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 |
|---|---|---|---|---|---|---|
infinite_range_add_smul_iff | Mathlib/Algebra/Module/Torsion.lean | @[simp]
lemma infinite_range_add_smul_iff
[AddCommGroup M] [Ring R] [Module R M] [Infinite R] [NoZeroSMulDivisors R M] (x y : M) :
(Set.range <| fun r : R ↦ x + r • y).Infinite ↔ y ≠ 0 | R : Type u_1
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Ring R
inst✝² : Module R M
inst✝¹ : Infinite R
inst✝ : NoZeroSMulDivisors R M
x y : M
⊢ (Set.range fun r => x + r • y).Infinite ↔ y ≠ 0 | refine ⟨fun h hy ↦ by simp [hy] at h, fun h ↦ Set.infinite_range_of_injective fun r s hrs ↦ ?_⟩ | R : Type u_1
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Ring R
inst✝² : Module R M
inst✝¹ : Infinite R
inst✝ : NoZeroSMulDivisors R M
x y : M
h : y ≠ 0
r s : R
hrs : x + r • y = x + s • y
⊢ r = s | 102b66a5247196f4 |
Irreducible.isRelPrime_iff_not_dvd | Mathlib/Algebra/GroupWithZero/Associated.lean | /-- See also `Irreducible.coprime_iff_not_dvd`. -/
lemma Irreducible.isRelPrime_iff_not_dvd [Monoid M] {p n : M} (hp : Irreducible p) :
IsRelPrime p n ↔ ¬ p ∣ n | M : Type u_1
inst✝ : Monoid M
p n : M
hp : Irreducible p
d : M
hdp : d ∣ p
hdn : d ∣ n
hpn : ¬IsUnit d
⊢ p ~ᵤ d | exact (hp.dvd_iff.mp hdp).resolve_left hpn | no goals | 7649a8e0b6517ae9 |
CategoryTheory.Limits.SequentialProduct.functorMap_epi | Mathlib/CategoryTheory/Limits/Shapes/SequentialProduct.lean | lemma functorMap_epi (n : ℕ) : Epi (functorMap f n) | case inst
C : Type u_1
M N : ℕ → C
inst✝⁵ : Category.{u_2, u_1} C
f : (n : ℕ) → M n ⟶ N n
inst✝⁴ : HasProductsOfShape ℕ C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
inst✝¹ : HasCountableProducts C
inst✝ : ∀ (n : ℕ), Epi (f n)
n : ℕ
⊢ Epi
(Pi.map fun i =>
if h : ↑i < n then eqToHom ⋯ else if h'... | apply ( config := { allowSynthFailures := true } ) Pi.map_epi | case inst
C : Type u_1
M N : ℕ → C
inst✝⁵ : Category.{u_2, u_1} C
f : (n : ℕ) → M n ⟶ N n
inst✝⁴ : HasProductsOfShape ℕ C
inst✝³ : HasZeroMorphisms C
inst✝² : HasFiniteBiproducts C
inst✝¹ : HasCountableProducts C
inst✝ : ∀ (n : ℕ), Epi (f n)
n : ℕ
⊢ ∀ (j : { x // x < n + 1 }),
Epi (if h : ↑j < n then eqToHom ⋯ else... | ff7fccff98cea99f |
HomologicalComplex.homologyMap_comp | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | @[reassoc]
lemma homologyMap_comp : homologyMap (φ ≫ ψ) i = homologyMap φ i ≫ homologyMap ψ i | C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
ψ : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
⊢ ShortComplex.homologyMap ((shortComplexFunctor C c i).map (φ ≫ ψ)) =
ShortCompl... | rw [Functor.map_comp, ShortComplex.homologyMap_comp] | no goals | 6455d6761e52a238 |
Complex.Gamma_mul_Gamma_add_half | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(√π) | s : ℂ
h1 : AnalyticOnNhd ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
h3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)
⊢ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π | refine AnalyticOnNhd.eq_of_frequently_eq h1 h2 (h3.frequently ?_) | s : ℂ
h1 : AnalyticOnNhd ℂ (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ
h2 : AnalyticOnNhd ℂ (fun z => (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ
h3 : Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)
⊢ ∃ᶠ (x : ℝ) in 𝓝[≠] 1, (Gamma ↑x)⁻¹ * (Gamma (↑x + 1 / 2))⁻¹ = (Gamma (2 * ↑x))⁻¹ * 2 ^ (2 * ↑x - 1) / ↑√π | 12960a3c3095c2a8 |
IntermediateField.adjoin.range_algebraMap_subset | Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.lean | theorem adjoin.range_algebraMap_subset : Set.range (algebraMap F E) ⊆ adjoin F S | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
x : E
hx : x ∈ Set.range ⇑(algebraMap F E)
⊢ x ∈ ↑(adjoin F S) | obtain ⟨f, hf⟩ := hx | case intro
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
x : E
f : F
hf : (algebraMap F E) f = x
⊢ x ∈ ↑(adjoin F S) | 8125e536e0a4de76 |
MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem eq_of_posPart_eq_posPart {j₁ j₂ : JordanDecomposition α}
(hj : j₁.posPart = j₂.posPart) (hj' : j₁.toSignedMeasure = j₂.toSignedMeasure) : j₁ = j₂ | case negPart
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.posPart = j₂.posPart
hj' : j₁.posPart.toSignedMeasure - j₁.negPart.toSignedMeasure = j₂.posPart.toSignedMeasure - j₂.negPart.toSignedMeasure
⊢ j₁.negPart.toSignedMeasure = j₂.negPart.toSignedMeasure | simp_rw [hj, sub_right_inj] at hj' | case negPart
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.posPart = j₂.posPart
hj' : j₁.negPart.toSignedMeasure = j₂.negPart.toSignedMeasure
⊢ j₁.negPart.toSignedMeasure = j₂.negPart.toSignedMeasure | f270db3000901bda |
LipschitzOnWith.ediam_image2_le | Mathlib/Topology/EMetricSpace/Lipschitz.lean | theorem ediam_image2_le (f : α → β → γ) {K₁ K₂ : ℝ≥0} (s : Set α) (t : Set β)
(hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (f · b) s) (hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t) :
EMetric.diam (Set.image2 f s t) ≤ ↑K₁ * EMetric.diam s + ↑K₂ * EMetric.diam t | α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
f : α → β → γ
K₁ K₂ : ℝ≥0
s : Set α
t : Set β
hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (fun x => f x b) s
hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t
⊢ EMetric.diam (image2 f s t) ≤ ↑K₁ * EMetric.diam s + ↑K... | simp only [EMetric.diam_le_iff, forall_mem_image2] | α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : PseudoEMetricSpace β
inst✝ : PseudoEMetricSpace γ
f : α → β → γ
K₁ K₂ : ℝ≥0
s : Set α
t : Set β
hf₁ : ∀ b ∈ t, LipschitzOnWith K₁ (fun x => f x b) s
hf₂ : ∀ a ∈ s, LipschitzOnWith K₂ (f a) t
⊢ ∀ x ∈ s, ∀ y ∈ t, ∀ x_1 ∈ s, ∀ y_1 ∈ t, edist (f x y) (... | fc0fbcf768dfe50a |
integral_withDensity_eq_integral_smul | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem integral_withDensity_eq_integral_smul {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E) :
∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ | case pos.refine_2.hg
X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
f_meas : Measurable f
g : X → E
hE : CompleteSpace E
hg : Integrable g (μ.withDensity fun x => ↑(f x))
u u' : X → E
a✝ : Disjoint (support u) (support u')
u_int : Int... | exact (integrable_withDensity_iff_integrable_smul f_meas).1 u'_int | no goals | cd239c0816bb7a0c |
Vector.mapIdx_push | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/MapIdx.lean | theorem mapIdx_push {l : Vector α n} {a : α} :
mapIdx f (l.push a) = (mapIdx f l).push (f l.size a) | α : Type u_1
n : Nat
α✝ : Type u_2
f : Nat → α → α✝
l : Vector α n
a : α
⊢ mapIdx f (l.push a) = (mapIdx f l).push (f l.size a) | simp [← append_singleton, mapIdx_append] | no goals | 45fbc3444931ff68 |
CategoryTheory.ActionCategory.cases' | Mathlib/CategoryTheory/Action.lean | lemma cases' ⦃a' b' : ActionCategory G X⦄ (f : a' ⟶ b') :
∃ (a b : X) (g : G) (ha : a' = a) (hb : b' = b) (hg : a = g⁻¹ • b),
f = eqToHom (by rw [ha, hg]) ≫ homOfPair b g ≫ eqToHom (by rw [hb]) | X : Type u
G : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
⊢ ∀ ⦃a' b' : ActionCategory G X⦄ (f : a' ⟶ b'),
∃ a b g, ∃ (ha : a' = ⟨(), a⟩) (hb : b' = ⟨(), b⟩) (hg : a = g⁻¹ • b), f = eqToHom ⋯ ≫ homOfPair b g ≫ eqToHom ⋯ | exact ActionCategory.cases (fun t g => ⟨g⁻¹ • t, t, g, rfl, rfl, rfl, by simp⟩) | no goals | dfe025127d7a3454 |
Std.Sat.AIG.mkOrCached_le_size | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedGatesLemmas.lean | theorem mkOrCached_le_size (aig : AIG α) (input : BinaryInput aig) :
aig.decls.size ≤ (aig.mkOrCached input).aig.decls.size | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤ (aig.mkOrCached input).aig.decls.size | simp only [mkOrCached] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.BinaryInput
⊢ aig.decls.size ≤
(((aig.mkGateCached (input.asGateInput true true)).aig.mkConstCached true).aig.mkGateCached
{
lhs :=
{ ref := ((aig.mkGateCached (input.asGateInput true true)).aig.mkCo... | c56fb0dc153c637e |
Ordinal.cof_eq_one_iff_is_succ | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem cof_eq_one_iff_is_succ {o} : cof.{u} o = 1 ↔ ∃ a, o = succ a :=
⟨inductionOn o fun α r _ z => by
rcases cof_eq r with ⟨S, hl, e⟩; rw [z] at e
obtain ⟨a⟩ := mk_ne_zero_iff.1 (by rw [e]; exact one_ne_zero)
refine
⟨typein r a,
Eq.symm <|
Quotient.sound
... | case intro.intro
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
⊢ ∃ a, type r = succ a | obtain ⟨a⟩ := mk_ne_zero_iff.1 (by rw [e]; exact one_ne_zero) | case intro.intro.intro
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
a : ↑S
⊢ ∃ a, type r = succ a | 7afdd5212ddba93b |
AkraBazziRecurrence.eventually_deriv_rpow_p_mul_one_sub_smoothingFn | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma eventually_deriv_rpow_p_mul_one_sub_smoothingFn (p : ℝ) :
deriv (fun z => z ^ p * (1 - ε z))
=ᶠ[atTop] fun z => p * z ^ (p-1) * (1 - ε z) + z ^ (p-1) / (log z ^ 2) := calc
deriv (fun x => x ^ p * (1 - ε x))
=ᶠ[atTop] fun x => deriv (· ^ p) x * (1 - ε x) + x ^ p * deriv (1 - ε ·) x | case h
p x : ℝ
hx : 0 < x
⊢ p * x ^ (p - 1) * (1 - ε x) + x ^ p * (x⁻¹ / log x ^ 2) = p * x ^ (p - 1) * (1 - ε x) + x ^ (p - 1) / log x ^ 2 | rw [mul_div, ← Real.rpow_neg_one, ← Real.rpow_add (by positivity), sub_eq_add_neg] | no goals | 51fc6dc849755feb |
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
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 = Fintype... | rw [card_map] | case intro.refine_2
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 = Fintype... | 3ec57b8d04698567 |
Affine.Simplex.reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem reflection_circumcenter_eq_affineCombination_of_pointsWithCircumcenter {n : ℕ}
(s : Simplex ℝ P n) {i₁ i₂ : Fin (n + 1)} (h : i₁ ≠ i₂) :
reflection (affineSpan ℝ (s.points '' {i₁, i₂})) s.circumcenter =
(univ : Finset (PointsWithCircumcenterIndex n)).affineCombination ℝ s.pointsWithCircumcenter
... | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
⊢ ↑((EuclideanGeometry.orthogonalProjection W... | apply eq_orthogonalProjection_of_eq_subspace | case h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P n
i₁ i₂ : Fin (n + 1)
h : i₁ ≠ i₂
hc : #{i₁, i₂} = 2
W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})
⊢ W = affineSpan ℝ (Set.range (s.face ... | bf7f74a93392e097 |
Nat.maxPowDiv.le_of_dvd | Mathlib/Data/Nat/MaxPowDiv.lean | theorem le_of_dvd {p n pow : ℕ} (hp : 1 < p) (hn : 0 < n) (h : p ^ pow ∣ n) :
pow ≤ p.maxPowDiv n | case a
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
⊢ c ≠ 0 | intro h' | case a
p n pow : ℕ
hp : 1 < p
hn : 0 < n
h : p ^ pow ∣ n
c : ℕ
hc : n = p ^ pow * c
h' : c = 0
⊢ False | ebf577af9efbf263 |
Complex.stolzCone_subset_stolzSet_aux' | Mathlib/Analysis/Complex/AbelLimit.lean | private lemma stolzCone_subset_stolzSet_aux' (s : ℝ) :
∃ M ε, 0 < M ∧ 0 < ε ∧ ∀ x y, 0 < x → x < ε → |y| < s * x →
sqrt (x ^ 2 + y ^ 2) < M * (1 - sqrt ((1 - x) ^ 2 + y ^ 2)) | s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ x ^ 2 + y ^ 2 ≤ x ^ 2 + (s * x) ^ 2 | rw [← sq_abs y] | s x y : ℝ
hx₀ : 0 < x
hx₁ : x < 1 / (1 + s ^ 2)
hy : |y| < s * x
H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2
⊢ x ^ 2 + |y| ^ 2 ≤ x ^ 2 + (s * x) ^ 2 | 9323596596941863 |
exists_dist_lt_lt | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem exists_dist_lt_lt (hδ : 0 < δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
∃ y, dist x y < δ ∧ dist y z < ε | E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z < ε + δ
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x z / (ε + δ) * δ < δ ∧ dist x z / (ε + δ) * ε < ε | rw [← div_lt_one (add_pos hε hδ)] at h | E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x z : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
h : dist x z / (ε + δ) < 1
y : E
hy : dist x y = δ / (ε + δ) * dist x z ∧ dist y z = ε / (ε + δ) * dist x z
⊢ dist x z / (ε + δ) * δ < δ ∧ dist x z / (ε + δ) * ε < ε | 3c2b87a2394cdde0 |
integral_exp_mul_complex | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_exp_mul_complex {c : ℂ} (hc : c ≠ 0) :
(∫ x in a..b, Complex.exp (c * x)) = (Complex.exp (c * b) - Complex.exp (c * a)) / c | a b : ℝ
c : ℂ
hc : c ≠ 0
x : ℝ
⊢ HasDerivAt (fun y => Complex.exp (c * ↑y) / c) (Complex.exp (c * ↑x) * c / c) x | apply ((Complex.hasDerivAt_exp _).comp x _).div_const c | a b : ℝ
c : ℂ
hc : c ≠ 0
x : ℝ
⊢ HasDerivAt (fun x => c * ↑x) c x | e4e37ec6a00be6b9 |
Prop.exists | Mathlib/Logic/Lemmas.lean | lemma Prop.exists {f : Prop → Prop} : (∃ p, f p) ↔ f True ∨ f False :=
⟨fun ⟨p, h⟩ ↦ by refine (em p).imp ?_ ?_ <;> intro H <;> convert h <;> simp [H],
by rintro (h | h) <;> exact ⟨_, h⟩⟩
| f : Prop → Prop
x✝ : ∃ p, f p
p : Prop
h : f p
⊢ f True ∨ f False | refine (em p).imp ?_ ?_ <;> intro H <;> convert h <;> simp [H] | no goals | cc5c151bc7b6f380 |
Batteries.RBNode.Any_reverse | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem Any_reverse {t : RBNode α} : t.reverse.Any p ↔ t.Any p | α : Type u_1
p : α → Prop
t : RBNode α
⊢ Any p t.reverse ↔ Any p t | simp [Any_def] | no goals | dcfb4a70dbaefbcc |
RelSeries.mem_snoc | Mathlib/Order/RelSeries.lean | lemma mem_snoc {p : RelSeries r} {newLast : α} {rel : r p.last newLast} {x : α} :
x ∈ p.snoc newLast rel ↔ x ∈ p ∨ x = newLast | case mpr
α : Type u_1
r : Rel α α
p : RelSeries r
newLast : α
rel : r p.last newLast
x : α
h : (∃ y, p.toFun y = x) ∨ x = newLast
⊢ ∃ y, Fin.append p.toFun (singleton r newLast).toFun y = x | rcases h with (⟨i, rfl⟩ | rfl) | case mpr.inl.intro
α : Type u_1
r : Rel α α
p : RelSeries r
newLast : α
rel : r p.last newLast
i : Fin (p.length + 1)
⊢ ∃ y, Fin.append p.toFun (singleton r newLast).toFun y = p.toFun i
case mpr.inr
α : Type u_1
r : Rel α α
p : RelSeries r
x : α
rel : r p.last x
⊢ ∃ y, Fin.append p.toFun (singleton r x).toFun y = x | df1eaf588a8790d3 |
FirstOrder.Language.BoundedFormula.realize_constantsVarsEquiv | Mathlib/ModelTheory/Semantics.lean | theorem realize_constantsVarsEquiv [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
{n} {φ : L[[α]].BoundedFormula β n} {v : β → M} {xs : Fin n → M} :
(constantsVarsEquiv φ).Realize (Sum.elim (fun a => ↑(L.con a)) v) xs ↔ φ.Realize v xs | case a.inr
L : Language
M : Type w
inst✝² : L.Structure M
α : Type u'
β : Type v'
inst✝¹ : L[[α]].Structure M
inst✝ : (L.lhomWithConstants α).IsExpansionOn M
n✝ : ℕ
φ : L[[α]].BoundedFormula β n✝
v : β → M
xs✝ : Fin n✝ → M
n : ℕ
xs : Fin n → M
R : (constantsOn α).Relations n
⊢ RelMap
((L.lhomWithConstants α).onRe... | exact isEmptyElim R | no goals | 2140c1fcba2365f1 |
RingHom.surjective_ofLocalizationSpan | Mathlib/RingTheory/RingHom/Surjective.lean | theorem surjective_ofLocalizationSpan : OfLocalizationSpan surjective | case H.intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
e : Ideal.span s = ⊤
H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)
this : Algebra R S := f.toAlgebra
x : S
r : ↑s
a : Localization.Away ↑r
e' : (Localization.awayMap f... | obtain ⟨b, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.mk'_surjective (Submonoid.powers (r : R)) a | case H.intro.intro.intro.mk.intro
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
e : Ideal.span s = ⊤
H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)
this : Algebra R S := f.toAlgebra
x : S
r : ↑s
b : R
n : ℕ
e' :
(Localization... | 907595718e16894a |
normalizerCondition_of_isNilpotent | Mathlib/GroupTheory/Nilpotent.lean | theorem normalizerCondition_of_isNilpotent [h : IsNilpotent G] : NormalizerCondition G | G : Type u_1
inst✝¹ : Group G
inst✝ : IsNilpotent G
ih : ∀ (H : Subgroup (G ⧸ center G)), H.normalizer = H → H = ⊤
H : Subgroup G
hH : H.normalizer = H
hch : center G ≤ H
hkh : (mk' (center G)).ker ≤ H
hsur : Function.Surjective ⇑(mk' (center G))
H' : Subgroup (G ⧸ center G) := Subgroup.map (mk' (center G)) H
⊢ H'.norm... | apply comap_injective hsur | case a
G : Type u_1
inst✝¹ : Group G
inst✝ : IsNilpotent G
ih : ∀ (H : Subgroup (G ⧸ center G)), H.normalizer = H → H = ⊤
H : Subgroup G
hH : H.normalizer = H
hch : center G ≤ H
hkh : (mk' (center G)).ker ≤ H
hsur : Function.Surjective ⇑(mk' (center G))
H' : Subgroup (G ⧸ center G) := Subgroup.map (mk' (center G)) H
⊢ ... | 5bbd76002dba81f3 |
Matrix.cramer_one | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | theorem cramer_one : cramer (1 : Matrix n n α) = 1 | n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
⊢ cramer 1 = 1 | ext i j | case h.h.h
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
i j : n
⊢ (cramer 1 ∘ₗ LinearMap.single α (fun i => α) i) 1 j = (1 ∘ₗ LinearMap.single α (fun i => α) i) 1 j | 82419a2865041e60 |
CategoryTheory.Enriched.FunctorCategory.enrichedId_π | Mathlib/CategoryTheory/Enriched/FunctorCategory.lean | @[reassoc (attr := simp)]
lemma enrichedId_π (j : J) : enrichedId V F₁ ≫ end_.π _ j = eId V (F₁.obj j) | V : Type u₁
inst✝⁵ : Category.{v₁, u₁} V
inst✝⁴ : MonoidalCategory V
C : Type u₂
inst✝³ : Category.{v₂, u₂} C
J : Type u₃
inst✝² : Category.{v₃, u₃} J
inst✝¹ : EnrichedOrdinaryCategory V C
F₁ : J ⥤ C
inst✝ : HasEnrichedHom V F₁ F₁
j : J
⊢ enrichedId V F₁ ≫ end_.π (diagram V F₁ F₁) j = eId V (F₁.obj j) | simp [enrichedId] | no goals | 28463e675b605800 |
Polynomial.eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le | Mathlib/Algebra/Polynomial/Div.lean | lemma eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le {R} [CommRing R] {p q : R[X]}
(hp : p.Monic) (hdiv : p ∣ q) (hdeg : q.natDegree ≤ p.natDegree) :
q = C q.leadingCoeff * p | case intro.inr
R : Type u_1
inst✝ : CommRing R
p q : R[X]
hp : p.Monic
hdeg : q.natDegree ≤ p.natDegree
r : R[X]
hr : q = p * r
hq : q ≠ 0
rzero : r ≠ 0
⊢ q = C q.leadingCoeff * p | rw [hr, natDegree_mul'] at hdeg | case intro.inr
R : Type u_1
inst✝ : CommRing R
p q : R[X]
hp : p.Monic
r : R[X]
hdeg : p.natDegree + r.natDegree ≤ p.natDegree
hr : q = p * r
hq : q ≠ 0
rzero : r ≠ 0
⊢ q = C q.leadingCoeff * p
case intro.inr
R : Type u_1
inst✝ : CommRing R
p q : R[X]
hp : p.Monic
r : R[X]
hdeg : (p * r).natDegree ≤ p.natDegree
hr : q... | 695053f04ba4918f |
List.head?_take | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem head?_take {l : List α} {n : Nat} :
(l.take n).head? = if n = 0 then none else l.head? | α : Type u_1
l : List α
n : Nat
⊢ (take n l).head? = if n = 0 then none else l.head? | simp [head?_eq_getElem?, getElem?_take] | α : Type u_1
l : List α
n : Nat
⊢ (if 0 < n then l[0]? else none) = if n = 0 then none else l[0]? | 5003cc8f68172873 |
cauchySeq_bdd | Mathlib/Topology/MetricSpace/Cauchy.lean | theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R | α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R | rcases Metric.cauchySeq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩ | case intro
α : Type u
inst✝ : PseudoMetricSpace α
u : ℕ → α
hu : CauchySeq u
N : ℕ
hN : ∀ n ≥ N, dist (u n) (u N) < 1
⊢ ∃ R > 0, ∀ (m n : ℕ), dist (u m) (u n) < R | 7d0a907ebf6342f0 |
HurwitzKernelBounds.f_le_g_nat | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | lemma f_le_g_nat (k : ℕ) {a t : ℝ} (ha : 0 ≤ a) (ht : 0 < t) (n : ℕ) :
‖f_nat k a t n‖ ≤ g_nat k a t n | k : ℕ
a t : ℝ
ha : 0 ≤ a
ht : 0 < t
n : ℕ
⊢ 0 ≤ (↑n + a) ^ k | positivity | no goals | eb511bb4633a4ae0 |
SimpleGraph.Subgraph.IsMatching.exists_of_universalVerts | Mathlib/Combinatorics/SimpleGraph/UniversalVerts.lean | lemma Subgraph.IsMatching.exists_of_universalVerts [Finite V] {s : Set V}
(h : Disjoint G.universalVerts s) (hc : s.ncard ≤ G.universalVerts.ncard) :
∃ t ⊆ G.universalVerts, ∃ (M : Subgraph G), M.verts = s ∪ t ∧ M.IsMatching | V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
s : Set V
h : Disjoint G.universalVerts s
hc : s.ncard ≤ G.universalVerts.ncard
⊢ ∃ t ⊆ G.universalVerts, ∃ M, M.verts = s ∪ t ∧ M.IsMatching | obtain ⟨t, ht⟩ := Set.exists_subset_card_eq hc | case intro
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
s : Set V
h : Disjoint G.universalVerts s
hc : s.ncard ≤ G.universalVerts.ncard
t : Set V
ht : t ⊆ G.universalVerts ∧ t.ncard = s.ncard
⊢ ∃ t ⊆ G.universalVerts, ∃ M, M.verts = s ∪ t ∧ M.IsMatching | 1113651ddb826df4 |
LinearPMap.mem_adjoint_domain_of_exists | Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | theorem mem_adjoint_domain_of_exists (y : F) (h : ∃ w : E, ∀ x : T.domain, ⟪w, x⟫ = ⟪y, T x⟫) :
y ∈ T†.domain | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
y : F
w : E
hw : ∀ (x : ↥T.domain), inner w ↑x = inner y (↑T x)
⊢ Continuous ⇑(((innerSL 𝕜) w).c... | fun_prop | no goals | 32ad3c57235e7e53 |
Turing.TM1to0.tr_supports | Mathlib/Computability/PostTuringMachine.lean | theorem tr_supports {S : Finset Λ} (ss : TM1.Supports M S) :
TM0.Supports (tr M) ↑(trStmts M S) | case right.mk.some.mk
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
q' : Option (TM1.Stmt Γ Λ σ)
v' : σ
h₁ : ((q', v'), s) ∈ tr M (some q, v) a
h₂ : (some q, v).1 ... | simp only [Finset.mem_univ, and_true] at h₂ ⊢ | case right.mk.some.mk
Γ : Type u_1
Λ : Type u_2
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → TM1.Stmt Γ Λ σ
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : TM0.Stmt Γ
v : σ
q : TM1.Stmt Γ Λ σ
q' : Option (TM1.Stmt Γ Λ σ)
v' : σ
h₁ : ((q', v'), s) ∈ tr M (some q, v) a
h₂ : some q ∈ TM1.s... | c5e09604fd205550 |
RingHom.toMorphismProperty_respectsIso_iff | Mathlib/RingTheory/RingHomProperties.lean | lemma toMorphismProperty_respectsIso_iff :
RespectsIso P ↔ (toMorphismProperty P).RespectsIso | case refine_2
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
X Y Z : CommRingCat
e : Y ≅ Z
f : X ⟶ Y
hf : toMorphismProperty (fun {R S} [CommRing R] [CommRing S] => P) f
⊢ toMorphismProperty (fun {R S} [CommRing R] [CommRing S... | exact h.left f.hom e.commRingCatIsoToRingEquiv hf | no goals | 66ff7804f60642f6 |
MeasureTheory.FiniteMeasure.limsup_measure_closed_le_of_tendsto | Mathlib/MeasureTheory/Measure/Portmanteau.lean | theorem FiniteMeasure.limsup_measure_closed_le_of_tendsto {Ω ι : Type*} {L : Filter ι}
[MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω]
[OpensMeasurableSpace Ω] {μ : FiniteMeasure Ω}
{μs : ι → FiniteMeasure Ω} (μs_lim : Tendsto μs L (𝓝 μ)) {F : Set Ω} (F_closed : IsClosed F) :
(L.limsu... | case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F ... | have key₁ : Tendsto (fun n ↦ ∫⁻ ω, (fs n ω : ℝ≥0∞) ∂μ) atTop (𝓝 ((μ : Measure Ω) F)) :=
HasOuterApproxClosed.tendsto_lintegral_apprSeq F_closed (μ : Measure Ω) | case inr.h
Ω : Type u_1
ι : Type u_2
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
μs : ι → FiniteMeasure Ω
μs_lim : Tendsto μs L (𝓝 μ)
F : Set Ω
F_closed : IsClosed F
hne : L.NeBot
ε : ℝ≥0
ε_pos : 0 < ε
a✝ : ↑μ F ... | ffd329bb427f57b4 |
NNReal.Lp_add_le_tsum | Mathlib/Analysis/MeanInequalities.lean | theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) | case refine_1
ι : Type u
f g : ι → ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : Summable fun i => f i ^ p
hg : Summable fun i => g i ^ p
pos : 0 < p
s : Finset ι
⊢ Summable fun i => f i ^ p
case refine_2
ι : Type u
f g : ι → ℝ≥0
p : ℝ
hp : 1 ≤ p
hf : Summable fun i => f i ^ p
hg : Summable fun i => g i ^ p
pos : 0 < p
s : Finset ι
⊢ Sum... | exacts [hf, hg] | no goals | 1e4d77366ccfd4cd |
CategoryTheory.Abelian.has_injective_coseparator | Mathlib/CategoryTheory/Generator/Abelian.lean | theorem has_injective_coseparator [HasLimits C] [EnoughInjectives C] (G : C) (hG : IsSeparator G) :
∃ G : C, Injective G ∧ IsCoseparator G | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasLimits C
inst✝ : EnoughInjectives C
G : C
hG : IsSeparator G
this✝ : WellPowered.{v, v, u} C
this : HasProductsOfShape (Subobject (op G)) C
⊢ ∃ G, Injective G ∧ IsCoseparator G | let T : C := Injective.under (piObj fun P : Subobject (op G) => unop P) | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasLimits C
inst✝ : EnoughInjectives C
G : C
hG : IsSeparator G
this✝ : WellPowered.{v, v, u} C
this : HasProductsOfShape (Subobject (op G)) C
T : C := Injective.under (∏ᶜ fun P => unop (Subobject.underlying.obj P))
⊢ ∃ G, Injective G ∧ IsCoseparator G | 6b085b71b2f6e5e0 |
Algebra.Generators.H1Cotangent.δ_eq_δ | Mathlib/RingTheory/Kaehler/JacobiZariski.lean | lemma δ_eq_δ (Q : Generators.{u₁} S T) (P : Generators.{u₂} R S)
(P' : Generators.{u₃} R S) :
δ Q P = δ Q P' | case h.mk.intro
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type uT
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : Generators S T
P : Generators R S
P' : Generators R S
x : ↥Q.toExtension.ker
hx : Extension.Cotangent.mk x ∈ Linear... | rw [δ_eq_δAux, δ_eq_δAux] | no goals | bc59960258c9e766 |
isNoetherian_iff_fg_wellFounded | Mathlib/RingTheory/Noetherian/Defs.lean | theorem isNoetherian_iff_fg_wellFounded :
IsNoetherian R M ↔ WellFoundedGT { N : Submodule R M // N.FG } | R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
α : Type (max 0 u_2) := { N // N.FG }
⊢ IsNoetherian R M ↔ WellFoundedGT { N // N.FG } | constructor | case mp
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
α : Type (max 0 u_2) := { N // N.FG }
⊢ IsNoetherian R M → WellFoundedGT { N // N.FG }
case mpr
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
α : Type (max 0 u_2) := { N // N.FG... | 3acc6e2f9e0e2255 |
AhlswedeZhang.infSum_compls_add_supSum | Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | /-- The **Ahlswede-Zhang Identity**. -/
lemma infSum_compls_add_supSum (𝒜 : Finset (Finset α)) :
infSum 𝒜ᶜˢ + supSum 𝒜 = card α * ∑ k ∈ range (card α), (k : ℚ)⁻¹ + 1 | α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Nonempty α
𝒜 : Finset (Finset α)
⊢ ∑ s : Finset α, ↑(#(𝒜ᶜˢ.truncatedInf s)) / (↑(#s) * ↑((card α).choose #s)) +
∑ s : Finset α, ↑(#(𝒜.truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) =
↑(card α) * ∑ k ∈ range (card α), (↑k)⁻¹ + ... | rw [← @map_univ_of_surjective (Finset α) _ _ _ ⟨compl, compl_injective⟩ compl_surjective, sum_map] | α : Type u_1
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : Nonempty α
𝒜 : Finset (Finset α)
⊢ ∑ x : Finset α,
↑(#(𝒜ᶜˢ.truncatedInf ({ toFun := compl, inj' := ⋯ } x))) /
(↑(#({ toFun := compl, inj' := ⋯ } x)) * ↑((card α).choose #({ toFun := compl, inj' := ⋯ } x))) +
∑ s ∈ map { toFun := co... | 95719bf69b03a019 |
List.zipWithAll_go_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean | theorem zipWithAll_go_toArray (as : List α) (bs : List β) (f : Option α → Option β → γ) (i : Nat) (cs : Array γ) :
zipWithAll.go f as.toArray bs.toArray i cs = cs ++ (List.zipWithAll f (as.drop i) (bs.drop i)).toArray | case isTrue
α : Type u_1
β : Type u_2
γ : Type u_3
as : List α
bs : List β
f : Option α → Option β → γ
i : Nat
cs : Array γ
h : i < max as.length bs.length
⊢ cs ++ (f as[i]? bs[i]? :: zipWithAll f (drop (i + 1) as) (drop (i + 1) bs)).toArray =
cs ++ (zipWithAll f (drop i as) (drop i bs)).toArray | if ha : i < as.length then
if hb : i < bs.length then
rw [List.drop_eq_getElem_cons ha, List.drop_eq_getElem_cons hb]
simp only [ha, hb, getElem?_eq_getElem, zipWithAll_cons_cons]
else
simp only [Nat.not_lt] at hb
rw [List.drop_eq_getElem_cons ha]
rw [(drop_eq_nil_iff (l := bs)).mpr (by omega), ... | no goals | b1432ecd63078bf6 |
CategoryTheory.Functor.id_comp | Mathlib/CategoryTheory/Functor/Basic.lean | theorem id_comp (F : C ⥤ D) : 𝟭 C ⋙ F = F | case mk
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
toPrefunctor✝ : C ⥤q D
map_id✝ : ∀ (X : C), toPrefunctor✝.map (𝟙 X) = 𝟙 (toPrefunctor✝.obj X)
map_comp✝ : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), toPrefunctor✝.map (f ≫ g) = toPrefunctor✝.map f ≫ toPrefunctor✝.map g
⊢ 𝟭 C ⋙ { toP... | rfl | no goals | c189fbba7d6d5519 |
CategoryTheory.Abelian.Pseudoelement.pseudo_pullback | Mathlib/CategoryTheory/Abelian/Pseudoelements.lean | theorem pseudo_pullback {P Q R : C} {f : P ⟶ R} {g : Q ⟶ R} {p : P} {q : Q} :
f p = g q →
∃ s, pullback.fst f g s = p ∧ pullback.snd f g s = q :=
Quotient.inductionOn₂ p q fun x y h => by
obtain ⟨Z, a, b, ea, eb, comm⟩ := Quotient.exact h
obtain ⟨l, hl₁, hl₂⟩ := @pullback.lift' _ _ _ _ _ _ f g _ (a ... | case intro.intro.intro.intro.intro.mk.intro
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasPullbacks C
P Q R : C
f : P ⟶ R
g : Q ⟶ R
p : Pseudoelement P
q : Pseudoelement Q
x : Over P
y : Over Q
h : pseudoApply f ⟦x⟧ = pseudoApply g ⟦y⟧
Z : C
a : Z ⟶ ((fun g => app f g) x).left
b : Z ⟶ ((fun g_1 =>... | exact ⟨l, ⟨Quotient.sound ⟨Z, 𝟙 Z, a, inferInstance, ea, by rwa [Category.id_comp]⟩,
Quotient.sound ⟨Z, 𝟙 Z, b, inferInstance, eb, by rwa [Category.id_comp]⟩⟩⟩ | no goals | 92b3b5dba8e659f2 |
entry_norm_bound_of_unitary | Mathlib/Analysis/CStarAlgebra/Matrix.lean | theorem entry_norm_bound_of_unitary {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜)
(i j : n) : ‖U i j‖ ≤ 1 | 𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
U : Matrix n n 𝕜
hU : U ∈ Matrix.unitaryGroup n 𝕜
i j : n
norm_sum : ‖U i j‖ ^ 2 ≤ ∑ x : n, ‖U i x‖ ^ 2
⊢ (U * Uᴴ) i i = ↑(∑ x : n, ‖U i x‖ ^ 2) | simp only [Matrix.mul_apply, Matrix.conjTranspose_apply, ← starRingEnd_apply, RCLike.mul_conj,
RCLike.normSq_eq_def', RCLike.ofReal_pow] | 𝕜 : Type u_1
n : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
U : Matrix n n 𝕜
hU : U ∈ Matrix.unitaryGroup n 𝕜
i j : n
norm_sum : ‖U i j‖ ^ 2 ≤ ∑ x : n, ‖U i x‖ ^ 2
⊢ ∑ x : n, ↑‖U i x‖ ^ 2 = ↑(∑ x : n, ‖U i x‖ ^ 2) | 21b1f0485d73594d |
Matrix.blockDiagonal_mul | Mathlib/Data/Matrix/Block.lean | theorem blockDiagonal_mul [Fintype n] [Fintype o] [NonUnitalNonAssocSemiring α]
(M : o → Matrix m n α) (N : o → Matrix n p α) :
(blockDiagonal fun k => M k * N k) = blockDiagonal M * blockDiagonal N | case a.mk.mk
m : Type u_2
n : Type u_3
o : Type u_4
p : Type u_5
α : Type u_12
inst✝³ : DecidableEq o
inst✝² : Fintype n
inst✝¹ : Fintype o
inst✝ : NonUnitalNonAssocSemiring α
M : o → Matrix m n α
N : o → Matrix n p α
i : m
k : o
j : p
k' : o
⊢ (if k = k' then ∑ j_1 : n, M k i j_1 * N k j_1 j else 0) =
∑ x : n, ∑ x... | split_ifs with h <;> simp [h] | no goals | 8bc2b75c4fc72a49 |
CategoryTheory.Functor.IsFibered.isStronglyCartesian_of_exists_isCartesian | Mathlib/CategoryTheory/FiberedCategory/Fibered.lean | /-- In a category which admits strongly cartesian pullbacks, any cartesian morphism is
strongly cartesian. This is a helper-lemma for the fact that admitting strongly cartesian pullbacks
implies being fibered. -/
lemma isStronglyCartesian_of_exists_isCartesian (p : 𝒳 ⥤ 𝒮) (h : ∀ (a : 𝒳) (R : 𝒮)
(f : R ⟶ p.obj a... | case universal_property'.map
𝒮 : Type u₁
𝒳 : Type u₂
inst✝² : Category.{v₁, u₁} 𝒮
inst✝¹ : Category.{v₂, u₂} 𝒳
p : 𝒳 ⥤ 𝒮
h : ∀ (a : 𝒳) (R : 𝒮) (f : R ⟶ p.obj a), ∃ b φ, p.IsStronglyCartesian f φ
a b : 𝒳
φ : a ⟶ b
c : 𝒳
φ' : c ⟶ b
g : p.obj c ⟶ p.obj a
inst✝ : p.IsCartesian (p.map φ) φ
hφ' : p.IsHomLift (g ≫ p... | obtain ⟨a', ψ, hψ⟩ := h _ _ (p.map φ) | case universal_property'.map.intro.intro
𝒮 : Type u₁
𝒳 : Type u₂
inst✝² : Category.{v₁, u₁} 𝒮
inst✝¹ : Category.{v₂, u₂} 𝒳
p : 𝒳 ⥤ 𝒮
h : ∀ (a : 𝒳) (R : 𝒮) (f : R ⟶ p.obj a), ∃ b φ, p.IsStronglyCartesian f φ
a b : 𝒳
φ : a ⟶ b
c : 𝒳
φ' : c ⟶ b
g : p.obj c ⟶ p.obj a
inst✝ : p.IsCartesian (p.map φ) φ
hφ' : p.IsHo... | 022b837ace85f955 |
List.getElem_idxOf | Mathlib/Data/List/Basic.lean | theorem getElem_idxOf [DecidableEq α] {a : α} : ∀ {l : List α} (h : idxOf a l < l.length),
l[idxOf a l] = a
| b :: l, h => by
by_cases h' : b = a <;>
simp [h', if_pos, if_false, getElem_idxOf]
| α : Type u
inst✝ : DecidableEq α
a b : α
l : List α
h : idxOf a (b :: l) < (b :: l).length
⊢ (b :: l)[idxOf a (b :: l)] = a | by_cases h' : b = a <;>
simp [h', if_pos, if_false, getElem_idxOf] | no goals | 662118d45f8b96ef |
lineMap_le_lineMap_iff_of_lt | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r ≤ lineMap a b r' ↔ a ≤ b | k : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField k
inst✝² : OrderedAddCommGroup E
inst✝¹ : Module k E
inst✝ : OrderedSMul k E
a b : E
r r' : k
h : r < r'
⊢ (lineMap a b) r ≤ (lineMap a b) r' ↔ a ≤ b | simp only [lineMap_apply_module] | k : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField k
inst✝² : OrderedAddCommGroup E
inst✝¹ : Module k E
inst✝ : OrderedSMul k E
a b : E
r r' : k
h : r < r'
⊢ (1 - r) • a + r • b ≤ (1 - r') • a + r' • b ↔ a ≤ b | fdc18652dbecc7e3 |
List.Nodup.rotate_congr_iff | Mathlib/Data/List/Rotate.lean | theorem Nodup.rotate_congr_iff {l : List α} (hl : l.Nodup) {i j : ℕ} :
l.rotate i = l.rotate j ↔ i % l.length = j % l.length ∨ l = [] | α : Type u
l : List α
hl : l.Nodup
i j : ℕ
⊢ l.rotate i = l.rotate j ↔ i % l.length = j % l.length ∨ l = [] | rcases eq_or_ne l [] with rfl | hn | case inl
α : Type u
i j : ℕ
hl : [].Nodup
⊢ [].rotate i = [].rotate j ↔ i % [].length = j % [].length ∨ [] = []
case inr
α : Type u
l : List α
hl : l.Nodup
i j : ℕ
hn : l ≠ []
⊢ l.rotate i = l.rotate j ↔ i % l.length = j % l.length ∨ l = [] | 965e918a3e97caa0 |
HomologicalComplex.isZero_stupidTrunc_iff | Mathlib/Algebra/Homology/Embedding/StupidTrunc.lean | lemma isZero_stupidTrunc_iff :
IsZero (K.stupidTrunc e) ↔ K.IsStrictlySupportedOutside e | case mp
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsRelIff
⊢ IsZero (K.stupidTrunc e) → K.IsStrictlySupportedOutside e | exact fun h ↦ ⟨fun i ↦
((eval _ _ (e.f i)).map_isZero h).of_iso (K.stupidTruncXIso e rfl).symm⟩ | no goals | a94ebe185c2b9f31 |
GenLoop.transAt_distrib | Mathlib/Topology/Homotopy/HomotopyGroup.lean | theorem transAt_distrib {i j : N} (h : i ≠ j) (a b c d : Ω^ N X x) :
transAt i (transAt j a b) (transAt j c d) = transAt j (transAt i a c) (transAt i b d) | case H
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : i ≠ j
a b c d : ↑(Ω^ N X x)
y✝ : N → ↑I
⊢ (transAt i (transAt j a b) (transAt j c d)) y✝ = (transAt j (transAt i a c) (transAt i b d)) y✝ | simp_rw [transAt, coe_copy, Function.update_apply, if_neg h, if_neg h.symm] | case H
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i j : N
h : i ≠ j
a b c d : ↑(Ω^ N X x)
y✝ : N → ↑I
⊢ (if ↑(y✝ i) ≤ 1 / 2 then
if ↑(y✝ j) ≤ 1 / 2 then
a
(Function.update (Function.update y✝ i (Set.projIcc 0 1 transAt.proof_2 (2 * ↑(y✝ i)))) j
... | 573e215630fe8b61 |
Orientation.cos_oangle_add_right_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem cos_oangle_add_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.cos (o.oangle x (x + y)) = ‖x‖ / ‖x + y‖ | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
⊢ (o.oangle x (x + y)).cos = ‖x‖ / ‖x + y‖ | have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ (o.oangle x (x + y)).cos = ‖x‖ / ‖x + y‖ | 18961bae4b270cd5 |
MeasureTheory.smulInvariantMeasure_tfae | Mathlib/MeasureTheory/Group/Action.lean | theorem smulInvariantMeasure_tfae :
List.TFAE
[SMulInvariantMeasure G α μ,
∀ (c : G) (s), MeasurableSet s → μ ((c • ·) ⁻¹' s) = μ s,
∀ (c : G) (s), MeasurableSet s → μ (c • s) = μ s,
∀ (c : G) (s), μ ((c • ·) ⁻¹' s) = μ s,
∀ (c : G) (s), μ (c • s) = μ s,
∀ c : G, Measur... | G : Type u
α : Type w
m : MeasurableSpace α
inst✝³ : Group G
inst✝² : MulAction G α
μ : Measure α
inst✝¹ : MeasurableSpace G
inst✝ : MeasurableSMul G α
tfae_1_iff_2 : SMulInvariantMeasure G α μ ↔ ∀ (c : G) (s : Set α), MeasurableSet s → μ ((fun x => c • x) ⁻¹' s) = μ s
tfae_1_to_6 : SMulInvariantMeasure G α μ → ∀ (c : ... | tfae_finish | no goals | 02a2e87d938c5d06 |
Algebra.FormallyUnramified.of_map_maximalIdeal | Mathlib/RingTheory/Unramified/LocalRing.lean | @[stacks 02FM]
lemma FormallyUnramified.of_map_maximalIdeal
[Algebra.IsSeparable (ResidueField R) (ResidueField S)]
(H : (maximalIdeal R).map (algebraMap R S) = maximalIdeal S) :
Algebra.FormallyUnramified R S | case subsingleton_kaehlerDifferential.intro.intro.mk
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)
H : Idea... | replace hx : x ∈ Ideal.map (algebraMap R S) (maximalIdeal R) := by simpa [H] using hx | case subsingleton_kaehlerDifferential.intro.intro.mk
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : IsLocalRing R
inst✝³ : IsLocalRing S
inst✝² : IsLocalHom (algebraMap R S)
inst✝¹ : EssFiniteType R S
inst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)
H : Idea... | 74b4dfddeffd2023 |
YoungDiagram.mem_iff_lt_rowLen | Mathlib/Combinatorics/Young/YoungDiagram.lean | theorem mem_iff_lt_rowLen {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ ↔ j < μ.rowLen i | μ : YoungDiagram
i j : ℕ
⊢ (i, j) ∈ μ ↔ ∀ m ≤ j, ¬(i, m) ∉ μ | push_neg | μ : YoungDiagram
i j : ℕ
⊢ (i, j) ∈ μ ↔ ∀ m ≤ j, (i, m) ∈ μ | 1c292af67d970a3d |
IsPrimitiveRoot.dvd_of_isCyclotomicExtension | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem dvd_of_isCyclotomicExtension [IsCyclotomicExtension {n} ℚ K] {ζ : K}
{l : ℕ} (hζ : IsPrimitiveRoot ζ l) (hl : l ≠ 0) : l ∣ 2 * n | n : ℕ+
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
ζ : K
l : ℕ
hζ : IsPrimitiveRoot ζ l
hl✝ : l ≠ 0
hl : NeZero l
⊢ l ∣ 2 * ↑n | have hroot := IsCyclotomicExtension.zeta_spec n ℚ K | n : ℕ+
K : Type u
inst✝² : Field K
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {n} ℚ K
ζ : K
l : ℕ
hζ : IsPrimitiveRoot ζ l
hl✝ : l ≠ 0
hl : NeZero l
hroot : IsPrimitiveRoot (zeta n ℚ K) ↑n
⊢ l ∣ 2 * ↑n | a9e7cdd1282ef8c8 |
ENNReal.log_strictMono | Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean | theorem log_strictMono : StrictMono log | case inr.inr
x y : ℝ≥0∞
h : x < y
x_real : 0 < x.toReal
⊢ (if x = 0 then ⊥ else if x = ⊤ then ⊤ else ↑(Real.log x.toReal)) <
if y = 0 then ⊥ else if y = ⊤ then ⊤ else ↑(Real.log y.toReal) | simp only [(ENNReal.toReal_pos_iff.1 x_real).1.ne',
(ENNReal.toReal_pos_iff.1 x_real).2.ne, if_false] | case inr.inr
x y : ℝ≥0∞
h : x < y
x_real : 0 < x.toReal
⊢ ↑(Real.log x.toReal) < if y = 0 then ⊥ else if y = ⊤ then ⊤ else ↑(Real.log y.toReal) | 42b90fcfa0c95640 |
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul | Mathlib/MeasureTheory/Integral/Layercake.lean | theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
(f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
(g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * E... | case intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᶠ[ae (volume.rest... | obtain ⟨F, F_mble, F_nn, f_eq_F⟩ : ∃ F : α → ℝ, Measurable F ∧ 0 ≤ F ∧ f =ᵐ[μ] F := by
refine ⟨fun ω ↦ max (f_mble.mk f ω) 0, f_mble.measurable_mk.max measurable_const,
fun ω ↦ le_max_right _ _, ?_⟩
filter_upwards [f_mble.ae_eq_mk, f_nn] with ω hω h'ω
rw [← hω]
exact (max_eq_left h'ω).symm | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g ... | 4669fe42c637d7b2 |
Ordinal.toPGame_nmul | Mathlib/SetTheory/Game/Ordinal.lean | theorem toPGame_nmul (a b : Ordinal) : (a ⨳ b).toPGame ≈ a.toPGame * b.toPGame | a b : Ordinal.{u_1}
i✝ : (a.toPGame * b.toPGame).LeftMoves
i : a.toPGame.LeftMoves
j : b.toPGame.LeftMoves
⊢ (a.toPGame * b.toPGame).moveLeft (toLeftMovesMul (Sum.inl (i, j))) ⧏ (a ⨳ b).toPGame | rw [mul_moveLeft_inl, toPGame_moveLeft', toPGame_moveLeft', lf_iff_game_lf,
quot_sub, quot_add, ← Game.not_le, le_sub_iff_add_le] | a b : Ordinal.{u_1}
i✝ : (a.toPGame * b.toPGame).LeftMoves
i : a.toPGame.LeftMoves
j : b.toPGame.LeftMoves
⊢ ¬⟦(a ⨳ b).toPGame⟧ + ⟦(↑(toLeftMovesToPGame.symm i)).toPGame * (↑(toLeftMovesToPGame.symm j)).toPGame⟧ ≤
⟦(↑(toLeftMovesToPGame.symm i)).toPGame * b.toPGame⟧ + ⟦a.toPGame * (↑(toLeftMovesToPGame.symm j)).t... | 18382d4f22ac2836 |
CategoryTheory.MorphismProperty.multiplicativeClosure_eq_multiplicativeClosure' | Mathlib/CategoryTheory/MorphismProperty/Composition.lean | lemma multiplicativeClosure_eq_multiplicativeClosure' :
W.multiplicativeClosure = W.multiplicativeClosure' :=
le_antisymm
((multiplicativeClosure_le_iff _ _).mpr (fun _ _ f hf ↦ .of f hf)) <|
fun x y f hf ↦ by induction hf with
| of _ h => exact .of _ h
| id x => exact .id x
| of_comp f ... | case of_comp
C : Type u
inst✝ : Category.{v, u} C
W : MorphismProperty C
x y : C
f✝ : x ⟶ y
x✝ y✝ z✝ : C
f : x✝ ⟶ y✝
g : y✝ ⟶ z✝
hf : W f
hg : W.multiplicativeClosure' g
hr : W.multiplicativeClosure g
⊢ W.multiplicativeClosure (f ≫ g) | exact W.multiplicativeClosure.comp_mem f g (.of f hf) hr | no goals | fa66a6ce5053b234 |
Complex.norm_max_aux₁ | Mathlib/Analysis/Complex/AbsMax.lean | theorem norm_max_aux₁ [CompleteSpace F] {f : ℂ → F} {z w : ℂ}
(hd : DiffContOnCl ℂ f (ball z (dist w z)))
(hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ | case refine_2
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
hsub : sphere z r ⊆ closedBall z r
⊢ ∀ ζ ∈ sphe... | rintro ζ (hζ : ‖ζ - z‖ = r) | case refine_2
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
hsub : sphere z r ⊆ closedBall z r
ζ : ℂ
hζ : ‖... | 5ca8e31b32469418 |
Filter.ptendsto'_of_ptendsto | Mathlib/Order/Filter/Partial.lean | theorem ptendsto'_of_ptendsto {f : α →. β} {l₁ : Filter α} {l₂ : Filter β} (h : f.Dom ∈ l₁) :
PTendsto f l₁ l₂ → PTendsto' f l₁ l₂ | α : Type u
β : Type v
f : α →. β
l₁ : Filter α
l₂ : Filter β
h : f.Dom ∈ l₁
h' : ∀ s ∈ l₂, f.core s ∈ l₁
s : Set β
sl₂ : s ∈ l₂
⊢ f.preimage s ∈ l₁ | rw [PFun.preimage_eq] | α : Type u
β : Type v
f : α →. β
l₁ : Filter α
l₂ : Filter β
h : f.Dom ∈ l₁
h' : ∀ s ∈ l₂, f.core s ∈ l₁
s : Set β
sl₂ : s ∈ l₂
⊢ f.core s ∩ f.Dom ∈ l₁ | b4b8ddb37e1f7aa0 |
SimpleGraph.Subgraph.induce_self_verts | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | theorem induce_self_verts : G'.induce G'.verts = G' | case Adj.h.h.a
V : Type u
G : SimpleGraph V
G' : G.Subgraph
x✝¹ x✝ : V
⊢ (G'.induce G'.verts).Adj x✝¹ x✝ ↔ G'.Adj x✝¹ x✝ | constructor <;>
simp +contextual only [induce_adj, imp_true_iff, and_true] | case Adj.h.h.a.mpr
V : Type u
G : SimpleGraph V
G' : G.Subgraph
x✝¹ x✝ : V
⊢ G'.Adj x✝¹ x✝ → x✝¹ ∈ G'.verts ∧ x✝ ∈ G'.verts | 22737c60a58dbd6b |
Finset.prod_update_of_not_mem | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem prod_update_of_not_mem [DecidableEq α] {s : Finset α} {i : α} (h : i ∉ s) (f : α → β)
(b : β) : ∏ x ∈ s, Function.update f i b x = ∏ x ∈ s, f x | α : Type u_3
β : Type u_4
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s : Finset α
i : α
h : i ∉ s
f : α → β
b : β
j : α
hj : j ∈ s
this : j ≠ i
⊢ update f i b j = f j | simp [this] | no goals | 09131c9f87281507 |
four_functions_theorem_aux | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | private lemma four_functions_theorem_aux (h₁ : 0 ≤ f₁) (h₂ : 0 ≤ f₂) (h₃ : 0 ≤ f₃) (h₄ : 0 ≤ f₄)
(h : ∀ s t, f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)) (𝒜 ℬ : Finset (Finset α)) :
(∑ s ∈ 𝒜, f₁ s) * ∑ s ∈ ℬ, f₂ s ≤ (∑ s ∈ 𝒜 ⊼ ℬ, f₃ s) * ∑ s ∈ 𝒜 ⊻ ℬ, f₄ s | α : Type u_1
β : Type u_2
inst✝³ : DecidableEq α
inst✝² : LinearOrderedCommSemiring β
f₁ f₂ f₃ f₄ : Finset α → β
inst✝¹ : ExistsAddOfLE β
inst✝ : Fintype α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (s t : Finset α), f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
⊢ (∑ s ∈ 𝒜, f₁ s) * ∑ s ∈ ℬ,... | refine univ.four_functions_theorem h₁ h₂ h₃ h₄ ?_ ?_ ?_ <;> simp [h] | no goals | 2cc57ecd51568506 |
List.foldr_reverse | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem foldr_reverse (l : List α) (f : α → β → β) (b) :
l.reverse.foldr f b = l.foldl (fun x y => f y x) b :=
(foldl_reverse ..).symm.trans <| by simp
| α : Type u_1
β : Type u_2
l : List α
f : α → β → β
b : β
⊢ foldl (fun y x => f x y) b l.reverse.reverse = foldl (fun x y => f y x) b l | simp | no goals | f39cc1958ee6eb49 |
Finset.mulConst_le_divConst_sq | Mathlib/Combinatorics/Additive/DoublingConst.lean | /-- If `A` has small difference, then it has small doubling, with the constant squared.
This is a consequence of the Ruzsa triangle inequality. -/
@[to_additive
"If `A` has small difference, then it has small doubling, with the constant squared.
This is a consequence of the Ruzsa triangle inequality."]
lemma mulConst... | G : Type u_1
inst✝¹ : CommGroup G
inst✝ : DecidableEq G
A : Finset G
hA' : A.Nonempty
⊢ ↑(#(A / A)) * ↑(#(A / A)) = δₘ[A] ^ 2 * (↑(#A) * ↑(#A)) | rw [← divConst_mul_card] | G : Type u_1
inst✝¹ : CommGroup G
inst✝ : DecidableEq G
A : Finset G
hA' : A.Nonempty
⊢ δₘ[A] * ↑(#A) * (δₘ[A] * ↑(#A)) = δₘ[A] ^ 2 * (↑(#A) * ↑(#A)) | fe98c19de07a422d |
Nat.multinomial_two_mul_le_mul_multinomial | Mathlib/Data/Nat/Choose/Multinomial.lean | lemma multinomial_two_mul_le_mul_multinomial :
multinomial s (fun i ↦ 2 * f i) ≤ ((∑ i ∈ s, f i) ^ ∑ i ∈ s, f i) * multinomial s f | case bc.h
ι : Type u_1
s : Finset ι
f : ι → ℕ
i✝ : ι
a✝ : i✝ ∈ s
⊢ (2 * f i✝)‼ ≤ (2 * f i✝)! | exact doubleFactorial_le_factorial _ | no goals | f44b98bbd43a1101 |
CategoryTheory.Localization.hasSmallLocalizedHom_of_isLocalization | Mathlib/CategoryTheory/Localization/SmallHom.lean | lemma hasSmallLocalizedHom_of_isLocalization :
HasSmallLocalizedHom.{v₂} W X Y | C : Type u₁
inst✝² : Category.{v₁, u₁} C
W : MorphismProperty C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
L : C ⥤ D
inst✝ : L.IsLocalization W
X Y : C
⊢ Small.{v₂, v₂} (L.obj X ⟶ L.obj Y) | infer_instance | no goals | 77d11c69fd348aef |
rieszContentAux_image_nonempty | Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean | theorem rieszContentAux_image_nonempty (K : Compacts X) :
(Λ '' { f : C_c(X, ℝ≥0) | ∀ x ∈ K, (1 : ℝ≥0) ≤ f x }).Nonempty | case h
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hf... | intro x hx | case h
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hf... | 2a4990d43927e6b7 |
CategoryTheory.ShortComplex.exact_and_epi_g_iff_g_is_cokernel | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | lemma exact_and_epi_g_iff_g_is_cokernel [S.HasHomology] :
S.Exact ∧ Epi S.g ↔ Nonempty (IsColimit (CokernelCofork.ofπ S.g S.zero)) | case mp
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
S : ShortComplex C
inst✝¹ : Balanced C
inst✝ : S.HasHomology
hS : S.Exact
right✝ : Epi S.g
⊢ Nonempty (IsColimit (CokernelCofork.ofπ S.g ⋯)) | exact ⟨hS.gIsCokernel⟩ | no goals | a7042e22c003432f |
CategoryTheory.zigzag_isPreconnected | Mathlib/CategoryTheory/IsConnected.lean | theorem zigzag_isPreconnected (h : ∀ j₁ j₂ : J, Zigzag j₁ j₂) : IsPreconnected J | case h
J : Type u₁
inst✝ : Category.{v₁, u₁} J
α : Type u₁
F : J → α
hF : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂
j j' : J
h : Zigzag j j'
⊢ F j = F j' | induction' h with j₁ j₂ _ hj ih | case h.refl
J : Type u₁
inst✝ : Category.{v₁, u₁} J
α : Type u₁
F : J → α
hF : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂
j j' : J
⊢ F j = F j
case h.tail
J : Type u₁
inst✝ : Category.{v₁, u₁} J
α : Type u₁
F : J → α
hF : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → F j₁ = F j₂
j j' j₁ j₂ : J
a✝ : Relation.ReflTransGen Zag j j₁
hj : Zag j₁... | d14312c70bbcfa69 |
Function.not_surjective_Type | Mathlib/Logic/Function/Basic.lean | theorem not_surjective_Type {α : Type u} (f : α → Type max u v) : ¬Surjective f | case intro
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
⊢ False | let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩ | case intro
α : Type u
f : α → Type (max u v)
hf : Surjective f
T : Type (max u v) := Sigma f
U : α
hU : f U = Set T
g : Set T → T := fun s => ⟨U, cast ⋯ s⟩
⊢ False | 6bedcf21fa60c989 |
IsMulFreimanHom.mono | Mathlib/Combinatorics/Additive/FreimanHom.lean | @[to_additive]
lemma IsMulFreimanHom.mono (hmn : m ≤ n) (hf : IsMulFreimanHom n A B f) :
IsMulFreimanHom m A B f where
mapsTo := hf.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h | α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CancelCommMonoid β
A : Set α
B : Set β
f : α → β
m n : ℕ
hmn : m ≤ n
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = m
ht : t.card = m
h : s.prod = t.prod
⊢ (map f s).prod = (map f t).prod | obtain rfl | hm := m.eq_zero_or_pos | case inl
α : Type u_2
β : Type u_3
inst✝¹ : CommMonoid α
inst✝ : CancelCommMonoid β
A : Set α
B : Set β
f : α → β
n : ℕ
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
h : s.prod = t.prod
hmn : 0 ≤ n
hs : s.card = 0
ht : t.card = 0
⊢ (map f s).prod = (map f t)... | b2311f9c87a69848 |
RingHom.finiteType_ofLocalizationSpan | Mathlib/RingTheory/RingHom/FiniteType.lean | theorem finiteType_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.FiniteType | ⊢ OfLocalizationFiniteSpan @FiniteType | introv R hs H | R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), (Localization.awayMap f ↑r).FiniteType
⊢ f.FiniteType | 99d09e8c0e75eabf |
Option.forIn'_congr | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Option/Monadic.lean | theorem forIn'_congr [Monad m] [LawfulMonad m] {as bs : Option α} (w : as = bs)
{b b' : β} (hb : b = b')
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
(h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
forIn' as b f = forIn' bs b' g | case some.none
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
b b' : β
hb : b = b'
val✝ : α
f : (a' : α) → a' ∈ some val✝ → β → m (ForInStep β)
g : (a' : α) → a' ∈ none → β → m (ForInStep β)
w : some val✝ = none
h : ∀ (a : α) (m_1 : a ∈ none) (b : β), f a ⋯ b = g a m_1 b
⊢ forI... | simp at w | no goals | 7ca8cc842aac1a36 |
Nat.mul_max_mul_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem mul_max_mul_right (a b c : Nat) : max (a * c) (b * c) = max a b * c | case zero
c b : Nat
⊢ max (0 * c) (b * c) = max 0 b * c | simp | no goals | 78155b4c3c95a55e |
Polynomial.one_scaleRoots | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | @[simp]
lemma one_scaleRoots (r : R) :
(1 : R[X]).scaleRoots r = 1 | R : Type u_1
inst✝ : Semiring R
r : R
⊢ scaleRoots 1 r = 1 | ext | case a
R : Type u_1
inst✝ : Semiring R
r : R
n✝ : ℕ
⊢ (scaleRoots 1 r).coeff n✝ = coeff 1 n✝ | f8df34978e5753e9 |
Sum.isPreconnected_iff | Mathlib/Topology/Connected/Clopen.lean | theorem Sum.isPreconnected_iff [TopologicalSpace β] {s : Set (α ⊕ β)} :
IsPreconnected s ↔
(∃ t, IsPreconnected t ∧ s = Sum.inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = Sum.inr '' t | case refine_1
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : Set (α ⊕ β)
hs : IsPreconnected s
⊢ (∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t | obtain rfl | h := s.eq_empty_or_nonempty | case refine_1.inl
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
hs : IsPreconnected ∅
⊢ (∃ t, IsPreconnected t ∧ ∅ = inl '' t) ∨ ∃ t, IsPreconnected t ∧ ∅ = inr '' t
case refine_1.inr
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : Set (α ⊕ β)
hs : IsPrec... | 82411909fca8a188 |
Fin.foldrM_loop_succ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Fold.lean | theorem foldrM_loop_succ [Monad m] (f : Fin n → α → m α) (x) (h : i < n) :
foldrM.loop n f ⟨i+1, h⟩ x = f ⟨i, h⟩ x >>= foldrM.loop n f ⟨i, Nat.le_of_lt h⟩ | m : Type u_1 → Type u_2
n : Nat
α : Type u_1
i : Nat
inst✝ : Monad m
f : Fin n → α → m α
x : α
h : i < n
⊢ foldrM.loop n f ⟨i + 1, h⟩ x = f ⟨i, h⟩ x >>= foldrM.loop n f ⟨i, ⋯⟩ | rw [foldrM.loop] | no goals | 802bb73e0414a204 |
linearIndependent_monoidHom | Mathlib/LinearAlgebra/LinearIndependent/Basic.lean | theorem linearIndependent_monoidHom (G : Type*) [Monoid G] (L : Type*) [CommRing L]
[NoZeroDivisors L] : LinearIndependent L (M := G → L) (fun f => f : (G →* L) → G → L) | case h₀
G : Type u_6
inst✝² : Monoid G
L : Type u_7
inst✝¹ : CommRing L
inst✝ : NoZeroDivisors L
this✝ : DecidableEq (G →* L) := Classical.decEq (G →* L)
this : MulAction L L := DistribMulAction.toMulAction
s✝ : Finset (G →* L)
a : G →* L
s : Finset (G →* L)
has : a ∉ s
ih : ∀ (g : (G →* L) → L), ∑ i ∈ s, g i • ⇑i = 0 ... | intro i his hia | case h₀
G : Type u_6
inst✝² : Monoid G
L : Type u_7
inst✝¹ : CommRing L
inst✝ : NoZeroDivisors L
this✝ : DecidableEq (G →* L) := Classical.decEq (G →* L)
this : MulAction L L := DistribMulAction.toMulAction
s✝ : Finset (G →* L)
a : G →* L
s : Finset (G →* L)
has : a ∉ s
ih : ∀ (g : (G →* L) → L), ∑ i ∈ s, g i • ⇑i = 0 ... | 6e04f1bf1aff8114 |
integral_one_div_one_add_sq | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_one_div_one_add_sq :
(∫ x : ℝ in a..b, ↑1 / (↑1 + x ^ 2)) = arctan b - arctan a | a b : ℝ
⊢ ∫ (x : ℝ) in a..b, 1 / (1 + x ^ 2) = arctan b - arctan a | refine integral_deriv_eq_sub' _ Real.deriv_arctan (fun _ _ => differentiableAt_arctan _)
(continuous_const.div ?_ fun x => ?_).continuousOn | case refine_1
a b : ℝ
⊢ Continuous fun x => 1 + x ^ 2
case refine_2
a b x : ℝ
⊢ 1 + x ^ 2 ≠ 0 | 307fef01e5ca042f |
List.prod_map_lt_prod_map | Mathlib/Algebra/Order/BigOperators/GroupWithZero/List.lean | theorem prod_map_lt_prod_map {ι : Type*} {s : List ι} (hs : s ≠ [])
(f : ι → R) (g : ι → R) (h0 : ∀ i ∈ s, 0 < f i) (h : ∀ i ∈ s, f i < g i) :
(map f s).prod < (map g s).prod | case h₂.h0.hab
R : Type u_1
inst✝⁴ : CommMonoidWithZero R
inst✝³ : PartialOrder R
inst✝² : ZeroLEOneClass R
inst✝¹ : PosMulStrictMono R
inst✝ : NeZero 1
ι : Type u_2
s✝ : List ι
f g : ι → R
a : ι
s : List ι
hs : a :: s ≠ []
h0 : ∀ (i : ι), i ∈ a :: s → 0 < f i
h : ∀ (i : ι), i ∈ a :: s → f i < g i
this : MulPosStrictMo... | apply h0 | case h₂.h0.hab.a
R : Type u_1
inst✝⁴ : CommMonoidWithZero R
inst✝³ : PartialOrder R
inst✝² : ZeroLEOneClass R
inst✝¹ : PosMulStrictMono R
inst✝ : NeZero 1
ι : Type u_2
s✝ : List ι
f g : ι → R
a : ι
s : List ι
hs : a :: s ≠ []
h0 : ∀ (i : ι), i ∈ a :: s → 0 < f i
h : ∀ (i : ι), i ∈ a :: s → f i < g i
this : MulPosStrict... | 8b9e81fb8e539e8a |
MeasureTheory.Measure.map_apply₀ | Mathlib/MeasureTheory/Measure/Map.lean | lemma map_apply₀ {f : α → β} (hf : AEMeasurable f μ) {s : Set β}
(hs : NullMeasurableSet s (map f μ)) : μ.map f s = μ (f ⁻¹' s) | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
s : Set β
hs : NullMeasurableSet s ((liftLinear (OuterMeasure.map (AEMeasurable.mk f hf)) ⋯) μ)
⊢ ((liftLinear (OuterMeasure.map (AEMeasurable.mk f hf)) ⋯) μ) s = μ (f ⁻¹' s) | rw [liftLinear_apply₀ _ hs, measure_congr (hf.ae_eq_mk.preimage s)] | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
s : Set β
hs : NullMeasurableSet s ((liftLinear (OuterMeasure.map (AEMeasurable.mk f hf)) ⋯) μ)
⊢ ((OuterMeasure.map (AEMeasurable.mk f hf)) μ.toOuterMeasure) s = μ (AEMeasurable.mk f hf ⁻¹' s) | ecdc6285e37a6cc4 |
CategoryTheory.IsUniversalColimit.map_reflective | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem IsUniversalColimit.map_reflective
{Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
{F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
(H : IsUniversalColimit c)
[∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
[∀ X (f : X ⟶ Gl.obj c.pt), PreservesLimit (cospan (Gr.map ... | case refine_3.h₀
J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullba... | dsimp [β] | case refine_3.h₀
J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullba... | 0ba4821bd37956f9 |
CategoryTheory.ShortComplex.exact_iff_isZero_homology | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | lemma exact_iff_isZero_homology [S.HasHomology] :
S.Exact ↔ IsZero S.homology | case mp.mk.intro
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
h' : S.HomologyData
z : IsZero h'.left.H
⊢ IsZero S.homology | exact IsZero.of_iso z h'.left.homologyIso | no goals | fee328dd43339724 |
List.modifyHead_modifyHead | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean | theorem modifyHead_modifyHead {l : List α} {f g : α → α} :
(l.modifyHead f).modifyHead g = l.modifyHead (g ∘ f) | α : Type u_1
l : List α
f g : α → α
⊢ modifyHead g (modifyHead f l) = modifyHead (g ∘ f) l | cases l <;> simp [modifyHead] | no goals | 6385f7b35d363bd4 |
NoetherNormalization.degreeOf_zero_t | Mathlib/RingTheory/NoetherNormalization.lean | private lemma degreeOf_zero_t {a : k} (ha : a ≠ 0) : ((T f) (monomial v a)).degreeOf 0 =
∑ i : Fin (n + 1), (r i) * v i | k : Type u_1
inst✝ : Field k
n : ℕ
f : MvPolynomial (Fin (n + 1)) k
v : Fin (n + 1) →₀ ℕ
a : k
ha : a ≠ 0
h : ∀ (i : Fin n), (Polynomial.C (MvPolynomial.X i) + Polynomial.X ^ r i.succ) ^ v i.succ ≠ 0
⊢ Polynomial.X ^ v 0 ≠ 0 | simp | no goals | c85c6fd1d2284785 |
ascPochhammer_eval_eq_zero_iff | Mathlib/RingTheory/Polynomial/Pochhammer.lean | theorem ascPochhammer_eval_eq_zero_iff [IsDomain R]
(n : ℕ) (r : R) : (ascPochhammer R n).eval r = 0 ↔ ∃ k < n, k = -r | case refine_1.succ.inl.intro.intro
R : Type u
inst✝¹ : Ring R
inst✝ : IsDomain R
r : R
n : ℕ
ih : eval r (ascPochhammer R n) = 0 → ∃ k < n, ↑k = -r
h : eval r (ascPochhammer R n) = 0
rn : ℕ
hrn : rn < n
rrn : ↑rn = -r
⊢ ∃ k < n + 1, ↑k = -r | exact ⟨rn, by omega, rrn⟩ | no goals | 2471204549ab0950 |
Matroid.IsRestriction.eq_or_isStrictRestriction | Mathlib/Data/Matroid/Restrict.lean | theorem IsRestriction.eq_or_isStrictRestriction (h : N ≤r M) : N = M ∨ N <r M | α : Type u_1
M N : Matroid α
h : N ≤r M
⊢ N = M ∨ N <r M | simpa using eq_or_lt_of_le (ofMatroid_le_iff.2 h) | no goals | 05f35abb4ca2400c |
PSet.le_succ_rank_sUnion | Mathlib/SetTheory/ZFC/Rank.lean | theorem le_succ_rank_sUnion (x : PSet) : rank x ≤ succ (rank (⋃₀ x)) | case h
x z : PSet.{u_1}
a✝ : z ∈ x
⊢ ∀ ⦃z_1 : PSet.{u_1}⦄, z_1 ∈ z → z_1 ∈ ⋃₀ x | intro _ _ | case h
x z : PSet.{u_1}
a✝¹ : z ∈ x
z✝ : PSet.{u_1}
a✝ : z✝ ∈ z
⊢ z✝ ∈ ⋃₀ x | 319762a811a41216 |
divRadical_dvd_derivative | Mathlib/RingTheory/Polynomial/Radical.lean | theorem divRadical_dvd_derivative (a : k[X]) : divRadical a ∣ derivative a | k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
a : k[X]
⊢ divRadical a ∣ derivative a | induction a using induction_on_coprime | case h0
k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
⊢ divRadical 0 ∣ derivative 0
case h1
k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
x✝ : k[X]
a✝ : IsUnit x✝
⊢ divRadical x✝ ∣ derivative x✝
case hpr
k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
p✝ : k[X]
i✝ : ℕ
a✝ : Prime p✝
⊢ divRadical (p✝ ^ ... | 51efdd72c654425d |
Equiv.Perm.Basis.card_ofPermHom_support | Mathlib/GroupTheory/Perm/Centralizer.lean | theorem card_ofPermHom_support :
(ofPermHom a τ).support.card =
∑ c ∈ (τ : Perm g.cycleFactorsFinset).support, c.val.support.card | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
τ : ↥(range_toPermHom' g)
⊢ ∀ x ∈ (↑τ).support, ∀ y ∈ (↑τ).support, x ≠ y → _root_.Disjoint (↑x).support (↑y).support | intro c _ d _ h | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
τ : ↥(range_toPermHom' g)
c : { x // x ∈ g.cycleFactorsFinset }
a✝¹ : c ∈ (↑τ).support
d : { x // x ∈ g.cycleFactorsFinset }
a✝ : d ∈ (↑τ).support
h : c ≠ d
⊢ _root_.Disjoint (↑c).support (↑d).support | fc1c5fa3f0bd505f |
AkraBazziRecurrence.isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma isEquivalent_deriv_rpow_p_mul_one_add_smoothingFn {p : ℝ} (hp : p ≠ 0) :
deriv (fun z => z ^ p * (1 + ε z)) ~[atTop] fun z => p * z ^ (p-1) := calc
deriv (fun z => z ^ p * (1 + ε z))
=ᶠ[atTop] fun z => p * z ^ (p-1) * (1 + ε z) - z ^ (p-1) / (log z ^ 2) :=
eventually_deriv_rpow_p_mul_one_add_smo... | p : ℝ
hp : p ≠ 0
⊢ (fun z => p * z ^ (p - 1) * (1 + ε z) - z ^ (p - 1) / log z ^ 2) ~[atTop] fun z => p * z ^ (p - 1) | refine IsEquivalent.add_isLittleO ?one ?two | case one
p : ℝ
hp : p ≠ 0
⊢ (fun z => p * z ^ (p - 1) * (1 + ε z)) ~[atTop] fun z => p * z ^ (p - 1)
case two
p : ℝ
hp : p ≠ 0
⊢ (fun z => -(z ^ (p - 1) / log z ^ 2)) =o[atTop] fun z => p * z ^ (p - 1) | 79a4691c4020ee10 |
Finset.toDual_min' | Mathlib/Data/Finset/Max.lean | theorem toDual_min' {s : Finset α} (hs : s.Nonempty) :
toDual (min' s hs) = max' (s.image toDual) (hs.image _) | α : Type u_2
inst✝ : LinearOrder α
s : Finset α
hs : s.Nonempty
⊢ s.sup (WithBot.some ∘ fun x => toDual x) = s.sup ((WithBot.some ∘ fun x => x) ∘ ⇑toDual) | rfl | no goals | 1b1f0b1321d26f49 |
DivisorChain.eq_second_of_chain_of_prime_dvd | Mathlib/RingTheory/ChainOfDivisors.lean | theorem eq_second_of_chain_of_prime_dvd {p q r : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c)
(h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i) (hp : Prime p) (hr : r ∣ q) (hp' : p ∣ r) :
p = c 1 | case succ.intro.refine_1
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
q r : Associates M
hr : r ∣ q
n : ℕ
hn : n + 1 ≠ 0
c : Fin (n + 1 + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hp : Prime (c 0)
hp' : c 0 ∣ r
⊢ False | exact hp.not_unit (first_of_chain_isUnit h₁ @h₂) | no goals | ab445c4bc010329e |
Basis.ofIsLocalizedModule_span | Mathlib/RingTheory/Localization/Module.lean | theorem Basis.ofIsLocalizedModule_span :
span R (Set.range (b.ofIsLocalizedModule Rₛ S f)) = LinearMap.range f | R : Type u_1
Rₛ : Type u_2
inst✝¹⁰ : CommSemiring R
S : Submonoid R
inst✝⁹ : CommSemiring Rₛ
inst✝⁸ : Algebra R Rₛ
inst✝⁷ : IsLocalization S Rₛ
M : Type u_3
Mₛ : Type u_4
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module R M
inst✝⁴ : AddCommMonoid Mₛ
inst✝³ : Module R Mₛ
inst✝² : Module Rₛ Mₛ
inst✝¹ : IsScalarTower R Rₛ Mₛ
f : ... | rw [Submodule.map_span] | no goals | 80d21e87910cc595 |
Sym2.two_mul_card_image_offDiag | Mathlib/Data/Sym/Card.lean | lemma two_mul_card_image_offDiag (s : Finset α) : 2 * #(s.offDiag.image Sym2.mk) = #s.offDiag | case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
x y : α
a : α × α
ha₁ : a.1 ∈ s
ha₂ : a.2 ∈ s
ha : a.1 ≠ a.2
h : Rel α a (x, y)
hx : x ∈ s
hy : y ∈ s
hxy : x ≠ y
hxy' : y ≠ x
this : filter (fun z => Sym2.mk z = s(x, y)) s.offDiag = {(x, y), (y, x)}
⊢ x = y → ¬y = x | exact fun _ => hxy' | no goals | 54700382534a202b |
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