ a < c.ord → derivFamily f a < c.ord | a8993783bfefcfe4 |
Rat.num_lt_succ_floor_mul_den | Mathlib/Data/Rat/Floor.lean | theorem num_lt_succ_floor_mul_den (q : ℚ) : q.num < (⌊q⌋ + 1) * q.den | q : ℚ
⊢ 0 < 1 - fract q | have : fract q < 1 := fract_lt_one q | q : ℚ
this : fract q < 1
⊢ 0 < 1 - fract q | 891cb1b04cc4d94e |
CStarAlgebra.rpow_neg_one_le_rpow_neg_one | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean | lemma rpow_neg_one_le_rpow_neg_one {a b : A} (ha : 0 ≤ a) (hab : a ≤ b) (hau : IsUnit a) :
b ^ (-1 : ℝ) ≤ a ^ (-1 : ℝ) | case intro
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : 0 ≤ a
hau : IsUnit a
b : Aˣ
hab : a ≤ ↑b
⊢ ↑b ^ (-1) ≤ a ^ (-1) | lift a to Aˣ using hau | case intro.intro
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
b a : Aˣ
ha : 0 ≤ ↑a
hab : ↑a ≤ ↑b
⊢ ↑b ^ (-1) ≤ ↑a ^ (-1) | 3d0bcf27da9959ca |
CategoryTheory.Limits.zeroProdIso_inv_snd | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | theorem zeroProdIso_inv_snd (X : C) : (zeroProdIso X).inv ≫ prod.snd = 𝟙 X | C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
X : C
⊢ (zeroProdIso X).inv ≫ prod.snd = 𝟙 X | dsimp [zeroProdIso, binaryFanZeroLeft] | C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
X : C
⊢ (limit.isoLimitCone { cone := BinaryFan.mk 0 (𝟙 X), isLimit := binaryFanZeroLeftIsLimit X }).inv ≫ prod.snd = 𝟙 X | 4240bc38c382b05a |
fourierCoeffOn_of_hasDeriv_right | Mathlib/Analysis/Fourier/AddCircle.lean | theorem fourierCoeffOn_of_hasDeriv_right {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ}
{n : ℤ} (hn : n ≠ 0)
(hf : ContinuousOn f [[a, b]])
(hff' : ∀ x, x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x)
(hf' : IntervalIntegrable f' volume a b) :
fourierCoeffOn hab f n = 1 / (-2 * π * I * n)... | a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
⊢ fourierCoeffOn hab f n =
1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - (↑b - ↑a) * fourierCoeffOn hab f' n) | rw [← ofReal_sub] | a b : ℝ
hab : a < b
f f' : ℝ → ℂ
n : ℤ
hn : n ≠ 0
hf : ContinuousOn f [[a, b]]
hff' : ∀ x ∈ Ioo (a ⊓ b) (a ⊔ b), HasDerivWithinAt f (f' x) (Ioi x) x
hf' : IntervalIntegrable f' volume a b
⊢ fourierCoeffOn hab f n =
1 / (-2 * ↑π * I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - ↑(b - a) * fourierCoeffOn hab f' n) | 0be290c2ece914fe |
Dynamics.coverMincard_univ | Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean | lemma coverMincard_univ (T : X → X) {F : Set X} (h : F.Nonempty) (n : ℕ) :
coverMincard T F univ n = 1 | case intro
X : Type u_1
T : X → X
F : Set X
n : ℕ
x : X
h✝ : x ∈ F
this : IsDynCoverOf T F univ n ↑{x}
⊢ ↑{x}.card = 1 | rw [Finset.card_singleton, Nat.cast_one] | no goals | 1832f6ac09cf7d05 |
CategoryTheory.rightAdjointMate_comp | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | theorem rightAdjointMate_comp {X Y Z : C} [HasRightDual X] [HasRightDual Y] {f : X ⟶ Y}
{g : Xᘁ ⟶ Z} :
fᘁ ≫ g =
(ρ_ (Yᘁ)).inv ≫
_ ◁ η_ X (Xᘁ) ≫ _ ◁ (f ⊗ g) ≫ (α_ (Yᘁ) Y Z).inv ≫ ε_ Y (Yᘁ) ▷ _ ≫ (λ_ Z).hom :=
calc
_ = 𝟙 _ ⊗≫ (Yᘁ : C) ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ⊗≫ (ε_ Y Yᘁ ▷ Xᘁ ≫ 𝟙_ C ◁ g) ⊗≫ �... | C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
X Y Z : C
inst✝¹ : HasRightDual X
inst✝ : HasRightDual Y
f : X ⟶ Y
g : Xᘁ ⟶ Z
⊢ fᘁ ≫ g = 𝟙 Yᘁ ⊗≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ⊗≫ (ε_ Y Yᘁ ▷ Xᘁ ≫ 𝟙_ C ◁ g) ⊗≫ 𝟙 Z | dsimp only [rightAdjointMate] | C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
X Y Z : C
inst✝¹ : HasRightDual X
inst✝ : HasRightDual Y
f : X ⟶ Y
g : Xᘁ ⟶ Z
⊢ ((ρ_ Yᘁ).inv ≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ≫ (α_ Yᘁ Y Xᘁ).inv ≫ ε_ Y Yᘁ ▷ Xᘁ ≫ (λ_ Xᘁ).hom) ≫ g =
𝟙 Yᘁ ⊗≫ Yᘁ ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ⊗≫ (ε_ Y Yᘁ ▷ Xᘁ ≫ 𝟙_ C ◁ g) ⊗≫ �... | 3d277c4849bedc73 |
Ordering.isLT_swap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Ord.lean | theorem isLT_swap {o : Ordering} : o.swap.isLT = o.isGT | o : Ordering
⊢ o.swap.isLT = o.isGT | cases o <;> simp | no goals | f24cc939e399dba4 |
Turing.ToPartrec.cont_eval_fix | Mathlib/Computability/TMConfig.lean | theorem cont_eval_fix {f k v} (fok : Code.Ok f) :
Turing.eval step (stepNormal f (Cont.fix f k) v) =
f.fix.eval v >>= fun v => Turing.eval step (Cfg.ret k v) | case pos
f : Code
k : Cont
v✝ : List ℕ
fok : f.Ok
x : Cfg
v'✝ : List ℕ
he✝¹ : v'✝ ∈ f.fix.eval v✝
hr✝ : x ∈ eval step (Cfg.ret k v'✝)
hr : x ∈ eval step (stepRet k v'✝)
v : List ℕ
he✝ : v'✝ ∈ f.fix.eval v
IH :
∀ (a'' : List ℕ),
Sum.inr a'' ∈ Part.map (fun v => if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tai... | cases he₂ | case pos.refl
f : Code
k : Cont
v✝ : List ℕ
fok : f.Ok
x : Cfg
v : List ℕ
IH :
∀ (a'' : List ℕ),
Sum.inr a'' ∈ Part.map (fun v => if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail) (f.eval v) →
x ∈ eval step (stepNormal f (Cont.fix f k) a'')
v' : List ℕ
he₁ : v' ∈ f.eval v
h : v'.headI = 0
he✝¹ : v'.ta... | 4a2318a05b9f1866 |
ENNReal.rpow_le_rpow_of_exponent_le | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | theorem rpow_le_rpow_of_exponent_le {x : ℝ≥0∞} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) :
x ^ y ≤ x ^ z | x : ℝ≥0∞
y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ x ^ y ≤ x ^ z | cases x | case top
y z : ℝ
hyz : y ≤ z
hx : 1 ≤ ⊤
⊢ ⊤ ^ y ≤ ⊤ ^ z
case coe
y z : ℝ
hyz : y ≤ z
x✝ : ℝ≥0
hx : 1 ≤ ↑x✝
⊢ ↑x✝ ^ y ≤ ↑x✝ ^ z | 54efd8cf616716a9 |
GromovHausdorff.hausdorffDist_optimal | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Nonempty X]
{Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) = ghDist X Y | case neg
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
inhabited_h✝ : Inhabited X
inhabited_h : Inhabited Y
A :
∀ (p q : NonemptyCompacts ↥(lp (fun n => ℝ) ⊤)),
⟦p⟧ = toGHSpace X →
⟦q⟧ = toGHSpace Y ... | calc
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤
HD (candidatesBDist X Y) :=
hausdorffDist_optimal_le_HD _ _ candidatesBDist_mem_candidatesB
_ ≤ diam (univ : Set X) + 1 + diam (univ : Set Y) := HD_candidatesBDist_le
_ ≤ hausdorffDist (p : Set ℓ_infty_ℝ) q := not_lt.1 h | no goals | e1b74305bb180259 |
ContMDiffWithinAt.mpullbackWithin_vectorField_of_eq' | Mathlib/Geometry/Manifold/VectorField.lean | /-- The pullback of a `C^m` vector field by a `C^n` function with invertible derivative and
with `m + 1 ≤ n` is `C^m`.
Version within a set at a point, with a set used for the pullback possibly larger. -/
protected lemma _root_.ContMDiffWithinAt.mpullbackWithin_vectorField_of_eq' {u : Set M}
(hV : ContMDiffWithinAt... | 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁴ : TopologicalSpace H
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
H' : Type u_5
inst✝⁹ : TopologicalSpace H'
E' : Type u_6
in... | exact ContMDiffWithinAt.mpullbackWithin_vectorField' hV hf hf' hx₀ hs hmn hst hu | no goals | 4ff92d9fe96f801d |
Stream'.WSeq.head_congr | Mathlib/Data/Seq/WSeq.lean | theorem head_congr : ∀ {s t : WSeq α}, s ~ʷ t → head s ~ head t | case intro.intro
α : Type u
s t : WSeq α
h : s ~ʷ t
o : Option α
ho : o ∈ s.head
ds : Option (α × WSeq α)
dsm : ds ∈ s.destruct
dse : Prod.fst <$> ds = o
⊢ Prod.fst <$> ds ∈ t.head | obtain ⟨l, r⟩ := destruct_congr h | case intro.intro.intro
α : Type u
s t : WSeq α
h : s ~ʷ t
o : Option α
ho : o ∈ s.head
ds : Option (α × WSeq α)
dsm : ds ∈ s.destruct
dse : Prod.fst <$> ds = o
l : ∀ {a : Option (α × WSeq α)}, a ∈ s.destruct → ∃ b, b ∈ t.destruct ∧ BisimO (fun x1 x2 => x1 ~ʷ x2) a b
r : ∀ {b : Option (α × WSeq α)}, b ∈ t.destruct → ∃ a... | 57a559acc372992b |
IsLocalization.localization_localization_surj | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | theorem localization_localization_surj [IsLocalization N T] (x : T) :
∃ y : R × localizationLocalizationSubmodule M N,
x * algebraMap R T y.2 = algebraMap R T y.1 | case intro.mk.intro.mk.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N ... | dsimp only at eq₁ eq₂ eq₃ | case intro.mk.intro.mk.intro.mk
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N ... | 2ce20bcc6beced05 |
Finset.subset_set_image₂ | Mathlib/Data/Finset/NAry.lean | theorem subset_set_image₂ {s : Set α} {t : Set β} (hu : ↑u ⊆ image2 f s t) :
∃ (s' : Finset α) (t' : Finset β), ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ image₂ f s' t' | α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
u : Finset γ
s : Set α
t : Set β
hu : ↑u ⊆ image2 f s t
⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ image₂ f s' t' | rw [← Set.image_prod, subset_set_image_iff] at hu | α : Type u_1
β : Type u_3
γ : Type u_5
inst✝ : DecidableEq γ
f : α → β → γ
u : Finset γ
s : Set α
t : Set β
hu : ∃ s', ↑s' ⊆ s ×ˢ t ∧ image (fun x => f x.1 x.2) s' = u
⊢ ∃ s' t', ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ image₂ f s' t' | c61cf2c3eef63961 |
FixedPoints.toAlgHom_bijective | Mathlib/FieldTheory/Fixed.lean | theorem toAlgHom_bijective [Finite G] [FaithfulSMul G F] :
Function.Bijective (MulSemiringAction.toAlgHom _ _ : G → F →ₐ[subfield G F] F) | case intro.right.a
G : Type u_1
F : Type u_2
inst✝⁴ : Group G
inst✝³ : Field F
inst✝² : MulSemiringAction G F
inst✝¹ : Finite G
inst✝ : FaithfulSMul G F
val✝ : Fintype G
⊢ Fintype.card (F →ₐ[↥(subfield G F)] F) ≤ finrank (↥(subfield G F)) F | exact LE.le.trans_eq (finrank_algHom _ F) (finrank_linearMap_self _ _ _) | no goals | 72fe18a2d5a5305b |
OrderIso.complementedLattice_iff | Mathlib/Order/Hom/Basic.lean | theorem OrderIso.complementedLattice_iff (f : α ≃o β) :
ComplementedLattice α ↔ ComplementedLattice β :=
⟨by intro; exact f.complementedLattice,
by intro; exact f.symm.complementedLattice⟩
| α : Type u_2
β : Type u_3
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : BoundedOrder α
inst✝ : BoundedOrder β
f : α ≃o β
⊢ ComplementedLattice β → ComplementedLattice α | intro | α : Type u_2
β : Type u_3
inst✝³ : Lattice α
inst✝² : Lattice β
inst✝¹ : BoundedOrder α
inst✝ : BoundedOrder β
f : α ≃o β
a✝ : ComplementedLattice β
⊢ ComplementedLattice α | 9af58b1bea5e94c4 |
AkraBazziRecurrence.GrowsPolynomially.add_isLittleO | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x | f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
x : ℝ
hf₁ : ∀ u ∈ Set.... | gcongr | no goals | dd533a6f58fe3819 |
CategoryTheory.MonoidalCategory.tensor_associativity | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
(tensorμ X₁ X₂ Y₁ Y₂ ▷ (Z₁ ⊗ Z₂)) ≫
tensorμ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) Z₁ Z₂ ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =
(α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫
((X₁ ⊗ X₂) ◁ tensorμ Y₁ Y₂ Z₁ Z₂) ≫ tensorμ X₁ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C
⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ≫
X₁ ◁ (α_ X₂ Y₁ Y₂).inv ≫ X₁ ◁ (β_ X₂ Y₁).hom ▷ Y₂ ≫ X₁ ◁ (α_ Y₁ X₂ Y₂).hom ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv) ▷
(Z₁ ⊗ Z₂) ≫
((α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z... | simp only [braiding_tensor_left, braiding_tensor_right] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C
⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ≫
X₁ ◁ (α_ X₂ Y₁ Y₂).inv ≫ X₁ ◁ (β_ X₂ Y₁).hom ▷ Y₂ ≫ X₁ ◁ (α_ Y₁ X₂ Y₂).hom ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv) ▷
(Z₁ ⊗ Z₂) ≫
((α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z... | 7602b5d4aa70aebc |
CategoryTheory.ShortComplex.isIso_homologyπ | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | lemma isIso_homologyπ (hf : S.f = 0) [S.HasHomology] :
IsIso S.homologyπ | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
hf : S.f = 0
inst✝ : S.HasHomology
this : IsIso S.leftHomologyπ
⊢ IsIso (S.leftHomologyπ ≫ S.leftHomologyIso.hom) | infer_instance | no goals | cdbbd4b054e83436 |
Complex.arctan_tan | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | theorem arctan_tan {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re) (h₂ : z.re ≤ π / 2) :
arctan (tan z) = z | z : ℂ
h₀ : z ≠ ↑π / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ -I / 2 * log ((cos z + sin z * I) / (cos (-z) + sin (-z) * I)) = z | rw [← exp_mul_I, ← exp_mul_I, ← exp_sub, show z * I - -z * I = 2 * (I * z) by ring, log_exp,
show -I / 2 * (2 * (I * z)) = -(I * I) * z by ring, I_mul_I, neg_neg, one_mul] | case hx₁
z : ℂ
h₀ : z ≠ ↑π / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ -π < (2 * (I * z)).im
case hx₂
z : ℂ
h₀ : z ≠ ↑π / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ (2 * (I * z)).im ≤ π | d18e3a0fad89d01b |
List.product_eq_productTR | Mathlib/.lake/packages/batteries/Batteries/Data/List/Basic.lean | theorem product_eq_productTR : @product = @productTR | case h.h.h.h
α : Type u_2
β : Type u_1
l₁ : List α
l₂ : List β
⊢ flatMap (fun a => map (Prod.mk a) l₂) l₁ = #[].toList ++ flatMap ?h.h.h.h.G l₁
case h.h.h.h.G
α : Type u_2
β : Type u_1
l₁ : List α
l₂ : List β
⊢ α → List (α × β)
case h.h.h.h.H
α : Type u_2
β : Type u_1
l₁ : List α
l₂ : List β
⊢ ∀ (acc : Array (α × β))... | rfl | case h.h.h.h.H
α : Type u_2
β : Type u_1
l₁ : List α
l₂ : List β
⊢ ∀ (acc : Array (α × β)) (a : α),
(foldl (fun acc b => acc.push (a, b)) acc l₂).toList = acc.toList ++ map (Prod.mk a) l₂ | 3fd4a1b4ab1aff1b |
TopologicalSpace.separableSpace_iff_countable | Mathlib/Topology/Bases.lean | theorem separableSpace_iff_countable [DiscreteTopology α] : SeparableSpace α ↔ Countable α | α : Type u
t : TopologicalSpace α
inst✝ : DiscreteTopology α
⊢ SeparableSpace α ↔ Countable α | simp [separableSpace_iff, countable_univ_iff] | no goals | c297e395d36538c1 |
Configuration.HasLines.card_le | Mathlib/Combinatorics/Configuration.lean | theorem HasLines.card_le [HasLines P L] [Fintype P] [Fintype L] :
Fintype.card P ≤ Fintype.card L | case intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hc₂ : ¬Fintype.card P ≤ Fintype.card L
f : L → P
hf₁ : Function.Injective f
hf₂ : ∀ (l : L), f l ∉ l
p : P
hp : ¬∃ a, f a = p
⊢ ∑ p ∈ map { toFun := f, inj' := hf₁ } univ, lineCount L p < ∑ p : P, lin... | refine sum_lt_sum_of_subset (subset_univ _) (mem_univ p) ?_ ?_ fun p _ _ ↦ zero_le _ | case intro.refine_1
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hc₂ : ¬Fintype.card P ≤ Fintype.card L
f : L → P
hf₁ : Function.Injective f
hf₂ : ∀ (l : L), f l ∉ l
p : P
hp : ¬∃ a, f a = p
⊢ p ∉ map { toFun := f, inj' := hf₁ } univ
case intro.refine_2
P... | 2f43c1eb0f46b0fe |
Set.Ico_union_Ici' | Mathlib/Order/Interval/Set/Basic.lean | theorem Ico_union_Ici' (h₁ : c ≤ b) : Ico a b ∪ Ici c = Ici (min a c) | case pos
α : Type u_1
inst✝ : LinearOrder α
a b c : α
h₁ : c ≤ b
x : α
hc : c ≤ x
⊢ a ≤ x ∧ x < b ∨ c ≤ x ↔ a ≤ x ∨ c ≤ x | simp only [hc, or_true] | no goals | 60f2b347afb4f106 |
VitaliFamily.withDensity_limRatioMeas_eq | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem withDensity_limRatioMeas_eq : μ.withDensity (v.limRatioMeas hρ) = ρ | case h.refine_2
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
⊢ ρ s ≤ (μ.withDensity (v.... | have :
Tendsto (fun t : ℝ≥0 => (t : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s) (𝓝[>] 1)
(𝓝 ((1 : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s)) := by
refine ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) ?_
simp only [ENNReal.coe_one, true_or, Ne, not_false_iff, one_ne_zero] | case h.refine_2
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
this :
Tendsto (fun t =>... | c98b00e271c474ed |
exists_forall_closed_ball_dist_add_le_two_mul_sub | Mathlib/Analysis/Convex/Uniform.lean | theorem exists_forall_closed_ball_dist_add_le_two_mul_sub (hε : 0 < ε) (r : ℝ) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ | case inr.intro.intro
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
r : ℝ
hr : 0 < r
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y : E⦄, ‖y‖ ≤ 1 → ε / r ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
x : E
hx : ‖r⁻¹ • x‖ ≤ 1
y : E
hy : ‖r⁻¹ • y‖ ≤ 1
hxy : ε ≤ ‖x - y‖... | have := h hx hy | case inr.intro.intro
E : Type u_1
inst✝² : SeminormedAddCommGroup E
inst✝¹ : UniformConvexSpace E
ε : ℝ
inst✝ : NormedSpace ℝ E
hε : 0 < ε
r : ℝ
hr : 0 < r
δ : ℝ
hδ : 0 < δ
h : ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y : E⦄, ‖y‖ ≤ 1 → ε / r ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
x : E
hx : ‖r⁻¹ • x‖ ≤ 1
y : E
hy : ‖r⁻¹ • y‖ ≤ 1
hxy : ε ≤ ‖x - y‖... | f878a0f9eb4a1c37 |
Real.rpow_eq_zero_iff_of_nonneg | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 | x y : ℝ
hx : 0 ≤ x
⊢ x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 | simp only [rpow_def_of_nonneg hx] | x y : ℝ
hx : 0 ≤ x
⊢ (if x = 0 then if y = 0 then 1 else 0 else rexp (log x * y)) = 0 ↔ x = 0 ∧ y ≠ 0 | 44817a059d6596ab |
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