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 |
|---|---|---|---|---|---|---|
CategoryTheory.Functor.IsLocalization.of_equivalence_target | Mathlib/CategoryTheory/Localization/Predicate.lean | theorem of_equivalence_target {E : Type*} [Category E] (L' : C ⥤ E) (eq : D ≌ E)
[L.IsLocalization W] (e : L ⋙ eq.functor ≅ L') : L'.IsLocalization W | C : Type u_1
D : Type u_2
inst✝³ : Category.{u_6, u_1} C
inst✝² : Category.{u_7, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_4
inst✝¹ : Category.{u_5, u_4} E
L' : C ⥤ E
eq : D ≌ E
inst✝ : L.IsLocalization W
e : L ⋙ eq.functor ≅ L'
h : W.IsInvertedBy L'
F₁ : W.Localization ⥤ D := Construction.lift L ⋯
F₂ : W.Loca... | let e' : F₁ ⋙ eq.functor ≅ F₂ := liftNatIso W.Q W (L ⋙ eq.functor) L' _ _ e | C : Type u_1
D : Type u_2
inst✝³ : Category.{u_6, u_1} C
inst✝² : Category.{u_7, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_4
inst✝¹ : Category.{u_5, u_4} E
L' : C ⥤ E
eq : D ≌ E
inst✝ : L.IsLocalization W
e : L ⋙ eq.functor ≅ L'
h : W.IsInvertedBy L'
F₁ : W.Localization ⥤ D := Construction.lift L ⋯
F₂ : W.Loca... | 4ebbdd2d47052214 |
not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux | Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | theorem not_integrableOn_of_tendsto_norm_atTop_of_deriv_isBigO_filter_aux
[CompleteSpace E] {f : ℝ → E} {g : ℝ → F}
{k : Set ℝ} (l : Filter ℝ) [NeBot l] [TendstoIxxClass Icc l l]
(hl : k ∈ l) (hd : ∀ᶠ x in l, DifferentiableAt ℝ f x) (hf : Tendsto (fun x => ‖f x‖) l atTop)
(hfg : deriv f =O[l] g) : ¬Inte... | case intro.intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x... | have h : ∀ᶠ x : ℝ × ℝ in l ×ˢ l,
∀ y ∈ [[x.1, x.2]], (DifferentiableAt ℝ f y ∧ ‖deriv f y‖ ≤ C * ‖g y‖) ∧ y ∈ k :=
(tendsto_fst.uIcc tendsto_snd).eventually ((hd.and hC.bound).and hl).smallSets | case intro.intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : CompleteSpace E
f : ℝ → E
g : ℝ → F
k : Set ℝ
l : Filter ℝ
inst✝¹ : l.NeBot
inst✝ : TendstoIxxClass Icc l l
hl : k ∈ l
hd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x
hf : Tendsto (fun x... | 5860b58ad524edb0 |
LinearMap.hasEigenvalue_zero_tfae | Mathlib/LinearAlgebra/Eigenspace/Zero.lean | lemma hasEigenvalue_zero_tfae (φ : Module.End K M) :
List.TFAE [
Module.End.HasEigenvalue φ 0,
IsRoot (minpoly K φ) 0,
constantCoeff φ.charpoly = 0,
LinearMap.det φ = 0,
⊥ < ker φ,
∃ (m : M), m ≠ 0 ∧ φ m = 0 ] | K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
tfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0
tfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0
tfae_3_to_4 : constantCoeff (charpoly φ) = 0 → LinearMap.det φ = 0
... | contrapose! | K : Type u_2
M : Type u_3
inst✝³ : Field K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
φ : End K M
tfae_1_iff_2 : φ.HasEigenvalue 0 ↔ (minpoly K φ).IsRoot 0
tfae_2_to_3 : (minpoly K φ).IsRoot 0 → constantCoeff (charpoly φ) = 0
tfae_3_to_4 : constantCoeff (charpoly φ) = 0 → LinearMap.det φ = 0
... | 06b5e7f7bb2990f7 |
LucasLehmer.sMod_lt | Mathlib/NumberTheory/LucasLehmer.lean | theorem sMod_lt (p : ℕ) (hp : p ≠ 0) (i : ℕ) : sMod p i < 2 ^ p - 1 | p : ℕ
hp : p ≠ 0
i : ℕ
⊢ |2 ^ p - 1| = 2 ^ p - 1 | exact abs_of_nonneg (mersenne_int_pos hp).le | no goals | 8b4643d36ff8b345 |
MeasureTheory.Measure.isEverywherePos_everywherePosSubset_of_measure_ne_top | Mathlib/MeasureTheory/Measure/EverywherePos.lean | /-- In a space with an inner regular measure for finite measure sets, the everywhere positive subset
of a measurable set of finite measure is itself everywhere positive. This is not obvious as
`μ.everywherePosSubset s` is defined as the points whose neighborhoods intersect `s` along positive
measure subsets, but this d... | case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegularCompactLTTop
hs : MeasurableSet s
h's : μ s ≠ ⊤
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈... | have A : 0 < μ (u ∩ s) := by
have : u ∩ s ∈ 𝓝[s] x := by rw [inter_comm]; exact inter_mem_nhdsWithin s u_mem
exact hx.2 _ this | case intro.intro
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegularCompactLTTop
hs : MeasurableSet s
h's : μ s ≠ ⊤
x : α
hx : x ∈ μ.everywherePosSubset s
n : Set α
hn : n ∈ 𝓝[μ.everywherePosSubset s] x
u : Set α
u_mem : u ∈... | fddce268d4beba5a |
NNReal.coe_iInf | Mathlib/Data/NNReal/Defs.lean | theorem coe_iInf {ι : Sort*} (s : ι → ℝ≥0) : (↑(⨅ i, s i) : ℝ) = ⨅ i, ↑(s i) | ι : Sort u_2
s : ι → ℝ≥0
⊢ sInf (Set.range (toReal ∘ fun i => s i)) = sInf (Set.range fun i => ↑(s i)) | rfl | no goals | 93ca1be55cb194ca |
Complex.tendsto_tsum_powerSeries_nhdsWithin_stolzSet | Mathlib/Analysis/Complex/AbelLimit.lean | theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
(h : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)) {M : ℝ} :
Tendsto (fun z ↦ ∑' n, f n * z ^ n) (𝓝[stolzSet M] 1) (𝓝 l) | case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / ... | simp_rw [Metric.tendsto_atTop, dist_eq_norm, norm_sub_rev] at p | case right
f : ℕ → ℂ
l : ℂ
h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)
M : ℝ
hM : 1 < M
s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i
g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n
hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε
ε : ℝ
εpos : ε > 0
B₁ : ℕ
hB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / ... | e1f8d437480849f9 |
hyperoperation_two | Mathlib/Data/Nat/Hyperoperation.lean | theorem hyperoperation_two : hyperoperation 2 = (· * ·) | case h.h.succ
m bn : ℕ
bih : hyperoperation 2 m bn = m * bn
⊢ (fun x1 x2 => x1 + x2) m (m * bn) = m * (bn + 1) | dsimp only | case h.h.succ
m bn : ℕ
bih : hyperoperation 2 m bn = m * bn
⊢ m + m * bn = m * (bn + 1) | 2701f86d00b32a67 |
contDiff_infty_iff_fderiv | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem contDiff_infty_iff_fderiv :
ContDiff 𝕜 ∞ f ↔ Differentiable 𝕜 f ∧ ContDiff 𝕜 ∞ (fderiv 𝕜 f) | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
⊢ ContDiff 𝕜 ∞ f ↔ Differentiable 𝕜 f ∧ ContDiff 𝕜 ∞ (fderiv 𝕜 f) | rw [← ENat.coe_top_add_one, contDiff_succ_iff_fderiv] | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
⊢ Differentiable 𝕜 f ∧ (∞ = ω → AnalyticOnNhd 𝕜 f univ) ∧ ContDiff 𝕜 ∞ (fderiv 𝕜 f) ↔
Differentiable 𝕜 f ∧ ContDiff 𝕜... | c8d3a25da4462275 |
CategoryTheory.Limits.Sigma.ι_reindex_hom | Mathlib/CategoryTheory/Limits/Shapes/Products.lean | theorem Sigma.ι_reindex_hom (b : β) :
Sigma.ι (f ∘ ε) b ≫ (Sigma.reindex ε f).hom = Sigma.ι f (ε b) | case p
β : Type w
C : Type u
inst✝² : Category.{v, u} C
γ : Type w'
ε : β ≃ γ
f : γ → C
inst✝¹ : HasCoproduct f
inst✝ : HasCoproduct (f ∘ ⇑ε)
b : β
h :
(Discrete.functor f).map (Discrete.eqToHom' ⋯) ≫ colimit.ι (Discrete.functor f) { as := ε b } =
colimit.ι (Discrete.functor f) { as := ε (ε.symm (ε b)) }
⊢ { as :... | simp | no goals | 533bcd754ca46240 |
Matrix.det_eq_prod_roots_charpoly_of_splits | Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | theorem det_eq_prod_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
A.det = (Matrix.charpoly A).roots.prod | n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
R : Type u_2
inst✝ : Field R
A : Matrix n n R
hAps : Splits (RingHom.id R) A.charpoly
⊢ A.det = A.charpoly.roots.prod | rw [det_eq_sign_charpoly_coeff, ← charpoly_natDegree_eq_dim A,
Polynomial.prod_roots_eq_coeff_zero_of_monic_of_splits A.charpoly_monic hAps, ← mul_assoc,
← pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul] | no goals | d2ceff898861a9fc |
List.merge_stable | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean | theorem merge_stable : ∀ (xs ys) (_ : ∀ x y, x ∈ xs → y ∈ ys → x.2 ≤ y.2),
(merge xs ys (zipIdxLE le)).map (·.1) = merge (xs.map (·.1)) (ys.map (·.1)) le
| [], ys, _ => by simp [merge]
| xs, [], _ => by simp [merge]
| (i, x) :: xs, (j, y) :: ys, h => by
simp only [merge, zipIdxLE, map_cons]
split <;> ... | case isTrue.x
α : Type u_1
le : α → α → Bool
i : α
x : Nat
xs : List (α × Nat)
j : α
y : Nat
ys : List (α × Nat)
h : ∀ (x_1 y_1 : α × Nat), x_1 ∈ (i, x) :: xs → y_1 ∈ (j, y) :: ys → x_1.snd ≤ y_1.snd
w : le i j = true
⊢ ∀ (x y_1 : α × Nat), x ∈ xs → y_1 ∈ (j, y) :: ys → x.snd ≤ y_1.snd | exact fun x' y' mx my => h x' y' (mem_cons_of_mem (i, x) mx) my | no goals | 4501691da509f498 |
Std.DHashMap.Internal.exists_bucket_of_uset | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Model.lean | theorem exists_bucket_of_uset [BEq α] [Hashable α]
(self : Array (AssocList α β)) (i : USize) (hi : i.toNat < self.size) (d : AssocList α β) :
∃ l, Perm (toListModel self) (self[i.toNat].toList ++ l) ∧
Perm (toListModel (self.uset i d hi)) (d.toList ++ l) ∧
(∀ [LawfulHashable α], IsHashSelf self →
... | case intro.intro.intro.intro.refine_3
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
self : Array (AssocList α β)
i : USize
hi : i.toNat < self.size
d : AssocList α β
h₀ : 0 < self.size
l₁ l₂ : List (AssocList α β)
h₁ : self.toList = l₁ ++ self[i] :: l₂
h₂ : l₁.length = i.toNat
h₃ : (self.uset i d hi).toL... | simp only [containsKey_append, Bool.or_eq_false_iff] | case intro.intro.intro.intro.refine_3
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : Hashable α
self : Array (AssocList α β)
i : USize
hi : i.toNat < self.size
d : AssocList α β
h₀ : 0 < self.size
l₁ l₂ : List (AssocList α β)
h₁ : self.toList = l₁ ++ self[i] :: l₂
h₂ : l₁.length = i.toNat
h₃ : (self.uset i d hi).toL... | 6de04445a420b4fd |
ContDiffWithinAt.comp | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem ContDiffWithinAt.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F} (x : E)
(hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x) (st : MapsTo f s t) :
ContDiffWithinAt 𝕜 n (g ∘ f) s x | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : WithTop ℕ∞... | let w := insert x s ∩ (u ∩ f ⁻¹' v) | case intro.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : WithTop ℕ∞... | 005b47312591acda |
MeasureTheory.setIntegral_condExpL1CLM_of_measure_ne_top | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | theorem setIntegral_condExpL1CLM_of_measure_ne_top (f : α →₁[μ] F') (hs : MeasurableSet[m] s)
(hμs : μ s ≠ ∞) : ∫ x in s, condExpL1CLM F' hm μ f x ∂μ = ∫ x in s, f x ∂μ | case refine_2
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
inst✝¹ : CompleteSpace F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
s : Set α
f✝ : ↥(Lp F' 1 μ)
hs : MeasurableSet s
hμs : μ s ≠ ⊤
f g : α → F'
hf_Lp : MemLp f 1 μ
hg_Lp : MemLp g 1... | rw [integral_add (L1.integrable_coeFn _).integrableOn (L1.integrable_coeFn _).integrableOn,
integral_add (L1.integrable_coeFn _).integrableOn (L1.integrable_coeFn _).integrableOn, hf,
hg] | no goals | 2606e5a544f6a48a |
SimpContFract.determinant_aux | Mathlib/Algebra/ContinuedFractions/Determinant.lean | theorem determinant_aux (hyp : n = 0 ∨ ¬(↑s : GenContFract K).TerminatedAt (n - 1)) :
((↑s : GenContFract K).contsAux n).a * ((↑s : GenContFract K).contsAux (n + 1)).b -
((↑s : GenContFract K).contsAux n).b * ((↑s : GenContFract K).contsAux (n + 1)).a =
(-1) ^ n | K : Type u_1
inst✝ : Field K
s : SimpContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬(↑s).TerminatedAt (n + 1 - 1)
g : GenContFract K := ↑s
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH : n = 0 ∨ ¬(↑s).Termi... | rw [gp_a_eq_one, this.symm] | K : Type u_1
inst✝ : Field K
s : SimpContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬(↑s).TerminatedAt (n + 1 - 1)
g : GenContFract K := ↑s
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH : n = 0 ∨ ¬(↑s).Termi... | 5478ed7427a774c4 |
HomologicalComplex.mem_quasiIso_iff | Mathlib/Algebra/Homology/QuasiIso.lean | @[simp]
lemma mem_quasiIso_iff (f : K ⟶ L) : quasiIso C c f ↔ QuasiIso f | ι : Type u_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
inst✝ : CategoryWithHomology C
f : K ⟶ L
⊢ quasiIso C c f ↔ QuasiIso f | rfl | no goals | 5cd7456addbf2f6e |
MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_top | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | theorem tendstoInMeasure_of_tendsto_eLpNorm_top {E} [NormedAddCommGroup E] {f : ι → α → E}
{g : α → E} {l : Filter ι} (hfg : Tendsto (fun n => eLpNorm (f n - g) ∞ μ) l (𝓝 0)) :
TendstoInMeasure μ f l g | α : Type u_1
ι : Type u_2
m : MeasurableSpace α
μ : Measure α
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ι → α → E
g : α → E
l : Filter ι
δ : ℝ
hδ : 0 < δ
hfg : Tendsto (fun n => essSup (fun x => ‖(f n - g) x‖ₑ) μ) l (𝓝 0)
⊢ Tendsto (fun i => μ {x | δ ≤ dist (f i x) (g x)}) l (𝓝 0) | rw [ENNReal.tendsto_nhds_zero] at hfg ⊢ | α : Type u_1
ι : Type u_2
m : MeasurableSpace α
μ : Measure α
E : Type u_5
inst✝ : NormedAddCommGroup E
f : ι → α → E
g : α → E
l : Filter ι
δ : ℝ
hδ : 0 < δ
hfg : ∀ ε > 0, ∀ᶠ (x : ι) in l, essSup (fun x_1 => ‖(f x - g) x_1‖ₑ) μ ≤ ε
⊢ ∀ ε > 0, ∀ᶠ (x : ι) in l, μ {x_1 | δ ≤ dist (f x x_1) (g x_1)} ≤ ε | aeddaca91d5c7b38 |
Batteries.TransCmp.compareOfLessAndEq | Mathlib/.lake/packages/batteries/Batteries/Classes/Order.lean | theorem TransCmp.compareOfLessAndEq
[LT α] [DecidableRel (LT.lt (α := α))] [DecidableEq α]
(lt_irrefl : ∀ x : α, ¬x < x)
(lt_trans : ∀ {x y z : α}, x < y → y < z → x < z)
(lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y) :
TransCmp (α := α) (compareOfLessAndEq · ·) | α : Type u_1
inst✝² : LT α
inst✝¹ : DecidableRel LT.lt
inst✝ : DecidableEq α
lt_irrefl : ∀ (x : α), ¬x < x
lt_trans : ∀ {x y z : α}, x < y → y < z → x < z
lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y
this : OrientedCmp fun x1 x2 => compareOfLessAndEq x1 x2
x y z : α
yx : ¬y < x
zy : ¬z < y
zx : z < x
xy : ¬x < y
... | exact zy (lt_antisymm yx xy ▸ zx) | no goals | 265051c307a250c5 |
OreLocalization.cardinalMk_le_lift_cardinalMk_of_commute | Mathlib/GroupTheory/OreLocalization/Cardinality.lean | theorem cardinalMk_le_lift_cardinalMk_of_commute (hc : ∀ s s' : S, Commute s s') :
#(OreLocalization S X) ≤ lift.{u} #X | R : Type u
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type v
inst✝ : MulAction R X
hc✝ : ∀ (s s' : ↥S), Commute s s'
h✝ : Infinite X
x : X
s s' : ↥S
h : s • x = s' • x
hc : Commute s s'
⊢ ∃ u v, u • x = v • x ∧ ↑u * ↑s' = v * ↑s | refine ⟨s, s'.1, h, ?_⟩ | R : Type u
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type v
inst✝ : MulAction R X
hc✝ : ∀ (s s' : ↥S), Commute s s'
h✝ : Infinite X
x : X
s s' : ↥S
h : s • x = s' • x
hc : Commute s s'
⊢ ↑s * ↑s' = ↑s' * ↑s | 8eaf46585280267c |
Nat.ordProj_dvd | Mathlib/Data/Nat/Factorization/Defs.lean | theorem ordProj_dvd (n p : ℕ) : ordProj[p] n ∣ n | n p : ℕ
hp : Prime p
⊢ p ^ count p n.primeFactorsList ∣ n | apply dvd_of_primeFactorsList_subperm (pow_ne_zero _ hp.ne_zero) | n p : ℕ
hp : Prime p
⊢ (p ^ count p n.primeFactorsList).primeFactorsList <+~ n.primeFactorsList | bd7926bf85841618 |
FDerivMeasurableAux.D_subset_differentiable_set | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L ... | exact norm_sub_le_of_mem_A hc P P I1 I2 | no goals | 28c357cacce7671e |
DFinsupp.lex_fibration | Mathlib/Data/DFinsupp/WellFounded.lean | theorem lex_fibration [∀ (i) (s : Set ι), Decidable (i ∈ s)] :
Fibration (InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd) (DFinsupp.Lex r s)
fun x => piecewise x.2.1 x.2.2 x.1 | case neg.refine_2
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
⊢ s i (if ... | split_ifs with hi | case pos
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
hi : r i i
⊢ s i (x... | ce70f659976abe0e |
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | case h.a₄
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 :=... | simp [ha₄, ha₄'] | no goals | 5599f8482dc6841e |
MeasureTheory.lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | theorem lpMeas.ae_eq_zero_of_forall_setIntegral_eq_zero (hm : m ≤ m0) (f : lpMeas E' 𝕜 m p μ)
(hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn (f : Lp E' p μ) s μ)
(hf_zero : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, (f : Lp E' p μ) x ∂μ... | case intro.intro.refine_2
α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ... | rw [integral_congr_ae hfg_restrict.symm] | case intro.intro.refine_2
α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f : ↥(lpMeas E' 𝕜 m p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ... | 8642d149ae85dbf7 |
MeasureTheory.addHaar_image_le_lintegral_abs_det_fderiv | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem addHaar_image_le_lintegral_abs_det_fderiv (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
μ (f '' s) ≤ ∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ | 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
u : ℕ → Set E := fun n =>... | conv_lhs => rw [A, image_iUnion] | 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
u : ℕ → Set E := fun n =>... | c1fb95a438d94fbc |
Equiv.Perm.OnCycleFactors.kerParam_range_eq | Mathlib/GroupTheory/Perm/Centralizer.lean | theorem kerParam_range_eq :
(kerParam g).range = (toPermHom g).ker.map (Subgroup.subtype _) | case a.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
hp : p ∈ ↑(toPermHom g).ker
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
⊢ ↑p ∈ (kerParam g).range | simp only [SetLike.mem_coe, mem_ker_toPermHom_iff, IsCycle.forall_commute_iff] at hp | case a.intro.intro
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
p : ↥(centralizer {g})
u : Perm ↑(Function.fixedPoints ⇑g) := (↑p).subtypePerm ⋯
hp :
∀ c ∈ g.cycleFactorsFinset,
∃ (hc : ∀ (x : α), x ∈ c.support ↔ ↑p x ∈ c.support), ofSubtype ((↑p).subtypePerm hc) ∈ zpowers c
⊢ ↑p ∈ (kerParam g... | b07be7bc19c84053 |
MeasureTheory.L1.setToL1_add_left' | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToL1_add_left' (hT : DominatedFinMeasAdditive μ T C)
(hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
(h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁[μ] E) :
setToL1 hT'' f = setToL1 hT f + setToL1 hT' f | α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T T' T'' : Set α → E →L[ℝ] F
C C' C'' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ ... | congr | no goals | 71e927bfd5937149 |
Surreal.Multiplication.P1_of_ih | Mathlib/SetTheory/Surreal/Multiplication.lean | theorem P1_of_ih (ih : ∀ a, ArgsRel a (Args.P1 x y) → P124 a) (hx : x.Numeric) (hy : y.Numeric) :
(x * y).Numeric | case refine_1.right
x y : PGame
ih : ∀ (a : Args), ArgsRel a (Args.P1 x y) → P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
⊢ ∀ (i_1 : (-x).LeftMoves) (j : y.LeftMoves), ⟦(x * y).moveLeft i⟧ < -⟦(-x).mulOption y i_1 j⟧ | intro j l | case refine_1.right
x y : PGame
ih : ∀ (a : Args), ArgsRel a (Args.P1 x y) → P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
j : (-x).LeftMoves
l : y.LeftMoves
⊢ ⟦(x * y).moveLeft i⟧ < -⟦(-x).mulOption y j l⟧ | 134128a4f42d0bc3 |
CategoryTheory.GrothendieckTopology.whiskerRight_toSheafify_sheafifyCompIso_hom | Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean | theorem whiskerRight_toSheafify_sheafifyCompIso_hom :
whiskerRight (J.toSheafify _) _ ≫ (J.sheafifyCompIso F P).hom = J.toSheafify _ | C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E... | dsimp [sheafifyCompIso] | C : Type u
inst✝⁸ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w₁
inst✝⁷ : Category.{max v u, w₁} D
E : Type w₂
inst✝⁶ : Category.{max v u, w₂} E
F : D ⥤ E
inst✝⁵ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) D
inst✝⁴ : ∀ (J : MulticospanShape), HasLimitsOfShape (WalkingMulticospan J) E... | 507204ea288dc9b1 |
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0) | f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
⊢ ∀ n ≥ 1,
(∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)) →
∀ z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2... | intro n hn hyp_ind z hz | f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
n : ℕ
hn : n ≥ 1
hyp_ind : ∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
z : ℝ
hz : z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2))
⊢ f z... | 69674f7264744f71 |
mem_affineSpan_iff_eq_affineCombination | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem mem_affineSpan_iff_eq_affineCombination [Nontrivial k] {p1 : P} {p : ι → P} :
p1 ∈ affineSpan k (Set.range p) ↔
∃ (s : Finset ι) (w : ι → k), ∑ i ∈ s, w i = 1 ∧ p1 = s.affineCombination k p w | case mp
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
p1 : P
p : ι → P
⊢ p1 ∈ affineSpan k (Set.range p) → ∃ s w, ∑ i ∈ s, w i = 1 ∧ p1 = (Finset.affineCombination k s p) w | exact eq_affineCombination_of_mem_affineSpan | no goals | 0958d0945311e8cf |
Stream'.Seq.ofStream_cons | Mathlib/Data/Seq/Seq.lean | theorem ofStream_cons (a : α) (s) : ofStream (a::s) = cons a (ofStream s) | α : Type u
a : α
s : Stream' α
⊢ ↑(a :: s) = cons a ↑s | apply Subtype.eq | case a
α : Type u
a : α
s : Stream' α
⊢ ↑↑(a :: s) = ↑(cons a ↑s) | 83329d52b9e3fbd1 |
Vector.mapFinIdx_singleton | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/MapIdx.lean | theorem mapFinIdx_singleton {a : α} {f : (i : Nat) → α → (h : i < 1) → β} :
#v[a].mapFinIdx f = #v[f 0 a (by simp)] | α : Type u_1
β : Type u_2
a : α
f : (i : Nat) → α → i < 1 → β
⊢ { toArray := #[a], size_toArray := ⋯ }.mapFinIdx f = { toArray := #[f 0 a ⋯], size_toArray := ⋯ } | simp | no goals | 5d3c2f3be3e767c5 |
Urysohns.CU.approx_le_one | Mathlib/Topology/UrysohnsLemma.lean | theorem approx_le_one (c : CU P) (n : ℕ) (x : X) : c.approx n x ≤ 1 | case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), approx n c x ≤ 1
c : CU P
this : approx n c.left x + approx n c.right x ≤ 1 + 1
⊢ (approx n c.left x + approx n c.right x) / 2 ≤ 1 | norm_num at this | case succ
X : Type u_1
inst✝ : TopologicalSpace X
P : Set X → Prop
x : X
n : ℕ
ihn : ∀ (c : CU P), approx n c x ≤ 1
c : CU P
this : approx n c.left x + approx n c.right x ≤ 2
⊢ (approx n c.left x + approx n c.right x) / 2 ≤ 1 | d26963483e1d7dcc |
SimpleGraph.chromaticNumber_le_iff_colorable | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem chromaticNumber_le_iff_colorable {n : ℕ} : G.chromaticNumber ≤ n ↔ G.Colorable n | V : Type u
G : SimpleGraph V
n : ℕ
h : G.chromaticNumber ≤ ↑n
this : G.chromaticNumber ≠ ⊤
⊢ G.Colorable n | rw [chromaticNumber_ne_top_iff_exists] at this | V : Type u
G : SimpleGraph V
n : ℕ
h : G.chromaticNumber ≤ ↑n
this : ∃ n, G.Colorable n
⊢ G.Colorable n | 4ec1c55975e623fc |
Int.cooper_resolution_left | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Cooper.lean | theorem cooper_resolution_left
{a b p q : Int} (a_pos : 0 < a) (b_pos : 0 < b) :
(∃ x, p ≤ a * x ∧ b * x ≤ q) ↔
(∃ k : Int, 0 ≤ k ∧ k < a ∧ b * k + b * p ≤ a * q ∧ a ∣ k + p) | a b p q : Int
a_pos : 0 < a
b_pos : 0 < b
h : (∃ x, p ≤ a * x ∧ b * x ≤ q) ↔ ∃ k, 0 ≤ k ∧ k < a ∧ b * k + b * p ≤ a * q ∧ a ∣ k + p
⊢ (∃ x, p ≤ a * x ∧ b * x ≤ q) ↔ ∃ k, 0 ≤ k ∧ k < a ∧ b * k + b * p ≤ a * q ∧ a ∣ k + p | exact h | no goals | 6379a7963bf757db |
Configuration.HasLines.lineCount_eq_pointCount | Mathlib/Combinatorics/Configuration.lean | theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
(hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
lineCount L p = pointCount P l | case h
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
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := {i | i.1 ∈ i.2}.toFinset
step1 ... | all_goals simp_rw [s, Finset.mem_univ, true_and, Set.mem_toFinset]; exact fun p => Iff.rfl | no goals | 55b04453afa8ecf4 |
Finset.le_card_diffs_mul_card_diffs | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | /-- A slight generalisation of the **Marica-Schönheim Inequality**. -/
lemma Finset.le_card_diffs_mul_card_diffs (s t : Finset α) :
#s * #t ≤ #(s \\ t) * #(t \\ s) | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : GeneralizedBooleanAlgebra α
s✝ t✝ s t : Finset α
⊢ image (⇑{ toFun := ⇑liftLatticeHom, inj' := ⋯ }) (s \\ t) =
image (⇑{ toFun := ⇑liftLatticeHom, inj' := ⋯ }) s \\ image (⇑{ toFun := ⇑liftLatticeHom, inj' := ⋯ }) t | exact image_image₂_distrib fun a b ↦ rfl | no goals | 8b40ab7672e58807 |
List.isInfix_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Sublist.lean | theorem isInfix_iff : l₁ <:+: l₂ ↔
∃ k, l₁.length + k ≤ l₂.length ∧ ∀ i (h : i < l₁.length), l₂[i + k]? = some l₁[i] | α✝ : Type u_1
l₁ l₂ : List α✝
h : l₁ <:+: l₂
t : List α✝
p : l₁ <:+ t
s : t <+: l₂
this : l₁.length ≤ t.length
⊢ l₁.length + (t.length - l₁.length) ≤ l₂.length | have := s.length_le | α✝ : Type u_1
l₁ l₂ : List α✝
h : l₁ <:+: l₂
t : List α✝
p : l₁ <:+ t
s : t <+: l₂
this✝ : l₁.length ≤ t.length
this : t.length ≤ l₂.length
⊢ l₁.length + (t.length - l₁.length) ≤ l₂.length | 8c836fc6d0b14b98 |
RingHom.locally_stableUnderComposition | Mathlib/RingTheory/RingHom/Locally.lean | /-- If `P` preserves localizations, then `Locally P` is stable under composition if `P` is. -/
lemma locally_stableUnderComposition (hPi : RespectsIso P) (hPl : LocalizationPreserves P)
(hPc : StableUnderComposition P) :
StableUnderComposition (Locally P) | P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
hPl : LocalizationPreserves fun {R S} [CommRing R] [CommRing S] => P
hPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] => P
R S T : Type u
inst✝² : CommRing R
ins... | ext x | case a
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
hPl : LocalizationPreserves fun {R S} [CommRing R] [CommRing S] => P
hPc : StableUnderComposition fun {R S} [CommRing R] [CommRing S] => P
R S T : Type u
inst✝² : CommRin... | c02901a417ec447b |
Algebra.FormallyUnramified.range_eq_top_of_isPurelyInseparable | Mathlib/RingTheory/Unramified/Field.lean | theorem range_eq_top_of_isPurelyInseparable
[IsPurelyInseparable K L] : (algebraMap K L).range = ⊤ | case neg.intro.intro.intro
K : Type u_1
L : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FormallyUnramified K L
inst✝¹ : EssFiniteType K L
inst✝ : IsPurelyInseparable K L
this✝² : Nontrivial (L ⊗[K] L)
x : L
a✝ : x ∈ ⊤
n : ℕ
hn : x ^ ringExpChar K ^ n ∈ (algebraMap K L).range
this✝¹ : ExpCha... | use (-this.unit⁻¹ * a) | case h
K : Type u_1
L : Type u_3
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FormallyUnramified K L
inst✝¹ : EssFiniteType K L
inst✝ : IsPurelyInseparable K L
this✝² : Nontrivial (L ⊗[K] L)
x : L
a✝ : x ∈ ⊤
n : ℕ
hn : x ^ ringExpChar K ^ n ∈ (algebraMap K L).range
this✝¹ : ExpChar (L ⊗[K] L) (ringEx... | 85771a0deda9ae28 |
countable_setOf_covBy_right | Mathlib/Topology/Order/Basic.lean | theorem countable_setOf_covBy_right [OrderTopology α] [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, x ⋖ y } | α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x}... | intro x hx | α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
a✝ : Nontrivial α
s : Set α := {x | ∃ y, x ⋖ y}
y : α → α
hy : ∀ x ∈ s, x ⋖ y x
Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x
a : Set α
ha : IsOpen a
t : Set α := {x | x ∈ s ∧ x ∈ a ∧ y x ∉ a ∧ ¬IsBot x}... | 7a450f28627ecdc7 |
FreeAlgebra.adjoin_range_ι | Mathlib/Algebra/FreeAlgebra.lean | theorem adjoin_range_ι : Algebra.adjoin R (Set.range (ι R : X → FreeAlgebra R X)) = ⊤ | R : Type u_1
inst✝ : CommSemiring R
X : Type u_2
⊢ Algebra.adjoin R (Set.range (ι R)) = ⊤ | set S := Algebra.adjoin R (Set.range (ι R : X → FreeAlgebra R X)) | R : Type u_1
inst✝ : CommSemiring R
X : Type u_2
S : Subalgebra R (FreeAlgebra R X) := Algebra.adjoin R (Set.range (ι R))
⊢ S = ⊤ | cec84737beae3660 |
Stream'.WSeq.head_terminates_of_head_tail_terminates | Mathlib/Data/Seq/WSeq.lean | theorem head_terminates_of_head_tail_terminates (s : WSeq α) [T : Terminates (head (tail s))] :
Terminates (head s) :=
(head_terminates_iff _).2 <| by
rcases (head_terminates_iff _).1 T with ⟨⟨a, h⟩⟩
simp? [tail] at h says simp only [tail, destruct_flatten, bind_map_left] at h
rcases exists_of_mem_bin... | α : Type u
s : WSeq α
T : s.tail.head.Terminates
⊢ s.destruct.Terminates | rcases (head_terminates_iff _).1 T with ⟨⟨a, h⟩⟩ | case mk.intro
α : Type u
s : WSeq α
T : s.tail.head.Terminates
a : Option (α × WSeq α)
h : a ∈ s.tail.destruct
⊢ s.destruct.Terminates | 48c28738910f67e0 |
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 |
List.eraseIdx_modifyHead_of_pos | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean | theorem eraseIdx_modifyHead_of_pos {f : α → α} {l : List α} {n} (h : 0 < n) :
(l.modifyHead f).eraseIdx n = (l.eraseIdx n).modifyHead f | α : Type u_1
f : α → α
l : List α
n : Nat
h : 0 < n
⊢ (modifyHead f l).eraseIdx n = modifyHead f (l.eraseIdx n) | cases l <;> cases n <;> simp_all | no goals | b3479c19338ba769 |
LipschitzOnWith.extend_real | Mathlib/Topology/MetricSpace/Lipschitz.lean | theorem LipschitzOnWith.extend_real {f : α → ℝ} {s : Set α} {K : ℝ≥0} (hf : LipschitzOnWith K f s) :
∃ g : α → ℝ, LipschitzWith K g ∧ EqOn f g s | case intro
α : Type u
inst✝ : PseudoMetricSpace α
f : α → ℝ
s : Set α
K : ℝ≥0
hf : LipschitzOnWith K f s
this : Nonempty ↑s
g : α → ℝ := fun y => ⨅ x, f ↑x + ↑K * dist y ↑x
y z : α
hz : z ∈ s
⊢ f z - ↑K * dist y z ∈ lowerBounds (range fun x => f ↑x + ↑K * dist y ↑x) | rintro w ⟨t, rfl⟩ | case intro.intro
α : Type u
inst✝ : PseudoMetricSpace α
f : α → ℝ
s : Set α
K : ℝ≥0
hf : LipschitzOnWith K f s
this : Nonempty ↑s
g : α → ℝ := fun y => ⨅ x, f ↑x + ↑K * dist y ↑x
y z : α
hz : z ∈ s
t : ↑s
⊢ f z - ↑K * dist y z ≤ (fun x => f ↑x + ↑K * dist y ↑x) t | 509beb34bd5ad882 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRat | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem sat_of_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (p : PosFin n → Bool)
(pf : p ⊨ f) :
(insertRatUnits f (negate c)).2 = true → p ⊨ c | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inl.intro.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFi... | apply Or.inl | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro.inl.intro.intro.h
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : Pos... | c0b439908684a182 |
Polynomial.splits_of_algHom | Mathlib/Algebra/Polynomial/Splits.lean | theorem splits_of_algHom {f : R[X]} (h : Splits (algebraMap R K) f) (e : K →ₐ[R] L) :
Splits (algebraMap R L) f | R : Type u_1
K : Type v
L : Type w
inst✝⁴ : CommRing R
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra R K
inst✝ : Algebra R L
f : R[X]
h : Splits (algebraMap R K) f
e : K →ₐ[R] L
⊢ Splits ((↑e).comp (algebraMap R K)) f | exact splits_comp_of_splits _ _ h | no goals | 500b097bba0776b6 |
AffineIndependent.convexHull_inter | Mathlib/Analysis/Convex/Combination.lean | /-- Two simplices glue nicely if the union of their vertices is affine independent. -/
lemma AffineIndependent.convexHull_inter (hs : AffineIndependent R ((↑) : s → E))
(ht₁ : t₁ ⊆ s) (ht₂ : t₂ ⊆ s) :
convexHull R (t₁ ∩ t₂ : Set E) = convexHull R t₁ ∩ convexHull R t₂ | case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s t₁ t₂ : Finset E
ht₁ : t₁ ⊆ s
ht₂ : t₂ ⊆ s
x : E
w₁ : E → R
h₁w₁ : ∀ y ∈ t₁, 0 ≤ w₁ y
h₂w₁ : ∑ y ∈ t₁, w₁ y = 1
h₃w₁ : ∑ y ∈ t₁, w₁ y • y = x
w₂ : E → R
h₂w₂ : ∑ y ∈ t₂, w₂... | have ht (x) (hx₁ : x ∈ t₁) (hx₂ : x ∉ t₂) : w₁ x = 0 := by
simpa [w, hx₁, hx₂] using hs _ (ht₁ hx₁) | case intro.intro.intro.intro.intro.intro.intro
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
s t₁ t₂ : Finset E
ht₁ : t₁ ⊆ s
ht₂ : t₂ ⊆ s
x : E
w₁ : E → R
h₁w₁ : ∀ y ∈ t₁, 0 ≤ w₁ y
h₂w₁ : ∑ y ∈ t₁, w₁ y = 1
h₃w₁ : ∑ y ∈ t₁, w₁ y • y = x
w₂ : E → R
h₂w₂ : ∑ y ∈ t₂, w₂... | c5414b8b46240fda |
Nat.Prime.emultiplicity_choose_prime_pow_add_emultiplicity | Mathlib/Data/Nat/Multiplicity.lean | theorem emultiplicity_choose_prime_pow_add_emultiplicity (hp : p.Prime) (hkn : k ≤ p ^ n)
(hk0 : k ≠ 0) : emultiplicity p (choose (p ^ n) k) + emultiplicity p k = n :=
le_antisymm
(by
have hdisj :
Disjoint {i ∈ Ico 1 n.succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}
{i ∈ Ico 1 n.succ |... | p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
⊢ emultiplicity p ((p ^ n).choose k) + emultiplicity p k ≤ ↑n | have hdisj :
Disjoint {i ∈ Ico 1 n.succ | p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i}
{i ∈ Ico 1 n.succ | p ^ i ∣ k} := by
simp +contextual [disjoint_right, *, dvd_iff_mod_eq_zero,
Nat.mod_lt _ (pow_pos hp.pos _)] | p n k : ℕ
hp : Prime p
hkn : k ≤ p ^ n
hk0 : k ≠ 0
hdisj :
Disjoint (filter (fun i => p ^ i ≤ k % p ^ i + (p ^ n - k) % p ^ i) (Ico 1 n.succ))
(filter (fun i => p ^ i ∣ k) (Ico 1 n.succ))
⊢ emultiplicity p ((p ^ n).choose k) + emultiplicity p k ≤ ↑n | 6fc8e6052d4cd478 |
List.exists_perm_sublist | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Perm.lean | theorem exists_perm_sublist {l₁ l₂ l₂' : List α} (s : l₁ <+ l₂) (p : l₂ ~ l₂') :
∃ l₁', l₁' ~ l₁ ∧ l₁' <+ l₂' | case swap
α : Type u_1
l₂ l₂' : List α
x y : α
l' l₁ : List α
s : l₁ <+ y :: x :: l'
⊢ ∃ l₁', l₁' ~ l₁ ∧ l₁' <+ x :: y :: l' | match s with
| .cons _ (.cons _ s) => exact ⟨_, .rfl, (s.cons _).cons _⟩
| .cons _ (.cons₂ _ s) => exact ⟨x :: _, .rfl, (s.cons _).cons₂ _⟩
| .cons₂ _ (.cons _ s) => exact ⟨y :: _, .rfl, (s.cons₂ _).cons _⟩
| .cons₂ _ (.cons₂ _ s) => exact ⟨x :: y :: _, .swap .., (s.cons₂ _).cons₂ _⟩ | no goals | b87d2baff04d68a4 |
fermatLastTheoremThree_case_1 | Mathlib/NumberTheory/FLT/Three.lean | theorem fermatLastTheoremThree_case_1 {a b c : ℤ} (hdvd : ¬ 3 ∣ a * b * c) :
a ^ 3 + b ^ 3 ≠ c ^ 3 | a b c : ℤ
hdvd : ¬3 ∣ a * b * c
⊢ a ^ 3 + b ^ 3 ≠ c ^ 3 | simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd | a b c : ℤ
hdvd : (¬3 ∣ a ∧ ¬3 ∣ b) ∧ ¬3 ∣ c
⊢ a ^ 3 + b ^ 3 ≠ c ^ 3 | 78210b4a95551d14 |
Polynomial.Monic.irreducible_iff_natDegree' | Mathlib/Algebra/Polynomial/Monic.lean | lemma Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) | case h.mpr
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f g : R[X]
hf : f.Monic
hg : g.Monic
hp : (f * g).Monic
h : f.natDegree ≠ 0 ∧ g.natDegree ≠ 0
⊢ ∃ f_1 g_1,
f_1.Monic ∧
g_1.Monic ∧ f_1 * g_1 = f * g ∧ g_1.natDegree ≠ 0 ∧ g_1.natDegree + g_1.natDegree ≤ f.natDegree + g.natDegree | obtain hl | hl := le_total f.natDegree g.natDegree | case h.mpr.inl
R : Type u
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f g : R[X]
hf : f.Monic
hg : g.Monic
hp : (f * g).Monic
h : f.natDegree ≠ 0 ∧ g.natDegree ≠ 0
hl : f.natDegree ≤ g.natDegree
⊢ ∃ f_1 g_1,
f_1.Monic ∧
g_1.Monic ∧ f_1 * g_1 = f * g ∧ g_1.natDegree ≠ 0 ∧ g_1.natDegree + g_1.natDegree ≤ f... | 995646a876615ee1 |
StrictConvexOn.lt_slope_of_hasDerivWithinAt_Ioi | Mathlib/Analysis/Convex/Deriv.lean | /-- If `f : ℝ → ℝ` is strictly convex on `S` and right-differentiable at `x ∈ S`, then the slope of
any secant line with left endpoint at `x` is strictly greater than the right derivative of `f` at
`x`. -/
lemma lt_slope_of_hasDerivWithinAt_Ioi (hfc : StrictConvexOn ℝ S f)
(hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (h... | case intro.intro
S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : StrictConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Ioi x) x
u : ℝ
hxu : x < u
huy : u < y
hu : u ∈ S
this : (f u - f x) / (u - x) < (f y - f x) / (y - x)
⊢ f' < slope f x y | simp only [← slope_def_field] at this | case intro.intro
S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : StrictConvexOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hf' : HasDerivWithinAt f f' (Ioi x) x
u : ℝ
hxu : x < u
huy : u < y
hu : u ∈ S
this : slope f x u < slope f x y
⊢ f' < slope f x y | f9eece9e1c330b0b |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.clear_insert_inductive_case | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem clear_insert_inductive_case {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(units : Array (Literal (PosFin n))) (units_nodup : ∀ i : Fin units.size, ∀ j : Fin units.size, i ≠ j → units[i] ≠ units[j])
(idx : Fin units.size) (assignments : Array Assignment)
(ih : ClearI... | case intro.inr.inl.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hs... | by_cases idx = j | case pos
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i... | c90be518eb885cb1 |
MeasureTheory.setIntegral_tilted | Mathlib/MeasureTheory/Measure/Tilted.lean | lemma setIntegral_tilted [SFinite μ] (f : α → ℝ) (g : α → E) (s : Set α) :
∫ x in s, g x ∂(μ.tilted f) = ∫ x in s, (exp (f x) / ∫ x, exp (f x) ∂μ) • (g x) ∂μ | case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x => rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ | rw [integral_undef hf'] | case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite μ
f : α → ℝ
g : α → E
s : Set α
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x => rexp (f x)) μ
⊢ 0 = ∫ (x : α) in s, (rexp (f x) / 0) • g x ∂μ | 9479368488db4a2b |
MeasureTheory.Measure.exists_null_set_measure_lt_of_disjoint | Mathlib/MeasureTheory/Measure/MutuallySingular.lean | lemma exists_null_set_measure_lt_of_disjoint (h : Disjoint μ ν) {ε : ℝ≥0} (hε : 0 < ε) :
∃ s, μ s = 0 ∧ ν sᶜ ≤ 2 * ε | α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
t : ℕ → Set α
ht₂ : ∀ (n : ℕ), μ (t n) + ν (t n)ᶜ < ↑ε * (1 / 2) ^ n
⊢ ∃ s, μ s = 0 ∧ ν sᶜ ≤ 2 * ↑ε | refine ⟨⋂ n, t n, ?_, ?_⟩ | case refine_1
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥0
hε : 0 < ε
h₁ : sInf {m | ∃ t, m = μ t + ν tᶜ} = 0
t : ℕ → Set α
ht₂ : ∀ (n : ℕ), μ (t n) + ν (t n)ᶜ < ↑ε * (1 / 2) ^ n
⊢ μ (⋂ n, t n) = 0
case refine_2
α : Type u_1
m0 : MeasurableSpace α
μ ν : Measure α
h : Disjoint μ ν
ε : ℝ≥... | 46dcd329efc0e15f |
Cardinal.mk_multiset_of_isEmpty | Mathlib/SetTheory/Cardinal/Finsupp.lean | theorem mk_multiset_of_isEmpty (α : Type u) [IsEmpty α] : #(Multiset α) = 1 :=
Multiset.toFinsupp.toEquiv.cardinal_eq.trans (by simp)
| α : Type u
inst✝ : IsEmpty α
⊢ #(α →₀ ℕ) = 1 | simp | no goals | aca62b8ebc201fe2 |
Real.cos_lt_one_div_sqrt_sq_add_one | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | theorem cos_lt_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2)
(hx3 : x ≠ 0) : cos x < (1 / √(x ^ 2 + 1) : ℝ) | case inl
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
hy3 : 0 < y ^ 2 + 1
hy2' : y < π / 2
⊢ cos y < 1 / √(y ^ 2 + 1) | have hy4 : 0 < cos y := cos_pos_of_mem_Ioo ⟨by linarith, hy2'⟩ | case inl
x : ℝ
hx1 : -(3 * π / 2) ≤ x
hx2 : x ≤ 3 * π / 2
hx3 : x ≠ 0
y : ℝ
hy1 : 0 < y
hy2 : y ≤ 3 * π / 2
hy3 : 0 < y ^ 2 + 1
hy2' : y < π / 2
hy4 : 0 < cos y
⊢ cos y < 1 / √(y ^ 2 + 1) | f405817ae92adc5b |
orderOf_abs_ne_one | Mathlib/GroupTheory/OrderOfElement.lean | theorem orderOf_abs_ne_one (h : |x| ≠ 1) : orderOf x = 0 | case inl
G : Type u_1
inst✝ : LinearOrderedRing G
x : G
h✝ : |x| ≠ 1
n : ℕ
hn : 0 < n
hx : |x| ^ n = 1
h : |x| < 1
⊢ False | exact ((pow_lt_one₀ (abs_nonneg x) h hn.ne').ne hx).elim | no goals | 19b05ba116765651 |
Polynomial.content_X_mul | Mathlib/RingTheory/Polynomial/Content.lean | theorem content_X_mul {p : R[X]} : content (X * p) = content p | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : (X * p).support = Finset.map { toFun := Nat.succ, inj' := Nat.succ_injective } p.support
⊢ Multiset.map (fun x => (X * p).coeff x.succ) p.support.val = Multiset.map p.coeff p.support.val | refine congr (congr rfl ?_) rfl | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : (X * p).support = Finset.map { toFun := Nat.succ, inj' := Nat.succ_injective } p.support
⊢ (fun x => (X * p).coeff x.succ) = p.coeff | ac071a5a6c3a3683 |
Pell.IsFundamental.eq_pow_of_nonneg | Mathlib/NumberTheory/Pell.lean | theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
(hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n | case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 = a.y
⊢ a = 1 | ext <;> simp only [x_one, y_one] | case intro.h.inl.hx
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 = a.y
⊢ a.x = 1
case intro.h.inl.hy
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ ... | e2cabc4b76df1e43 |
EuclideanGeometry.existsUnique_dist_eq_of_insert | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem existsUnique_dist_eq_of_insert {s : AffineSubspace ℝ P}
[HasOrthogonalProjection s.direction] {ps : Set P} (hnps : ps.Nonempty) {p : P} (hps : ps ⊆ s)
(hp : p ∉ s) (hu : ∃! cs : Sphere P, cs.center ∈ s ∧ ps ⊆ (cs : Set P)) :
∃! cs₂ : Sphere P,
cs₂.center ∈ affineSpan ℝ (insert p (s : Set P)) ∧... | case h.left.right.inl
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ... | rw [hpo,
dist_sq_smul_orthogonal_vadd_smul_orthogonal_vadd (orthogonalProjection_mem p) hcc _ _
(vsub_orthogonalProjection_mem_direction_orthogonal s p),
← dist_eq_norm_vsub V p, dist_comm _ cc] | case h.left.right.inl
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : HasOrthogonalProjection s.direction
ps : Set P
hnps : ps.Nonempty
p : P
hps : ps ⊆ ↑s
hp : p ∉ s
this : Nonempty ↥s
cc : P
cr : ... | 6d31314b3548c9e7 |
sum_Ioo_inv_sq_le | Mathlib/Analysis/PSeries.lean | theorem sum_Ioo_inv_sq_le (k n : ℕ) : (∑ i ∈ Ioo k n, (i ^ 2 : α)⁻¹) ≤ 2 / (k + 1) :=
calc
(∑ i ∈ Ioo k n, ((i : α) ^ 2)⁻¹) ≤ ∑ i ∈ Ioc k (max (k + 1) n), ((i : α) ^ 2)⁻¹ | α : Type u_1
inst✝ : LinearOrderedField α
k n : ℕ
⊢ (↑k.succ ^ 2)⁻¹ + ∑ k ∈ Ico (k.succ + 1) ((k + 1) ⊔ n).succ, (↑k ^ 2)⁻¹ ≤
((↑k + 1) ^ 2)⁻¹ + ∑ i ∈ Ioc k.succ ((k + 1) ⊔ n), (↑i ^ 2)⁻¹ | rw [Nat.Ico_succ_right, Nat.Icc_succ_left, Nat.cast_succ] | no goals | 3d762faeae1fa149 |
OmegaCompletePartialOrder.ωScottContinuous.isLUB | Mathlib/Order/OmegaCompletePartialOrder.lean | lemma ωScottContinuous.isLUB {c : Chain α} (hf : ωScottContinuous f) :
IsLUB (Set.range (c.map ⟨f, hf.monotone⟩)) (f (ωSup c)) | α : Type u_2
β : Type u_3
inst✝¹ : OmegaCompletePartialOrder α
inst✝ : OmegaCompletePartialOrder β
f : α → β
c : Chain α
hf : ωScottContinuous f
⊢ IsLUB (Set.range ⇑(c.map { toFun := f, monotone' := ⋯ })) (f (ωSup c)) | simpa [map_coe, OrderHom.coe_mk, Set.range_comp]
using hf (by simp) (Set.range_nonempty _) (isChain_range c).directedOn (isLUB_range_ωSup c) | no goals | 3e9e62d15cccc954 |
padicValRat.mul | Mathlib/NumberTheory/Padics/PadicVal/Basic.lean | theorem mul {q r : ℚ} (hq : q ≠ 0) (hr : r ≠ 0) :
padicValRat p (q * r) = padicValRat p q + padicValRat p r | p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hq : q ≠ 0
hr : r ≠ 0
this : q * r = q.num * r.num /. (↑q.den * ↑r.den)
hq' : q.num /. ↑q.den ≠ 0
hr' : r.num /. ↑r.den ≠ 0
hp' : _root_.Prime ↑p
⊢ FiniteMultiplicity (↑p) (q.num * r.num) | simp [finite_int_prime_iff, hq, hr] | no goals | 692ee1817eaa4fad |
WeierstrassCurve.variableChange_c₄ | Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.lean | @[simp]
lemma variableChange_c₄ : (W.variableChange C).c₄ = C.u⁻¹ ^ 4 * W.c₄ | R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
C : VariableChange R
⊢ (↑C.u⁻¹ ^ 2 * (W.b₂ + 12 * C.r)) ^ 2 - 24 * (↑C.u⁻¹ ^ 4 * (W.b₄ + C.r * W.b₂ + 6 * C.r ^ 2)) =
↑C.u⁻¹ ^ 4 * (W.b₂ ^ 2 - 24 * W.b₄) | ring1 | no goals | 42dcb77fb4eff309 |
MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
(h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)) (t : Set E)
(ht : MeasurableSet t) (h''t : μ t ≠ ∞) :
Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
... | exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists | no goals | 8e4b11ebb89db50e |
List.get_eq_get_rotate | Mathlib/Data/List/Rotate.lean | theorem get_eq_get_rotate (l : List α) (n : ℕ) (k : Fin l.length) :
l.get k = (l.rotate n).get ⟨(l.length - n % l.length + k) % l.length,
(Nat.mod_lt _ (k.1.zero_le.trans_lt k.2)).trans_eq (length_rotate _ _).symm⟩ | α : Type u
l : List α
n : ℕ
k : Fin l.length
⊢ ↑k < l.length
α : Type u
l : List α
n : ℕ
k : Fin l.length
⊢ n % l.length ≤ l.length | exacts [k.2, (mod_lt _ (k.1.zero_le.trans_lt k.2)).le] | no goals | a0cf818b1b7a9d14 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftLeftConst.go_denote_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftLeft.lean | theorem go_denote_eq (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(assign : α → Bool) (curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx1 : idx < w),
curr ≤ idx
→
⟦
(go aig input distance curr hcurr s).aig,
(go aig input dis... | case isTrue.inl.isTrue.isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
assign : α → Bool
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx1 : idx < w
hidx2 : curr ≤ idx
res : RefVecEntry α w
h✝² : curr < w
heq : curr = idx
h✝¹ : curr < dist... | omega | no goals | 625f9e09be737d85 |
ZMod.erdos_ginzburg_ziv_prime | Mathlib/Combinatorics/Additive/ErdosGinzburgZiv.lean | theorem ZMod.erdos_ginzburg_ziv_prime (a : ι → ZMod p) (hs : #s = 2 * p - 1) :
∃ t ⊆ s, #t = p ∧ ∑ i ∈ t, a i = 0 | case intro.refine_2.refine_1
ι : Type u_1
p : ℕ
inst✝ : Fact (Nat.Prime p)
s : Finset ι
a : ι → ZMod p
hs : #s = 2 * p - 1
this : NeZero p
N : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }
zero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩
hN₀ : 0 < N
hs' : 2 * p - 1 ... | exact hx.imp (fun a ha ↦ mem_filter.2 ⟨Finset.mem_attach _ _, ha⟩) | no goals | 3ec57b8d04698567 |
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 | 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)
⊢ (f.noncommCoprod g comm).range = f.range ⊔ g.range | apply le_antisymm | case a
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)
⊢ (f.noncommCoprod g comm).range ≤ f.range ⊔ g.range
case a
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 ... | a43981614b1c4850 |
List.cons_diff | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem cons_diff (a : α) (l₁ l₂ : List α) :
(a :: l₁).diff l₂ = if a ∈ l₂ then l₁.diff (l₂.erase a) else a :: l₁.diff l₂ | α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a b : α
l₂ : List α
ih : ∀ (l₁ : List α), (a :: l₁).diff l₂ = if a ∈ l₂ then l₁.diff (l₂.erase a) else a :: l₁.diff l₂
l₁ : List α
h : ¬a = b
⊢ (a :: l₁).diff (b :: l₂) = if a ∈ b :: l₂ then l₁.diff ((b :: l₂).erase a) else a :: l₁.diff (b :: l₂) | have := Ne.symm h | α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a b : α
l₂ : List α
ih : ∀ (l₁ : List α), (a :: l₁).diff l₂ = if a ∈ l₂ then l₁.diff (l₂.erase a) else a :: l₁.diff l₂
l₁ : List α
h : ¬a = b
this : b ≠ a
⊢ (a :: l₁).diff (b :: l₂) = if a ∈ b :: l₂ then l₁.diff ((b :: l₂).erase a) else a :: l₁.diff (b :: l₂) | 3ae4b2e0ad22eea9 |
Nat.nth_eq_zero | Mathlib/Data/Nat/Nth.lean | theorem nth_eq_zero {n} :
nth p n = 0 ↔ p 0 ∧ n = 0 ∨ ∃ hf : (setOf p).Finite, #hf.toFinset ≤ n | case refine_2
p : ℕ → Prop
n : ℕ
⊢ (p 0 ∧ n = 0 ∨ ∃ (hf : (setOf p).Finite), #hf.toFinset ≤ n) → nth p n = 0 | rintro (⟨h₀, rfl⟩ | ⟨hf, hle⟩) | case refine_2.inl.intro
p : ℕ → Prop
h₀ : p 0
⊢ nth p 0 = 0
case refine_2.inr.intro
p : ℕ → Prop
n : ℕ
hf : (setOf p).Finite
hle : #hf.toFinset ≤ n
⊢ nth p n = 0 | cc87bef2873e0539 |
SimpleGraph.not_isUniform_zero | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | theorem not_isUniform_zero : ¬G.IsUniform (0 : 𝕜) s t := fun h =>
(abs_nonneg _).not_lt <| h (empty_subset _) (empty_subset _) (by simp) (by simp)
| α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
s t : Finset α
h : G.IsUniform 0 s t
⊢ ↑(#t) * 0 ≤ ↑(#∅) | simp | no goals | b2c832a8e1306f2d |
WittVector.constantCoeff_wittNeg | Mathlib/RingTheory/WittVector/Defs.lean | theorem constantCoeff_wittNeg (n : ℕ) : constantCoeff (wittNeg p n) = 0 | p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ constantCoeff (-X 0) = 0 | simp only [neg_zero, RingHom.map_neg, constantCoeff_X] | no goals | b3bfcf4755852422 |
Cardinal.mk_toNat_of_infinite | Mathlib/SetTheory/Cardinal/ToNat.lean | theorem mk_toNat_of_infinite [h : Infinite α] : toNat #α = 0 | α : Type u
h : Infinite α
⊢ toNat #α = 0 | simp | no goals | 7e829a2ad2d4fde3 |
mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | theorem mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt {B : PowerBasis K L}
(hp : Prime p) (hBint : IsIntegral R B.gen) {z : L} (hzint : IsIntegral R z)
(hz : p • z ∈ adjoin R ({B.gen} : Set L)) (hei : (minpoly R B.gen).IsEisensteinAt 𝓟) :
z ∈ adjoin R ({B.gen} : Set L) | case h
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K ... | rw [h] at hk | case h
R : Type u
K : Type v
L : Type z
p : R
inst✝¹⁰ : CommRing R
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : Algebra R K
inst✝⁴ : IsScalarTower R K L
inst✝³ : Algebra.IsSeparable K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K ... | fa4f10e8cbdbb4c7 |
MeasurableEmbedding.essSup_map_measure | Mathlib/MeasureTheory/Function/EssSup.lean | theorem MeasurableEmbedding.essSup_map_measure (hf : MeasurableEmbedding f) :
essSup g (Measure.map f μ) = essSup (g ∘ f) μ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
hf : MeasurableEmbedding f
c : β
h_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c
⊢ ∀ᵐ (a : γ) ∂Measure.map f μ, g a ≤ c | exact hf.ae_map_iff.mpr h_le | no goals | 99a011d48a3b55e4 |
BitVec.some_getElem_eq_getElem? | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem some_getElem_eq_getElem? (l : BitVec w) (i : Nat) (h : i < w) :
(some l[i] = l[i]?) ↔ True | w : Nat
l : BitVec w
i : Nat
h : i < w
⊢ some l[i] = l[i]? ↔ True | simp [h] | no goals | c13177415c5b2ca8 |
isIso_left_of_isIso_biprod_map | Mathlib/CategoryTheory/Limits/Shapes/BinaryBiproducts.lean | theorem isIso_left_of_isIso_biprod_map {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z)
[IsIso (biprod.map f g)] : IsIso f :=
⟨⟨biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst,
⟨by
have t := congrArg (fun p : W ⊞ X ⟶ W ⊞ X => biprod.inl ≫ p ≫ biprod.fst)
(IsIso.hom_inv_id (biprod.map f g))
sim... | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasBinaryBiproducts C
W X Y Z : C
f : W ⟶ Y
g : X ⟶ Z
inst✝ : IsIso (biprod.map f g)
t : f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = biprod.inl ≫ biprod.fst
⊢ f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = 𝟙 W | simp [t] | no goals | 065c81a972284a54 |
Computation.terminates_parallel.aux | Mathlib/Data/Seq/Parallel.lean | theorem terminates_parallel.aux :
∀ {l : List (Computation α)} {S c},
c ∈ l → Terminates c → Terminates (corec parallel.aux1 (l, S)) | case cons.inr.inr.inr
α : Type u
lem1 :
∀ (l : List (Computation α)) (S : WSeq (Computation α)),
(∃ a, parallel.aux2 l = Sum.inl a) → (corec parallel.aux1 (l, S)).Terminates
c✝ : Computation α
T : c✝.Terminates
s : Computation α
IH : ∀ {l : List (Computation α)} {S : WSeq (Computation α)}, s ∈ l → (corec parallel... | simp [this] | no goals | 933ed16bf89826a9 |
continuous_bool_rng | Mathlib/Topology/Constructions.lean | lemma continuous_bool_rng [TopologicalSpace X] {f : X → Bool} (b : Bool) :
Continuous f ↔ IsClopen (f ⁻¹' {b}) | X : Type u
inst✝ : TopologicalSpace X
f : X → Bool
b : Bool
⊢ Continuous f ↔ IsClopen (f ⁻¹' {b}) | rw [continuous_discrete_rng, Bool.forall_bool' b, IsClopen, ← isOpen_compl_iff, ← preimage_compl,
Bool.compl_singleton, and_comm] | no goals | 66d9cd62cfae156a |
MeasureTheory.ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real {f : α → ℝ}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) {t : Set α}
(ht : MeasurableSet t) (hμt : μ t ≠ ∞) : f =ᵐ[μ.restrict t] 0 | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ 0 ≤ ∫ (x :... | simp_rw [Pi.neg_apply] | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ⊤
s : Set α
hs : MeasurableSet s
hμs : μ s < ⊤
⊢ 0 ≤ ∫ (x :... | e453198f8b45794c |
Field.primitive_element_inf_aux | Mathlib/FieldTheory/PrimitiveElement.lean | theorem primitive_element_inf_aux [Algebra.IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField := algeb... | ring | no goals | c759c025da469f0d |
Vector.lawfulBEq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem lawfulBEq_iff [BEq α] [NeZero n] : LawfulBEq (Vector α n) ↔ LawfulBEq α | α : Type u_1
n✝ : Nat
inst✝¹ : BEq α
inst✝ : NeZero n✝
n : Nat
x✝ : n + 1 ≠ 0
h : LawfulBEq (Vector α (n + 1))
a : α
this : (n + 1 == 0 || a == a) = true
⊢ (a == a) = true | simpa | no goals | fe7d41c34f58a0c2 |
Polynomial.Sequence.span | Mathlib/Algebra/Polynomial/Sequence.lean | /-- A polynomial sequence spans `R[X]` if all of its elements' leading coefficients are units. -/
protected lemma span (hCoeff : ∀ i, IsUnit (S i).leadingCoeff) : span R (Set.range S) = ⊤ | R : Type u_1
inst✝ : Ring R
S : Sequence R
hCoeff : ∀ (i : ℕ), IsUnit (↑S i).leadingCoeff
a✝ : Nontrivial R
n : ℕ
ih : ∀ m < n, ∀ (P : R[X]), P.natDegree = m → P ∈ span R (Set.range ↑S)
P : R[X]
hp : P.natDegree = n
p_ne_zero : ¬P = 0
u : R
leftinv : (↑S n).leadingCoeff * u = 1
rightinv : u * (↑S n).leadingCoeff = 1
he... | have in_span : S n ∈ span R (Set.range S) := subset_span (by simp) | R : Type u_1
inst✝ : Ring R
S : Sequence R
hCoeff : ∀ (i : ℕ), IsUnit (↑S i).leadingCoeff
a✝ : Nontrivial R
n : ℕ
ih : ∀ m < n, ∀ (P : R[X]), P.natDegree = m → P ∈ span R (Set.range ↑S)
P : R[X]
hp : P.natDegree = n
p_ne_zero : ¬P = 0
u : R
leftinv : (↑S n).leadingCoeff * u = 1
rightinv : u * (↑S n).leadingCoeff = 1
he... | 1a688d34c785ae90 |
mem_list_primes_of_dvd_prod | Mathlib/Data/List/Prime.lean | theorem mem_list_primes_of_dvd_prod {p : M} (hp : Prime p) {L : List M} (hL : ∀ q ∈ L, Prime q)
(hpL : p ∣ L.prod) : p ∈ L | case intro.intro
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : Subsingleton Mˣ
p : M
hp : Prime p
L : List M
hL : ∀ q ∈ L, Prime q
hpL : p ∣ L.prod
x : M
hx1 : x ∈ L
hx2 : p ∣ x
⊢ p ∈ L | rwa [(prime_dvd_prime_iff_eq hp (hL x hx1)).mp hx2] | no goals | 007b2111fe7671e1 |
Submodule.inf_comap_le_comap_add | Mathlib/Algebra/Module/Submodule/Map.lean | theorem inf_comap_le_comap_add (f₁ f₂ : M →ₛₗ[τ₁₂] M₂) :
comap f₁ q ⊓ comap f₂ q ≤ comap (f₁ + f₂) q | R : Type u_1
R₂ : Type u_3
M : Type u_5
M₂ : Type u_7
inst✝⁵ : Semiring R
inst✝⁴ : Semiring R₂
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂
q : Submodule R₂ M₂
f₁ f₂ : M →ₛₗ[τ₁₂] M₂
m : M
h : f₁ m ∈ q ∧ f₂ m ∈ q
⊢ f₁ m + f₂ m ∈ q | apply q.add_mem h.1 h.2 | no goals | eee85a1daac4f5a6 |
SameRay.trans | Mathlib/LinearAlgebra/Ray.lean | theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
SameRay R x z | case inr.inr.inr.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : StrictOrderedCommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x y z : M
hxy : SameRay R x y
hyz : SameRay R y z
hy✝ : y = 0 → x = 0 ∨ z = 0
hx : x ≠ 0
hz : z ≠ 0
hy : y ≠ 0
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h... | rw [mul_smul, mul_smul, h₁, ← h₂, smul_comm] | no goals | 58153a221638fd7f |
Set.mulIndicator_iInter_apply | Mathlib/Algebra/Order/Group/Indicator.lean | @[to_additive]
lemma mulIndicator_iInter_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α → M) (x : α) :
mulIndicator (⋂ i, s i) f x = ⨅ i, mulIndicator (s i) f x | case neg.intro
ι : Sort u_1
α : Type u_2
M : Type u_3
inst✝² : CompleteLattice M
inst✝¹ : One M
inst✝ : Nonempty ι
h1 : ⊥ = 1
s : ι → Set α
f : α → M
x : α
j : ι
hj : x ∉ s j
⊢ ⨅ i, (s i).mulIndicator f x ≤ 1 | simpa [mulIndicator_of_not_mem hj] using (iInf_le (fun i ↦ (s i).mulIndicator f) j) x | no goals | 1f86a6e9fd525ad5 |
Finset.small_alternating_pow_of_small_tripling | Mathlib/Combinatorics/Additive/SmallTripling.lean | /-- If `A` has small tripling, say with constant `K`, then `A` has small alternating powers, in the
sense that `|A^±1 * ... * A^±1|` is at most `|A|` times a constant exponential in the number of
terms in the product.
When `A` is symmetric (`A⁻¹ = A`), the base of the exponential can be lowered from `K ^ 3` to `K`,
wh... | case inr.intro.inl.intro.inr.inl
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
m : ℕ
hm : 3 ≤ m
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
ε : Fin m → ℤ
hε : ∀ (i : Fin m), |ε i| = 1
hm₀ : m ≠ 0
hε₀ : ∀ (i : Fin m), ε i ≠ 0
hA₀ : A.Nonempty
hK₁ : 1 ≤ K
δ : Fin 3 → ℤ
this✝ : K ≤ K ^ 3
this : K ^ 2 ≤ K ^ 3
hδ₀... | nlinarith [small_pos_neg_pos_mul hA] | no goals | 647fdb3ae1591fd5 |
Std.DHashMap.Internal.List.getEntry?_eraseKey_of_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getEntry?_eraseKey_of_false [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)}
{k a : α} (hka : (k == a) = false) : getEntry? a (eraseKey k l) = getEntry? a l | case false
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
k a : α
hka : (k == a) = false
k' : α
v' : β k'
t : List ((a : α) × β a)
ih : getEntry? a (eraseKey k t) = getEntry? a t
h' : (k' == k) = false
⊢ getEntry? a (⟨k', v'⟩ :: eraseKey k t) = getEntry? a (⟨k', v'⟩ :: t) | cases h'' : k' == a | case false.false
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
k a : α
hka : (k == a) = false
k' : α
v' : β k'
t : List ((a : α) × β a)
ih : getEntry? a (eraseKey k t) = getEntry? a t
h' : (k' == k) = false
h'' : (k' == a) = false
⊢ getEntry? a (⟨k', v'⟩ :: eraseKey k t) = getEntry? a (⟨k', v'⟩ :: ... | 91f3eedba4da607e |
MeasureTheory.exists_measurable_le_setLintegral_eq_of_integrable | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem exists_measurable_le_setLintegral_eq_of_integrable {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞) :
∃ (g : α → ℝ≥0∞), Measurable g ∧ g ≤ f ∧ ∀ s : Set α, MeasurableSet s →
∫⁻ a in s, f a ∂μ = ∫⁻ a in s, g a ∂μ | case right
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤
g : α → ℝ≥0∞
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α) in sᶜ, g x ∂μ ≠ ⊤ | rw [hifg] at hf | case right
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : ∫⁻ (a : α), g a ∂μ ≠ ⊤
hmg : Measurable g
hgf : g ≤ f
hifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ
s : Set α
hms : MeasurableSet s
⊢ ∫⁻ (x : α) in sᶜ, g x ∂μ ≠ ⊤ | 95fc19ee505cdcac |
Equiv.Perm.zpow_eq_zpow_on_iff | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem zpow_eq_zpow_on_iff [DecidableEq α] [Fintype α]
(g : Perm α) {m n : ℤ} {x : α} (hx : g x ≠ x) :
(g ^ m) x = (g ^ n) x ↔ m % #(g.cycleOf x).support = n % #(g.cycleOf x).support | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ (m - n)) x = x ↔ (m - n) % ↑(#(g.cycleOf x).support) = 0 | rw [← Int.dvd_iff_emod_eq_zero] | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
m n : ℤ
x : α
hx : g x ≠ x
⊢ (g ^ (m - n)) x = x ↔ ↑(#(g.cycleOf x).support) ∣ m - n | 8f1720eac6149a76 |
fderivWithin_comp_derivWithin | Mathlib/Analysis/Calculus/Deriv/Comp.lean | theorem fderivWithin_comp_derivWithin {t : Set F} (hl : DifferentiableWithinAt 𝕜 l t (f x))
(hf : DifferentiableWithinAt 𝕜 f s x) (hs : MapsTo f s t) :
derivWithin (l ∘ f) s x = (fderivWithin 𝕜 l t (f x) : F → E) (derivWithin f s x) | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs : MapsTo ... | rcases uniqueDiffWithinAt_or_nhdsWithin_eq_bot s x with hxs | hxs | case inl
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type w
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
l : F → E
t : Set F
hl : DifferentiableWithinAt 𝕜 l t (f x)
hf : DifferentiableWithinAt 𝕜 f s x
hs ... | 0d62fad7b734d9f8 |
MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
(f : ℝⁿ⁺¹ → Eⁿ⁺¹)
(f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
(s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
(Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
(Hi : IntegrableOn (∑ i, f' · (e i) i) (B... | case intro.intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)... | have Hic : ∀ k, IntegrableOn (fun x => f (i.insertNth (c k) x) i) (Box.Icc (I.face i)) := fun k =>
(Box.continuousOn_face_Icc ((continuous_apply i).comp_continuousOn Hc) (hc k)).integrableOn_Icc | case intro.intro.intro
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)... | d25a0c4e5a28a1f3 |
Profinite.NobelingProof.GoodProducts.span_iff_products | Mathlib/Topology/Category/Profinite/Nobeling.lean | theorem GoodProducts.span_iff_products [WellFoundedLT I] :
⊤ ≤ Submodule.span ℤ (Set.range (eval C)) ↔
⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C)) | I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
⊢ Set.range (Products.eval C) ⊆ ↑(Submodule.span ℤ (Set.range (eval C))) | rintro f ⟨l, rfl⟩ | case intro
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
h : ⊤ ≤ Submodule.span ℤ (Set.range (Products.eval C))
l : Products I
⊢ Products.eval C l ∈ ↑(Submodule.span ℤ (Set.range (eval C))) | e9eecc3b629e2248 |
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