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 |
|---|---|---|---|---|---|---|
EuclideanGeometry.Sphere.secondInter_dist | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | theorem Sphere.secondInter_dist (s : Sphere P) (p : P) (v : V) :
dist (s.secondInter p v) s.center = dist p s.center | case neg
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
s : Sphere P
p : P
v : V
hv : ¬v = 0
⊢ dist ((-2 * inner v (p -ᵥ s.center) / inner v v) • v +ᵥ p) s.center = dist p s.center | rw [dist_smul_vadd_eq_dist _ _ hv] | case neg
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
s : Sphere P
p : P
v : V
hv : ¬v = 0
⊢ -2 * inner v (p -ᵥ s.center) / inner v v = 0 ∨
-2 * inner v (p -ᵥ s.center) / inner v v = -2 * inner v (p -ᵥ s.center) / inner v v | 1d3cdcba3601ff61 |
CategoryTheory.Pretriangulated.Triangle.eqToHom_hom₃ | Mathlib/CategoryTheory/Triangulated/Basic.lean | lemma Triangle.eqToHom_hom₃ {A B : Triangle C} (h : A = B) :
(eqToHom h).hom₃ = eqToHom (by subst h; rfl) | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
T₁ T₂ T₃ A : Triangle C
⊢ A.obj₃ = A.obj₃ | rfl | no goals | 98c2147d38434045 |
Turing.TM2to1.tr_respects_aux₂ | Mathlib/Computability/TuringMachine.lean | theorem tr_respects_aux₂ [DecidableEq K] {k : K} {q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ} {v : σ}
{S : ∀ k, List (Γ k)} {L : ListBlank (∀ k, Option (Γ k))}
(hL : ∀ k, L.map (proj k) = ListBlank.mk ((S k).map some).reverse) (o : StAct K Γ σ k) :
let v' := stVar v (S k) o
let Sk' := stWrite v (S k) o
le... | K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
k : K
q : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
f : σ → Option (Γ k) → σ
e : S k = []
⊢ (bi... | rw [addBottom_head_fst, cond] | no goals | 77a7f2ed036af218 |
Int.ofNat_le | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem ofNat_le {m n : Nat} : (↑m : Int) ≤ ↑n ↔ m ≤ n :=
⟨fun h =>
let ⟨k, hk⟩ := le.dest h
Nat.le.intro <| Int.ofNat.inj <| (Int.ofNat_add m k).trans hk,
fun h =>
let ⟨k, (hk : m + k = n)⟩ := Nat.le.dest h
le.intro k (by rw [← hk]; rfl)⟩
| m n : Nat
h : m ≤ n
k : Nat
hk : m + k = n
⊢ ↑m + ↑k = ↑(m + k) | rfl | no goals | 7966cd977341202b |
Equiv.Perm.IsCycle.eq_swap_of_apply_apply_eq_self | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem IsCycle.eq_swap_of_apply_apply_eq_self {α : Type*} [DecidableEq α] {f : Perm α}
(hf : IsCycle f) {x : α} (hfx : f x ≠ x) (hffx : f (f x) = x) : f = swap x (f x) :=
Equiv.ext fun y =>
let ⟨z, hz⟩ := hf
let ⟨i, hi⟩ := hz.2 hfx
if hyx : y = x then by simp [hyx]
else
if hfyx : y = f x th... | case intro.inl
α : Type u_4
inst✝ : DecidableEq α
f : Perm α
hf : f.IsCycle
x : α
hfx : f x ≠ x
hffx : f (f x) = x
y z : α
hz : f z ≠ z ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle z y
i : ℤ
hi : (f ^ i) z = x
hyx : ¬y = x
hfyx : ¬y = f x
hy : ¬f y = y
j : ℤ
hj : (f ^ (j - i)) x = y
hji : (f ^ (j - i)) x = x
⊢ False | rw [← hj, hji] at hyx | case intro.inl
α : Type u_4
inst✝ : DecidableEq α
f : Perm α
hf : f.IsCycle
x : α
hfx : f x ≠ x
hffx : f (f x) = x
y z : α
hz : f z ≠ z ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle z y
i : ℤ
hi : (f ^ i) z = x
hfyx : ¬y = f x
hy : ¬f y = y
j : ℤ
hyx : ¬x = x
hj : (f ^ (j - i)) x = y
hji : (f ^ (j - i)) x = x
⊢ False | e1c9ac989638c398 |
ae_eq_const_or_norm_average_lt_of_norm_le_const | Mathlib/Analysis/Convex/Integral.lean | theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E]
(h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖⨍ x, f x ∂μ‖ < C | case inl
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
C : ℝ
inst✝ : StrictConvexSpace ℝ E
h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
hC0 : C ≤ 0
this : f =ᶠ[ae μ] 0
⊢ f =ᶠ[ae μ] const α 0 ∨ ‖0‖ < C | exact Or.inl this | no goals | ad5ef237f6de5f77 |
DenseRange.piMap | Mathlib/Topology/ContinuousOn.lean | theorem DenseRange.piMap {ι : Type*} {X Y : ι → Type*} [∀ i, TopologicalSpace (Y i)]
{f : (i : ι) → (X i) → (Y i)} (hf : ∀ i, DenseRange (f i)):
DenseRange (Pi.map f) | ι : Type u_5
X : ι → Type u_6
Y : ι → Type u_7
inst✝ : (i : ι) → TopologicalSpace (Y i)
f : (i : ι) → X i → Y i
hf : ∀ (i : ι), DenseRange (f i)
⊢ DenseRange (Pi.map f) | rw [DenseRange, Set.range_piMap] | ι : Type u_5
X : ι → Type u_6
Y : ι → Type u_7
inst✝ : (i : ι) → TopologicalSpace (Y i)
f : (i : ι) → X i → Y i
hf : ∀ (i : ι), DenseRange (f i)
⊢ Dense (univ.pi fun i => range (f i)) | 77800a84bb5030c3 |
Metric.forall_of_forall_mem_ball | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | theorem forall_of_forall_mem_ball (p : α → Prop) (x : α)
(H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ ball x R, p y) (y : α) : p y | case intro.intro
α : Type u
inst✝ : PseudoMetricSpace α
p : α → Prop
x : α
H : ∃ᶠ (R : ℝ) in atTop, ∀ y ∈ ball x R, p y
y : α
R : ℝ
hR : R > dist y x
h : ∀ z ∈ ball x R, p z
⊢ p y | exact h _ hR | no goals | b3eba102e0317dc3 |
Vector.map_prod_right_eq_zip | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Zip.lean | theorem map_prod_right_eq_zip {l : Vector α n} (f : α → β) :
(l.map fun x => (f x, x)) = (l.map f).zip l | case mk.e_w
α : Type u_1
β : Type u_2
f : α → β
l : Array α
⊢ map (fun x => x) { toArray := l, size_toArray := ⋯ } = { toArray := l, size_toArray := ⋯ } | simp | no goals | a0d714b70680bda6 |
Real.mul_add_one_le_add_one_pow | Mathlib/Data/Real/Basic.lean | lemma mul_add_one_le_add_one_pow {a : ℝ} (ha : 0 ≤ a) (b : ℕ) : a * b + 1 ≤ (a + 1) ^ b | case inr
a : ℝ
ha : 0 ≤ a
b : ℕ
ha' : 0 < a
⊢ a * ↑b + 1 ≤ (a + 1) ^ b | clear ha | case inr
a : ℝ
b : ℕ
ha' : 0 < a
⊢ a * ↑b + 1 ≤ (a + 1) ^ b | cc47c81a5d85b08c |
Vector.mapFinIdx_eq_singleton_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/MapIdx.lean | theorem mapFinIdx_eq_singleton_iff {l : Vector α 1} {f : (i : Nat) → α → (h : i < 1) → β} {b : β} :
l.mapFinIdx f = #v[b] ↔ ∃ (a : α), l = #v[a] ∧ f 0 a (by omega) = b | case mk.mp
α : Type u_1
β : Type u_2
f : (i : Nat) → α → i < 1 → β
b : β
l : Array α
h : l.size = 1
⊢ (∃ a w, f 0 a ⋯ = b) → ∃ a, l = #[a] ∧ f 0 a ⋯ = b | rintro ⟨a, rfl, rfl⟩ | case mk.mp.intro.intro
α : Type u_1
β : Type u_2
f : (i : Nat) → α → i < 1 → β
a : α
h : #[a].size = 1
⊢ ∃ a_1, #[a] = #[a_1] ∧ f 0 a_1 ⋯ = f 0 a ⋯ | 91321782d8f8ef28 |
ClassGroup.equiv_mk | Mathlib/RingTheory/ClassGroup.lean | theorem ClassGroup.equiv_mk (K' : Type*) [Field K'] [Algebra R K'] [IsFractionRing R K']
(I : (FractionalIdeal R⁰ K)ˣ) :
ClassGroup.equiv K' (ClassGroup.mk I) =
QuotientGroup.mk' _ (Units.mapEquiv (↑(FractionalIdeal.canonicalEquiv R⁰ K K')) I) | R : Type u_1
K : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : Field K
inst✝⁵ : Algebra R K
inst✝⁴ : IsFractionRing R K
inst✝³ : IsDomain R
K' : Type u_3
inst✝² : Field K'
inst✝¹ : Algebra R K'
inst✝ : IsFractionRing R K'
I : (FractionalIdeal R⁰ K)ˣ
⊢ (equiv K') (mk I) = (QuotientGroup.mk' (toPrincipalIdeal R K').range) ((Unit... | erw [ClassGroup.equiv, ClassGroup.mk, MonoidHom.comp_apply, QuotientGroup.congr_mk'] | R : Type u_1
K : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : Field K
inst✝⁵ : Algebra R K
inst✝⁴ : IsFractionRing R K
inst✝³ : IsDomain R
K' : Type u_3
inst✝² : Field K'
inst✝¹ : Algebra R K'
inst✝ : IsFractionRing R K'
I : (FractionalIdeal R⁰ K)ˣ
⊢ (QuotientGroup.mk' (toPrincipalIdeal R K').range)
((Units.mapEquiv (ca... | 213fd075714f484a |
MeasureTheory.AEStronglyMeasurable.integral_kernel_comp | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_comp [NormedSpace ℝ E]
⦃f : γ → E⦄ (hf : AEStronglyMeasurable f ((η ∘ₖ κ) a)) :
AEStronglyMeasurable (fun x ↦ ∫ y, f y ∂η x) (κ a) :=
⟨fun x ↦ ∫ y, hf.mk f y ∂η x, hf.stronglyMeasurable_mk.integral_kernel, by
filter_upwards [ae_ae_of_ae_com... | α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
f : γ → E
hf : AEStronglyMeasurable f ((η ∘ₖ κ) a)
⊢ (fun x => ∫ (y : γ), f y ∂η x) =ᶠ[ae (κ a)] fun x => ∫ (... | filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk] with _ hx using integral_congr_ae hx | no goals | 2070171ea82af84a |
KuratowskiEmbedding.embeddingOfSubset_isometry | Mathlib/Topology/MetricSpace/Kuratowski.lean | theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) | α : Type u
inst✝ : MetricSpace α
x : ℕ → α
H : DenseRange x
a b : α
e : ℝ
epos : 0 < e
n : ℕ
hn : dist a (x n) < e / 2
C : dist b (x n) - dist a (x n) = ↑(embeddingOfSubset x b) n - ↑(embeddingOfSubset x a) n
⊢ 2 * dist a (x n) + (dist b (x n) - dist a (x n)) ≤ 2 * dist a (x n) + |dist b (x n) - dist a (x n)| | apply_rules [add_le_add_left, le_abs_self] | no goals | a61e80bcf82ba877 |
MeasureTheory.Lp.lintegral_rpow_tsum_coe_enorm_sub_le_tsum | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem lintegral_rpow_tsum_coe_enorm_sub_le_tsum {f : ℕ → α → E}
(hf : ∀ n, AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞}
(h : ∀ n, ∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' i, B i) ^ p) :
(∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i... | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
⊢ ∀ (g : ℕ →... | intro g | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
h : ∀ (n : ℕ), ∫⁻ (a : α), (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' (i : ℕ), B i) ^ p
hp_pos : 0 < p
g : ℕ → ℝ≥0∞... | ebb94e8857d3711d |
mem_posTangentConeAt_of_frequently_mem | Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | theorem mem_posTangentConeAt_of_frequently_mem (h : ∃ᶠ t : ℝ in 𝓝[>] 0, x + t • y ∈ s) :
y ∈ posTangentConeAt s x | case h
E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : ∃ᶠ (t : ℝ) in 𝓝[>] 0, x + t • y ∈ s
a : ℕ → ℝ
ha : Tendsto a atTop (𝓝[>] 0)
has : ∀ (n : ℕ), x + a n • y ∈ s
n : ℕ
hn : 0 < a n
⊢ y = a⁻¹ n • a n • y | simp [ne_of_gt hn] | no goals | 2805a654cfb28ce8 |
SimpleGraph.Walk.length_append | Mathlib/Combinatorics/SimpleGraph/Walk.lean | theorem length_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :
(p.append q).length = p.length + q.length | case nil
V : Type u
G : SimpleGraph V
u v w u✝ : V
q : G.Walk u✝ w
⊢ (nil.append q).length = nil.length + q.length | simp | no goals | ca530b6734dc2936 |
cfc_tsub | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean | theorem cfc_tsub {A : Type*} [TopologicalSpace A] [Ring A] [PartialOrder A] [StarRing A]
[StarOrderedRing A] [Algebra ℝ A] [IsTopologicalRing A] [T2Space A]
[ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
[NonnegSpectrumClass ℝ A] (f g : ℝ≥0 → ℝ≥0)
(a : A) (hfg : ∀ x ∈ spectrum ℝ≥0 a, g x ≤ ... | case refine_1
A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : Ring A
inst✝⁷ : PartialOrder A
inst✝⁶ : StarRing A
inst✝⁵ : StarOrderedRing A
inst✝⁴ : Algebra ℝ A
inst✝³ : IsTopologicalRing A
inst✝² : T2Space A
inst✝¹ : ContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f g : ℝ≥0 → ℝ≥0
a : A
h... | all_goals
exact continuous_subtype_val.comp_continuousOn <|
ContinuousOn.comp ‹_› continuous_real_toNNReal.continuousOn <| ha'.image ▸ Set.mapsTo_image .. | no goals | bfcc5c40ea36abec |
Real.cauchy_mul | Mathlib/Data/Real/Basic.lean | theorem cauchy_mul : ∀ a b, (a * b : ℝ).cauchy = a.cauchy * b.cauchy
| ⟨a⟩, ⟨b⟩ => show (mul _ _).cauchy = _ by rw [mul_def]
| a b : Cauchy abs
⊢ (Real.mul { cauchy := a } { cauchy := b }).cauchy = { cauchy := a }.cauchy * { cauchy := b }.cauchy | rw [mul_def] | no goals | b732c03270213869 |
NormedSpace.norm_expSeries_summable_of_mem_ball' | Mathlib/Analysis/Normed/Algebra/Exponential.lean | theorem norm_expSeries_summable_of_mem_ball' (x : 𝔸)
(hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
Summable fun n => ‖(n !⁻¹ : 𝕂) • x ^ n‖ | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : NontriviallyNormedField 𝕂
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
x : 𝔸
hx : x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
⊢ Summable (norm ∘ fun n => (expSeries 𝕂 𝔸 n) fun x_1 => x) | exact norm_expSeries_summable_of_mem_ball x hx | no goals | dcbdca1bba3f2303 |
LinearMap.continuousAt_zero_of_locally_bounded | Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean | theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
(hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 | case refine_2
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : UniformSpace F
inst✝⁶ : FirstCountableTopology E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : RCLike 𝕜'
ins... | intro n hn | case refine_2
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : UniformSpace E
inst✝⁹ : UniformAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : UniformSpace F
inst✝⁶ : FirstCountableTopology E
inst✝⁵ : RCLike 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : RCLike 𝕜'
ins... | eb01391c7362c5dd |
Int.emultiplicity_pow_sub_pow | Mathlib/NumberTheory/Multiplicity.lean | theorem Int.emultiplicity_pow_sub_pow {x y : ℤ} (hxy : ↑p ∣ x - y) (hx : ¬↑p ∣ x) (n : ℕ) :
emultiplicity (↑p) (x ^ n - y ^ n) = emultiplicity (↑p) (x - y) + emultiplicity p n | case succ.intro.intro.hn
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
k : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * k
⊢ ¬p ∣ k | rintro ⟨c, rfl⟩ | case succ.intro.intro.hn.intro
p : ℕ
hp : Prime ↑p
hp1 : Odd p
x y : ℤ
hxy : ↑p ∣ x - y
hx : ¬↑p ∣ x
n : ℕ
h : FiniteMultiplicity p (n + 1)
hpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1
c : ℕ
hk : n + 1 = p ^ multiplicity p (n + 1) * (p * c)
⊢ False | a22ac8f0646dd0b8 |
Sym2.card_image_offDiag | Mathlib/Data/Sym/Card.lean | theorem card_image_offDiag (s : Finset α) : #(s.offDiag.image Sym2.mk) = (#s).choose 2 | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
⊢ #(image Sym2.mk s.offDiag) = (#s).choose 2 | rw [Nat.choose_two_right, Nat.mul_sub_left_distrib, mul_one, ← offDiag_card,
Nat.div_eq_of_eq_mul_right Nat.zero_lt_two (two_mul_card_image_offDiag s).symm] | no goals | 721747a43c5e102d |
Batteries.RBNode.mem_insert_of_mem | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem mem_insert_of_mem {t : RBNode α} (ht : Balanced t c n) (h : v' ∈ t) :
v' ∈ t.insert cmp v ∨ cmp v v' = .eq | α : Type u_1
c : RBColor
n : Nat
v' : α
cmp : α → α → Ordering
v : α
t : RBNode α
ht : t.Balanced c n
h : v' ∈ t
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
p : Path α
e : zoom (cmp v) t = (node c✝ l✝ v✝ r✝, p)
w✝¹ w✝ : List α
h₁ : t.toList = w✝¹ ++ v✝ :: w✝
h₂ : (insert cmp t v).toList = w✝¹ ++ v :: w✝
⊢ v' ∈ inse... | simp [← mem_toList, h₁] at h | α : Type u_1
c : RBColor
n : Nat
v' : α
cmp : α → α → Ordering
v : α
t : RBNode α
ht : t.Balanced c n
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
p : Path α
e : zoom (cmp v) t = (node c✝ l✝ v✝ r✝, p)
w✝¹ w✝ : List α
h₁ : t.toList = w✝¹ ++ v✝ :: w✝
h₂ : (insert cmp t v).toList = w✝¹ ++ v :: w✝
h : v' ∈ w✝¹ ∨ v' = v✝... | bb0a162495bf6167 |
Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_isLocalRing | Mathlib/RingTheory/Unramified/Field.lean | theorem bijective_of_isAlgClosed_of_isLocalRing
[IsAlgClosed K] [IsLocalRing A] :
Function.Bijective (algebraMap K A) | K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
⊢ IsNilpotent (IsLocalRing.maximalIdeal A) | rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥] | K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : CommRing A
inst✝⁴ : Algebra K A
inst✝³ : FormallyUnramified K A
inst✝² : EssFiniteType K A
inst✝¹ : IsAlgClosed K
inst✝ : IsLocalRing A
this✝ : Module.Finite K A
this : IsArtinianRing A
⊢ IsNilpotent ⊥.jacobson
K : Type u_1
A : Type u_2
inst✝⁶ : Field K
inst✝⁵ : Comm... | 5099a67fd835b73a |
Array.eq_mkArray_of_mem | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem eq_mkArray_of_mem {a : α} {l : Array α} (h : ∀ (b) (_ : b ∈ l), b = a) : l = mkArray l.size a | α : Type u_1
a : α
l : Array α
h : ∀ (b : α), b ∈ l → b = a
⊢ ∀ (b : α), b ∈ l.toList → b = a | simpa using h | no goals | 1b5adbe8aea8862b |
EulerProduct.one_sub_inv_eq_geometric_of_summable_norm | Mathlib/NumberTheory/EulerProduct/Basic.lean | lemma one_sub_inv_eq_geometric_of_summable_norm {f : ℕ →*₀ F} {p : ℕ} (hp : p.Prime)
(hsum : Summable fun x ↦ ‖f x‖) :
(1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e) | F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
p : ℕ
hp : Nat.Prime p
hsum : Summable fun x => ‖f x‖
⊢ (1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e) | simp only [map_pow] | F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
p : ℕ
hp : Nat.Prime p
hsum : Summable fun x => ‖f x‖
⊢ (1 - f p)⁻¹ = ∑' (e : ℕ), f p ^ e | bde940928d778afb |
Int.sign_nonneg | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem sign_nonneg : 0 ≤ sign x ↔ 0 ≤ x | x : Int
n✝ : Nat
⊢ 0 ≤ (ofNat (n✝ + 1)).sign ↔ 0 ≤ ofNat (n✝ + 1) | simp +decide only [sign, true_iff] | x : Int
n✝ : Nat
⊢ 0 ≤ ofNat (n✝ + 1) | dfbafae3313ed376 |
Besicovitch.exists_closedBall_covering_tsum_measure_le | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ]
[Measure.OuterRegular μ] {ε : ℝ≥0∞} (hε : ε ≠ 0) (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧
(s ⊆ ⋃ x ∈ t, closedBa... | case h
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u... | apply Subset.trans (closedBall_subset_ball (hr0 x hx).2.2) (hR x (t0s hx)).2 | no goals | 181682c2ed74f98b |
IsDedekindDomain.HeightOneSpectrum.intValuation_exists_uniformizer | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | theorem intValuation_exists_uniformizer :
∃ π : R, v.intValuationDef π = Multiplicative.ofAdd (-1 : ℤ) | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
hv : Irreducible (Associates.mk v.asIdeal)
⊢ DvdNotUnit v.asIdeal (v.asIdeal ^ 2) | exact
⟨v.ne_bot, v.asIdeal, (not_congr Ideal.isUnit_iff).mpr (Ideal.IsPrime.ne_top v.isPrime),
sq v.asIdeal⟩ | no goals | 2c1cd36bab64f29a |
Multiset.foldl_induction' | Mathlib/Data/Multiset/MapFold.lean | theorem foldl_induction' (f : β → α → β) [RightCommutative f] (x : β) (q : α → Prop)
(p : β → Prop) (s : Multiset α) (hpqf : ∀ a b, q a → p b → p (f b a)) (px : p x)
(q_s : ∀ a ∈ s, q a) : p (foldl f x s) | α : Type u_1
β : Type v
f : β → α → β
inst✝ : RightCommutative f
x : β
q : α → Prop
p : β → Prop
s : Multiset α
hpqf : ∀ (a : α) (b : β), q a → p b → p (f b a)
px : p x
q_s : ∀ a ∈ s, q a
⊢ p (foldr (fun x y => f y x) x s) | exact foldr_induction' (fun x y => f y x) x q p s hpqf px q_s | no goals | c8f91e98d4f935d7 |
Computation.terminates_parallel | Mathlib/Data/Seq/Parallel.lean | theorem terminates_parallel {S : WSeq (Computation α)} {c} (h : c ∈ S) [T : Terminates c] :
Terminates (parallel S) | case a
α : Type u
S✝ : WSeq (Computation α)
c✝ : Computation α
h✝ : c✝ ∈ S✝
T✝ : c✝.Terminates
n : ℕ
IH :
∀ (l : List (Computation α)) (S : Stream'.Seq (Option (Computation α))) (c : Computation α),
c ∈ l ∨ some (some c) = S.get? n → c.Terminates → (corec parallel.aux1 (l, S)).Terminates
l : List (Computation α)
... | simp only [rmap] | no goals | b95c6ab968566afe |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRup | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem sat_of_insertRup {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f) (c : DefaultClause n)
(p : PosFin n → Bool) (pf : p ⊨ f) :
(insertRupUnits f (negate c)).2 = true → p ⊨ c | case intro.inr.inr.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false = true → ∃ i, f.as... | have i_true_in_insertUnit_fold :=
mem_insertUnit_fold_units #[] f.assignments false (c.clause.map Literal.negate) (i, true) i_true_in_insertUnit_fold | case intro.inr.inr.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.rupUnits, f.assignments, false) c.negate).2.snd = true
false_imp : false = true → ∃ i, f.as... | 6b3be4f74b21096f |
Order.Icc_subset_Ico_succ_right_of_not_isMax | Mathlib/Order/SuccPred/Basic.lean | theorem Icc_subset_Ico_succ_right_of_not_isMax (hb : ¬IsMax b) : Icc a b ⊆ Ico a (succ b) | α : Type u_1
inst✝¹ : Preorder α
inst✝ : SuccOrder α
a b : α
hb : ¬IsMax b
⊢ Ici a ∩ Iic b ⊆ Ici a ∩ Iio (succ b) | gcongr | case H
α : Type u_1
inst✝¹ : Preorder α
inst✝ : SuccOrder α
a b : α
hb : ¬IsMax b
⊢ Iic b ⊆ Iio (succ b) | bdc6aac34a4c2c00 |
Ordinal.isClosed_iff_bsup | Mathlib/SetTheory/Ordinal/Topology.lean | theorem isClosed_iff_bsup :
IsClosed s ↔
∀ {o : Ordinal}, o ≠ 0 → ∀ f : ∀ a < o, Ordinal,
(∀ i hi, f i hi ∈ s) → bsup.{u, u} o f ∈ s | s : Set Ordinal.{u}
⊢ IsClosed s ↔
∀ {o : Ordinal.{u}},
o ≠ 0 →
∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s | rw [isClosed_iff_iSup] | s : Set Ordinal.{u}
⊢ (∀ {ι : Type u}, Nonempty ι → ∀ (f : ι → Ordinal.{u}), (∀ (i : ι), f i ∈ s) → ⨆ i, f i ∈ s) ↔
∀ {o : Ordinal.{u}},
o ≠ 0 →
∀ (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), (∀ (i : Ordinal.{u}) (hi : i < o), f i hi ∈ s) → o.bsup f ∈ s | f5377d407ff3a5c7 |
uniform_extend_subtype | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | theorem uniform_extend_subtype [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β}
{s : Set α} (hf : UniformContinuous fun x : Subtype p => f x.val) (he : IsUniformEmbedding e)
(hd : ∀ x : β, x ∈ closure (range e)) (hb : closure (e '' s) ∈ 𝓝 b) (hs : IsClosed s)
(hp : ∀ x ∈ s, p x) : ∃ c, Tend... | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
inst✝ : CompleteSpace γ
p : α → Prop
e : α → β
f : α → γ
b : β
s : Set α
hf : UniformContinuous fun x => f ↑x
he : IsUniformEmbedding e
hd : ∀ (x : β), x ∈ closure (range e)
hb : closure (e '' s) ∈ 𝓝 b
hs : I... | simpa only [nhds_subtype_eq_comap, comap_comap, IsDenseEmbedding.subtypeEmb_coe] using hc | no goals | 9fe3a2e44291b4d4 |
AlgebraicGeometry.isOpenImmersion_sigmaDesc | Mathlib/AlgebraicGeometry/Limits.lean | lemma isOpenImmersion_sigmaDesc
{X : Scheme} (α : ∀ i, f i ⟶ X) [∀ i, IsOpenImmersion (α i)]
(hα : Pairwise (Disjoint on (Set.range <| α · |>.base))) :
IsOpenImmersion (Sigma.desc α) | case left
ι : Type u
f : ι → Scheme
X : Scheme
α : (i : ι) → f i ⟶ X
inst✝ : ∀ (i : ι), IsOpenImmersion (α i)
hα : _root_.Pairwise (Disjoint on fun x => Set.range ⇑(ConcreteCategory.hom (α x).base))
⊢ Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (Sigma.desc α).base) | suffices Topology.IsOpenEmbedding ((Sigma.desc α).base ∘ sigmaMk f) by
convert this.comp (sigmaMk f).symm.isOpenEmbedding; ext; simp | case left
ι : Type u
f : ι → Scheme
X : Scheme
α : (i : ι) → f i ⟶ X
inst✝ : ∀ (i : ι), IsOpenImmersion (α i)
hα : _root_.Pairwise (Disjoint on fun x => Set.range ⇑(ConcreteCategory.hom (α x).base))
⊢ Topology.IsOpenEmbedding (⇑(ConcreteCategory.hom (Sigma.desc α).base) ∘ ⇑(sigmaMk f)) | a3962f9239181cbe |
Mon_.Mon_tensor_mul_assoc | Mathlib/CategoryTheory/Monoidal/Mon_.lean | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
((tensorμ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul)) ▷ (M.X ⊗ N.X)) ≫
tensorμ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
((M.X ⊗ N.X) ◁ (tensorμ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul))) ≫
tensorμ M.X N.X M.X N.X... | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ tensorμ M.X N.X M.X N.X ▷ (M.X ⊗ N.X) ≫
tensorμ (M.X ⊗ M.X) (N.X ⊗ N.X) M.X N.X ≫
((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom) ≫ (M.X ◁ M.mul ⊗ N.X ◁ N.mul) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X... | slice_lhs 1 3 => rw [tensor_associativity] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
M N : Mon_ C
⊢ (((α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
(M.X ⊗ N.X) ◁ tensorμ M.X N.X M.X N.X ≫ tensorμ M.X N.X (M.X ⊗ M.X) (N.X ⊗ N.X)) ≫
(M.X ◁ M.mul ⊗ N.X ◁ N.mul)) ≫
(M.mul ⊗ N.mul) =
(... | 720e587f4afb9552 |
Filter.isCountablyGenerated_seq | Mathlib/Order/Filter/CountablyGenerated.lean | theorem isCountablyGenerated_seq [Countable ι'] (x : ι' → Set α) :
IsCountablyGenerated (⨅ i, 𝓟 (x i)) | α : Type u_1
ι' : Sort u_5
inst✝ : Countable ι'
x : ι' → Set α
⊢ (⨅ i, 𝓟 (x i)).IsCountablyGenerated | use range x, countable_range x | case right
α : Type u_1
ι' : Sort u_5
inst✝ : Countable ι'
x : ι' → Set α
⊢ ⨅ i, 𝓟 (x i) = generate (range x) | b85418f4749ff8fd |
CategoryTheory.Adjunction.LeftAdjointCommShift.compatibilityUnit_iso | Mathlib/CategoryTheory/Shift/Adjunction.lean | /--
The commutation isomorphisms of `Adjunction.LeftAdjointCommShift.iso` are compatible with
the unit of the adjunction.
-/
lemma compatibilityUnit_iso (a : A) :
CommShift.CompatibilityUnit adj (iso adj a) (G.commShiftIso a) | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : G.CommShift A
a : A
X✝ : C
⊢ (shiftFunctor C a).map (adj.unit.app X✝) =
(shiftFunctor C a).map (adj.unit.app X✝... | rw [shift_shiftFunctorCompIsoId_inv_app, ← Functor.comp_map,
(shiftFunctorCompIsoId C _ _ (neg_add_cancel a)).hom.naturality_assoc] | C : Type u_1
D : Type u_2
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : G.CommShift A
a : A
X✝ : C
⊢ (shiftFunctor C a).map (adj.unit.app X✝) =
(shiftFunctor C a).map (adj.unit.app X✝... | c804dc2cb607e547 |
sdiff_sdiff_right' | Mathlib/Order/BooleanAlgebra.lean | theorem sdiff_sdiff_right' : x \ (y \ z) = x \ y ⊔ x ⊓ z :=
calc
x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z := sdiff_sdiff_right
_ = z ⊓ x ⊓ y ⊔ x \ y | α : Type u
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x \ y ⊔ x ⊓ y ⊓ z = z ⊓ x ⊓ y ⊔ x \ y | ac_rfl | no goals | cdc6425268a979c6 |
Cardinal.cantor | Mathlib/SetTheory/Cardinal/Basic.lean | theorem cantor (a : Cardinal.{u}) : a < 2 ^ a | case h.intro.mk
α : Type u
f : Set α → α
hf : Injective f
⊢ False | exact cantor_injective f hf | no goals | 217c601a61335676 |
FreeGroup.injective_lift_of_ping_pong | Mathlib/GroupTheory/CoprodI.lean | theorem _root_.FreeGroup.injective_lift_of_ping_pong : Function.Injective (FreeGroup.lift a) | case hpp.inr
ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆... | have h1n : n ≤ -1 := by
apply Int.le_of_lt_add_one
simpa using hgt | case hpp.inr
ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆... | 7d73ea3c881ec9f6 |
Batteries.BinomialHeap.Imp.Heap.WF.merge' | Mathlib/.lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean | theorem Heap.WF.merge' (h₁ : s₁.WF le n) (h₂ : s₂.WF le n) :
(merge le s₁ s₂).WF le n ∧ ((s₁.rankGT n ↔ s₂.rankGT n) → (merge le s₁ s₂).rankGT n) | case h_3
α✝ : Type u_1
le : α✝ → α✝ → Bool
n : Nat
x✝¹ x✝ : Heap α✝
r₁✝ : Nat
a₁✝ : α✝
n₁✝ : HeapNode α✝
t₁✝ : Heap α✝
r₂✝ : Nat
a₂✝ : α✝
n₂✝ : HeapNode α✝
t₂✝ : Heap α✝
h₁ : WF le n (cons r₁✝ a₁✝ n₁✝ t₁✝)
h₂ : WF le n (cons r₂✝ a₂✝ n₂✝ t₂✝)
⊢ WF le n
(if r₁✝ < r₂✝ then cons r₁✝ a₁✝ n₁✝ (merge le t₁✝ (cons r₂✝ a₂... | rename_i r₁ a₁ n₁ t₁ r₂ a₂ n₂ t₂ | case h_3
α✝ : Type u_1
le : α✝ → α✝ → Bool
n : Nat
x✝¹ x✝ : Heap α✝
r₁ : Nat
a₁ : α✝
n₁ : HeapNode α✝
t₁ : Heap α✝
r₂ : Nat
a₂ : α✝
n₂ : HeapNode α✝
t₂ : Heap α✝
h₁ : WF le n (cons r₁ a₁ n₁ t₁)
h₂ : WF le n (cons r₂ a₂ n₂ t₂)
⊢ WF le n
(if r₁ < r₂ then cons r₁ a₁ n₁ (merge le t₁ (cons r₂ a₂ n₂ t₂))
else
... | dbba1f19a50f69f9 |
sameRay_or_ne_zero_and_sameRay_neg_iff_not_linearIndependent | Mathlib/LinearAlgebra/Ray.lean | theorem sameRay_or_ne_zero_and_sameRay_neg_iff_not_linearIndependent {x y : M} :
SameRay R x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] | R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
⊢ SameRay R x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] | rw [← sameRay_or_sameRay_neg_iff_not_linearIndependent] | R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
⊢ SameRay R x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay R x (-y) ↔ SameRay R x y ∨ SameRay R x (-y) | 2ef13ed9bc876873 |
HomologicalComplex.mapBifunctor₂₃.ι_D₃ | Mathlib/Algebra/Homology/BifunctorAssociator.lean | @[reassoc (attr := simp)]
lemma ι_D₃ :
ι F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j h ≫ D₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ j j' =
d₃ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j' | C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²³ : Category.{u_15, u_1} C₁
inst✝²² : Category.{u_17, u_2} C₂
inst✝²¹ : Category.{u_16, u_5} C₃
inst✝²⁰ : Category.{u_13, u_6} C₄
inst✝¹⁹ : Category.{u_14, u_4} C₂₃
inst✝¹⁸ : HasZeroMorphisms C₁
inst✝¹⁷ : HasZeroMorphisms C₂
inst✝¹⁶ : HasZeroM... | simp [D₃] | no goals | 00b428dda1c7deb8 |
MeasureTheory.tendsto_Lp_of_tendsto_ae_of_meas | Mathlib/MeasureTheory/Function/UnifTight.lean | theorem tendsto_Lp_of_tendsto_ae_of_meas (hp : 1 ≤ p) (hp' : p ≠ ∞)
{f : ℕ → α → β} {g : α → β} (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
(hg' : MemLp g p μ) (hui : UnifIntegrable f p μ) (hut : UnifTight f p μ)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) :
Tendsto (... | case neg.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x ... | obtain ⟨Ef, hmEf, hμEf, hfε⟩ := hut.exists_measurableSet_indicator hε'.ne' | case neg.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ⊤
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight ... | 77334843f098ce11 |
FirstOrder.Language.StrongHomClass.realize_boundedFormula | Mathlib/ModelTheory/Semantics.lean | theorem realize_boundedFormula (φ : L.BoundedFormula α n) {v : α → M}
{xs : Fin n → M} : φ.Realize (g ∘ v) (g ∘ xs) ↔ φ.Realize v xs | case falsum
L : Language
M : Type w
N : Type u_1
inst✝³ : L.Structure M
inst✝² : L.Structure N
α : Type u'
n : ℕ
F : Type u_4
inst✝¹ : EquivLike F M N
inst✝ : L.StrongHomClass F M N
g : F
v : α → M
n✝ : ℕ
xs : Fin n✝ → M
⊢ BoundedFormula.falsum.Realize (⇑g ∘ v) (⇑g ∘ xs) ↔ BoundedFormula.falsum.Realize v xs | rfl | no goals | a5c390dd0b13cfc7 |
Complex.arg_eq_pi_div_two_iff | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | case neg.mp
z : ℂ
h₀ : ¬z = 0
h : z.arg = π / 2
⊢ z.re = 0 ∧ 0 < z.im | rw [← norm_mul_cos_add_sin_mul_I z, h] | case neg.mp
z : ℂ
h₀ : ¬z = 0
h : z.arg = π / 2
⊢ (↑‖z‖ * (cos ↑(π / 2) + sin ↑(π / 2) * I)).re = 0 ∧ 0 < (↑‖z‖ * (cos ↑(π / 2) + sin ↑(π / 2) * I)).im | d10f1f15dbe0a05c |
ProbabilityTheory.Kernel.apply_eq_measure_condKernel_of_compProd_eq | Mathlib/Probability/Kernel/Disintegration/Unique.lean | lemma Kernel.apply_eq_measure_condKernel_of_compProd_eq
{ρ : Kernel α (β × Ω)} [IsFiniteKernel ρ] {κ : Kernel (α × β) Ω} [IsFiniteKernel κ]
(hκ : Kernel.fst ρ ⊗ₖ κ = ρ) (a : α) :
(fun b ↦ κ (a, b)) =ᵐ[Kernel.fst ρ a] (ρ a).condKernel | case h
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
ρ : Kernel α (β × Ω)
inst✝¹ : IsFiniteKernel ρ
κ : Kernel (α × β) Ω
inst✝ : IsFiniteKernel κ
hκ : ρ.fst ⊗ₖ κ = ρ
a : α
this : ρ a = (ρ a).fst ⊗ₘ κ.coma... | rw [← hb, Kernel.comap_apply] | no goals | 3feca3af78b81662 |
Int.mul_pos_of_neg_of_neg | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem mul_pos_of_neg_of_neg {a b : Int} (ha : a < 0) (hb : b < 0) : 0 < a * b | a b : Int
ha : a < 0
hb : b < 0
this : 0 * b < a * b
⊢ 0 < a * b | rwa [Int.zero_mul] at this | no goals | c193aa4bcca4bfd8 |
AlgebraicGeometry.UniversallyClosed.of_comp_of_isSeparated | Mathlib/AlgebraicGeometry/Morphisms/Proper.lean | @[stacks 01W6 "(1)"]
lemma UniversallyClosed.of_comp_of_isSeparated [UniversallyClosed (f ≫ g)] [IsSeparated g] :
UniversallyClosed f | X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : UniversallyClosed (f ≫ g)
inst✝ : IsSeparated g
this : UniversallyClosed (Limits.pullback.snd (f ≫ g) g)
⊢ UniversallyClosed (Limits.pullback.lift (𝟙 X) f ⋯ ≫ Limits.pullback.snd (f ≫ g) g) | infer_instance | no goals | 0f3e173a4fa4ff02 |
Submodule.fg_of_fg_map_of_fg_inf_ker | Mathlib/RingTheory/Finiteness/Finsupp.lean | theorem fg_of_fg_map_of_fg_inf_ker (f : M →ₗ[R] P) {s : Submodule R M}
(hs1 : (s.map f).FG)
(hs2 : (s ⊓ LinearMap.ker f).FG) : s.FG | case intro.intro.intro
R : Type u_1
M : Type u_2
P : Type u_4
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] P
s : Submodule R M
this✝¹ : DecidableEq R
this✝ : DecidableEq M
this : DecidableEq P
t1 : Finset P
ht1 : span R ↑t1 = map f s
t2 : Finset M
ht... | apply le_antisymm | case intro.intro.intro.a
R : Type u_1
M : Type u_2
P : Type u_4
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup P
inst✝ : Module R P
f : M →ₗ[R] P
s : Submodule R M
this✝¹ : DecidableEq R
this✝ : DecidableEq M
this : DecidableEq P
t1 : Finset P
ht1 : span R ↑t1 = map f s
t2 : Finset M
... | db8c10624993c4e9 |
AlgebraicGeometry.Scheme.germ_stalkClosedPointTo | Mathlib/AlgebraicGeometry/Stalk.lean | @[reassoc]
lemma germ_stalkClosedPointTo (U : Opens X) (hU : f.base (closedPoint R) ∈ U) :
X.presheaf.germ U _ hU ≫ stalkClosedPointTo f = f.app U ≫
((Spec R).presheaf.mapIso (eqToIso (preimage_eq_top_of_closedPoint_mem f hU).symm).op ≪≫
ΓSpecIso R).hom | case e_a
X : Scheme
R : CommRingCat
inst✝ : IsLocalRing ↑R
f : Spec R ⟶ X
U : X.Opens
hU : (ConcreteCategory.hom f.base) (closedPoint ↑R) ∈ U
⊢ (Spec R).presheaf.germ (f ⁻¹ᵁ U) (closedPoint ↑R) hU ≫ (stalkClosedPointIso R).hom =
(Functor.mapIso (Spec R).presheaf (eqToIso ⋯).op).hom ≫ (ΓSpecIso R).hom | rw [← Iso.eq_comp_inv, Category.assoc, ΓSpecIso_hom_stalkClosedPointIso_inv] | case e_a
X : Scheme
R : CommRingCat
inst✝ : IsLocalRing ↑R
f : Spec R ⟶ X
U : X.Opens
hU : (ConcreteCategory.hom f.base) (closedPoint ↑R) ∈ U
⊢ (Spec R).presheaf.germ (f ⁻¹ᵁ U) (closedPoint ↑R) hU =
(Functor.mapIso (Spec R).presheaf (eqToIso ⋯).op).hom ≫ (Spec R).presheaf.germ ⊤ (closedPoint ↑R) trivial | 6e682145e6ef3b18 |
tendsto_tsum_of_dominated_convergence | Mathlib/Analysis/Normed/Group/Tannery.lean | /-- **Tannery's theorem**: topological sums commute with termwise limits, when the norms of the
summands are eventually uniformly bounded by a summable function.
(This is the special case of the Lebesgue dominated convergence theorem for the counting measure
on a discrete set. However, we prove it under somewhat weake... | α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le :... | calc _ ≤ ‖∑' (k : (Tᶜ : Set β)), bound k‖ := Real.le_norm_self _
_ = ‖S - ∑ b ∈ T, bound b‖ := congrArg _ ?_
_ < ε / 3 := by rwa [dist_eq_norm, norm_sub_rev] at hT | α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le :... | d4198172206cf97e |
Filter.mem_prod_iff | Mathlib/Order/Filter/Prod.lean | theorem mem_prod_iff {s : Set (α × β)} {f : Filter α} {g : Filter β} :
s ∈ f ×ˢ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s | case mp
α : Type u_1
β : Type u_2
s : Set (α × β)
f : Filter α
g : Filter β
⊢ s ∈ f ×ˢ g → ∃ t₁ ∈ f, ∃ t₂ ∈ g, t₁ ×ˢ t₂ ⊆ s | rintro ⟨t₁, ⟨s₁, hs₁, hts₁⟩, t₂, ⟨s₂, hs₂, hts₂⟩, rfl⟩ | case mp.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
f : Filter α
g : Filter β
t₁ : Set (α × β)
s₁ : Set α
hs₁ : s₁ ∈ f
hts₁ : Prod.fst ⁻¹' s₁ ⊆ t₁
t₂ : Set (α × β)
s₂ : Set β
hs₂ : s₂ ∈ g
hts₂ : Prod.snd ⁻¹' s₂ ⊆ t₂
⊢ ∃ t₁_1 ∈ f, ∃ t₂_1 ∈ g, t₁_1 ×ˢ t₂_1 ⊆ t₁ ∩ t₂ | 743ac113f24bf37b |
MeasureTheory.LevyProkhorov.continuous_equiv_probabilityMeasure | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | /-- The identity map `LevyProkhorov (ProbabilityMeasure Ω) → ProbabilityMeasure Ω` is continuous. -/
lemma LevyProkhorov.continuous_equiv_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)) | case neg.h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := (equiv (ProbabilityMeasure Ω)) ν
Ps : ℕ → ProbabilityMeasure Ω :... | intro δ δ_pos | case neg.h
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := (equiv (ProbabilityMeasure Ω)) ν
Ps : ℕ → ProbabilityMeasure Ω :... | e3f8df53302ac07c |
sup_inf_self | Mathlib/Order/Lattice.lean | theorem sup_inf_self : a ⊔ a ⊓ b = a | α : Type u
inst✝ : Lattice α
a b : α
⊢ a ⊔ a ⊓ b = a | simp | no goals | 8ab3d91ac2bc5c1f |
CFC.exists_pos_algebraMap_le_iff | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean | /-- In a unital `ℝ`-algebra `A` with a continuous functional calculus, an element `a : A` is larger
than some `algebraMap ℝ A r` if and only if every element of the `ℝ`-spectrum is nonnegative. -/
lemma CFC.exists_pos_algebraMap_le_iff {A : Type*} [TopologicalSpace A] [Ring A] [StarRing A]
[PartialOrder A] [StarOrd... | case refine_1
A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Algebra ℝ A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : Nontrivial A
inst✝ : ContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha : autoParam (IsSelfAdjoint a) _auto✝
h_c... | rintro ⟨r, hr, hr_le⟩ | case refine_1.intro.intro
A : Type u_1
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : PartialOrder A
inst✝⁴ : StarOrderedRing A
inst✝³ : Algebra ℝ A
inst✝² : NonnegSpectrumClass ℝ A
inst✝¹ : Nontrivial A
inst✝ : ContinuousFunctionalCalculus ℝ IsSelfAdjoint
a : A
ha : autoParam (IsSelfAdjoint a... | e9e557be1398cc06 |
exists_bijective_map_powers | Mathlib/Algebra/Module/FinitePresentation.lean | /--
Let `M` be a finite `R`-module, and `N` be a finitely presented `R`-module.
If `l : M →ₗ[R] N` is a linear map whose localization at `S : Submonoid R` is bijective,
then `l` is already bijective under the localization at some `r ∈ S`.
-/
lemma exists_bijective_map_powers [Module.Finite R M] [Module.FinitePresentati... | R : Type u_3
M : Type u_4
N : Type u_5
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
S : Submonoid R
M' : Type u_1
inst✝⁷ : AddCommGroup M'
inst✝⁶ : Module R M'
f : M →ₗ[R] M'
inst✝⁵ : IsLocalizedModule S f
N' : Type u_2
inst✝⁴ : AddCommGroup N'
inst✝³ : ... | obtain ⟨l', s₀, H⟩ := Module.FinitePresentation.exists_lift_of_isLocalizedModule S f
(e.symm.toLinearMap.comp g) | case intro.intro
R : Type u_3
M : Type u_4
N : Type u_5
inst✝¹² : CommRing R
inst✝¹¹ : AddCommGroup M
inst✝¹⁰ : Module R M
inst✝⁹ : AddCommGroup N
inst✝⁸ : Module R N
S : Submonoid R
M' : Type u_1
inst✝⁷ : AddCommGroup M'
inst✝⁶ : Module R M'
f : M →ₗ[R] M'
inst✝⁵ : IsLocalizedModule S f
N' : Type u_2
inst✝⁴ : AddCommG... | 8598e5ad0b5e6c9d |
RCLike.lt_iff_re_im | Mathlib/Analysis/RCLike/Basic.lean | theorem lt_iff_re_im : z < w ↔ re z < re w ∧ im z = im w | case mp.intro.intro
K : Type u_1
inst✝ : RCLike K
z w : K
heq : z ≠ w
hr : re z ≤ re w
hi : im z = im w
⊢ (re z ≤ re w ∧ re z ≠ re w) ∧ im z = im w | exact ⟨⟨hr, mt (fun hreq => ext hreq hi) heq⟩, hi⟩ | no goals | 8626b25d0a5f2479 |
AlgebraicGeometry.HasRingHomProperty.stalkwise | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | lemma stalkwise {P} (hP : RingHom.RespectsIso P) :
HasRingHomProperty (stalkwise P) fun {_ S _ _} φ ↦
∀ (p : Ideal S) (_ : p.IsPrime), P (Localization.localRingHom _ p φ rfl) | case h.e'_2.h.h.h.h.h.a
P : {R S : Type u_1} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : RingHom.RespectsIso P
this✝ : IsLocalAtTarget (AlgebraicGeometry.stalkwise fun {R S} [CommRing R] [CommRing S] => P)
this : IsLocalAtSource (AlgebraicGeometry.stalkwise fun {R S} [CommRing R] [CommRing S] ... | exact (stalkwise_Spec_map_iff hP (CommRingCat.ofHom φ)).symm | no goals | 8b3cd2c3752dc955 |
HomologicalComplex.HomologySequence.composableArrows₃_exact | Mathlib/Algebra/Homology/HomologySequence.lean | /-- The diagram `K.homology i ⟶ K.opcycles i ⟶ K.cycles j ⟶ K.homology j` is exact
when `c.Rel i j`. -/
lemma composableArrows₃_exact [CategoryWithHomology C] :
(composableArrows₃ K i j).Exact | C : Type u_1
ι✝ : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι✝
K : HomologicalComplex C c
i j : ι✝
hij : c.Rel i j
inst✝ : CategoryWithHomology C
S : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯
S' : ShortComplex C := ShortComplex.mk (K.homologyι i) ... | let T := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) (by simp) | C : Type u_1
ι✝ : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι✝
K : HomologicalComplex C c
i j : ι✝
hij : c.Rel i j
inst✝ : CategoryWithHomology C
S : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯
S' : ShortComplex C := ShortComplex.mk (K.homologyι i) ... | 916349ac32ed1fb7 |
GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | theorem integral_cexp_neg_mul_sq_add_real_mul_I (hb : 0 < b.re) (c : ℝ) :
∫ x : ℝ, cexp (-b * (x + c * I) ^ 2) = (π / b) ^ (1 / 2 : ℂ) | case h
b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T => ∫ (y : ℝ) in 0 ..c, cexp (-b * (↑T + ↑y * I) ^ 2)
I₅ : ℝ → ... | rw [mul_sub, ← C] | case h
b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T => ∫ (y : ℝ) in 0 ..c, cexp (-b * (↑T + ↑y * I) ^ 2)
I₅ : ℝ → ... | 8f46f2d622ac75ef |
Matroid.comap_isBasis_iff | Mathlib/Data/Matroid/Map.lean | @[simp] lemma comap_isBasis_iff {I X : Set α} :
(N.comap f).IsBasis I X ↔ N.IsBasis (f '' I) (f '' X) ∧ I.InjOn f ∧ I ⊆ X | case refine_1.intro
α : Type u_1
β : Type u_2
f : α → β
N : Matroid β
I X : Set α
h : (N.comap f).IsBasis I X
hI : N.Indep (f '' I)
hinj : InjOn f I
⊢ N.IsBasis (f '' I) (f '' X) ∧ InjOn f I ∧ I ⊆ X | refine ⟨hI.isBasis_of_forall_insert (image_subset f h.subset) fun e he ↦ ?_, hinj, h.subset⟩ | case refine_1.intro
α : Type u_1
β : Type u_2
f : α → β
N : Matroid β
I X : Set α
h : (N.comap f).IsBasis I X
hI : N.Indep (f '' I)
hinj : InjOn f I
e : β
he : e ∈ f '' X \ f '' I
⊢ N.Dep (insert e (f '' I)) | 066e348ba5181ee3 |
List.nodup_replicate | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Pairwise.lean | theorem nodup_replicate {n : Nat} {a : α} :
(replicate n a).Nodup ↔ n ≤ 1 | α : Type u_1
n : Nat
a : α
⊢ (replicate n a).Nodup ↔ n ≤ 1 | simp [Nodup] | no goals | ebde18a7870c6a54 |
Nat.coprime_fintype_prod_right_iff | Mathlib/Data/Nat/GCD/BigOperators.lean | theorem coprime_fintype_prod_right_iff [Fintype ι] {x : ℕ} {s : ι → ℕ} :
Coprime x (∏ i, s i) ↔ ∀ i, Coprime x (s i) | ι : Type u_1
inst✝ : Fintype ι
x : ℕ
s : ι → ℕ
⊢ x.Coprime (∏ i : ι, s i) ↔ ∀ (i : ι), x.Coprime (s i) | simp [coprime_prod_right_iff] | no goals | 9c7d1da0094b7b1f |
LinearMap.IsSymmetric.continuous | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | theorem IsSymmetric.continuous [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) :
Continuous T | 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
⊢ inner (y - T x) (y - T x) = 0 | have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ := by
intro k
rw [← T.map_sub, hT] | 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
u : ℕ → E
x y : E
hu : Filter.Tendsto u Filter.atTop (nhds x)
hTu : Filter.Tendsto (⇑T ∘ u) Filter.atTop (nhds y)
hlhs : ∀ (k : ℕ), inner (T (u k) - T x) ... | 52353d41424a9736 |
EisensteinSeries.eisensteinSeries_tendstoLocallyUniformly | Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean | theorem eisensteinSeries_tendstoLocallyUniformly {k : ℤ} (hk : 3 ≤ k) {N : ℕ} (a : Fin 2 → ZMod N) :
TendstoLocallyUniformly (fun (s : Finset (gammaSet N a)) ↦ (∑ x ∈ s, eisSummand k x ·))
(eisensteinSeries a k ·) Filter.atTop | k : ℤ
hk : 3 ≤ k
N : ℕ
a : Fin 2 → ZMod N
hk' : 2 < ↑k
p_sum : Summable fun x => ‖↑x‖ ^ (-k)
K : Set ℍ
hK : IsCompact K
A B : ℝ
hB : 0 < B
HABK : K ⊆ verticalStrip A B
p : ↑(gammaSet N a)
z : ℍ
hz : z ∈ verticalStrip A B
⊢ 0 ≤ ↑k | positivity | no goals | 25804a9281a9f714 |
NormedAddGroupHom.ker_completion | Mathlib/Analysis/Normed/Group/HomCompletion.lean | theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : ℝ}
(h : f.SurjectiveOnWith f.range C) :
(f.completion.ker : Set <| Completion G) = closure (toCompl.comp <| incl f.ker).range | G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
⊢ hatg ∈ closure ↑(toCompl.comp (incl f.ker)).range | rw [SeminormedAddCommGroup.mem_closure_iff] | G : Type u_1
inst✝¹ : SeminormedAddCommGroup G
H : Type u_2
inst✝ : SeminormedAddCommGroup H
f : NormedAddGroupHom G H
C : ℝ
h : f.SurjectiveOnWith f.range C
hatg : Completion G
hatg_in : f.completion hatg = 0
⊢ ∀ (ε : ℝ), 0 < ε → ∃ b ∈ ↑(toCompl.comp (incl f.ker)).range, ‖hatg - b‖ < ε | f0c87529fe1bb1ed |
Matrix.StdBasisMatrix.trace_eq | Mathlib/LinearAlgebra/Matrix/Trace.lean | theorem trace_eq : trace (stdBasisMatrix i i c) = c | n : Type u_10
α : Type u_12
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : AddCommMonoid α
i : n
c : α
⊢ (stdBasisMatrix i i c).trace = c | simp [trace, -diag_apply] | no goals | 949c64ffdb2e0e46 |
Complex.integral_boundary_rect_of_hasFDerivAt_real_off_countable | Mathlib/Analysis/Complex/CauchyIntegral.lean | theorem integral_boundary_rect_of_hasFDerivAt_real_off_countable (f : ℂ → E) (f' : ℂ → ℂ →L[ℝ] E)
(z w : ℂ) (s : Set ℂ) (hs : s.Countable)
(Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[z.im, w.im]]))
(Hd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min z.im w.im) (max z.im w.im) \ s,
HasFDerivAt f ... | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : s.Countable
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])
Hd : ∀ x ∈ Ioo (z.re ⊓ w.re) (z.re ⊔ w.re) ×ℂ Ioo (w.im ⊓ z.im) (w.im ⊔ z.im) \ s, HasFDerivAt f (f' x) x
Hi : ... | have htd :
∀ p ∈ Ioo (min z.re w.re) (max z.re w.re) ×ˢ Ioo (min w.im z.im) (max w.im z.im) \ t,
HasFDerivAt F (F' p) p :=
fun p hp => (Hd (e p) hp).comp p e.hasFDerivAt | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f : ℂ → E
f' : ℂ → ℂ →L[ℝ] E
z w : ℂ
s : Set ℂ
hs : s.Countable
Hc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])
Hd : ∀ x ∈ Ioo (z.re ⊓ w.re) (z.re ⊔ w.re) ×ℂ Ioo (w.im ⊓ z.im) (w.im ⊔ z.im) \ s, HasFDerivAt f (f' x) x
Hi : ... | 0798a56cd56be0be |
not_irrational_exists_rep | Mathlib/Data/Real/Pi/Irrational.lean | /-- If `x` is rational, it can be written as `a / b` with `a : ℤ` and `b : ℕ` satisfying `b > 0`. -/
private lemma not_irrational_exists_rep {x : ℝ} :
¬Irrational x → ∃ (a : ℤ) (b : ℕ), 0 < b ∧ x = a / b | q : ℚ
⊢ ↑q = ↑q.num / ↑q.den | exact_mod_cast (Rat.num_div_den _).symm | no goals | 1f3a2c9ccb198e1f |
AlgebraicGeometry.isIntegral_appTop_of_universallyClosed | Mathlib/AlgebraicGeometry/Morphisms/Proper.lean | theorem isIntegral_appTop_of_universallyClosed (f : X ⟶ Y) [UniversallyClosed f] [IsAffine Y] :
f.appTop.hom.IsIntegral | case surj
X Y : Scheme
f : X ⟶ Y
inst✝¹ : UniversallyClosed f
inst✝ : IsAffine Y
this✝¹ : CompactSpace ↑↑X.toPresheafedSpace
this✝ : UniversallyClosed (X.toSpecΓ ≫ Spec.map (Scheme.Hom.appTop f))
this : UniversallyClosed X.toSpecΓ
⊢ Function.Surjective ⇑(ConcreteCategory.hom X.toSpecΓ.base) | apply surjective_of_isClosed_range_of_injective | case surj.hfcl
X Y : Scheme
f : X ⟶ Y
inst✝¹ : UniversallyClosed f
inst✝ : IsAffine Y
this✝¹ : CompactSpace ↑↑X.toPresheafedSpace
this✝ : UniversallyClosed (X.toSpecΓ ≫ Spec.map (Scheme.Hom.appTop f))
this : UniversallyClosed X.toSpecΓ
⊢ IsClosed (Set.range ⇑(ConcreteCategory.hom X.toSpecΓ.base))
case surj.hfinj
X Y :... | 0f196145409a30ed |
TopCat.Presheaf.stalkToFiber_injective | Mathlib/Topology/Sheaves/Sheafify.lean | theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) | case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U... | dsimp at wUx | case w.intro.intro.intro.intro.intro.intro.intro.intro
X : TopCat
F : Presheaf (Type v) X
x : ↑X
U V : OpenNhds x
fU : (y : ↥U.obj) → F.stalk ↑y
hU : (Sheafify.isLocallyGerm F).pred fU
fV : (y : ↥V.obj) → F.stalk ↑y
hV : (Sheafify.isLocallyGerm F).pred fV
e : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
U' : Opens ↑X
mU : ↑⟨x, ⋯⟩ ∈ U'
iU : U... | c3bddb63a4c72896 |
Nat.ModEq.mul_right_cancel' | Mathlib/Data/Nat/ModEq.lean | theorem mul_right_cancel' {a b c m : ℕ} (hc : c ≠ 0) :
a * c ≡ b * c [MOD m * c] → a ≡ b [MOD m] | a b c m : ℕ
hc : c ≠ 0
⊢ a * c ≡ b * c [MOD m * c] → a ≡ b [MOD m] | simp only [modEq_iff_dvd, Int.natCast_mul, ← Int.sub_mul] | a b c m : ℕ
hc : c ≠ 0
⊢ ↑m * ↑c ∣ (↑b - ↑a) * ↑c → ↑m ∣ ↑b - ↑a | 5eb0a3f32184f1c0 |
Ordinal.infinite_pigeonhole | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem infinite_pigeonhole {β α : Type u} (f : β → α) (h₁ : ℵ₀ ≤ #β) (h₂ : #α < (#β).ord.cof) :
∃ a : α, #(f ⁻¹' {a}) = #β | case intro
β α : Type u
f : β → α
h₁ : ℵ₀ ≤ #β
h₂ : #α < (#β).ord.cof
x : α
h : #β ≤ #↑(f ⁻¹' {x})
⊢ ∃ p, #↑p = #↑(f ⁻¹' {x}) | exact ⟨_, rfl⟩ | no goals | 8fa0d4d26a8c4da8 |
Path.trans_range | Mathlib/Topology/Path.lean | theorem trans_range {a b c : X} (γ₁ : Path a b) (γ₂ : Path b c) :
range (γ₁.trans γ₂) = range γ₁ ∪ range γ₂ | X : Type u_1
inst✝ : TopologicalSpace X
a b c : X
γ₁ : Path a b
γ₂ : Path b c
x : X
t : ℝ
ht0 : 0 ≤ t
ht1 : t ≤ 1
hxt :
{ toFun := (fun t => if t ≤ 1 / 2 then γ₁.extend (2 * t) else γ₂.extend (2 * t - 1)) ∘ Subtype.val,
continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⟨t, ⋯⟩ =
x
h : ¬t ≤ 1 / 2
⊢ ... | linarith | no goals | 4045b3b0d5c91c52 |
Combinatorics.exists_mono_homothetic_copy | Mathlib/Combinatorics/HalesJewett.lean | theorem exists_mono_homothetic_copy {M κ : Type*} [AddCommMonoid M] (S : Finset M) [Finite κ]
(C : M → κ) : ∃ a > 0, ∃ (b : M) (c : κ), ∀ s ∈ S, C (a • s + b) = c | case intro.intro.intro.intro.refine_3.e_a.e_a
M : Type u_5
κ : Type u_6
inst✝¹ : AddCommMonoid M
S : Finset M
inst✝ : Finite κ
C : M → κ
ι : Type
_inst : Fintype ι
l : Line { x // x ∈ S } ι
c : κ
s : Finset ι := Finset.filter (fun i => l.idxFun i = none) Finset.univ
hs : s = Finset.filter (fun i => l.idxFun i = none) F... | apply Finset.sum_congr rfl | case intro.intro.intro.intro.refine_3.e_a.e_a
M : Type u_5
κ : Type u_6
inst✝¹ : AddCommMonoid M
S : Finset M
inst✝ : Finite κ
C : M → κ
ι : Type
_inst : Fintype ι
l : Line { x // x ∈ S } ι
c : κ
s : Finset ι := Finset.filter (fun i => l.idxFun i = none) Finset.univ
hs : s = Finset.filter (fun i => l.idxFun i = none) F... | 7ae8871d3dcacc2f |
Std.DHashMap.Internal.Raw₀.Const.isEmpty_alter | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem isEmpty_alter (h : m.1.WF) {k : α} {f : Option β → Option β} :
(Const.alter m k f).1.isEmpty = ((m.1.isEmpty || (m.1.size == 1 && m.contains k)) &&
(f (Const.get? m k)).isNone) | α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
m : Raw₀ α fun x => β
h : m.val.WF
k : α
f : Option β → Option β
⊢ (alter m k f).val.isEmpty = ((m.val.isEmpty || m.val.size == 1 && m.contains k) && (f (get? m k)).isNone) | simp_to_model [Const.alter] using List.Const.isEmpty_alterKey | no goals | a37802e0d33a421b |
Nat.factorization_choose_of_lt_three_mul | Mathlib/Data/Nat/Choose/Factorization.lean | theorem factorization_choose_of_lt_three_mul (hp' : p ≠ 2) (hk : p ≤ k) (hk' : p ≤ n - k)
(hn : n < 3 * p) : (choose n k).factorization p = 0 | case inr.inl
p n k : ℕ
hp' : p ≠ 2
hk : p ≤ k
hk' : p ≤ n - k
hn : n < 3 * p
hp : Prime p
hnk : n < k
⊢ (n.choose k).factorization p = 0 | simp [choose_eq_zero_of_lt hnk] | no goals | 4fee1cd51a2f2fa2 |
StieltjesFunction.outer_Ioc | Mathlib/MeasureTheory/Measure/Stieltjes.lean | theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a) | case intro.intro.intro.intro
f : StieltjesFunction
a b : ℝ
s : ℕ → Set ℝ
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ⊤
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : ℝ
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : ∀ (i : ℕ), ∃ p, ... | choose g hg using this | case intro.intro.intro.intro
f : StieltjesFunction
a b : ℝ
s : ℕ → Set ℝ
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ⊤
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : ℝ
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
g : ℕ → ℝ × ℝ
hg : ∀ (i... | c3b0d539cbbc47aa |
mfderivWithin_eq_mfderiv | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | theorem mfderivWithin_eq_mfderiv (hs : UniqueMDiffWithinAt I s x) (h : MDifferentiableAt I I' f x) :
mfderivWithin I I' f s x = mfderiv I I' f x | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | rw [← mfderivWithin_univ] | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpac... | 6007d8c519f3acfb |
ProbabilityTheory.Kernel.tendsto_densityProcess_atTop_empty_of_antitone | Mathlib/Probability/Kernel/Disintegration/Density.lean | lemma tendsto_densityProcess_atTop_empty_of_antitone (κ : Kernel α (γ × β)) (ν : Kernel α γ)
[IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ)
(seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
(hseq_meas : ∀ m, MeasurableSet (seq m)) :
Tendsto (fun m ↦ densityProcess κ ν n a x (seq m)) atT... | case hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
inst✝ : IsFiniteKernel κ
n : ℕ
a : α
x : γ
seq : ℕ → Set β
hseq : Antitone seq
hseq_iInter : ⋂ i, seq i = ∅
hseq_meas : ∀ (m : ℕ), Measura... | measurability | no goals | 1d0b83eb9b0fedd0 |
Nat.two_mul_sq_add_one_le_two_pow_two_mul | Mathlib/Data/Nat/Cast/Order/Ring.lean | lemma two_mul_sq_add_one_le_two_pow_two_mul (k : ℕ) : 2 * k ^ 2 + 1 ≤ 2 ^ (2 * k) | k : ℕ
⊢ 2 * k ^ 2 + 1 ≤ 2 ^ (2 * k) | induction k with
| zero => simp
| succ k hk =>
rw [add_pow_two, one_pow, mul_one, add_assoc, mul_add, add_right_comm]
refine (add_le_add_right hk _).trans ?_
rw [mul_add 2 k, pow_add, mul_one, pow_two, ← mul_assoc, mul_two, mul_two, add_assoc]
gcongr
rw [← two_mul, ← pow_succ']
exact le_add_of_le_right (mul... | no goals | 51018307cf316205 |
Int.nonneg_or_nonpos_of_mul_nonneg | Mathlib/Data/Int/Order/Basic.lean | theorem nonneg_or_nonpos_of_mul_nonneg {a b : ℤ} : 0 ≤ a * b → 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 | case pos
a b : ℤ
h : 0 ≤ a * b
ha : 0 ≤ a
hb : 0 ≤ b
⊢ 0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0 | exact .inl ⟨ha, hb⟩ | no goals | e98d5bbf69c5d15d |
Real.qaryEntropy_strictAntiOn | Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean | /-- Qary entropy is strictly decreasing in the interval [1 - q⁻¹, 1]. -/
lemma qaryEntropy_strictAntiOn (qLe2 : 2 ≤ q) :
StrictAntiOn (qaryEntropy q) (Icc (1 - 1/q) 1) | q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
⊢ 0 < ↑q | positivity | no goals | 37758adf34f3dcdd |
Int.sq_of_gcd_eq_one | Mathlib/RingTheory/Int/Basic.lean | theorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :
∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 | case intro.intro
a b c : ℤ
h : a.gcd b = 1
heq : a * b = c ^ 2
h' : IsUnit (GCDMonoid.gcd a b)
d : ℤ
u : ℤˣ
hu : d ^ 2 * ↑u = a
⊢ ∃ a0, a = a0 ^ 2 ∨ a = -a0 ^ 2 | use d | case h
a b c : ℤ
h : a.gcd b = 1
heq : a * b = c ^ 2
h' : IsUnit (GCDMonoid.gcd a b)
d : ℤ
u : ℤˣ
hu : d ^ 2 * ↑u = a
⊢ a = d ^ 2 ∨ a = -d ^ 2 | de99b93abef95ba2 |
tendsto_nhds_Prop | Mathlib/Topology/Order.lean | theorem tendsto_nhds_Prop {l : Filter α} {p : α → Prop} {q : Prop} :
Tendsto p l (𝓝 q) ↔ (q → ∀ᶠ x in l, p x) | α : Type u_1
l : Filter α
p : α → Prop
q : Prop
⊢ Tendsto p l (𝓝 q) ↔ q → ∀ᶠ (x : α) in l, p x | by_cases q <;> simp [*] | no goals | c9ff38cff542c5c1 |
Lean.Order.Option.monotone_bindM | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_bindM (f : γ → α → m (Option β)) (xs : Option α) (hmono : monotone f) :
monotone (fun x => xs.bindM (f x)) | case some.hmono₁
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → m (Option β)
hmono : monotone f
x : α
⊢ monotone fun x_1 => f x_1 x | apply monotone_apply | case some.hmono₁.h
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → m (Option β)
hmono : monotone f
x : α
⊢ monotone f | b937dac3f62d8aa6 |
AkraBazziRecurrence.eventually_deriv_one_add_smoothingFn | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma eventually_deriv_one_add_smoothingFn :
deriv (fun x => 1 + ε x) =ᶠ[atTop] fun x => -x⁻¹ / (log x ^ 2) := calc
deriv (fun x => 1 + ε x) =ᶠ[atTop] deriv ε | case h
x : ℝ
hx : 1 < x
⊢ deriv (fun x => 1 + ε x) x = deriv ε x | rw [deriv_add] <;> aesop | no goals | 3cc9ebd84b21879d |
sameRay_or_sameRay_neg_iff_not_linearIndependent | Mathlib/LinearAlgebra/Ray.lean | theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y] | case neg.refine_2.intro.intro.inr.inr.inr.inr
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ 0).succ → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : 0 < m 0
hm1 : 0 < m 1
⊢ Sa... | refine Or.inr (Or.inr (Or.inr ⟨m 0, m 1, hm0, hm1, ?_⟩)) | case neg.refine_2.intro.intro.inr.inr.inr.inr
R : Type u_1
inst✝³ : LinearOrderedCommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
x y : M
hx : ¬x = 0
hy : ¬y = 0
m : Fin (Nat.succ 0).succ → R
hm : m 0 • x = -(m 1 • y)
hmne : ∃ i, m i ≠ 0
hm0 : 0 < m 0
hm1 : 0 < m 1
⊢ m ... | 6d16c99f1722f5f4 |
Diffeomorph.sumComm_inl | Mathlib/Geometry/Manifold/Diffeomorph.lean | lemma sumComm_inl : (Diffeomorph.sumComm I M n M') ∘ Sum.inl = Sum.inr | case h
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_5
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_9
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
n : WithTop ℕ∞
M' : Type u_13
inst✝¹ : TopologicalSpace M'... | exact Sum.swap_inl | no goals | eb806b01d33d9225 |
MeasureTheory.exists_setLintegral_compl_lt | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem exists_setLintegral_compl_lt {f : α → ℝ≥0∞} (hf : ∫⁻ a, f a ∂μ ≠ ∞)
{ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ s : Set α, MeasurableSet s ∧ μ s < ∞ ∧ ∫⁻ a in sᶜ, f a ∂μ < ε | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
hf₀ : ¬∫⁻ (a : α), f a ∂μ = 0
g : α → ℝ≥0∞
hgf : g ≤ f
hg_meas : Measurable g
hgsupp : μ (support g) < ⊤
hgε : ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ
a✝ : α
⊢ a✝ ∈ (support g)ᶜ → f a✝ = f a✝ - g a✝ | simp_all | no goals | 3d82ee275fc876fb |
Std.Sat.AIG.RefVec.map_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean | theorem map_decl_eq {aig : AIG α} (target : MapTarget aig len) :
∀ idx (h1 : idx < aig.decls.size) (h2),
(map aig target).1.decls[idx]'h2
=
aig.decls[idx]'h1 | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
aig : AIG α
target : MapTarget aig len
idx✝ : Nat
h1✝ : idx✝ < aig.decls.size
h2✝ : idx✝ < (map aig target).aig.decls.size
⊢ (map.go aig 0 ⋯ empty target.vec target.func).aig.decls[idx✝] = aig.decls[idx✝] | apply map.go_decl_eq | no goals | a5e5d29bd9848809 |
TendstoUniformlyOnFilter.tendsto_of_eventually_tendsto | Mathlib/Topology/UniformSpace/UniformConvergence.lean | theorem TendstoUniformlyOnFilter.tendsto_of_eventually_tendsto
(h1 : TendstoUniformlyOnFilter F f p p') (h2 : ∀ᶠ i in p, Tendsto (F i) p' (𝓝 (L i)))
(h3 : Tendsto L p (𝓝 ℓ)) : Tendsto f p' (𝓝 ℓ) | case intro.intro
α : Type u
β : Type v
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
inst✝ : p.NeBot
L : ι → β
ℓ : β
h1 : TendstoUniformlyOnFilter F f p p'
h2 : ∀ᶠ (i : ι) in p, Tendsto (F i) p' (𝓝 (L i))
h3 : Tendsto L p (𝓝 ℓ)
s : Set (β × β)
hs : s ∈ 𝓤 β
t : Set (β × β)
ht :... | have p1 : ∀ᶠ i in p, (L i, ℓ) ∈ t := tendsto_nhds_left.mp h3 ht | case intro.intro
α : Type u
β : Type v
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
inst✝ : p.NeBot
L : ι → β
ℓ : β
h1 : TendstoUniformlyOnFilter F f p p'
h2 : ∀ᶠ (i : ι) in p, Tendsto (F i) p' (𝓝 (L i))
h3 : Tendsto L p (𝓝 ℓ)
s : Set (β × β)
hs : s ∈ 𝓤 β
t : Set (β × β)
ht :... | 38aac9494fde1a73 |
Nat.fermat_primeFactors_one_lt | Mathlib/NumberTheory/Fermat.lean | lemma fermat_primeFactors_one_lt (n p : ℕ) (hn : 1 < n) (hp : p.Prime)
(hpdvd : p ∣ fermatNumber n) :
∃ k, p = k * 2 ^ (n + 2) + 1 | case intro
n p : ℕ
hn : 1 < n
hp : Prime p
hpdvd : p ∣ n.fermatNumber
this : Fact (Prime p)
hp2 : p ≠ 2
hp8 : p % 8 = 1
a : ZMod p
ha : 2 = a * a
⊢ ∃ k, p = k * 2 ^ (n + 2) + 1 | suffices h : p ∣ a.val ^ (2 ^ (n + 1)) + 1 by
exact pow_pow_add_primeFactors_one_lt hp hp2 h | case intro
n p : ℕ
hn : 1 < n
hp : Prime p
hpdvd : p ∣ n.fermatNumber
this : Fact (Prime p)
hp2 : p ≠ 2
hp8 : p % 8 = 1
a : ZMod p
ha : 2 = a * a
⊢ p ∣ a.val ^ 2 ^ (n + 1) + 1 | d91b28813044d95b |
IsScalarTower.algebraMap_apply | Mathlib/Algebra/Algebra/Tower.lean | theorem algebraMap_apply (x : R) : algebraMap R A x = algebraMap S A (algebraMap R S x) | R : Type u
S : Type v
A : Type w
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Semiring A
inst✝³ : Algebra R S
inst✝² : Algebra S A
inst✝¹ : Algebra R A
inst✝ : IsScalarTower R S A
x : R
⊢ (algebraMap R A) x = (algebraMap S A) ((algebraMap R S) x) | rw [algebraMap_eq R S A, RingHom.comp_apply] | no goals | d2e8bd32d6be6b7e |
StarConvex.affine_image | Mathlib/Analysis/Convex/Star.lean | theorem StarConvex.affine_image (f : E →ᵃ[𝕜] F) {s : Set E} (hs : StarConvex 𝕜 x s) :
StarConvex 𝕜 (f x) (f '' s) | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x : E
f : E →ᵃ[𝕜] F
s : Set E
hs : StarConvex 𝕜 x s
⊢ StarConvex 𝕜 (f x) (⇑f '' s) | rintro y ⟨y', ⟨hy', hy'f⟩⟩ a b ha hb hab | case intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x : E
f : E →ᵃ[𝕜] F
s : Set E
hs : StarConvex 𝕜 x s
y : F
y' : E
hy' : y' ∈ s
hy'f : f y' = y
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ a • f... | 6ae8fa9f6d288806 |
EReal.continuousAt_mul | Mathlib/Topology/Instances/EReal/Lemmas.lean | theorem continuousAt_mul {p : EReal × EReal} (h₁ : p.1 ≠ 0 ∨ p.2 ≠ ⊥)
(h₂ : p.1 ≠ 0 ∨ p.2 ≠ ⊤) (h₃ : p.1 ≠ ⊥ ∨ p.2 ≠ 0) (h₄ : p.1 ≠ ⊤ ∨ p.2 ≠ 0) :
ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) p | case mk.h_top.h_real
a✝ : ℝ
h₁ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊥
h₂ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊤
h₃ : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ 0
h₄ : (⊤, ↑a✝).1 ≠ ⊤ ∨ (⊤, ↑a✝).2 ≠ 0
⊢ ContinuousAt (fun p => p.1 * p.2) (⊤, ↑a✝) | simp only [ne_eq, not_true_eq_false, EReal.coe_eq_zero, false_or] at h₄ | case mk.h_top.h_real
a✝ : ℝ
h₁ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊥
h₂ : (⊤, ↑a✝).1 ≠ 0 ∨ (⊤, ↑a✝).2 ≠ ⊤
h₃ : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ 0
h₄ : ¬a✝ = 0
⊢ ContinuousAt (fun p => p.1 * p.2) (⊤, ↑a✝) | 7c243b47bdc719c0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.