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 |
|---|---|---|---|---|---|---|
Polynomial.content_eq_gcd_range_of_lt | Mathlib/RingTheory/Polynomial/Content.lean | theorem content_eq_gcd_range_of_lt (p : R[X]) (n : ℕ) (h : p.natDegree < n) :
p.content = (Finset.range n).gcd p.coeff | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
n : ℕ
h : p.natDegree < n
⊢ p.content = (Finset.range n).gcd p.coeff | apply dvd_antisymm_of_normalize_eq normalize_content Finset.normalize_gcd | case hab
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
n : ℕ
h : p.natDegree < n
⊢ p.content ∣ (Finset.range n).gcd p.coeff
case hba
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
n : ℕ
h : p.natDegree < n
⊢ (Finset.range n).gcd... | ef04f5bb22efb7b8 |
AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux | Mathlib/Analysis/Analytic/Uniqueness.lean | theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f : E → F} {U : Set E}
(hf : AnalyticOnNhd 𝕜 f U) (hU : IsPreconnected U)
{z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
EqOn f 0 U | case intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ U
hfz₀ : f =... | rcases hf x xU with ⟨p, r, hp⟩ | case intro.intro.intro
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ ... | 3412e0fd598e7c40 |
LiouvilleWith.irrational | Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean | theorem irrational (h : LiouvilleWith p x) (hp : 1 < p) : Irrational x | case intro
p : ℝ
hp : 1 < p
r : ℚ
h : LiouvilleWith p ↑r
⊢ False | rcases eq_or_ne r 0 with (rfl | h0) | case intro.inl
p : ℝ
hp : 1 < p
h : LiouvilleWith p ↑0
⊢ False
case intro.inr
p : ℝ
hp : 1 < p
r : ℚ
h : LiouvilleWith p ↑r
h0 : r ≠ 0
⊢ False | 1cfab64e5175ef75 |
forall_imp_iff_exists_imp | Mathlib/Logic/Basic.lean | theorem forall_imp_iff_exists_imp {α : Sort*} {p : α → Prop} {b : Prop} [ha : Nonempty α] :
(∀ x, p x) → b ↔ ∃ x, p x → b | α : Sort u_3
p : α → Prop
b : Prop
ha : Nonempty α
⊢ (∀ (x : α), p x) → b ↔ ∃ x, p x → b | let ⟨a⟩ := ha | α : Sort u_3
p : α → Prop
b : Prop
ha : Nonempty α
a : α
⊢ (∀ (x : α), p x) → b ↔ ∃ x, p x → b | c40e6e7d13e74a31 |
EuclideanGeometry.collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | theorem collinear_iff_eq_or_eq_or_angle_eq_zero_or_angle_eq_pi {p₁ p₂ p₃ : P} :
Collinear ℝ ({p₁, p₂, p₃} : Set P) ↔ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π | case pos
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
p₁ p₂ p₃ : P
h : Wbtw ℝ p₁ p₂ p₃ ∨ Wbtw ℝ p₂ p₃ p₁ ∨ Wbtw ℝ p₃ p₁ p₂
h₁₂ : ¬p₁ = p₂
h₃₂ : p₃ = p₂
⊢ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π | exact Or.inr (Or.inl h₃₂) | no goals | 6d4fda70b0d8e02f |
prime_pow_iff | Mathlib/Algebra/GroupWithZero/Associated.lean | @[simp]
lemma prime_pow_iff [CancelCommMonoidWithZero M] {p : M} {n : ℕ} :
Prime (p ^ n) ↔ Prime p ∧ n = 1 | case succ.inr.intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : M
n : ℕ
hp : IsUnit p
hpn : Prime (p ^ n)
⊢ False | exact hpn.not_unit (hp.pow n) | no goals | ed9de7a07a4b8a76 |
AkraBazziRecurrence.isEquivalent_smoothingFn_sub_self | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma isEquivalent_smoothingFn_sub_self (i : α) :
(fun (n : ℕ) => ε (b i * n) - ε n) ~[atTop] fun n => -log (b i) / (log n)^2 | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
this : (fun n => log (b i) + log ↑n) = fun n => log ↑n + log (b i)
⊢ (fun n => log (b i) + log ↑n) ~[atTop] fun n => log ↑n | rw [this] | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
i : α
this : (fun n => log (b i) + log ↑n) = fun n => log ↑n + log (b i)
⊢ (fun n => log ↑n + log (b i)) ~[atTop] fun n => log ↑n | 058fd0dc7e6df047 |
iUnion_Iic_of_not_bddAbove_range | Mathlib/Order/Interval/Set/Disjoint.lean | theorem iUnion_Iic_of_not_bddAbove_range (hf : ¬ BddAbove (range f)) : ⋃ i, Iic (f i) = univ | ι : Sort u
α : Type v
inst✝ : LinearOrder α
f : ι → α
hf : ¬BddAbove (range f)
⊢ ⋃ i, Iic (f i) = univ | refine Set.eq_univ_of_subset ?_ (iUnion_Iio_eq_univ_iff.mpr hf) | ι : Sort u
α : Type v
inst✝ : LinearOrder α
f : ι → α
hf : ¬BddAbove (range f)
⊢ ⋃ i, Iio (f i) ⊆ ⋃ i, Iic (f i) | d922ebb0648dbe4b |
MeasureTheory.FiniteMeasure.tendsto_testAgainstNN_filter_of_le_const | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | theorem tendsto_testAgainstNN_filter_of_le_const {ι : Type*} {L : Filter ι}
[L.IsCountablyGenerated] {μ : FiniteMeasure Ω} {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
(fs_le_const : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂(μ : Measure Ω), fs i ω ≤ c) {f : Ω →ᵇ ℝ≥0}
(fs_lim : ∀ᵐ ω : Ω ∂(μ : Measure Ω), Tendsto (fun i ↦ fs i ω) L (𝓝 (f ω)))... | Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝ : L.IsCountablyGenerated
μ : FiniteMeasure Ω
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂↑μ, (fs i) ω ≤ c
f : Ω →ᵇ ℝ≥0
fs_lim : ∀ᵐ (ω : Ω) ∂↑μ, Tendsto (fun i => (... | apply (ENNReal.tendsto_toNNReal (f.lintegral_lt_top_of_nnreal (μ : Measure Ω)).ne).comp | Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
ι : Type u_2
L : Filter ι
inst✝ : L.IsCountablyGenerated
μ : FiniteMeasure Ω
fs : ι → Ω →ᵇ ℝ≥0
c : ℝ≥0
fs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂↑μ, (fs i) ω ≤ c
f : Ω →ᵇ ℝ≥0
fs_lim : ∀ᵐ (ω : Ω) ∂↑μ, Tendsto (fun i => (... | 3ca341adb5fa5528 |
Real.diam_Ioo | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem diam_Ioo {a b : ℝ} (h : a ≤ b) : Metric.diam (Ioo a b) = b - a | a b : ℝ
h : a ≤ b
⊢ Metric.diam (Ioo a b) = b - a | simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | no goals | f8c013df8e5105ce |
Finset.kruskal_katona_lovasz_form | Mathlib/Combinatorics/SetFamily/KruskalKatona.lean | theorem kruskal_katona_lovasz_form (hir : i ≤ r) (hrk : r ≤ k) (hkn : k ≤ n)
(h₁ : (𝒜 : Set (Finset (Fin n))).Sized r) (h₂ : k.choose r ≤ #𝒜) :
k.choose (r - i) ≤ #(∂^[i] 𝒜) | case h.e'_2.h.mp.intro.intro.intro.intro
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
hBk : B ⊆ ra... | rw [hB, ← Nat.add_sub_assoc hir, Nat.add_sub_cancel_left] at hcard | case h.e'_2.h.mp.intro.intro.intro.intro
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
B : Finset (Fin n)
hBk : B ⊆ ra... | edb5f2f186d17abf |
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✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K... | have hf : (Algebra.ofId _ _).toLinearMap ∘ₗ f = LinearMap.id := by
dsimp [f]
rw [← LinearMap.comp_assoc, ← comp_sec K A]
congr 1
apply LinearMap.restrictScalars_injective K
apply _root_.TensorProduct.ext'
intros r s
obtain ⟨s, rfl⟩ := e.surjective s
suffices s • (Ideal.Quotient.mk (IsLocalRing.maximalId... | 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✝¹ : Module.Finite K A
this✝ : IsArtinianRing A
hA : IsNilpotent (IsLocalRing.maximalIdeal A)
this : Function.Bijective ⇑(ofId K... | 5099a67fd835b73a |
CategoryTheory.ShortComplex.LeftHomologyData.homologyIso_leftHomologyData | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | @[simp]
lemma LeftHomologyData.homologyIso_leftHomologyData [S.HasHomology] :
S.leftHomologyData.homologyIso = S.leftHomologyIso.symm | case w
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ leftHomologyMap' (𝟙 S) S.homologyData.left S.leftHomologyData ≫
leftHomologyMap' (𝟙 S) S.leftHomologyData S.leftHomologyData =
leftHomologyMap' (𝟙 S) S.homologyData.left S.leftHomologyData | rw [← leftHomologyMap'_comp, comp_id] | no goals | b471bf4a052e6479 |
bergelson' | Mathlib/MeasureTheory/Function/Intersectivity.lean | /-- **Bergelson Intersectivity Lemma**: In a finite measure space, a sequence of events that have
measure at least `r` has an infinite subset whose finite intersections all have positive volume.
TODO: The infinity of `t` should be strengthened to `t` having positive natural density. -/
lemma bergelson' {s : ℕ → Set α}... | α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n) | let M (f : α → ℝ) : Set α := {x | eLpNormEssSup f μ < ‖f x‖₊} | α : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
r : ℝ≥0∞
s : ℕ → Set α
hs : ∀ (n : ℕ), MeasurableSet (s n)
hr₀ : r ≠ 0
hr : ∀ (n : ℕ), r ≤ μ (s n)
M : (α → ℝ) → Set α := fun f => {x | eLpNormEssSup f μ < ↑‖f x‖₊}
⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n) | 8eca484049265c21 |
Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd | Mathlib/Algebra/Prime/Lemmas.lean | theorem Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd [CancelCommMonoidWithZero M] {p a b : M}
{n : ℕ} (hp : Prime p) (hpow : p ^ n.succ ∣ a ^ n.succ * b ^ n) (hb : ¬p ^ 2 ∣ b) : p ∣ a | case inr.intro.intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p a : M
n : ℕ
hp : Prime p
x : M
hb : ¬p ^ 2 ∣ p * x
hbdiv : p ∣ (p * x) ^ n
y : M
hy : a ^ n.succ * (p * x) ^ n = p ^ n.succ * y
⊢ p ∣ a | have : a ^ n.succ * x ^ n = p * y := by
refine mul_left_cancel₀ (pow_ne_zero n hp.ne_zero) ?_
rw [← mul_assoc _ p, ← pow_succ, ← hy, mul_pow, ← mul_assoc (a ^ n.succ), mul_comm _ (p ^ n),
mul_assoc] | case inr.intro.intro
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p a : M
n : ℕ
hp : Prime p
x : M
hb : ¬p ^ 2 ∣ p * x
hbdiv : p ∣ (p * x) ^ n
y : M
hy : a ^ n.succ * (p * x) ^ n = p ^ n.succ * y
this : a ^ n.succ * x ^ n = p * y
⊢ p ∣ a | a203319e403717ea |
BitVec.append_assoc | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem append_assoc {x₁ : BitVec w₁} {x₂ : BitVec w₂} {x₃ : BitVec w₃} :
(x₁ ++ x₂) ++ x₃ = (x₁ ++ (x₂ ++ x₃)).cast (by omega) | w₂ w₃ n : Nat
x₁ : BitVec (n + 1)
x₂ : BitVec w₂
x₃ : BitVec w₃
ih : ∀ {x₂ : BitVec w₂} {x₃ : BitVec w₃}, setWidth n x₁ ++ x₂ ++ x₃ = BitVec.cast ⋯ (setWidth n x₁ ++ (x₂ ++ x₃))
j : Nat
h : j < n + 1 + w₂ + w₃
⊢ n + w₂ + w₃ = n + (w₂ + w₃) | omega | no goals | ac5627f045e44cbc |
Filter.subtype_coe_map_comap | Mathlib/Order/Filter/Map.lean | theorem subtype_coe_map_comap (s : Set α) (f : Filter α) :
map ((↑) : s → α) (comap ((↑) : s → α) f) = f ⊓ 𝓟 s | α : Type u_1
s : Set α
f : Filter α
⊢ map Subtype.val (comap Subtype.val f) = f ⊓ 𝓟 s | rw [map_comap, Subtype.range_coe] | no goals | ed34ebae10c60bcc |
Nat.max_add_left | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem max_add_left {a b : Nat} : max a (b + a) = b + a | a b : Nat
⊢ max a (b + a) = b + a | rw [Nat.max_def] | a b : Nat
⊢ (if a ≤ b + a then b + a else a) = b + a | dd1afedf8b581d16 |
ProbabilityTheory.Kernel.compProd_fst_borelMarkovFromReal_eq_comapRight_compProd | Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean | /-- For `κ' := map κ (Prod.map (id : β → β) e)`, the hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`.
The conclusion of the lemma is `fst κ ⊗ₖ borelMarkovFromReal Ω η = comapRight (fst κ' ⊗ₖ η) _`. -/
lemma compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
(κ : Kernel α (β × Ω)) [IsSFiniteKernel κ] (η : Kernel (α × β) ... | α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (... | simp only [preimage_compl, mem_univ, mk_preimage_prod_right] at h_compProd | α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (... | 903782bd03958013 |
CategoryTheory.ShortComplex.leftRightHomologyComparison'_fac | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | @[reassoc]
lemma leftRightHomologyComparison'_fac (h₁ : S.LeftHomologyData) (h₂ : S.RightHomologyData)
[S.HasHomology] :
leftRightHomologyComparison' h₁ h₂ = h₁.homologyIso.inv ≫ h₂.homologyIso.hom | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
inst✝ : S.HasHomology
⊢ leftRightHomologyComparison' h₁ h₂ = h₁.homologyIso.inv ≫ h₂.homologyIso.hom | rw [leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap'
S.homologyData h₁ h₂] | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
inst✝ : S.HasHomology
⊢ leftHomologyMap' (𝟙 S) h₁ S.homologyData.left ≫
S.homologyData.iso.hom ≫ rightHomologyMap' (𝟙 S) S.homologyData.right h₂ =
h₁.homologyIso.inv ≫ h₂.hom... | 69bc854b718d65a4 |
CategoryTheory.ComposableArrows.ext_succ | Mathlib/CategoryTheory/ComposableArrows.lean | lemma ext_succ {F G : ComposableArrows C (n + 1)} (h₀ : F.obj' 0 = G.obj' 0)
(h : F.δ₀ = G.δ₀) (w : F.map' 0 1 = eqToHom h₀ ≫ G.map' 0 1 ≫
eqToHom (Functor.congr_obj h.symm 0)) : F = G | C : Type u_1
inst✝ : Category.{u_2, u_1} C
n : ℕ
F G : ComposableArrows C (n + 1)
h₀ : F.obj' 0 ⋯ = G.obj' 0 ⋯
h : F.δ₀ = G.δ₀
w : F.map' 0 1 ⋯ ⋯ = eqToHom h₀ ≫ G.map' 0 1 ⋯ ⋯ ≫ eqToHom ⋯
i : ℕ
hi : i + 1 < n + 1 + 1
⊢ i < n + 1 | valid | no goals | 96bc6a50285454b6 |
MeasureTheory.hasPDF_of_pdf_ne_zero | Mathlib/Probability/Density.lean | theorem hasPDF_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E}
(hac : map X ℙ ≪ μ) (hpdf : ¬pdf X ℙ μ =ᵐ[μ] 0) : HasPDF X ℙ μ | case refine_2
Ω : Type u_1
E : Type u_2
inst✝ : MeasurableSpace E
m : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
X : Ω → E
hac : map X ℙ ≪ μ
hpdf : ¬(map X ℙ).HaveLebesgueDecomposition μ
this : pdf X ℙ μ = 0
⊢ pdf X ℙ μ =ᶠ[ae μ] 0 | filter_upwards using congrFun this | no goals | fe40d74ce6cc24be |
Ideal.inertiaDeg_eq_of_isGalois | Mathlib/NumberTheory/RamificationInertia/Galois.lean | theorem inertiaDeg_eq_of_isGalois [p.IsMaximal] [IsGalois K L] :
inertiaDeg p P = inertiaDeg p Q | case intro
A : Type u_1
B : Type u_2
inst✝²¹ : CommRing A
inst✝²⁰ : IsDomain A
inst✝¹⁹ : IsIntegrallyClosed A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : IsDomain B
inst✝¹⁶ : IsIntegrallyClosed B
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Module.Finite A B
p : Ideal A
P Q : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
hQp : Q.IsPrime
inst✝¹³ : Q... | rw [← hs] | case intro
A : Type u_1
B : Type u_2
inst✝²¹ : CommRing A
inst✝²⁰ : IsDomain A
inst✝¹⁹ : IsIntegrallyClosed A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : IsDomain B
inst✝¹⁶ : IsIntegrallyClosed B
inst✝¹⁵ : Algebra A B
inst✝¹⁴ : Module.Finite A B
p : Ideal A
P Q : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
hQp : Q.IsPrime
inst✝¹³ : Q... | eed5418a748b615b |
Real.one_half_lt_eulerMascheroniSeq_six | Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | lemma one_half_lt_eulerMascheroniSeq_six : 1 / 2 < eulerMascheroniSeq 6 | this : eulerMascheroniSeq 6 = 49 / 20 - log 7
⊢ 7 < ∑ i ∈ Finset.range 7, (49 / 20 - 1 / 2) ^ i / ↑i.factorial | simp_rw [Finset.sum_range_succ, Nat.factorial_succ] | this : eulerMascheroniSeq 6 = 49 / 20 - log 7
⊢ 7 <
∑ i ∈ Finset.range 0, (49 / 20 - 1 / 2) ^ i / ↑i.factorial + (49 / 20 - 1 / 2) ^ 0 / ↑(Nat.factorial 0) +
(49 / 20 - 1 / 2) ^ 1 / ↑((0 + 1) * Nat.factorial 0) +
(49 / 20 - 1 / 2) ^ 2 / ↑((1 + 1) * ((0 + 1) * Nat.factorial 0)) +
... | 89ee95ad7e3cf599 |
WeierstrassCurve.exists_variableChange_of_char_three | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_three (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | case intro.of_j_ne_zero.intro.of_j_eq_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
heq : E.j = E'.j
C : VariableChange F
inst✝¹ : (E.variableChange C).IsCharThreeJNeZeroNF
C' : VariableChange F
inst✝ : (E'.variableCha... | exact False.elim (h rfl) | no goals | ee4716f9fd7fa5bc |
MeasureTheory.IntegrableAtFilter.sup_iff | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | theorem IntegrableAtFilter.sup_iff {l l' : Filter α} :
IntegrableAtFilter f (l ⊔ l') μ ↔ IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ | α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f : α → E
μ : Measure α
l l' : Filter α
⊢ IntegrableAtFilter f (l ⊔ l') μ ↔ IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ | constructor | case mp
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f : α → E
μ : Measure α
l l' : Filter α
⊢ IntegrableAtFilter f (l ⊔ l') μ → IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ
case mpr
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
f : α → E... | 3db9bfca8a9ce9ce |
MeasureTheory.Measure.haar.chaar_self | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem chaar_self (K₀ : PositiveCompacts G) : chaar K₀ K₀.toCompacts = 1 | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
⊢ chaar K₀ K₀.toCompacts = 1 | have : Continuous eval := continuous_apply _ | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ chaar K₀ K₀.toCompacts = 1 | ac57553b2117b122 |
LieAlgebra.IsKilling.span_weight_eq_top | Mathlib/Algebra/Lie/Weights/Killing.lean | /-- Given a splitting Cartan subalgebra `H` of a finite-dimensional Lie algebra with non-singular
Killing form, the corresponding roots span the dual space of `H`. -/
@[simp]
lemma span_weight_eq_top :
span K (range (Weight.toLinear K H L)) = ⊤ | K : Type u_2
L : Type u_3
inst✝⁶ : LieRing L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
⊢ span K (range (Weight.toLinear K (↥H) L)) = ⊤ | refine eq_top_iff.mpr (le_trans ?_ (LieModule.range_traceForm_le_span_weight K H L)) | K : Type u_2
L : Type u_3
inst✝⁶ : LieRing L
inst✝⁵ : Field K
inst✝⁴ : LieAlgebra K L
inst✝³ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝² : H.IsCartanSubalgebra
inst✝¹ : IsKilling K L
inst✝ : IsTriangularizable K (↥H) L
⊢ ⊤ ≤ LinearMap.range (traceForm K (↥H) L) | b06b520a71c4215b |
Ordnode.balance_eq_balance' | Mathlib/Data/Ordmap/Ordset.lean | theorem balance_eq_balance' {l x r} (hl : Balanced l) (hr : Balanced r) (sl : Sized l)
(sr : Sized r) : @balance α l x r = balance' l x r | α : Type u_1
x : α
hl : nil.Balanced
sl : nil.Sized
rs : ℕ
rx : α
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
hr : ((node rls rll rlx rlr).node' rx nil).Balanced
sr : (node rs (node rls rll rlx rlr) rx nil).Sized
this : (node rls rll rlx rlr).size ≤ 1
⊢ rll.size = 0 ∧ rlr.size = 0 | rwa [size, sr.2.1.1, Nat.succ_le_succ_iff, Nat.le_zero, add_eq_zero] at this | no goals | 23a52200299a6569 |
lp.norm_sum_single | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem norm_sum_single (hp : 0 < p.toReal) (f : ∀ i, E i) (s : Finset α) :
‖∑ i ∈ s, lp.single p i (f i)‖ ^ p.toReal = ∑ i ∈ s, ‖f i‖ ^ p.toReal | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : DecidableEq α
hp : 0 < p.toReal
f : (i : α) → E i
s : Finset α
⊢ HasSum (fun i => ‖↑(∑ i ∈ s, lp.single p i (f i)) i‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal) | simp only [lp.coeFn_single, coeFn_sum, Finset.sum_apply, Finset.sum_pi_single] | α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : DecidableEq α
hp : 0 < p.toReal
f : (i : α) → E i
s : Finset α
⊢ HasSum (fun i => ‖if i ∈ s then f i else 0‖ ^ p.toReal) (∑ i ∈ s, ‖f i‖ ^ p.toReal) | 3207ba840ff2abcd |
AddLECancellable.tsub_lt_self_iff | Mathlib/Algebra/Order/Sub/Basic.lean | theorem tsub_lt_self_iff (ha : AddLECancellable a) : a - b < a ↔ 0 < a ∧ 0 < b | α : Type u_1
inst✝³ : LinearOrderedAddCommMonoid α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a : α
ha : AddLECancellable a
h : a - 0 < a
⊢ False | rw [tsub_zero] at h | α : Type u_1
inst✝³ : LinearOrderedAddCommMonoid α
inst✝² : CanonicallyOrderedAdd α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a : α
ha : AddLECancellable a
h : a < a
⊢ False | b8746531f98eaeb4 |
MeasureTheory.exists_lt_lintegral_simpleFunc_of_lt_lintegral | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : MeasurableSpace α} {μ : Measure α}
[SigmaFinite μ] {f : α → ℝ≥0} {L : ℝ≥0∞} (hL : L < ∫⁻ x, f x ∂μ) :
∃ g : α →ₛ ℝ≥0, (∀ x, g x ≤ f x) ∧ ∫⁻ x, g x ∂μ < ∞ ∧ L < ∫⁻ x, g x ∂μ | case h.e'_4
α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
f : α → ℝ≥0
L : ℝ≥0∞
g₀ : α →ₛ ℝ≥0
hg₀ : ∀ (x : α), ↑(g₀ x) ≤ ↑(f x)
g₀L : L < (SimpleFunc.map ofNNReal g₀).lintegral μ
⊢ ∫⁻ (x : α), ↑(g₀ x) ∂μ = ∫⁻ (a : α), (ofNNReal ∘ ⇑g₀) a ∂μ | simp only [Function.comp_apply] | no goals | 831464279e8b5079 |
MulAction.IsPreprimitive.of_prime_card | Mathlib/GroupTheory/GroupAction/Primitive.lean | theorem of_prime_card [hGX : IsPretransitive G X] (hp : Nat.Prime (Nat.card X)) :
IsPreprimitive G X | G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
hGX : IsPretransitive G X
hp : Nat.Prime (Nat.card X)
B : Set X
hB : IsBlock G B
hB' : B.Nontrivial
this : Finite X
⊢ B = Set.univ | rw [Set.eq_univ_iff_ncard, eq_comm, ← hp.dvd_iff_eq ((Set.one_lt_ncard).mpr hB').ne'] | G : Type u_1
X : Type u_2
inst✝¹ : Group G
inst✝ : MulAction G X
hGX : IsPretransitive G X
hp : Nat.Prime (Nat.card X)
B : Set X
hB : IsBlock G B
hB' : B.Nontrivial
this : Finite X
⊢ B.ncard ∣ Nat.card X | 64d5787e91ea51a9 |
Polynomial.reflect_add | Mathlib/Algebra/Polynomial/Reverse.lean | theorem reflect_add (f g : R[X]) (N : ℕ) : reflect N (f + g) = reflect N f + reflect N g | R : Type u_1
inst✝ : Semiring R
f g : R[X]
N : ℕ
⊢ reflect N (f + g) = reflect N f + reflect N g | ext | case a
R : Type u_1
inst✝ : Semiring R
f g : R[X]
N n✝ : ℕ
⊢ (reflect N (f + g)).coeff n✝ = (reflect N f + reflect N g).coeff n✝ | 16d403ee4b769130 |
MeasureTheory.lpMeasToLpTrim_smul | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | theorem lpMeasToLpTrim_smul (hm : m ≤ m0) (c : 𝕜) (f : lpMeas F 𝕜 m p μ) :
lpMeasToLpTrim F 𝕜 p μ hm (c • f) = c • lpMeasToLpTrim F 𝕜 p μ hm f | case h.refine_1
α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
c : 𝕜
f : ↥(lpMeas F 𝕜 m p μ)
⊢ StronglyMeasurable (c • ↑↑(lpMeasToLpTrim F 𝕜 p μ hm f)) | exact (Lp.stronglyMeasurable _).const_smul c | no goals | 6e0abadb66cc1334 |
gal_X_sub_C_isSolvable | Mathlib/FieldTheory/AbelRuffini.lean | theorem gal_X_sub_C_isSolvable (x : F) : IsSolvable (X - C x).Gal | F : Type u_1
inst✝ : Field F
x : F
⊢ IsSolvable (X - C x).Gal | infer_instance | no goals | 47b7353ae1841b74 |
List.prod_map_le_prod_map₀ | Mathlib/Algebra/Order/BigOperators/GroupWithZero/List.lean | theorem prod_map_le_prod_map₀ {ι : Type*} {s : List ι} (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 cons.h₂
R : Type u_1
inst✝³ : CommMonoidWithZero R
inst✝² : PartialOrder R
inst✝¹ : ZeroLEOneClass R
inst✝ : PosMulMono R
ι : Type u_2
f g : ι → R
a : ι
s : List ι
hind : (∀ (i : ι), i ∈ s → 0 ≤ f i) → (∀ (i : ι), i ∈ s → f i ≤ g i) → (map f s).prod ≤ (map g s).prod
h0 : ∀ (i : ι), i ∈ a :: s → 0 ≤ f i
h : ∀ (i : ... | apply hind | case cons.h₂.h0
R : Type u_1
inst✝³ : CommMonoidWithZero R
inst✝² : PartialOrder R
inst✝¹ : ZeroLEOneClass R
inst✝ : PosMulMono R
ι : Type u_2
f g : ι → R
a : ι
s : List ι
hind : (∀ (i : ι), i ∈ s → 0 ≤ f i) → (∀ (i : ι), i ∈ s → f i ≤ g i) → (map f s).prod ≤ (map g s).prod
h0 : ∀ (i : ι), i ∈ a :: s → 0 ≤ f i
h : ∀ (i... | f33d8aef43f74307 |
EulerSine.tendsto_integral_cos_pow_mul_div | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | theorem tendsto_integral_cos_pow_mul_div {f : ℝ → ℂ} (hf : ContinuousOn f (Icc 0 (π / 2))) :
Tendsto
(fun n : ℕ => (∫ x in (0 : ℝ)..π / 2, (cos x : ℂ) ^ n * f x) /
(∫ x in (0 : ℝ)..π / 2, cos x ^ n : ℝ))
atTop (𝓝 <| f 0) | f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
⊢ Tendsto
(fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)
atTop (𝓝 (f 0)) | have c_nonneg : ∀ x : ℝ, x ∈ Icc 0 (π / 2) → 0 ≤ cos x := fun x hx =>
cos_nonneg_of_mem_Icc ((Icc_subset_Icc_left (neg_nonpos_of_nonneg pi_div_two_pos.le)) hx) | f : ℝ → ℂ
hf : ContinuousOn f (Icc 0 (π / 2))
c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0
c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x
⊢ Tendsto
(fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)
atTop (𝓝 (f 0)) | 6e4d26a12747c680 |
PartENat.eq_top_iff_forall_lt | Mathlib/Data/Nat/PartENat.lean | theorem eq_top_iff_forall_lt (x : PartENat) : x = ⊤ ↔ ∀ n : ℕ, (n : PartENat) < x | case mp
n : ℕ
⊢ ↑n < ⊤ | exact natCast_lt_top _ | no goals | 695337830db755c7 |
Matrix.PosSemidef.posSemidef_sqrt | Mathlib/LinearAlgebra/Matrix/PosDef.lean | lemma posSemidef_sqrt : PosSemidef hA.sqrt | case hA
n : Type u_2
𝕜 : Type u_4
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.PosSemidef
i : n
⊢ 0 ≤ (RCLike.ofReal ∘ Real.sqrt ∘ ⋯.eigenvalues) i | rw [Function.comp_apply, RCLike.nonneg_iff] | case hA
n : Type u_2
𝕜 : Type u_4
inst✝² : Fintype n
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.PosSemidef
i : n
⊢ 0 ≤ RCLike.re ↑((Real.sqrt ∘ ⋯.eigenvalues) i) ∧ RCLike.im ↑((Real.sqrt ∘ ⋯.eigenvalues) i) = 0 | 5e2a412857806d13 |
MeasureTheory.integrable_map_equiv | Mathlib/MeasureTheory/Function/L1Space/Integrable.lean | theorem integrable_map_equiv (f : α ≃ᵐ δ) (g : δ → β) :
Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ | α : Type u_1
β : Type u_2
δ : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MeasurableSpace δ
inst✝ : NormedAddCommGroup β
f : α ≃ᵐ δ
g : δ → β
⊢ Integrable g (Measure.map (⇑f) μ) ↔ Integrable (g ∘ ⇑f) μ | simp_rw [← memLp_one_iff_integrable] | α : Type u_1
β : Type u_2
δ : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MeasurableSpace δ
inst✝ : NormedAddCommGroup β
f : α ≃ᵐ δ
g : δ → β
⊢ MemLp g 1 (Measure.map (⇑f) μ) ↔ MemLp (g ∘ ⇑f) 1 μ | 5fefe11189d7da78 |
List.sublist_insertionSort' | Mathlib/Data/List/Sort.lean | theorem sublist_insertionSort' {l c : List α} (hs : c.Sorted r) (hc : c <+~ l) :
c <+ insertionSort r l | case intro.intro.nil
α : Type u
r : α → α → Prop
inst✝³ : DecidableRel r
inst✝² : IsAntisymm α r
inst✝¹ : IsTotal α r
inst✝ : IsTrans α r
c : List α
hs : Sorted r c
d : List α
hc : d ~ c
hd : d <+ []
⊢ c <+ insertionSort r [] | simp_all only [sublist_nil, insertionSort, nil_perm] | no goals | 870111a403d3360d |
isAdic_iff | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | theorem isAdic_iff [top : TopologicalSpace R] [IsTopologicalRing R] {J : Ideal R} :
IsAdic J ↔
(∀ n : ℕ, IsOpen ((J ^ n : Ideal R) : Set R)) ∧
∀ s ∈ 𝓝 (0 : R), ∃ n : ℕ, ((J ^ n : Ideal R) : Set R) ⊆ s | case mpr.intro.h
R : Type u_1
inst✝¹ : CommRing R
top : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen ↑(J ^ n)
H₂ : ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
⊢ 𝓝 0 = 𝓝 0 | ext s | case mpr.intro.h.h
R : Type u_1
inst✝¹ : CommRing R
top : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen ↑(J ^ n)
H₂ : ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
s : Set R
⊢ s ∈ 𝓝 0 ↔ s ∈ 𝓝 0 | 5361f5c3aa3b31e4 |
AffineIndependent.existsUnique_dist_eq | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem _root_.AffineIndependent.existsUnique_dist_eq {ι : Type*} [hne : Nonempty ι] [Finite ι]
{p : ι → P} (ha : AffineIndependent ℝ p) :
∃! cs : Sphere P, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ (cs : Set P) | case intro.succ.succ
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max ... | rw [hr, ← affineSpan_insert_affineSpan] | case intro.succ.succ
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
m : ℕ
hn : Fintype.card ι = m + 1 + 1
i : ι
ι2 : Type (max ... | c2eb4c938c177320 |
HomotopyCategory.isZero_quotient_obj_iff | Mathlib/Algebra/Homology/HomotopyCategory.lean | lemma isZero_quotient_obj_iff (C : HomologicalComplex V c) :
IsZero ((quotient _ _).obj C) ↔ Nonempty (Homotopy (𝟙 C) 0) | case mp
ι : Type u_2
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C : HomologicalComplex V c
h : 𝟙 ((quotient V c).obj C) = 0
⊢ Nonempty (Homotopy (𝟙 C) 0) | exact ⟨(homotopyOfEq _ _ (by simp [h]))⟩ | no goals | 172e359745de60b3 |
ContDiffAt.isSymmSndFDerivAt | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | theorem ContDiffAt.isSymmSndFDerivAt {n : WithTop ℕ∞}
(hf : ContDiffAt 𝕜 n f x) (hn : minSmoothness 𝕜 2 ≤ n) : IsSymmSndFDerivAt 𝕜 f x | case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : WithTop ℕ∞
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
h : IsRCLikeNormedField 𝕜
v w : E
⊢ ((f... | apply second_derivative_symmetric_of_eventually (f := f) (f' := fderiv 𝕜 f) (x := x) | case pos.hf
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : WithTop ℕ∞
hf : ContDiffAt 𝕜 n f x
hn : minSmoothness 𝕜 2 ≤ n
h : IsRCLikeNormedField 𝕜
v w : E
⊢ ... | 66d1291a98a72262 |
Topology.IsUpperSet.closure_eq_lowerClosure | Mathlib/Topology/Order/UpperLowerSetTopology.lean | lemma closure_eq_lowerClosure {s : Set α} : closure s = lowerClosure s | α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
⊢ closure s ⊆ ↑(lowerClosure s) | apply closure_minimal subset_lowerClosure _ | α : Type u_1
inst✝² : Preorder α
inst✝¹ : TopologicalSpace α
inst✝ : Topology.IsUpperSet α
s : Set α
⊢ IsClosed ↑(lowerClosure s) | 1f292c00211206a2 |
Algebra.FinitePresentation.of_restrict_scalars_finitePresentation | Mathlib/RingTheory/FinitePresentation.lean | theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
[FinitePresentation.{w₁, w₃} R B] [FiniteType R A] :
FinitePresentation.{w₂, w₃} A B | case refine_1.inl.intro.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : S... | rw [MvPolynomial.algebraMap_eq, ← sub_add_cancel (MvPolynomial.C x)
(map (algebraMap R A) (t' ⟨x, hx⟩)), add_comm] | case refine_1.inl.intro.intro
R : Type w₁
A : Type w₂
B : Type w₃
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : CommRing B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FinitePresentation R B
inst✝ : FiniteType R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] B
hf : S... | 765dbb818bc3202d |
MeasureTheory.NullMeasurableSet.toMeasurable_ae_eq | Mathlib/MeasureTheory/Measure/NullMeasurable.lean | theorem toMeasurable_ae_eq (h : NullMeasurableSet s μ) : toMeasurable μ s =ᵐ[μ] s | α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
h : NullMeasurableSet s μ
⊢ toMeasurable μ s =ᶠ[ae μ] s | rw [toMeasurable_def, dif_pos] | α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
h : NullMeasurableSet s μ
⊢ Exists.choose ?hc =ᶠ[ae μ] s
case hc
α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
s : Set α
h : NullMeasurableSet s μ
⊢ ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s | 237ba8165d7a138a |
GenContFract.terminatedAt_iff_partDen_none | Mathlib/Algebra/ContinuedFractions/Translations.lean | theorem terminatedAt_iff_partDen_none : g.TerminatedAt n ↔ g.partDens.get? n = none | α : Type u_1
g : GenContFract α
n : ℕ
⊢ g.TerminatedAt n ↔ g.partDens.get? n = none | rw [terminatedAt_iff_s_none, partDen_none_iff_s_none] | no goals | 1381b32e71586503 |
CategoryTheory.Limits.pullbackAssoc_inv_fst_fst | Mathlib/CategoryTheory/Limits/Shapes/Pullback/Assoc.lean | theorem pullbackAssoc_inv_fst_fst [HasPullback ((pullback.snd _ _ : Z₁ ⟶ X₂) ≫ f₃) f₄]
[HasPullback f₁ ((pullback.fst _ _ : Z₂ ⟶ X₂) ≫ f₂)] :
(pullbackAssoc f₁ f₂ f₃ f₄).inv ≫ pullback.fst _ _ ≫ pullback.fst _ _ = pullback.fst _ _ | case e_a
C : Type u
inst✝⁴ : Category.{v, u} C
X₁ X₂ X₃ Y₁ Y₂ : C
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₁
f₃ : X₂ ⟶ Y₂
f₄ : X₃ ⟶ Y₂
inst✝³ : HasPullback f₁ f₂
inst✝² : HasPullback f₃ f₄
inst✝¹ : HasPullback (pullback.snd f₁ f₂ ≫ f₃) f₄
inst✝ : HasPullback f₁ (pullback.fst f₃ f₄ ≫ f₂)
⊢ (pullbackAssoc f₁ f₂ f₃ f₄).inv ≫ pullback.fst ... | exact IsLimit.conePointUniqueUpToIso_inv_comp _ _ WalkingCospan.left | no goals | 5f1de471186ad581 |
Squarefree.pow_dvd_of_squarefree_of_pow_succ_dvd_mul_right | Mathlib/Algebra/Squarefree/Basic.lean | theorem pow_dvd_of_squarefree_of_pow_succ_dvd_mul_right {k : ℕ}
(hx : Squarefree x) (hp : Prime p) (h : p ^ (k + 1) ∣ x * y) :
p ^ k ∣ y | case pos.intro
R : Type u_1
inst✝ : CancelCommMonoidWithZero R
y p : R
k : ℕ
hp : Prime p
x' : R
hx : Squarefree (p * x')
hx' : ¬p ∣ x'
h : p ^ k ∣ x' * y
⊢ p ^ k ∣ y | exact hp.pow_dvd_of_dvd_mul_left _ hx' h | no goals | aeac094447beb2ec |
Polynomial.coeff_hermite_succ_succ | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | theorem coeff_hermite_succ_succ (n k : ℕ) : coeff (hermite (n + 1)) (k + 1) =
coeff (hermite n) k - (k + 2) * coeff (hermite n) (k + 2) | n k : ℕ
⊢ (hermite (n + 1)).coeff (k + 1) = (hermite n).coeff k - (↑k + 2) * (hermite n).coeff (k + 2) | rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm] | n k : ℕ
⊢ (hermite n).coeff k - (↑(k + 1) + 1) * (hermite n).coeff (k + 1 + 1) =
(hermite n).coeff k - (↑k + 2) * (hermite n).coeff (k + 2) | c707a2fd9785fc9b |
Module.eq_of_isLocalized_span | Mathlib/RingTheory/LocalProperties/Submodule.lean | theorem Module.eq_of_isLocalized_span (x y : M) (h : ∀ r : s, f r x = f r y) : x = y | R : Type u_1
M : Type u_2
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
s : Set R
span_eq : span s = ⊤
Mₚ : ↑s → Type u_5
inst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)
inst✝¹ : (r : ↑s) → Module R (Mₚ r)
f : (r : ↑s) → M →ₗ[R] Mₚ r
inst✝ : ∀ (r : ↑s), IsLocalizedModule (Submonoid.powers ↑r) (f r)
x y ... | let r : s := ⟨r, hrs⟩ | R : Type u_1
M : Type u_2
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
s : Set R
span_eq : span s = ⊤
Mₚ : ↑s → Type u_5
inst✝² : (r : ↑s) → AddCommMonoid (Mₚ r)
inst✝¹ : (r : ↑s) → Module R (Mₚ r)
f : (r : ↑s) → M →ₗ[R] Mₚ r
inst✝ : ∀ (r : ↑s), IsLocalizedModule (Submonoid.powers ↑r) (f r)
x y ... | 302cee96503aaa37 |
Int.add_mul_ediv_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c = a / c + b :=
suffices ∀ {{a b c : Int}}, 0 < c → (a + b * c).ediv c = a.ediv c + b from
match Int.lt_trichotomy c 0 with
| Or.inl hlt => by
rw [← Int.neg_inj, ← Int.ediv_neg, Int.neg_add, ← Int.ediv_neg, ← Int.neg_mul_neg]
... | a b c : Int
H : c ≠ 0
this : ∀ ⦃a b c : Int⦄, 0 < c → (a + b * c).ediv c = a.ediv c + b
hlt : c < 0
⊢ (a + -b * -c) / -c = a / -c + -b | exact this (Int.neg_pos_of_neg hlt) | no goals | b62a45e15f46c8a0 |
hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt | Mathlib/Analysis/Calculus/MeanValue.lean | theorem hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt
(hder : ∀ᶠ y in 𝓝 x, HasFDerivAt f (f' y) y) (hcont : ContinuousAt f' x) :
HasStrictFDerivAt f (f' x) x | case intro.intro
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ... | refine ⟨ε, ε0, ?_⟩ | case intro.intro
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
H : Type u_5
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
f : G → H
f' : G → G →L[𝕜] H
x : G
hder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y
hcont : ContinuousAt f' x
c : ℝ
hc : 0 < c
ε : ... | 1d3d02e6864db095 |
List.reverse_inj | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem reverse_inj {xs ys : List α} : xs.reverse = ys.reverse ↔ xs = ys | α : Type u_1
xs ys : List α
⊢ xs.reverse = ys.reverse ↔ xs = ys | simp [reverse_eq_iff] | no goals | 987568ded5360d63 |
Int.dvd_bmod_sub_self | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem dvd_bmod_sub_self {x : Int} {m : Nat} : (m : Int) ∣ bmod x m - x | case isFalse
x : Int
m : Nat
h✝ : ¬x % ↑m < (↑m + 1) / 2
⊢ ↑m ∣ x % ↑m - x - ↑m | exact Int.dvd_sub dvd_emod_sub_self (Int.dvd_refl _) | no goals | 01ba4b1e096a860f |
Monoid.exists_orderOf_eq_exponent | Mathlib/GroupTheory/Exponent.lean | theorem exists_orderOf_eq_exponent (hG : ExponentExists G) : ∃ g : G, orderOf g = exponent G | case h.intro.intro.intro
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p
hk : ... | obtain ⟨a, ha⟩ := Nat.exists_eq_add_of_lt hpe | case h.intro.intro.intro.intro
G : Type u
inst✝ : CommMonoid G
hG : ExponentExists G
he : exponent G ≠ 0
hne : (Set.range orderOf).Nonempty
hfin : (Set.range orderOf).Finite
t : G
ht : orderOf t = sSup (Set.range orderOf)
p : ℕ
hp : Nat.Prime p
k : ℕ := (orderOf t).factorization p
hpe : k < (exponent G).factorization p... | c9d54a30c09c3ae9 |
Subgroup.exists_pow_mem_of_index_ne_zero | Mathlib/GroupTheory/Index.lean | @[to_additive]
lemma exists_pow_mem_of_index_ne_zero (h : H.index ≠ 0) (a : G) :
∃ n, 0 < n ∧ n ≤ H.index ∧ a ^ n ∈ H | case mk.intro.mk.intro
G : Type u_1
inst✝ : Group G
H : Subgroup G
h : H.index ≠ 0
a : G
hc : ∀ (x x_1 : ℕ), ¬(x ≠ x_1 ∧ x ≤ H.index ∧ x_1 ≤ H.index ∧ ↑(a ^ x_1) = ↑(a ^ x))
f : ↑(Set.Icc 0 H.index) → G ⧸ H := fun n => ↑(a ^ ↑n)
n₁ : ℕ
h₁ : 0 ≤ n₁
hle₁ : n₁ ≤ H.index
n₂ : ℕ
h₂ : 0 ≤ n₂
hle₂ : n₂ ≤ H.index
he : f ⟨n₁, ⋯... | dsimp only [f] at he | case mk.intro.mk.intro
G : Type u_1
inst✝ : Group G
H : Subgroup G
h : H.index ≠ 0
a : G
hc : ∀ (x x_1 : ℕ), ¬(x ≠ x_1 ∧ x ≤ H.index ∧ x_1 ≤ H.index ∧ ↑(a ^ x_1) = ↑(a ^ x))
f : ↑(Set.Icc 0 H.index) → G ⧸ H := fun n => ↑(a ^ ↑n)
n₁ : ℕ
h₁ : 0 ≤ n₁
hle₁ : n₁ ≤ H.index
n₂ : ℕ
h₂ : 0 ≤ n₂
hle₂ : n₂ ≤ H.index
he : ↑(a ^ n₁... | e678927c1f4a5913 |
Sum.lex_acc_inr | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Sum/Lemmas.lean | theorem lex_acc_inr (aca : ∀ a, Acc (Lex r s) (inl a)) {b} (acb : Acc s b) :
Acc (Lex r s) (inr b) | case intro.h.sep
α✝ : Type u_1
r : α✝ → α✝ → Prop
β✝ : Type u_2
s : β✝ → β✝ → Prop
aca : ∀ (a : α✝), Acc (Lex r s) (inl a)
b x✝ : β✝
h✝ : ∀ (y : β✝), s y x✝ → Acc s y
IH : ∀ (y : β✝), s y x✝ → Acc (Lex r s) (inr y)
a✝ : α✝
⊢ Acc (Lex r s) (inl a✝) | exact aca _ | no goals | c37e6030218e165e |
List.tails_eq_inits | Mathlib/Data/List/Infix.lean | theorem tails_eq_inits : ∀ l : List α, l.tails = (reverse <| map reverse <| inits <| reverse l)
| [] => by simp
| a :: l => by simp [tails_eq_inits l, append_left_inj]
| α : Type u_1
⊢ [].tails = (map reverse [].reverse.inits).reverse | simp | no goals | 899af2cf04c3ee69 |
krullTopology_t2 | Mathlib/FieldTheory/KrullTopology.lean | theorem krullTopology_t2 {K L : Type*} [Field K] [Field L] [Algebra K L]
[Algebra.IsIntegral K L] : T2Space (L ≃ₐ[K] L) :=
{ t2 := fun f g hfg => by
let φ := f⁻¹ * g
obtain ⟨x, hx⟩ := DFunLike.exists_ne hfg
have hφx : φ x ≠ x | case intro
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
f g : L ≃ₐ[K] L
hfg : f ≠ g
φ : L ≃ₐ[K] L := f⁻¹ * g
x : L
hx : f x ≠ g x
hφx : φ x ≠ x
⊢ (fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f g | let E : IntermediateField K L := IntermediateField.adjoin K {x} | case intro
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsIntegral K L
f g : L ≃ₐ[K] L
hfg : f ≠ g
φ : L ≃ₐ[K] L := f⁻¹ * g
x : L
hx : f x ≠ g x
hφx : φ x ≠ x
E : IntermediateField K L := K⟮x⟯
⊢ (fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) f ... | 49fcea39151d28eb |
UniformSpace.Completion.mem_uniformity_dist | Mathlib/Topology/MetricSpace/Completion.lean | theorem mem_uniformity_dist (s : Set (Completion α × Completion α)) :
s ∈ 𝓤 (Completion α) ↔ ∃ ε > 0, ∀ {a b}, dist a b < ε → (a, b) ∈ s | α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
ε : ℝ
εpos : ε > 0
hε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s
r : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}
this✝ : r ∈ 𝓤 ℝ
t1 : Set (Completion α × Completion α)
ht1 : t1 ∈ 𝓤 (Completion α)
t2 : Set (Completion α × Completion α)
ht2... | simp? [r, Completion.dist_self, Real.dist_eq, Completion.dist_comm] at I says
simp only [Real.dist_eq, mem_setOf_eq, preimage_setOf_eq, Completion.dist_self,
Completion.dist_comm, zero_sub, abs_neg, r] at I | α : Type u
inst✝ : PseudoMetricSpace α
s : Set (Completion α × Completion α)
ε : ℝ
εpos : ε > 0
hε : ∀ {a b : Completion α}, dist a b < ε → (a, b) ∈ s
r : Set (ℝ × ℝ) := {p | dist p.1 p.2 < ε}
this✝ : r ∈ 𝓤 ℝ
t1 : Set (Completion α × Completion α)
ht1 : t1 ∈ 𝓤 (Completion α)
t2 : Set (Completion α × Completion α)
ht2... | c4dfc92c04080ea5 |
Monoid.exponent_pi | Mathlib/GroupTheory/Exponent.lean | theorem Monoid.exponent_pi {ι : Type*} [Fintype ι] {M : ι → Type*} [∀ i, Monoid (M i)] :
exponent ((i : ι) → M i) = lcm univ (exponent <| M ·) | case refine_1.h
ι : Type u_1
inst✝¹ : Fintype ι
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
m : (i : ι) → M i
i : ι
⊢ (m ^ univ.lcm fun x => exponent (M x)) i = 1 i | rw [Pi.pow_apply, Pi.one_apply, ← orderOf_dvd_iff_pow_eq_one] | case refine_1.h
ι : Type u_1
inst✝¹ : Fintype ι
M : ι → Type u_2
inst✝ : (i : ι) → Monoid (M i)
m : (i : ι) → M i
i : ι
⊢ orderOf (m i) ∣ univ.lcm fun x => exponent (M x) | 8dfb64e9b5092485 |
ProbabilityTheory.setIntegral_stieltjesOfMeasurableRat_rat | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lemma setIntegral_stieltjesOfMeasurableRat_rat (hf : IsRatCondKernelCDF f κ ν) (a : α) (q : ℚ)
{s : Set β} (hs : MeasurableSet s) :
∫ b in s, stieltjesOfMeasurableRat f hf.measurable (a, b) q ∂(ν a)
= (κ a (s ×ˢ Iic (q : ℝ))).toReal | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ∀ᵐ (x : β) ∂ν a, x ∈ s → ↑(stieltjesOfMeasurableRat f ⋯ (a, x)) ↑q = (fun b => f (a, b) q) x | filter_upwards [stieltjesOfMeasurableRat_ae_eq hf a q] with b hb using fun _ ↦ hb | no goals | e322025a8c4331b9 |
LieAlgebra.hasTrivialRadical_and_of_isIrreducible_of_isFaithful | Mathlib/Algebra/Lie/Semisimple/Lemmas.lean | theorem hasTrivialRadical_and_of_isIrreducible_of_isFaithful
(h : ∀ x, LinearMap.trace k _ (toEnd k L M x) = 0) : HasTrivialRadical k L | case intro
k : Type u_1
L : Type u_2
M : Type u_3
inst✝¹² : Field k
inst✝¹¹ : CharZero k
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra k L
inst✝⁸ : Module.Finite k L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module k M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule k L M
inst✝³ : Module.Finite k M
inst✝² : LieModule.IsIrreducible k L ... | rw [hasTrivialRadical_iff, (hasCentralRadical_iff k L).mp inferInstance, LieSubmodule.eq_bot_iff] | case intro
k : Type u_1
L : Type u_2
M : Type u_3
inst✝¹² : Field k
inst✝¹¹ : CharZero k
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra k L
inst✝⁸ : Module.Finite k L
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module k M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule k L M
inst✝³ : Module.Finite k M
inst✝² : LieModule.IsIrreducible k L ... | a1ed9c83290ed046 |
Cycle.nontrivial_coe_nodup_iff | Mathlib/Data/List/Cycle.lean | theorem nontrivial_coe_nodup_iff {l : List α} (hl : l.Nodup) :
Nontrivial (l : Cycle α) ↔ 2 ≤ l.length | case cons.nil
α : Type u_1
hd : α
hl : [hd].Nodup
⊢ (∃ x y, x ≠ y ∧ x ∈ ↑[hd] ∧ y ∈ ↑[hd]) ↔ 2 ≤ [hd].length | simp | no goals | 2676bd88c89661d9 |
pow_right_strictMono' | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | theorem pow_right_strictMono' (ha : 1 < a) : StrictMono ((a ^ ·) : ℕ → M) :=
strictMono_nat_of_lt_succ fun n ↦ by rw [pow_succ]; exact lt_mul_of_one_lt_right' (a ^ n) ha
| M : Type u_3
inst✝² : Monoid M
inst✝¹ : Preorder M
inst✝ : MulLeftStrictMono M
a : M
ha : 1 < a
n : ℕ
⊢ a ^ n < a ^ n * a | exact lt_mul_of_one_lt_right' (a ^ n) ha | no goals | 9fdc6933be0ac873 |
AlgebraicGeometry.HasRingHomProperty.of_source_openCover | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | theorem of_source_openCover [IsAffine Y]
(𝒰 : X.OpenCover) [∀ i, IsAffine (𝒰.obj i)] (H : ∀ i, Q ((𝒰.map i ≫ f).appTop.hom)) :
P f | P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝² : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsAffine Y
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
H : ∀ (i : 𝒰.J), Q (CommRingCat.Hom.hom (Scheme.Hom.appTop (𝒰.map i ≫ f)... | let S i : X.affineOpens := ⟨_, isAffineOpen_opensRange (𝒰.map i)⟩ | P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝² : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsAffine Y
𝒰 : X.OpenCover
inst✝ : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
H : ∀ (i : 𝒰.J), Q (CommRingCat.Hom.hom (Scheme.Hom.appTop (𝒰.map i ≫ f)... | 14bf5a1ed5917276 |
Ideal.finrank_prime_pow_ramificationIdx | Mathlib/NumberTheory/RamificationInertia/Basic.lean | theorem finrank_prime_pow_ramificationIdx [IsDedekindDomain S] (hP0 : P ≠ ⊥)
[p.IsMaximal] [P.IsPrime] (he : e ≠ 0) :
finrank (R ⧸ p) (S ⧸ P ^ e) =
e *
@finrank (R ⧸ p) (S ⧸ P) _ _
(@Algebra.toModule _ _ _ _ <|
@Quotient.algebraQuotientOfRamificationIdxNeZero _ _ _ _ _ _ _ ⟨h... | R : Type u
inst✝⁵ : CommRing R
S : Type v
inst✝⁴ : CommRing S
p : Ideal R
P : Ideal S
inst✝³ : Algebra R S
inst✝² : IsDedekindDomain S
hP0 : P ≠ ⊥
inst✝¹ : p.IsMaximal
inst✝ : P.IsPrime
he : e ≠ 0
this : NeZero e := { out := he }
⊢ finrank (R ⧸ p) (S ⧸ P ^ e) = e * finrank (R ⧸ p) (S ⧸ P) | letI : Algebra (R ⧸ p) (S ⧸ P) := Quotient.algebraQuotientOfRamificationIdxNeZero p P | R : Type u
inst✝⁵ : CommRing R
S : Type v
inst✝⁴ : CommRing S
p : Ideal R
P : Ideal S
inst✝³ : Algebra R S
inst✝² : IsDedekindDomain S
hP0 : P ≠ ⊥
inst✝¹ : p.IsMaximal
inst✝ : P.IsPrime
he : e ≠ 0
this✝ : NeZero e := { out := he }
this : Algebra (R ⧸ p) (S ⧸ P) := Quotient.algebraQuotientOfRamificationIdxNeZero p P
⊢ f... | 19c75aae780d6275 |
integral_cos_pow_aux | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_cos_pow_aux :
(∫ x in a..b, cos x ^ (n + 2)) =
(cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (n + 1) * ∫ x in a..b, cos x ^ n) -
(n + 1) * ∫ x in a..b, cos x ^ (n + 2) | a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =
(cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1... | have hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x := fun x _ => hasDerivAt_sin x | a b : ℝ
n : ℕ
C : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt sin (cos x) x
⊢ ∫ (x : ℝ) in a..b, cos x ^ (n + 2) =
(cos b ^ (n + ... | 7aaa37e1ea8fae95 |
MeasureTheory.Measure.mutuallySingular_of_mutuallySingular_compProd | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | lemma mutuallySingular_of_mutuallySingular_compProd {ξ : Measure α}
[SFinite μ] [SFinite ν] [IsSFiniteKernel κ] [IsSFiniteKernel η]
(h : μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η) (hμ : ξ ≪ μ) (hν : ξ ≪ ν) :
∀ᵐ x ∂ξ, κ x ⟂ₘ η x | α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
ξ : Measure α
inst✝³ : SFinite μ
inst✝² : SFinite ν
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
h : μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η
hμ : ξ ≪ μ
hν : ξ ≪ ν
hs : MeasurableSet h.nullSet
hμ_zero : ∫⁻ (a : α), (κ a) (Prod.mk a... | exact (Kernel.measurable_kernel_prod_mk_left hs).aemeasurable | no goals | c600e4602686f158 |
List.sublist_orderedInsert | Mathlib/Data/List/Sort.lean | theorem sublist_orderedInsert (x : α) (xs : List α) : xs <+ xs.orderedInsert r x | α : Type u
r : α → α → Prop
inst✝ : DecidableRel r
x : α
xs : List α
⊢ xs <+ takeWhile (fun b => decide ¬r x b) xs ++ x :: dropWhile (fun b => decide ¬r x b) xs | refine Sublist.trans ?_ (.append_left (.cons _ (.refl _)) _) | α : Type u
r : α → α → Prop
inst✝ : DecidableRel r
x : α
xs : List α
⊢ xs <+ takeWhile (fun b => decide ¬r x b) xs ++ dropWhile (fun b => decide ¬r x b) xs | 356c8da478e9ed6e |
CategoryTheory.Functor.shiftIso_add_inv_app | Mathlib/CategoryTheory/Shift/ShiftSequence.lean | lemma shiftIso_add_inv_app (n m a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
(F.shiftIso (m + n) a a'' (by rw [add_assoc, ha', ha''])).inv.app X =
(shiftIso F m a' a'' ha'').inv.app X ≫
(shiftIso F n a a' ha').inv.app ((shiftFunctor C m).obj X) ≫
(shift F a).map ((shiftFu... | C : Type u_1
A : Type u_2
inst✝⁴ : Category.{u_6, u_1} C
inst✝³ : Category.{u_5, u_2} A
F : C ⥤ A
M : Type u_3
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso (m + n) a a'' ⋯).inv.app X =
(F.shiftIso m a' a'' ha'').inv.ap... | simp [F.shiftIso_add n m a a' a'' ha' ha''] | no goals | fbe800b23257b4b9 |
ContinuousLinearEquiv.subsingleton_or_norm_symm_pos | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | theorem subsingleton_or_norm_symm_pos [RingHomIsometric σ₁₂] (e : E ≃SL[σ₁₂] F) :
Subsingleton E ∨ 0 < ‖(e.symm : F →SL[σ₂₁] E)‖ | case inl
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝³ : RingHomInvPair σ₁₂ σ₂₁
inst... | left | case inl.h
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NontriviallyNormedField 𝕜₂
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝³ : RingHomInvPair σ₁₂ σ₂₁
in... | 46aefde766ab7d52 |
MeasureTheory.locallyIntegrable_map_homeomorph | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X ≃ₜ Y) {f : Y → E}
{μ : Measure X} : LocallyIntegrable f (Measure.map e μ) ↔ LocallyIntegrable (f ∘ e) μ | case refine_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ⇑e) μ
x : ... | simp only [Homeomorph.toMeasurableEquiv_coe] | case refine_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ⇑e) μ
x : ... | cbcd4eb2baca93b2 |
Algebra.FormallySmooth.exists_lift | Mathlib/RingTheory/Smooth/Basic.lean | theorem exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g | case h₂.intro.intro
R : Type u
inst✝⁵ : CommSemiring R
A : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B✝ : Type u
inst✝² : CommRing B✝
inst✝¹ : FormallySmooth R A
I✝ : Ideal B✝
hI : IsNilpotent I✝
B : Type u
inst✝ : CommRing B
I J : Ideal B
hIJ : I ≤ J
h₁ : ∀ [_RB : Algebra R B], Function.Surjective (Ideal.Quotien... | replace e := congr_arg this.toAlgHom.comp e | case h₂.intro.intro
R : Type u
inst✝⁵ : CommSemiring R
A : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B✝ : Type u
inst✝² : CommRing B✝
inst✝¹ : FormallySmooth R A
I✝ : Ideal B✝
hI : IsNilpotent I✝
B : Type u
inst✝ : CommRing B
I J : Ideal B
hIJ : I ≤ J
h₁ : ∀ [_RB : Algebra R B], Function.Surjective (Ideal.Quotien... | 76eb1b2fc562dd62 |
Module.lt_rank_of_lt_finrank | Mathlib/LinearAlgebra/Dimension/Finrank.lean | theorem lt_rank_of_lt_finrank {n : ℕ} (h : n < finrank R M) : ↑n < Module.rank R M | case hd
R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
h : ℵ₀ ≤ Module.rank R M
⊢ finrank R M ≤ n | rw [finrank, Cardinal.toNat_apply_of_aleph0_le h] | case hd
R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
h : ℵ₀ ≤ Module.rank R M
⊢ 0 ≤ n | c09c30d878869ec1 |
IsLocalization.surj'' | Mathlib/RingTheory/Localization/InvSubmonoid.lean | theorem surj'' (z : S) : ∃ (r : R) (m : M), z = r • (toInvSubmonoid M S m : S) | R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
z : S
⊢ ∃ r m, z = r • ↑((toInvSubmonoid M S) m) | rcases IsLocalization.surj M z with ⟨⟨r, m⟩, e : z * _ = algebraMap R S r⟩ | case intro.mk
R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S : Type u_2
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
z : S
r : R
m : ↥M
e : z * (algebraMap R S) ↑(r, m).2 = (algebraMap R S) r
⊢ ∃ r m, z = r • ↑((toInvSubmonoid M S) m) | de1a619505f23744 |
Bimod.whisker_assoc_bimod | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem whisker_assoc_bimod {W X Y Z : Mon_ C} (M : Bimod W X) {N N' : Bimod X Y} (f : N ⟶ N')
(P : Bimod Y Z) :
whiskerRight (whiskerLeft M f) P =
(associatorBimod M N P).hom ≫
whiskerLeft M (whiskerRight f P) ≫ (associatorBimod M N' P).inv | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M : Bimod W X
N N' : Bimod X Y
f :... | dsimp [AssociatorBimod.homAux] | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M : Bimod W X
N N' : Bimod X Y
f :... | 64d3d49872bb8a8d |
MeasureTheory.AEEqFun.compMeasurable_toGerm | Mathlib/MeasureTheory/Function/AEEqFun.lean | theorem compMeasurable_toGerm [MeasurableSpace β] [BorelSpace β] [PseudoMetrizableSpace β]
[PseudoMetrizableSpace γ] [SecondCountableTopology γ] [MeasurableSpace γ]
[OpensMeasurableSpace γ] (g : β → γ) (hg : Measurable g) (f : α →ₘ[μ] β) :
(compMeasurable g hg f).toGerm = f.toGerm.map g :=
induction_on f ... | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁹ : MeasurableSpace α
μ : Measure α
inst✝⁸ : TopologicalSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : PseudoMetrizableSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : MeasurableSpace γ
inst✝ :... | simp | no goals | 2058dfe15b8675aa |
Order.not_isPredLimit_iff | Mathlib/Order/SuccPred/Limit.lean | theorem not_isPredLimit_iff : ¬ IsPredLimit a ↔ IsMax a ∨ ¬ IsPredPrelimit a | α : Type u_1
a : α
inst✝ : Preorder α
⊢ ¬IsPredLimit a ↔ IsMax a ∨ ¬IsPredPrelimit a | rw [IsPredLimit, not_and_or, not_not] | no goals | 6e31cb92e0669b08 |
Nat.infinite_even_deficient | Mathlib/NumberTheory/FactorisationProperties.lean | theorem infinite_even_deficient : {n : ℕ | Even n ∧ n.Deficient}.Infinite | case h.right
n : ℕ
⊢ n < 2 ^ (n + 1) | calc
n ≤ 2 ^ n := Nat.le_of_lt n.lt_two_pow_self
_ < 2 ^ (n + 1) := (Nat.pow_lt_pow_iff_right (Nat.one_lt_two)).mpr (lt_add_one n) | no goals | 496dfe481af94a14 |
dite_comp_equiv_update | Mathlib/Logic/Equiv/Set.lean | theorem dite_comp_equiv_update {α : Type*} {β : Sort*} {γ : Sort*} {p : α → Prop}
(e : β ≃ Subtype p)
(v : β → γ) (w : α → γ) (j : β) (x : γ) [DecidableEq β] [DecidableEq α]
[∀ j, Decidable (p j)] :
(fun i : α => if h : p i then (Function.update v j x) (e.symm ⟨i, h⟩) else w i) =
Function.update (... | case neg
α : Type u_1
β : Sort u_2
γ : Sort u_3
p : α → Prop
e : β ≃ Subtype p
v : β → γ
w : α → γ
j : β
x : γ
inst✝² : DecidableEq β
inst✝¹ : DecidableEq α
inst✝ : (j : α) → Decidable (p j)
i : α
h : ¬p i
⊢ (if h : p i then update v j x (e.symm ⟨i, h⟩) else w i) =
update (fun i => if h : p i then v (e.symm ⟨i, h⟩)... | have : i ≠ e j := by
contrapose! h
have : p (e j : α) := (e j).2
rwa [← h] at this | case neg
α : Type u_1
β : Sort u_2
γ : Sort u_3
p : α → Prop
e : β ≃ Subtype p
v : β → γ
w : α → γ
j : β
x : γ
inst✝² : DecidableEq β
inst✝¹ : DecidableEq α
inst✝ : (j : α) → Decidable (p j)
i : α
h : ¬p i
this : i ≠ ↑(e j)
⊢ (if h : p i then update v j x (e.symm ⟨i, h⟩) else w i) =
update (fun i => if h : p i then... | 41ed01ad03da63ad |
Polynomial.separable_map | Mathlib/FieldTheory/Separable.lean | theorem separable_map {S} [CommRing S] [Nontrivial S] (f : F →+* S) {p : F[X]} :
(p.map f).Separable ↔ p.Separable | case intro
F : Type u
inst✝² : Field F
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Nontrivial S
f : F →+* S
p : F[X]
H : (map f p).Separable
m : Ideal S
hm : m.IsMaximal
⊢ p.Separable | have := Separable.map H (f := Ideal.Quotient.mk m) | case intro
F : Type u
inst✝² : Field F
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Nontrivial S
f : F →+* S
p : F[X]
H : (map f p).Separable
m : Ideal S
hm : m.IsMaximal
this : (map (Ideal.Quotient.mk m) (map f p)).Separable
⊢ p.Separable | afa3beb20fd74982 |
AlternatingGroup.card_of_cycleType_mul_eq | Mathlib/GroupTheory/SpecificGroups/Alternating/Centralizer.lean | theorem card_of_cycleType_mul_eq (m : Multiset ℕ) :
#({ g | g.val.cycleType = m} : Finset (alternatingGroup α)) *
((Fintype.card α - m.sum)! * m.prod * (∏ n ∈ m.toFinset, (m.count n)!)) =
if ((m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) ∧ Even (m.sum + Multiset.card m))
then (Fintype.card ... | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
m : Multiset ℕ
⊢ (if Even (m.sum + m.card) then
#(filter (fun g => g.cycleType = m) univ) *
((Fintype.card α - m.sum)! * m.prod * ∏ n ∈ m.toFinset, (Multiset.count n m)!)
else 0) =
if Even (m.sum + m.card) ∧ m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ... | rw [ite_and, Equiv.Perm.card_of_cycleType_mul_eq] | no goals | 0627f951ef54a608 |
List.exists_pw_disjoint_with_card | Mathlib/GroupTheory/Perm/Cycle/PossibleTypes.lean | theorem List.exists_pw_disjoint_with_card {α : Type*} [Fintype α]
{c : List ℕ} (hc : c.sum ≤ Fintype.card α) :
∃ o : List (List α),
o.map length = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise List.Disjoint o | case h.right.left.intro.intro.intro.intro.a
α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
hc'_lt :... | exact ranges_nodup hu | no goals | 463233383e1ded3a |
SetTheory.Game.small_setOf_birthday_lt | Mathlib/SetTheory/Game/Birthday.lean | theorem small_setOf_birthday_lt (o : Ordinal) : Small.{u} {x : Game.{u} // birthday x < o} | case h.inr.inr
o : Ordinal.{u}
IH : ∀ k < o, Small.{u, u + 1} { x // x.birthday < k }
S : Set Game := ⋃ a ∈ Set.Iio o, {x | x.birthday < a}
H : Small.{u, u + 1} ↑S := small_biUnion (Set.Iio o) fun a h => {x | x.birthday < a}
ho : o.IsLimit
⊢ Small.{u, u + 1} { x // x.birthday < o } | convert H | case h.e'_1.h.e'_2.h.a
o : Ordinal.{u}
IH : ∀ k < o, Small.{u, u + 1} { x // x.birthday < k }
S : Set Game := ⋃ a ∈ Set.Iio o, {x | x.birthday < a}
H : Small.{u, u + 1} ↑S := small_biUnion (Set.Iio o) fun a h => {x | x.birthday < a}
ho : o.IsLimit
x✝ : Game
⊢ x✝.birthday < o ↔ S x✝ | 942b1277300d8c20 |
AlgebraicGeometry.universally_isLocalAtTarget | Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean | theorem universally_isLocalAtTarget (P : MorphismProperty Scheme)
(hP₂ : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens)
(_ : IsOpenCover U), (∀ i, P (f ∣_ U i)) → P f) : IsLocalAtTarget P.universally | case of_sSup_eq_top.x
P : MorphismProperty Scheme
hP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
ι : Type u
U : ι → Y.Opens
hU : iSup U = ⊤
H : ∀ (i : ι), P.universally (f ∣_ U i)
X' Y' : Scheme
i₁ : X' ⟶ X
i₂ : Y' ⟶ Y
f' : X' ⟶ ... | simp only [IsOpenCover, ← top_le_iff] at hU ⊢ | case of_sSup_eq_top.x
P : MorphismProperty Scheme
hP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f
X Y : Scheme
f : X ⟶ Y
ι : Type u
U : ι → Y.Opens
H : ∀ (i : ι), P.universally (f ∣_ U i)
X' Y' : Scheme
i₁ : X' ⟶ X
i₂ : Y' ⟶ Y
f' : X' ⟶ Y'
h : IsPullbac... | ed7c38431ebe2f18 |
CategoryTheory.Limits.biprod_isZero_iff | Mathlib/CategoryTheory/Limits/Shapes/BinaryBiproducts.lean | @[simp]
lemma biprod_isZero_iff (A B : C) [HasBinaryBiproduct A B] :
IsZero (biprod A B) ↔ IsZero A ∧ IsZero B | case mpr.intro
C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
A B : C
inst✝ : HasBinaryBiproduct A B
hA : IsZero A
hB : IsZero B
⊢ 𝟙 (A ⊞ B) = 0 | apply biprod.hom_ext | case mpr.intro.h₀
C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
A B : C
inst✝ : HasBinaryBiproduct A B
hA : IsZero A
hB : IsZero B
⊢ 𝟙 (A ⊞ B) ≫ biprod.fst = 0 ≫ biprod.fst
case mpr.intro.h₁
C : Type uC
inst✝² : Category.{uC', uC} C
inst✝¹ : HasZeroMorphisms C
A B : C
inst✝ : HasBinaryBiproduc... | 5c852a34fafb21a3 |
ordinaryHypergeometricSeries_eq_zero_iff | Mathlib/Analysis/SpecialFunctions/OrdinaryHypergeometric.lean | /-- An iff variation on `ordinaryHypergeometricSeries_eq_zero_of_nonpos_int` for `[RCLike 𝕂]`. -/
lemma ordinaryHypergeometricSeries_eq_zero_iff (n : ℕ) :
ordinaryHypergeometricSeries 𝔸 a b c n = 0 ↔ ∃ k < n, k = -a ∨ k = -b ∨ k = -c | case refine_2
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
n : ℕ
zero : ∃ k < n, ↑k = -a ∨ ↑k = -b ∨ ↑k = -c
⊢ ordinaryHypergeometricSeries 𝔸 a b c n = 0 | obtain ⟨_, h, hn⟩ := zero | case refine_2.intro.intro
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
n w✝ : ℕ
h : w✝ < n
hn : ↑w✝ = -a ∨ ↑w✝ = -b ∨ ↑w✝ = -c
⊢ ordinaryHypergeometricSeries 𝔸 a b c n = 0 | a493be1d40c674b2 |
MeasureTheory.FiniteMeasure.testAgainstNN_lipschitz | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | theorem testAgainstNN_lipschitz (μ : FiniteMeasure Ω) :
LipschitzWith μ.mass fun f : Ω →ᵇ ℝ≥0 ↦ μ.testAgainstNN f | case left
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f₁ f₂ : Ω →ᵇ ℝ≥0
key : μ.testAgainstNN f₂ ≤ μ.testAgainstNN f₁ + nndist f₂ f₁ * μ.mass
⊢ -(↑μ.mass * dist f₁ f₂) ≤ ↑(μ.testAgainstNN f₁) - ↑(μ.testAgainstNN f₂) | rw [mul_comm] at key | case left
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
μ : FiniteMeasure Ω
f₁ f₂ : Ω →ᵇ ℝ≥0
key : μ.testAgainstNN f₂ ≤ μ.testAgainstNN f₁ + μ.mass * nndist f₂ f₁
⊢ -(↑μ.mass * dist f₁ f₂) ≤ ↑(μ.testAgainstNN f₁) - ↑(μ.testAgainstNN f₂) | 10c10f75c8222436 |
FirstOrder.Language.Substructure.FG.of_map_embedding | Mathlib/ModelTheory/FinitelyGenerated.lean | theorem FG.of_map_embedding {N : Type*} [L.Structure N] (f : M ↪[L] N) {s : L.Substructure M}
(hs : (s.map f.toHom).FG) : s.FG | case intro
L : Language
M : Type u_1
inst✝¹ : L.Structure M
N : Type u_2
inst✝ : L.Structure N
f : M ↪[L] N
s : L.Substructure M
t : Finset N
h : (closure L).toFun ↑t = Substructure.map f.toHom s
⊢ ∃ S, S.Finite ∧ (closure L).toFun S = s | refine ⟨f ⁻¹' t, t.finite_toSet.preimage f.injective.injOn, ?_⟩ | case intro
L : Language
M : Type u_1
inst✝¹ : L.Structure M
N : Type u_2
inst✝ : L.Structure N
f : M ↪[L] N
s : L.Substructure M
t : Finset N
h : (closure L).toFun ↑t = Substructure.map f.toHom s
⊢ (closure L).toFun (⇑f ⁻¹' ↑t) = s | e12b3164a9a419fb |
NumberField.mixedEmbedding.nnnorm_eq_sup_normAtPlace | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | theorem nnnorm_eq_sup_normAtPlace (x : mixedSpace K) :
‖x‖₊ = univ.sup fun w ↦ ⟨normAtPlace w x, normAtPlace_nonneg w x⟩ | case e_a.e_f
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
this : univ = image (fun w => ↑w) univ ∪ image (fun w => ↑w) univ
⊢ (fun b => ‖x.2 b‖₊) = (fun w => ⟨(normAtPlace w) x, ⋯⟩) ∘ fun w => ↑w | ext w | case e_a.e_f.h.a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
this : univ = image (fun w => ↑w) univ ∪ image (fun w => ↑w) univ
w : { w // w.IsComplex }
⊢ ↑‖x.2 w‖₊ = ↑(((fun w => ⟨(normAtPlace w) x, ⋯⟩) ∘ fun w => ↑w) w) | f04a0fc900651135 |
insert_mem_nhdsWithin_of_subset_insert | Mathlib/Topology/Separation/Basic.lean | theorem insert_mem_nhdsWithin_of_subset_insert [T1Space X] {x y : X} {s t : Set X}
(hu : t ⊆ insert y s) : insert x s ∈ 𝓝[t] x | case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
s t : Set X
hu : t ⊆ insert y s
h : x ≠ y
⊢ insert x s ∈ 𝓝[t] x | refine nhdsWithin_mono x hu ?_ | case inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
x y : X
s t : Set X
hu : t ⊆ insert y s
h : x ≠ y
⊢ insert x s ∈ 𝓝[insert y s] x | 0b5f6a1a0df0157d |
SimplexCategory.δ_comp_σ_self | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | theorem δ_comp_σ_self {n} {i : Fin (n + 1)} :
δ (Fin.castSucc i) ≫ σ i = 𝟙 ⦋n⦌ | case mk.a.h.h.mk.h
n i : ℕ
hi : i < n + 1
j : ℕ
hj : j < ⦋n⦌.len + 1
⊢ ↑((Hom.toOrderHom (δ ⟨i, hi⟩.castSucc ≫ σ ⟨i, hi⟩)) ⟨j, hj⟩) = ↑((Hom.toOrderHom (𝟙 ⦋n⦌)) ⟨j, hj⟩) | simp? at hj says simp only [len_mk] at hj | case mk.a.h.h.mk.h
n i : ℕ
hi : i < n + 1
j : ℕ
hj : j < n + 1
⊢ ↑((Hom.toOrderHom (δ ⟨i, hi⟩.castSucc ≫ σ ⟨i, hi⟩)) ⟨j, hj⟩) = ↑((Hom.toOrderHom (𝟙 ⦋n⦌)) ⟨j, hj⟩) | ce0410fc1dfae94d |
LinearOrder.bot_topologicalSpace_eq_generateFrom | Mathlib/Topology/Instances/Discrete.lean | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom {α} [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } | α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
⊢ ⊥ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} | let _ := Preorder.topology α | α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : PredOrder α
inst✝ : SuccOrder α
x✝ : TopologicalSpace α := Preorder.topology α
⊢ ⊥ = generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} | 90be8eb57698ca81 |
NumberField.Units.dirichletUnitTheorem.seq_next | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | theorem seq_next {x : 𝓞 K} (hx : x ≠ 0) :
∃ y : 𝓞 K, y ≠ 0 ∧
(∀ w, w ≠ w₁ → w y < w x) ∧
|Algebra.norm ℚ (y : K)| ≤ B | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
x : 𝓞 K
hx : x ≠ 0
hx' : (algebraMap (𝓞 K) K) x ≠ 0
f : InfinitePlace K → ℝ≥0 := fun w => ⟨w ↑x / 2, ⋯⟩
this : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
g : InfinitePlace K → ℝ≥0
h_geq... | rw [convexBodyLT_volume] | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
B : ℕ
hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B
x : 𝓞 K
hx : x ≠ 0
hx' : (algebraMap (𝓞 K) K) x ≠ 0
f : InfinitePlace K → ℝ≥0 := fun w => ⟨w ↑x / 2, ⋯⟩
this : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0
g : InfinitePlace K → ℝ≥0
h_geq... | 24174be95e6426c6 |
VectorField.eventuallyEq_mpullback_mpullbackWithin_extChartAt | Mathlib/Geometry/Manifold/VectorField.lean | lemma eventuallyEq_mpullback_mpullbackWithin_extChartAt (V : Π (x : M), TangentSpace I x) :
V =ᶠ[𝓝[s] x] mpullback I 𝓘(𝕜, E) (extChartAt I x)
(mpullbackWithin 𝓘(𝕜, E) I (extChartAt I x).symm V (range I)) | case h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
s : Set M
x : M
inst✝ : IsManifold I 1 M
V : (x : M) → Tan... | have A : (extChartAt I x).symm (extChartAt I x y) = y := (extChartAt I x).left_inv hy | case h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝⁵ : TopologicalSpace H
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
s : Set M
x : M
inst✝ : IsManifold I 1 M
V : (x : M) → Tan... | 9a25dd7cbbf98dfe |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.