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 |
|---|---|---|---|---|---|---|
SimpleGraph.cliqueSet_map_of_equiv | Mathlib/Combinatorics/SimpleGraph/Clique.lean | theorem cliqueSet_map_of_equiv (G : SimpleGraph α) (e : α ≃ β) (n : ℕ) :
(G.map e.toEmbedding).cliqueSet n = map e.toEmbedding '' G.cliqueSet n | case inl
α : Type u_1
β : Type u_2
G : SimpleGraph α
e : α ≃ β
⊢ (SimpleGraph.map e.toEmbedding G).cliqueSet 1 = map e.toEmbedding '' G.cliqueSet 1 | ext | case inl.h
α : Type u_1
β : Type u_2
G : SimpleGraph α
e : α ≃ β
x✝ : Finset β
⊢ x✝ ∈ (SimpleGraph.map e.toEmbedding G).cliqueSet 1 ↔ x✝ ∈ map e.toEmbedding '' G.cliqueSet 1 | bdf05c180515f25c |
zpow_eq_zpow_emod | Mathlib/Algebra/Group/Basic.lean | theorem zpow_eq_zpow_emod {x : G} (m : ℤ) {n : ℤ} (h : x ^ n = 1) :
x ^ m = x ^ (m % n) :=
calc
x ^ m = x ^ (m % n + n * (m / n)) | G : Type u_3
inst✝ : Group G
x : G
m n : ℤ
h : x ^ n = 1
⊢ x ^ m = x ^ (m % n + n * (m / n)) | rw [Int.emod_add_ediv] | no goals | 6e30c6d882c544d5 |
ConcaveOn.exists_lipschitzOnWith_of_isBounded | Mathlib/Analysis/Convex/Continuous.lean | lemma ConcaveOn.exists_lipschitzOnWith_of_isBounded (hf : ConcaveOn ℝ (ball x₀ r) f) (hr : r' < r)
(hf' : IsBounded (f '' ball x₀ r)) : ∃ K, LipschitzOnWith K f (ball x₀ r') | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
r r' : ℝ
hf : ConcaveOn ℝ (ball x₀ r) f
hr : r' < r
hf' : IsBounded (f '' ball x₀ r)
⊢ ∃ K, LipschitzOnWith K f (ball x₀ r') | replace hf' : IsBounded ((-f) '' ball x₀ r) := by convert hf'.neg; ext; simp [neg_eq_iff_eq_neg] | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
x₀ : E
r r' : ℝ
hf : ConcaveOn ℝ (ball x₀ r) f
hr : r' < r
hf' : IsBounded ((-f) '' ball x₀ r)
⊢ ∃ K, LipschitzOnWith K f (ball x₀ r') | c98c3139af73c787 |
tendsto_const_mul_zpow_atTop_nhds_iff | Mathlib/Topology/Algebra/Order/Field.lean | theorem tendsto_const_mul_zpow_atTop_nhds_iff {n : ℤ} {c d : 𝕜} (hc : c ≠ 0) :
Tendsto (fun x : 𝕜 => c * x ^ n) atTop (𝓝 d) ↔ n = 0 ∧ c = d ∨ n < 0 ∧ d = 0 | case refine_2
𝕜 : Type u_1
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d ∨ n < 0 ∧ d = 0
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d) | rcases h with h | h | case refine_2.inl
𝕜 : Type u_1
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
n : ℤ
c d : 𝕜
hc : c ≠ 0
h : n = 0 ∧ c = d
⊢ Tendsto (fun x => c * x ^ n) atTop (𝓝 d)
case refine_2.inr
𝕜 : Type u_1
inst✝² : LinearOrderedField 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology ... | 02629f67128f734b |
TypeVec.eq_nilFun | Mathlib/Data/TypeVec.lean | theorem eq_nilFun {α β : TypeVec 0} (f : α ⟹ β) : f = nilFun | α : TypeVec.{u_1} 0
β : TypeVec.{u_2} 0
f : α ⟹ β
⊢ f = nilFun | ext x | case a.h
α : TypeVec.{u_1} 0
β : TypeVec.{u_2} 0
f : α ⟹ β
x : Fin2 0
x✝ : α x
⊢ f x x✝ = nilFun x x✝ | c8a366cb28906070 |
GroupExtension.Section.exists_eq_mul_inl | Mathlib/GroupTheory/GroupExtension/Basic.lean | theorem exists_eq_mul_inl : ∃ n : N, σ g = σ' g * S.inl n | case intro
N : Type u_1
G : Type u_2
inst✝² : Group N
inst✝¹ : Group G
E : Type u_3
inst✝ : Group E
S : GroupExtension N E G
σ σ' : S.Section
g : G
n : N
hn : S.inl n = (σ' g)⁻¹ * σ g
⊢ ∃ n, σ g = σ' g * S.inl n | exact ⟨n, by rw [hn, mul_inv_cancel_left]⟩ | no goals | ad49194ca69659ad |
UpperHalfPlane.cosh_half_dist | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | theorem cosh_half_dist (z w : ℍ) :
cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * √(z.im * w.im)) | z w : ℍ
⊢ cosh (dist z w / 2) = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im)) | rw [← sq_eq_sq₀, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt] | z w : ℍ
⊢ (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =
dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))
z w : ℍ
⊢ 0 ≤ z.im * w.im
z w : ℍ
⊢ (2 * √(z.im * w.im)) ^ 2 ≠ 0
case ha
z w : ℍ
⊢ 0 ≤ cosh (dist z w / 2)
case hb
z w : ℍ
⊢ 0 ≤ dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.i... | 5c9fc747ec4d41a2 |
nilpotencyClass_eq_zero_of_subsingleton | Mathlib/RingTheory/Nilpotent/Defs.lean | @[simp] lemma nilpotencyClass_eq_zero_of_subsingleton [Subsingleton R] :
nilpotencyClass x = 0 | R : Type u_1
x : R
inst✝² : Zero R
inst✝¹ : Pow R ℕ
inst✝ : Subsingleton R
s : Set ℕ := {k | x ^ k = 0}
this : {k | x ^ k = 0} = univ
⊢ sInf {k | x ^ k = 0} = 0 | simp [this] | no goals | a12c1a51157c0cb3 |
PrimeSpectrum.isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | theorem isLocalization_away_iff_atPrime_of_basicOpen_eq_singleton [Algebra R S]
{f : R} {p : PrimeSpectrum R} (h : (basicOpen f).1 = {p}) :
IsLocalization.Away f S ↔ IsLocalization.AtPrime S p.1 :=
have : IsLocalization.AtPrime (Localization.Away f) p.1 | R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
f : R
p : PrimeSpectrum R
h : (basicOpen f).carrier = {p}
r : R
hr : ∀ m ∈ Submonoid.powers f, m ∉ Ideal.span {r}
q : Ideal R
prime : q.IsPrime
le : Ideal.span {r} ≤ q
disj : Disjoint ↑q ↑(Submonoid.powers f)
this : { asIdeal := q... | exact not_not.mpr (q.span_singleton_le_iff_mem.mp le) | no goals | a072e4e5ec37ba02 |
ZFSet.sUnion_lem | Mathlib/SetTheory/ZFC/Basic.lean | theorem sUnion_lem {α β : Type u} (A : α → PSet) (B : β → PSet) (αβ : ∀ a, ∃ b, Equiv (A a) (B b)) :
∀ a, ∃ b, Equiv ((sUnion ⟨α, A⟩).Func a) ((sUnion ⟨β, B⟩).Func b)
| ⟨a, c⟩ => by
let ⟨b, hb⟩ := αβ a
induction' ea : A a with γ Γ
induction' eb : B b with δ Δ
rw [ea, eb] at hb
obtain ⟨γδ, δγ⟩ ... | α β : Type u
A : α → PSet.{u}
B : β → PSet.{u}
αβ : ∀ (a : α), ∃ b, (A a).Equiv (B b)
a : (PSet.mk α A).Type
c✝ : ((PSet.mk α A).Func a).Type
b : β
γ : Type u
Γ : γ → PSet.{u}
A_ih✝¹ : ∀ (a_1 : γ), A a = Γ a_1 → ∃ b, ((⋃₀ PSet.mk α A).Func ⟨a, c✝⟩).Equiv ((⋃₀ PSet.mk β B).Func b)
ea : A a = PSet.mk γ Γ
δ : Type u
Δ : δ... | exact hd | no goals | 45858e0f18942367 |
AlgebraicGeometry.image_morphismRestrict_preimage | Mathlib/AlgebraicGeometry/Restrict.lean | theorem image_morphismRestrict_preimage {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (V : Opens U) :
(f ⁻¹ᵁ U).ι ''ᵁ ((f ∣_ U) ⁻¹ᵁ V) = f ⁻¹ᵁ (U.ι ''ᵁ V) | case h.e'_1
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
V : Opens ↑↑(↑U).toPresheafedSpace
x : ↑↑X.toPresheafedSpace
hx : x ∈ f ⁻¹ᵁ U
hx' : (ConcreteCategory.hom (f ∣_ U).base) ⟨x, hx⟩ ∈ V
⊢ ↑⟨(ConcreteCategory.hom f.base) x, hx⟩ = ↑((ConcreteCategory.hom (f ∣_ U).base) ⟨x, hx⟩) | exact (morphismRestrict_base_coe f U ⟨x, hx⟩).symm | no goals | 5fb629a9c7de0592 |
Real.exists_lt_of_strictMono | Mathlib/Data/NNReal/Defs.lean | theorem Real.exists_lt_of_strictMono [h : Nontrivial Γ₀ˣ] {f : Γ₀ →*₀ ℝ≥0} (hf : StrictMono f)
{r : ℝ} (hr : 0 < r) : ∃ d : Γ₀ˣ, (f d : ℝ) < r | Γ₀ : Type u_1
inst✝ : LinearOrderedCommGroupWithZero Γ₀
h : Nontrivial Γ₀ˣ
f : Γ₀ →*₀ ℝ≥0
hf : StrictMono ⇑f
r : ℝ
hr : 0 < r
s : ℝ≥0 := ⟨r, ⋯⟩
⊢ ∃ d, ↑(f ↑d) < r | have hs : 0 < s := hr | Γ₀ : Type u_1
inst✝ : LinearOrderedCommGroupWithZero Γ₀
h : Nontrivial Γ₀ˣ
f : Γ₀ →*₀ ℝ≥0
hf : StrictMono ⇑f
r : ℝ
hr : 0 < r
s : ℝ≥0 := ⟨r, ⋯⟩
hs : 0 < s
⊢ ∃ d, ↑(f ↑d) < r | 367092d092a92e2f |
MeasureTheory.Measure.measure_isMulLeftInvariant_eq_smul_of_ne_top | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | /-- **Uniqueness of left-invariant measures**:
Given two left-invariant measures which are finite on
compacts and inner regular for finite measure sets with respect to compact sets,
they coincide in the following sense: they give the same value to finite measure sets,
up to a multiplicative constant. -/
@[to_additive]
... | G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : IsTopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst... | have st : s ⊆ t := subset_inter (subset_toMeasurable μ' s) (subset_toMeasurable μ s) | G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : Group G
inst✝⁸ : IsTopologicalGroup G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : LocallyCompactSpace G
μ' μ : Measure G
inst✝⁴ : μ.IsHaarMeasure
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : μ'.IsMulLeftInvariant
inst✝¹ : μ.InnerRegularCompactLTTop
inst... | 3a873eec5efe2eb8 |
MeasureTheory.Supermartingale.smul_nonpos | Mathlib/Probability/Martingale/Basic.lean | theorem smul_nonpos {f : ι → Ω → F} {c : ℝ} (hc : c ≤ 0) (hf : Supermartingale f ℱ μ) :
Submartingale (c • f) ℱ μ | Ω : Type u_1
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ι m0
F : Type u_4
inst✝³ : NormedLatticeAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
inst✝ : OrderedSMul ℝ F
f : ι → Ω → F
c : ℝ
hc : c ≤ 0
hf : Supermartingale f ℱ μ
⊢ - -c • f = -(-c • f) | ext (i x) | case h.h
Ω : Type u_1
ι : Type u_3
inst✝⁴ : Preorder ι
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ι m0
F : Type u_4
inst✝³ : NormedLatticeAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : CompleteSpace F
inst✝ : OrderedSMul ℝ F
f : ι → Ω → F
c : ℝ
hc : c ≤ 0
hf : Supermartingale f ℱ μ
i : ι
x : Ω
⊢ (- -c • f) i... | 5b9e0d62d069da7c |
Real.Angle.eq_iff_sign_eq_and_abs_toReal_eq | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem eq_iff_sign_eq_and_abs_toReal_eq {θ ψ : Angle} :
θ = ψ ↔ θ.sign = ψ.sign ∧ |θ.toReal| = |ψ.toReal| | case refine_2.intro
θ ψ : Angle
hs : θ.sign = ψ.sign
hr : |θ.toReal| = |ψ.toReal|
⊢ θ = ψ | rw [abs_eq_abs] at hr | case refine_2.intro
θ ψ : Angle
hs : θ.sign = ψ.sign
hr : θ.toReal = ψ.toReal ∨ θ.toReal = -ψ.toReal
⊢ θ = ψ | 266df1502710853f |
ENNReal.le_of_top_imp_top_of_toNNReal_le | Mathlib/Data/ENNReal/Basic.lean | theorem le_of_top_imp_top_of_toNNReal_le {a b : ℝ≥0∞} (h : a = ⊤ → b = ⊤)
(h_nnreal : a ≠ ⊤ → b ≠ ⊤ → a.toNNReal ≤ b.toNNReal) : a ≤ b | a b : ℝ≥0∞
h : a = ⊤ → b = ⊤
h_nnreal : a ≠ ⊤ → b ≠ ⊤ → a.toNNReal ≤ b.toNNReal
hlt : b < a
⊢ False | lift b to ℝ≥0 using hlt.ne_top | case intro
a : ℝ≥0∞
b : ℝ≥0
h : a = ⊤ → ↑b = ⊤
h_nnreal : a ≠ ⊤ → ↑b ≠ ⊤ → a.toNNReal ≤ (↑b).toNNReal
hlt : ↑b < a
⊢ False | 5b7f8b6daf578fe8 |
Nat.primesBelow_card_eq_primeCounting' | Mathlib/NumberTheory/PrimeCounting.lean | theorem primesBelow_card_eq_primeCounting' (n : ℕ) : #n.primesBelow = primeCounting' n | n : ℕ
⊢ #n.primesBelow = π' n | simp only [primesBelow, primeCounting'] | n : ℕ
⊢ #(filter (fun p => Prime p) (range n)) = count Prime n | 98d77ddf83390b18 |
CategoryTheory.Abelian.Ext.covariant_sequence_exact₃' | Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.lean | /-- Alternative formulation of `covariant_sequence_exact₃` -/
lemma covariant_sequence_exact₃' :
(ShortComplex.mk (AddCommGrp.ofHom ((mk₀ S.g).postcomp X (add_zero n₀)))
(AddCommGrp.ofHom (hS.extClass.postcomp X h)) (by
ext x
dsimp
simp only [comp_assoc_of_second_deg_zero, ShortComplex... | case comm₁₂
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
Function.Exact
⇑(ConcreteCategory.hom
(ShortComplex.mk
(((preadditiveC... | ext x | case comm₁₂.h
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X : C
S : ShortComplex C
hS : S.ShortExact
n₀ n₁ : ℕ
h : n₀ + 1 = n₁
this✝ : HasDerivedCategory C := HasDerivedCategory.standard C
this :
Function.Exact
⇑(ConcreteCategory.hom
(ShortComplex.mk
(((preadditiv... | 944dbb4448b96008 |
EReal.div_le_iff_le_mul | Mathlib/Data/Real/EReal.lean | lemma div_le_iff_le_mul (h : 0 < b) (h' : b ≠ ⊤) : a / b ≤ c ↔ a ≤ b * c | a b c : EReal
h : 0 < b
h' : b ≠ ⊤
⊢ a / b ≤ c ↔ a ≤ b * c | nth_rw 1 [← @mul_div_cancel c b (ne_bot_of_gt h) h' (ne_of_gt h)] | a b c : EReal
h : 0 < b
h' : b ≠ ⊤
⊢ a / b ≤ b * (c / b) ↔ a ≤ b * c | b20bf9a85563cbdf |
Int.mul_mem_zero_one_two_three_four_iff | Mathlib/NumberTheory/Divisors.lean | /-- This lemma justifies its existence from its utility in crystallographic root system theory. -/
lemma mul_mem_zero_one_two_three_four_iff {a b : ℤ} (h₀ : a = 0 ↔ b = 0) :
a * b ∈ ({0, 1, 2, 3, 4} : Set ℤ) ↔ (a, b) ∈ ({
(0, 0),
(1, 1), (-1, -1),
(1, 2), (2, 1), (-1, -2), (-2, -1),
(1, 3), ... | a b : ℤ
h₀ : a = 0 ↔ b = 0
⊢ a * b = 0 ∨
(a, b) ∈ divisorsAntidiag 1 ∨
(a, b) ∈ divisorsAntidiag 2 ∨ (a, b) ∈ divisorsAntidiag 3 ∨ (a, b) ∈ divisorsAntidiag 4 ↔
(a, b) = (0, 0) ∨
(a, b) = (1, 1) ∨
(a, b) = (-1, -1) ∨
(a, b) = (1, 2) ∨
(a, b) = (2, 1) ∨
... | aesop | no goals | 1d92450ae40251b7 |
Module.End.invtSubmodule.isCompl_iff | Mathlib/Algebra/Module/Submodule/Invariant.lean | protected lemma isCompl_iff {p q : f.invtSubmodule} :
IsCompl p q ↔ IsCompl (p : Submodule R M) (q : Submodule R M) | case mk
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : End R M
q : ↥f.invtSubmodule
p : Submodule R M
hp : p ∈ f.invtSubmodule
⊢ IsCompl ⟨p, hp⟩ q ↔ IsCompl ↑⟨p, hp⟩ ↑q | obtain ⟨q, hq⟩ := q | case mk.mk
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : End R M
p : Submodule R M
hp : p ∈ f.invtSubmodule
q : Submodule R M
hq : q ∈ f.invtSubmodule
⊢ IsCompl ⟨p, hp⟩ ⟨q, hq⟩ ↔ IsCompl ↑⟨p, hp⟩ ↑⟨q, hq⟩ | 23c6e4f8d5ea227a |
IsLocalization.AtPrime.not_isField | Mathlib/RingTheory/DedekindDomain/Dvr.lean | theorem IsLocalization.AtPrime.not_isField {P : Ideal A} (hP : P ≠ ⊥) [pP : P.IsPrime] (Aₘ : Type*)
[CommRing Aₘ] [Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P] : ¬ IsField Aₘ | A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDomain A
P : Ideal A
hP : P ≠ ⊥
pP : P.IsPrime
Aₘ : Type u_2
inst✝² : CommRing Aₘ
inst✝¹ : Algebra A Aₘ
inst✝ : IsLocalization.AtPrime Aₘ P
h : IsField Aₘ
⊢ False | letI := h.toField | A : Type u_1
inst✝⁴ : CommRing A
inst✝³ : IsDomain A
P : Ideal A
hP : P ≠ ⊥
pP : P.IsPrime
Aₘ : Type u_2
inst✝² : CommRing Aₘ
inst✝¹ : Algebra A Aₘ
inst✝ : IsLocalization.AtPrime Aₘ P
h : IsField Aₘ
this : Field Aₘ := h.toField
⊢ False | 921327a5477174a5 |
Field.Emb.Cardinal.eq_bot_of_not_nonempty | Mathlib/FieldTheory/CardinalEmb.lean | lemma eq_bot_of_not_nonempty (hi : ¬ Nonempty (Iio i)) : filtration i = ⊥ | case top
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
rank_inf : Fact (ℵ₀ ≤ Module.rank F E)
inst✝ : Algebra.IsAlgebraic F E
hi✝ : IsSuccPrelimit ⊤
hi : ¬Nonempty ↑(range WithTop.some)
this : Nonempty (Module.rank F E).ord.toType
⊢ filtration ⊤ = ⊥ | exact (hi inferInstance).elim | no goals | ae4098a4d5d2c994 |
Ordinal.not_bddAbove_compl_of_small | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem Ordinal.not_bddAbove_compl_of_small (s : Set Ordinal.{u}) [hs : Small.{u} s] :
¬BddAbove sᶜ | s : Set Ordinal.{u}
hs : Small.{u, u + 1} ↑s
h : Small.{u, u + 1} ↑sᶜ
⊢ False | have := small_union s sᶜ | s : Set Ordinal.{u}
hs : Small.{u, u + 1} ↑s
h : Small.{u, u + 1} ↑sᶜ
this : Small.{u, u + 1} ↑(s ∪ sᶜ)
⊢ False | 88bf06c16280c8c3 |
tendsto_mul_cofinite_nhds_zero | Mathlib/Topology/Algebra/Monoid.lean | theorem tendsto_mul_cofinite_nhds_zero {f : α → M} {g : β → M}
(hf : Tendsto f cofinite (𝓝 0)) (hg : Tendsto g cofinite (𝓝 0)) :
Tendsto (fun i : α × β ↦ f i.1 * g i.2) cofinite (𝓝 0) | M : Type u_3
α : Type u_6
β : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
f : α → M
g : β → M
hf : Tendsto f cofinite (𝓝 0)
hg : Tendsto g cofinite (𝓝 0)
this : TopologicalSpace α := ⊥
⊢ Tendsto (fun i => f i.1 * g i.2) cofinite (𝓝 0) | haveI : DiscreteTopology α := discreteTopology_bot α | M : Type u_3
α : Type u_6
β : Type u_7
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
f : α → M
g : β → M
hf : Tendsto f cofinite (𝓝 0)
hg : Tendsto g cofinite (𝓝 0)
this✝ : TopologicalSpace α := ⊥
this : DiscreteTopology α
⊢ Tendsto (fun i => f i.1 * g i.2) cofinite (𝓝 0) | 5fa1372031606768 |
Polynomial.IsUnitTrinomial.irreducible_of_coprime' | Mathlib/Analysis/Complex/Polynomial/UnitTrinomial.lean | theorem irreducible_of_coprime' (hp : IsUnitTrinomial p)
(h : ∀ z : ℂ, ¬(aeval z p = 0 ∧ aeval z (mirror p) = 0)) : Irreducible p | p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
⊢ ¬0 < q.natDegree | intro hq'' | p : ℤ[X]
hp : p.IsUnitTrinomial
h : ∀ (z : ℂ), ¬((aeval z) p = 0 ∧ (aeval z) p.mirror = 0)
q : ℤ[X]
hq : q ∣ p
hq' : q ∣ p.mirror
hq'' : 0 < q.natDegree
⊢ False | eaa890db4bc6d654 |
Finset.fold_op_distrib | Mathlib/Data/Finset/Fold.lean | theorem fold_op_distrib {f g : α → β} {b₁ b₂ : β} :
(s.fold op (b₁ * b₂) fun x => f x * g x) = s.fold op b₁ f * s.fold op b₂ g | α : Type u_1
β : Type u_2
op : β → β → β
hc : Std.Commutative op
ha : Std.Associative op
s : Finset α
f g : α → β
b₁ b₂ : β
⊢ fold op (op b₁ b₂) (fun x => op (f x) (g x)) s = op (fold op b₁ f s) (fold op b₂ g s) | simp only [fold, fold_distrib] | no goals | 1c3e26495cac1aae |
IsVisible.rfl | Mathlib/Analysis/Convex/Visible.lean | @[simp, refl] lemma IsVisible.rfl : IsVisible 𝕜 s x x | 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : LinearOrderedField 𝕜
inst✝² : AddCommGroup V
inst✝¹ : Module 𝕜 V
inst✝ : AddTorsor V P
s : Set P
x : P
⊢ IsVisible 𝕜 s x x | simp [IsVisible] | no goals | f6660bde440c17f2 |
Submonoid.le_prod_iff | Mathlib/Algebra/Group/Submonoid/Operations.lean | theorem le_prod_iff {s : Submonoid M} {t : Submonoid N} {u : Submonoid (M × N)} :
u ≤ s.prod t ↔ u.map (fst M N) ≤ s ∧ u.map (snd M N) ≤ t | N : Type u_2
inst✝¹ : MulOneClass N
M : Type u_5
inst✝ : MulOneClass M
s : Submonoid M
t : Submonoid N
u : Submonoid (M × N)
⊢ u ≤ s.prod t ↔ map (fst M N) u ≤ s ∧ map (snd M N) u ≤ t | constructor | case mp
N : Type u_2
inst✝¹ : MulOneClass N
M : Type u_5
inst✝ : MulOneClass M
s : Submonoid M
t : Submonoid N
u : Submonoid (M × N)
⊢ u ≤ s.prod t → map (fst M N) u ≤ s ∧ map (snd M N) u ≤ t
case mpr
N : Type u_2
inst✝¹ : MulOneClass N
M : Type u_5
inst✝ : MulOneClass M
s : Submonoid M
t : Submonoid N
u : Submonoid (... | ddd3e6bb631fdb5c |
Char.card_pow_card | Mathlib/NumberTheory/GaussSum.lean | theorem Char.card_pow_card {F : Type*} [Field F] [Fintype F] {F' : Type*} [Field F'] [Fintype F']
{χ : MulChar F F'} (hχ₁ : χ ≠ 1) (hχ₂ : IsQuadratic χ)
(hch₁ : ringChar F' ≠ ringChar F) (hch₂ : ringChar F' ≠ 2) :
(χ (-1) * Fintype.card F) ^ (Fintype.card F' / 2) = χ (Fintype.card F') | case intro.intro.intro.intro.a
F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
F' : Type u_2
inst✝¹ : Field F'
inst✝ : Fintype F'
χ : MulChar F F'
hχ₁ : χ ≠ 1
hχ₂ : χ.IsQuadratic
hch₁ : ringChar F' ≠ ringChar F
hch₂ : ringChar F' ≠ 2
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
n' : ℕ+
hp' ... | have := Fact.mk hp' | case intro.intro.intro.intro.a
F : Type u_1
inst✝³ : Field F
inst✝² : Fintype F
F' : Type u_2
inst✝¹ : Field F'
inst✝ : Fintype F'
χ : MulChar F F'
hχ₁ : χ ≠ 1
hχ₂ : χ.IsQuadratic
hch₁ : ringChar F' ≠ ringChar F
hch₂ : ringChar F' ≠ 2
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
n' : ℕ+
hp' ... | e01707f133d9b7bd |
Vector.range'_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Range.lean | theorem range'_eq_append_iff : range' s (n + m) = xs ++ ys ↔ xs = range' s n ∧ ys = range' (s + n) m | s n m : Nat
xs : Vector Nat n
ys : Vector Nat m
⊢ (∃ k, k ≤ n + m ∧ xs.toArray = Array.range' s k ∧ ys.toArray = Array.range' (s + k) (n + m - k)) ↔
xs.toArray = Array.range' s n ∧ ys.toArray = Array.range' (s + n) m | constructor | case mp
s n m : Nat
xs : Vector Nat n
ys : Vector Nat m
⊢ (∃ k, k ≤ n + m ∧ xs.toArray = Array.range' s k ∧ ys.toArray = Array.range' (s + k) (n + m - k)) →
xs.toArray = Array.range' s n ∧ ys.toArray = Array.range' (s + n) m
case mpr
s n m : Nat
xs : Vector Nat n
ys : Vector Nat m
⊢ xs.toArray = Array.range' s n ∧... | fe298cc85f386ffd |
nilpotencyClass_eq_succ_iff | Mathlib/RingTheory/Nilpotent/Defs.lean | lemma nilpotencyClass_eq_succ_iff {k : ℕ} :
nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 | R : Type u_1
x : R
inst✝ : MonoidWithZero R
k : ℕ
s : Set ℕ := {k | x ^ k = 0}
⊢ nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 | have : ∀ k₁ k₂ : ℕ, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s := fun k₁ k₂ h_le hk₁ ↦ pow_eq_zero_of_le h_le hk₁ | R : Type u_1
x : R
inst✝ : MonoidWithZero R
k : ℕ
s : Set ℕ := {k | x ^ k = 0}
this : ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s
⊢ nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 | 88cf1fbdc3a16b39 |
Vector.comp_traverse | Mathlib/Data/Vector/Basic.lean | theorem comp_traverse (f : β → F γ) (g : α → G β) (x : Vector α n) :
Vector.traverse (Comp.mk ∘ Functor.map f ∘ g) x =
Comp.mk (Vector.traverse f <$> Vector.traverse g x) | case cons
n✝ : ℕ
F G : Type u → Type u
inst✝² : Applicative F
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
α β γ : Type u
f : β → F γ
g : α → G β
n : ℕ
x : α
xs : Vector α n
ih : Vector.traverse (Comp.mk ∘ Functor.map f ∘ g) xs = Comp.mk (Vector.traverse f <$> Vector.traverse g xs)
⊢ cons <$> (Comp.mk ∘ Functor.m... | simp [functor_norm, Function.comp_def] | no goals | 5d747e07894ea72a |
Real.inv_sign | Mathlib/Data/Real/Sign.lean | theorem inv_sign (r : ℝ) : (sign r)⁻¹ = sign r | case inl
r : ℝ
hn : r.sign = -1
⊢ (-1)⁻¹ = -1 | norm_num | no goals | be6545dd2dc24fd8 |
Nat.sub_le_iff_le_add' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem sub_le_iff_le_add' {a b c : Nat} : a - b ≤ c ↔ a ≤ b + c | a b c : Nat
⊢ a - b ≤ c ↔ a ≤ b + c | rw [Nat.add_comm, Nat.sub_le_iff_le_add] | no goals | c14988a0d300bea9 |
CategoryTheory.FinitaryPreExtensive.hasPullbacks_of_is_coproduct | Mathlib/CategoryTheory/Extensive.lean | lemma FinitaryPreExtensive.hasPullbacks_of_is_coproduct [FinitaryPreExtensive C] {ι : Type*}
[Finite ι] {F : Discrete ι ⥤ C} {c : Cocone F} (hc : IsColimit c) (i : Discrete ι) {X : C}
(g : X ⟶ _) : HasPullback g (c.ι.app i) | case mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
ι : Type u_1
inst✝ : Finite ι
X : C
f : ι → C
c : Cofan f
hc : IsColimit c
i : ι
g : X ⟶ ((Functor.const (Discrete ι)).obj c.pt).obj { as := i }
e : ∐ f ≅ f i ⨿ ∐ fun j => f ↑j :=
{
hom :=
Sigma.desc fun j => if h : j = i then eqT... | rw [← this] | case mk
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : FinitaryPreExtensive C
ι : Type u_1
inst✝ : Finite ι
X : C
f : ι → C
c : Cofan f
hc : IsColimit c
i : ι
g : X ⟶ ((Functor.const (Discrete ι)).obj c.pt).obj { as := i }
e : ∐ f ≅ f i ⨿ ∐ fun j => f ↑j :=
{
hom :=
Sigma.desc fun j => if h : j = i then eqT... | 89072af07eae1032 |
Nat.smoothNumbersUpTo_subset_image | Mathlib/NumberTheory/SmoothNumbers.lean | /-- The set of `k`-smooth numbers `≤ N` is contained in the set of numbers of the form `m^2 * P`,
where `m ≤ √N` and `P` is a product of distinct primes `< k`. -/
lemma smoothNumbersUpTo_subset_image (N k : ℕ) :
smoothNumbersUpTo N k ⊆ Finset.image (fun (s, m) ↦ m ^ 2 * (s.prod id))
(k.primesBelow.powerset ×ˢ... | case intro.intro.intro.intro.refine_2
N k n : ℕ
hn : n ∈ N.smoothNumbersUpTo k
hn₁ : n ≤ N
hn₂ : n ∈ k.smoothNumbers
s : Finset ℕ
hs : s ∈ k.primesBelow.powerset
m : ℕ
hm : n = m ^ 2 * s.prod id
⊢ m ^ 2 * 1 ≤ m ^ 2 * s.prod id | exact mul_le_mul_left' (Finset.one_le_prod' fun p hp ↦
(prime_of_mem_primesBelow <| Finset.mem_powerset.mp hs hp).one_lt.le) _ | no goals | 6ca0b940ee5775b9 |
WeierstrassCurve.Jacobian.dblXYZ_of_Z_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma dblXYZ_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
(hQz : Q z ≠ 0) (hx : P x * Q z ^ 2 = Q x * P z ^ 2) (hy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3) :
W.dblXYZ P = W.dblZ P •
![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
(W.toAffine.slope (P x / ... | F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3
⊢ W.dblXYZ P =
W.dblZ P •
![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ 2)
((toAffine W).slope (P x ... | have hZ {n : ℕ} : IsUnit <| W.dblZ P ^ n := (isUnit_dblZ_of_Y_ne' hP hQ hPz hx hy).pow n | F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3
hZ : ∀ {n : ℕ}, IsUnit (W.dblZ P ^ n)
⊢ W.dblXYZ P =
W.dblZ P •
![(toAffine W).addX (P x / P z ^ 2) (Q x / Q z ^ ... | 88fc2587d9539253 |
nonempty_unique | Mathlib/Logic/Unique.lean | theorem nonempty_unique (α : Sort u) [Subsingleton α] [Nonempty α] : Nonempty (Unique α) | α : Sort u
inst✝¹ : Subsingleton α
inst✝ : Nonempty α
⊢ Nonempty (Unique α) | inhabit α | α : Sort u
inst✝¹ : Subsingleton α
inst✝ : Nonempty α
inhabited_h : Inhabited α
⊢ Nonempty (Unique α) | ab39e6c695d6387b |
Algebra.Generators.H1Cotangent.δ_eq_δAux | Mathlib/RingTheory/Kaehler/JacobiZariski.lean | lemma δ_eq_δAux (x : Q.ker) (hx) :
δ Q P ⟨.mk x, hx⟩ = δAux R Q x.1 | 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
x : ↥Q.ker
hx : Extension.Cotangent.mk x ∈ LinearMap.ker Q.toExtension.cotangentComplex
y : (Q.co... | congr | case h.e_6.h.e_val
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
x : ↥Q.ker
hx : Extension.Cotangent.mk x ∈ LinearMap.ker Q.toExtension.cotange... | 9342de0dfa71b165 |
CoxeterSystem.isLeftInversion_of_mem_leftInvSeq | Mathlib/GroupTheory/Coxeter/Inversion.lean | theorem isLeftInversion_of_mem_leftInvSeq {ω : List B} (hω : cs.IsReduced ω) {t : W}
(ht : t ∈ lis ω) : cs.IsLeftInversion (π ω) t | case right.intro.intro
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
j : ℕ
hj : j < (cs.leftInvSeq ω).length
ht : (cs.leftInvSeq ω)[j] ∈ cs.leftInvSeq ω
⊢ cs.length ((cs.leftInvSeq ω)[j] * cs.wordProd ω) < cs.length (cs.wordProd ω) | rw [← List.getD_eq_getElem _ 1 hj, getD_leftInvSeq_mul_wordProd] | case right.intro.intro
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
hω : cs.IsReduced ω
j : ℕ
hj : j < (cs.leftInvSeq ω).length
ht : (cs.leftInvSeq ω)[j] ∈ cs.leftInvSeq ω
⊢ cs.length (cs.wordProd (ω.eraseIdx j)) < cs.length (cs.wordProd ω) | a33e4cfb3ae16223 |
Perfect.exists_nat_bool_injection | Mathlib/Topology/MetricSpace/Perfect.lean | theorem Perfect.exists_nat_bool_injection
(hC : Perfect C) (hnonempty : C.Nonempty) [CompleteSpace α] :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f | α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥... | apply mem_univ | no goals | ef9f9f33b845189a |
UV.mem_of_mem_compression | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem mem_of_mem_compression (ha : a ∈ 𝓒 u v s) (hva : v ≤ a) (hvu : v = ⊥ → u = ⊥) :
a ∈ s | case inr.intro.intro.intro
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
hva : v ≤ a
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hb : b ∈ s
h : compress u v b = a
⊢ a ∈ s | unfold compress at h | case inr.intro.intro.intro
α : Type u_1
inst✝³ : GeneralizedBooleanAlgebra α
inst✝² : DecidableRel Disjoint
inst✝¹ : DecidableRel fun x1 x2 => x1 ≤ x2
s : Finset α
u v a : α
inst✝ : DecidableEq α
hva : v ≤ a
hvu : v = ⊥ → u = ⊥
left✝ : a ∉ s
b : α
hb : b ∈ s
h : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \ v else b) = a
⊢ a... | ce92c86d3b6674d5 |
HasFPowerSeriesWithinOnBall.uniform_geometric_approx | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesWithinOnBall.uniform_geometric_approx {r' : ℝ≥0}
(hf : HasFPowerSeriesWithinOnBall f p s x r) (h : (r' : ℝ≥0∞) < r) :
∃ a ∈ Ioo (0 : ℝ) 1,
∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, x + y ∈ insert x s →
‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n | case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
r' : ℝ≥0
hf : HasFPowerSeriesWithi... | have := ha.1.le | case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
r' : ℝ≥0
hf : HasFPowerSeriesWithi... | 850491d5ad2aa45f |
IsClosed.exists_wbtw_isVisible | Mathlib/Analysis/Convex/Visible.lean | /-- If `s` is a closed set, then any point `x` sees some point of `s` in any direction where there
is something to see. -/
lemma IsClosed.exists_wbtw_isVisible (hs : IsClosed s) (hy : y ∈ s) (x : V) :
∃ z ∈ s, Wbtw ℝ x z y ∧ IsVisible ℝ s x z | case intro.intro.intro.intro.intro.intro
V : Type u_2
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
s : Set V
y : V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul ℝ V
hs : IsClosed s
hy : y ∈ s
x : V
t : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s
ht₁ : 1 ∈ t
ht : BddBelow t
δ : ℝ := sI... | replace hε₁ : ε < 1 := hε₁.lt_of_ne <| by rintro rfl; simp at h | case intro.intro.intro.intro.intro.intro
V : Type u_2
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
s : Set V
y : V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul ℝ V
hs : IsClosed s
hy : y ∈ s
x : V
t : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s
ht₁ : 1 ∈ t
ht : BddBelow t
δ : ℝ := sI... | b1e9329ed05e7997 |
Vector.range'_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Range.lean | theorem range'_eq_append_iff : range' s (n + m) = xs ++ ys ↔ xs = range' s n ∧ ys = range' (s + n) m | s n m : Nat
xs : Vector Nat n
ys : Vector Nat m
h₁ : xs.toArray = Array.range' s n
h₂ : ys.toArray = Array.range' (s + n) m
⊢ m = n + m - n | omega | no goals | fe298cc85f386ffd |
SimpleGraph.ConnectedComponent.map_comp | Mathlib/Combinatorics/SimpleGraph/Path.lean | theorem map_comp (C : G.ConnectedComponent) (φ : G →g G') (ψ : G' →g G'') :
(C.map φ).map ψ = C.map (ψ.comp φ) | V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
C : G.ConnectedComponent
φ : G →g G'
ψ : G' →g G''
⊢ ∀ (v : V), map ψ (map φ (G.connectedComponentMk v)) = map (ψ.comp φ) (G.connectedComponentMk v) | exact fun _ => rfl | no goals | 78353a11b790d3c2 |
SimpleGraph.IsAcyclic.path_unique | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | theorem IsAcyclic.path_unique {G : SimpleGraph V} (h : G.IsAcyclic) {v w : V} (p q : G.Path v w) :
p = q | case mk.mk.cons.inl.cons.inl.inr.intro
V : Type u
G : SimpleGraph V
v w v✝ w✝ : V
p : G.Walk v✝ w✝
ih : p.IsPath → ∀ (q : G.Walk v✝ w✝), q.IsPath → p = q
ph : G.Adj v✝ v✝
hp : p.IsPath ∧ v✝ ∉ p.support
h✝ : G.Adj v✝ v✝
q : G.Walk v✝ w✝
hq : q.IsPath ∧ v✝ ∉ q.support
⊢ cons ph p = cons h✝ q | simp at hq | no goals | c6bceaf8dd6ef474 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastArithShiftRight.twoPowShift_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem twoPowShift_eq (aig : AIG α) (target : TwoPowShiftTarget aig w) (lhs : BitVec w)
(rhs : BitVec target.n) (assign : α → Bool)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, target.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < target.n), ⟦aig, target.rhs.get idx hi... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦... | rw [hleft] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs : BitVec w
assign : α → Bool
idx : Nat
hidx : idx < w
res : RefVecEntry α w
n : Nat
lvec : aig.RefVec w
rvec : aig.RefVec n
pow : Nat
rhs : BitVec { n := n, lhs := lvec, rhs := rvec, pow := pow }.n
hleft :
∀ (idx : Nat) (hidx : idx < w),
⟦... | 6f686afc6bff74c7 |
CochainComplex.HomComplex.Cochain.leftUnshift_v | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | lemma leftUnshift_v {n' a : ℤ} (γ : Cochain (K⟦a⟧) L n') (n : ℤ) (hn : n + a = n')
(p q : ℤ) (hpq : p + n = q) (p' : ℤ) (hp' : p' + n' = q) :
(γ.leftUnshift n hn).v p q hpq = (a * n' + ((a * (a-1))/2)).negOnePow •
(K.shiftFunctorObjXIso a p' p (by omega)).inv ≫ γ.v p' q (by omega) | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
p q : ℤ
hpq : p + n = q
p' : ℤ
hp' : p' + n' = q
⊢ p' = p - a | omega | no goals | 871bb0093b336198 |
Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero | Mathlib/Analysis/Analytic/Uniqueness.lean | theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
(h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 | case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen t
δ : ℝ
δ_pos : δ... | have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
inv_mul_cancel_right₀ h₀.ne.symm ε ▸
mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀ | case intro.intro.intro.intro.intro.intro.intro.succ.intro.intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
y : E
c : ℝ
c_pos : c > 0
t : Set E
t_open : IsOpen t
δ : ℝ
δ_pos : δ... | a2f8ef35a556aa68 |
NormedAddGroupHom.mkNormedAddGroupHom_norm_le' | Mathlib/Analysis/Normed/Group/Hom.lean | theorem mkNormedAddGroupHom_norm_le' (f : V₁ →+ V₂) {C : ℝ} (h : ∀ x, ‖f x‖ ≤ C * ‖x‖) :
‖f.mkNormedAddGroupHom C h‖ ≤ max C 0 :=
opNorm_le_bound _ (le_max_right _ _) fun x =>
(h x).trans <| by gcongr; apply le_max_left
| V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : V₁ →+ V₂
C : ℝ
h : ∀ (x : V₁), ‖f x‖ ≤ C * ‖x‖
x : V₁
⊢ C * ‖x‖ ≤ (C ⊔ 0) * ‖x‖ | gcongr | case h
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : V₁ →+ V₂
C : ℝ
h : ∀ (x : V₁), ‖f x‖ ≤ C * ‖x‖
x : V₁
⊢ C ≤ C ⊔ 0 | 9103a32bfa4bec78 |
List.sizeOf_kerase | Mathlib/Data/List/Sigma.lean | theorem sizeOf_kerase [SizeOf (Sigma β)] (x : α)
(xs : List (Sigma β)) : SizeOf.sizeOf (List.kerase x xs) ≤ SizeOf.sizeOf xs | α : Type u
β : α → Type v
inst✝¹ : DecidableEq α
inst✝ : SizeOf (Sigma β)
x : α
xs : List (Sigma β)
⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x xs) ≤
rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) xs | induction' xs with y ys | case nil
α : Type u
β : α → Type v
inst✝¹ : DecidableEq α
inst✝ : SizeOf (Sigma β)
x : α
⊢ rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) (kerase x []) ≤
rec 1 (fun head tail tail_ih => 1 + sizeOf head + tail_ih) []
case cons
α : Type u
β : α → Type v
inst✝¹ : DecidableEq α
inst✝ : SizeOf (Sigma β)
x :... | 7ec14474a9ac018f |
List.injective_permutations'Aux | Mathlib/Data/List/Permutation.lean | theorem injective_permutations'Aux (x : α) : Function.Injective (permutations'Aux x) | α : Type u_1
x : α
⊢ Injective (permutations'Aux x) | intro s t h | α : Type u_1
x : α
s t : List α
h : permutations'Aux x s = permutations'Aux x t
⊢ s = t | c9337a8fada9365b |
WeierstrassCurve.ofJ_j | Mathlib/AlgebraicGeometry/EllipticCurve/ModelsWithJ.lean | lemma ofJ_j : (ofJ j).j = j | case pos
F : Type u_2
inst✝¹ : Field F
j : F
inst✝ : DecidableEq F
h0 : ¬j = 0
h1728 : j = 1728
h2 : 2 ≠ 0
this : Fact (IsUnit 2)
⊢ (ofJ j).j = j | simp_rw [h1728, ofJ_1728_of_two_ne_zero h2, ofJ1728_j] | no goals | 610a5c04ad621693 |
Rat.isSquare_iff | Mathlib/Data/Rat/Lemmas.lean | theorem isSquare_iff {q : ℚ} : IsSquare q ↔ IsSquare q.num ∧ IsSquare q.den | case mpr.intro.intro.intro
q : ℚ
nr : ℤ
hnr : q.num = nr * nr
dr : ℕ
hdr : q.den = dr * dr
⊢ q = ↑nr / ↑dr * (↑nr / ↑dr) | rw [div_mul_div_comm, ← Int.cast_mul, ← Nat.cast_mul, ← hnr, ← hdr, num_div_den] | no goals | 81d14c8d8fbc187d |
MvPolynomial.pderiv_monomial | Mathlib/Algebra/MvPolynomial/PDeriv.lean | theorem pderiv_monomial {i : σ} :
pderiv i (monomial s a) = monomial (s - single i 1) (a * s i) | R : Type u
σ : Type v
a : R
s : σ →₀ ℕ
inst✝ : CommSemiring R
i : σ
⊢ (s.sum fun a_1 b => (monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) =
(monomial (s - single i 1)) (a * ↑(s i)) | refine (Finset.sum_eq_single i (fun j _ hne => ?_) fun hi => ?_).trans ?_ | case refine_1
R : Type u
σ : Type v
a : R
s : σ →₀ ℕ
inst✝ : CommSemiring R
i j : σ
x✝ : j ∈ s.support
hne : j ≠ i
⊢ (fun a_1 b => (monomial (s - single a_1 1)) (a * ↑b) * Pi.single i 1 a_1) j (s j) = 0
case refine_2
R : Type u
σ : Type v
a : R
s : σ →₀ ℕ
inst✝ : CommSemiring R
i : σ
hi : i ∉ s.support
⊢ (fun a_1 b =>... | 290d2f3a8835d3f2 |
ExpGrowth.le_expGrowthInf_mul | Mathlib/Analysis/Asymptotics/ExpGrowth.lean | lemma le_expGrowthInf_mul {u v : ℕ → ℝ≥0∞} :
expGrowthInf u + expGrowthInf v ≤ expGrowthInf (u * v) | u v : ℕ → ℝ≥0∞
⊢ expGrowthInf u + expGrowthInf v ≤ expGrowthInf (u * v) | refine le_liminf_add.trans_eq (liminf_congr (Eventually.of_forall fun n ↦ ?_)) | u v : ℕ → ℝ≥0∞
n : ℕ
⊢ ((fun n => (u n).log / ↑n) + fun n => (v n).log / ↑n) n = ((u * v) n).log / ↑n | cbbd5b7d410dcda4 |
Multiset.card_powerset | Mathlib/Data/Multiset/Powerset.lean | theorem card_powerset (s : Multiset α) : card (powerset s) = 2 ^ card s :=
Quotient.inductionOn s <| by simp
| α : Type u_1
s : Multiset α
⊢ ∀ (a : List α), (powerset ⟦a⟧).card = 2 ^ card ⟦a⟧ | simp | no goals | d86c7c6019338e21 |
QPF.suppPreservation_iff_liftpPreservation | Mathlib/Data/QPF/Univariate/Basic.lean | theorem suppPreservation_iff_liftpPreservation : q.SuppPreservation ↔ q.LiftpPreservation | case mp.mk
F : Type u → Type u
q : QPF F
h : SuppPreservation
α : Type u
p : α → Prop
a : (P F).A
f : (P F).B a → α
⊢ Liftp p (abs ⟨a, f⟩) ↔ Liftp p ⟨a, f⟩ | have h' := h | case mp.mk
F : Type u → Type u
q : QPF F
h : SuppPreservation
α : Type u
p : α → Prop
a : (P F).A
f : (P F).B a → α
h' : SuppPreservation
⊢ Liftp p (abs ⟨a, f⟩) ↔ Liftp p ⟨a, f⟩ | b77afcefc3e32c1e |
hasDerivAt_sqrt_mul_log | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | theorem hasDerivAt_sqrt_mul_log {x : ℝ} (hx : x ≠ 0) :
HasDerivAt (fun x => √x * log x) ((2 + log x) / (2 * √x)) x | x : ℝ
hx : x ≠ 0
⊢ HasDerivAt (fun x => √x * log x) ((2 + log x) / (2 * √x)) x | convert (hasDerivAt_sqrt hx).mul (hasDerivAt_log hx) using 1 | case h.e'_9
x : ℝ
hx : x ≠ 0
⊢ (2 + log x) / (2 * √x) = 1 / (2 * √x) * log x + √x * x⁻¹ | 15e5fa89b07fa68c |
continuousAt_clog | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | theorem continuousAt_clog {x : ℂ} (h : x ∈ slitPlane) : ContinuousAt log x | case refine_1
x : ℂ
h : x ∈ slitPlane
⊢ ContinuousAt (fun x => ↑(Real.log ‖x‖)) x | refine continuous_ofReal.continuousAt.comp ?_ | case refine_1
x : ℂ
h : x ∈ slitPlane
⊢ ContinuousAt (fun x => Real.log ‖x‖) x | 3806bdd6f92461d6 |
OrderIso.supIrredLowerSet_symm_apply | Mathlib/Order/Birkhoff.lean | @[simp] lemma supIrredLowerSet_symm_apply (s : {s : LowerSet α // SupIrred s}) [Fintype s] :
supIrredLowerSet.symm s = (s.1 : Set α).toFinset.sup id | case mk
α : Type u_1
inst✝³ : SemilatticeSup α
inst✝² : OrderBot α
inst✝¹ : Finite α
s : LowerSet α
hs : SupIrred s
inst✝ : Fintype ↥↑⟨s, hs⟩
⊢ supIrredLowerSet.symm ⟨s, hs⟩ = (↑↑⟨s, hs⟩).toFinset.sup id | obtain ⟨a, rfl⟩ := supIrred_iff_of_finite.1 hs | case mk.intro
α : Type u_1
inst✝³ : SemilatticeSup α
inst✝² : OrderBot α
inst✝¹ : Finite α
a : α
hs : SupIrred (LowerSet.Iic a)
inst✝ : Fintype ↥↑⟨LowerSet.Iic a, hs⟩
⊢ supIrredLowerSet.symm ⟨LowerSet.Iic a, hs⟩ = (↑↑⟨LowerSet.Iic a, hs⟩).toFinset.sup id | ef4ee86f319a6f70 |
Real.volume_eq_stieltjes_id | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | theorem volume_eq_stieltjes_id : (volume : Measure ℝ) = StieltjesFunction.id.measure | ⊢ volume = StieltjesFunction.id.measure | haveI : IsAddLeftInvariant StieltjesFunction.id.measure :=
⟨fun a =>
Eq.symm <|
Real.measure_ext_Ioo_rat fun p q => by
simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo,
sub_sub_sub_cancel_right, StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim,
S... | this : StieltjesFunction.id.measure.IsAddLeftInvariant
⊢ volume = StieltjesFunction.id.measure | 27ebd7ab7498006a |
Nat.choose_mul_factorial_mul_factorial | Mathlib/Data/Nat/Choose/Basic.lean | theorem choose_mul_factorial_mul_factorial : ∀ {n k}, k ≤ n → choose n k * k ! * (n - k)! = n !
| 0, _, hk => by simp [Nat.eq_zero_of_le_zero hk]
| n + 1, 0, _ => by simp
| n + 1, succ k, hk => by
rcases lt_or_eq_of_le hk with hk₁ | hk₁
· have h : choose n k * k.succ ! * (n - k)! = (k + 1) * n ! | n k : ℕ
hk : k.succ ≤ n + 1
hk₁ : k.succ < n + 1
⊢ n.choose k * k.succ ! * (n - k)! = (k + 1) * n ! | rw [← choose_mul_factorial_mul_factorial (le_of_succ_le_succ hk)] | n k : ℕ
hk : k.succ ≤ n + 1
hk₁ : k.succ < n + 1
⊢ n.choose k * k.succ ! * (n - k)! = (k + 1) * (n.choose k * k ! * (n - k)!) | 994a3618b788ecd6 |
blimsup_cthickening_mul_ae_eq | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | theorem blimsup_cthickening_mul_ae_eq (p : ℕ → Prop) (s : ℕ → Set α) {M : ℝ} (hM : 0 < M)
(r : ℕ → ℝ) (hr : Tendsto r atTop (𝓝 0)) :
(blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) | α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
∀ (p : ℕ → Prop) {r ... | rw [blimsup_congr (Eventually.of_forall h₀), blimsup_congr (Eventually.of_forall h₁),
blimsup_congr (Eventually.of_forall h₂)] | α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
∀ (p : ℕ → Prop) {r ... | 8749d72cf710712d |
Complex.monotone_ofReal | Mathlib/Data/Complex/Order.lean | lemma monotone_ofReal : Monotone ofReal | x y : ℝ
hxy : x ≤ y
⊢ ↑x ≤ ↑y | simp only [ofRealHom_eq_coe, real_le_real, hxy] | no goals | 70016d4f019a4341 |
lowerSemicontinuousWithinAt_ciSup | Mathlib/Topology/Semicontinuous.lean | theorem lowerSemicontinuousWithinAt_ciSup {f : ι → α → δ'}
(bdd : ∀ᶠ y in 𝓝[s] x, BddAbove (range fun i => f i y))
(h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x | case inl
α : Type u_1
inst✝¹ : TopologicalSpace α
x : α
s : Set α
ι : Sort u_3
δ' : Type u_5
inst✝ : ConditionallyCompleteLinearOrder δ'
f : ι → α → δ'
bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)
h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x
h✝ : IsEmpty ι
⊢ LowerSemicontinuousWithinAt (fun x' =... | simpa only [iSup_of_empty'] using lowerSemicontinuousWithinAt_const | no goals | 83630254c05bce8e |
CategoryTheory.IsPullback.hasLiftingProperty | Mathlib/CategoryTheory/LiftingProperties/Limits.lean | lemma IsPullback.hasLiftingProperty (h : IsPullback s f g t)
{X' Y' : C} (f' : X' ⟶ Y') [HasLiftingProperty f' g] : HasLiftingProperty f' f where
sq_hasLift := fun {u v} sq ↦ by
have w : (u ≫ s) ≫ g = f' ≫ v ≫ t | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ ⋯.lift ≫ g = v ≫ t | rw [CommSq.fac_right] | no goals | 1937b89b23656811 |
MeasureTheory.AECover.integrable_of_integral_norm_bounded | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | theorem AECover.integrable_of_integral_norm_bounded [l.NeBot] [l.IsCountablyGenerated]
{φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
(hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ | α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, ∫ (x : α) in φ i, ‖f x‖ ∂μ ≤ I
h... | conv at hbounded in integral _ _ =>
rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun x => @norm_nonneg E _ (f x))
hfm.norm.restrict] | α : Type u_1
ι : Type u_2
E : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝² : NormedAddCommGroup E
inst✝¹ : l.NeBot
inst✝ : l.IsCountablyGenerated
φ : ι → Set α
hφ : AECover μ l φ
f : α → E
I : ℝ
hfi : ∀ (i : ι), IntegrableOn f (φ i) μ
hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ENNReal.ofRe... | 38f39dc46a6f6df3 |
norm_eq_iInf_iff_real_inner_eq_zero | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem norm_eq_iInf_iff_real_inner_eq_zero (K : Submodule ℝ F) {u : F} {v : F} (hv : v ∈ K) :
(‖u - v‖ = ⨅ w : (↑K : Set F), ‖u - w‖) ↔ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0 :=
Iff.intro
(by
intro h
have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
w' : F := w + v
this : w' ∈ K
h₁ : ⟪u - v, w' - v⟫_ℝ ≤ 0
h₂ : w' - v = w
⊢ ⟪u - v, w⟫_ℝ ≤ 0 | rw [h₂] at h₁ | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
h✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0
w : F
hw : w ∈ K
w' : F := w + v
this : w' ∈ K
h₁ : ⟪u - v, w⟫_ℝ ≤ 0
h₂ : w' - v = w
⊢ ⟪u - v, w⟫_ℝ ≤ 0 | cbfd395ad9083d60 |
FermatLastTheoremForThreeGen.Solution.lambda_pow_dvd_a_add_b | Mathlib/NumberTheory/FLT/Three.lean | /-- We have that `λ ^ (3*S.multiplicity-2)` divides `S.a + S.b`. -/
private lemma lambda_pow_dvd_a_add_b : λ ^ (3 * S.multiplicity - 2) ∣ S.a + S.b | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h :
λ ^ (3 * S.multiplicity) ∣
(S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)
⊢ λ ^ (3 * S.multiplicity - 2) ∣... | apply hζ.zeta_sub_one_prime'.pow_dvd_of_dvd_mul_left _ S.lambda_not_dvd_y | K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
h :
λ ^ (3 * S.multiplicity) ∣
(S.a + S.b) * (λ * FermatLastTheoremForThreeGen.Solution.y S) * (λ * FermatLastTheoremForThreeGen.Solution.z S)
⊢ λ ^ (3 * S.multiplicity - 2) ∣... | ca8f590be4efd444 |
Lean.Data.AC.insert_nonEmpty | Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean | theorem insert_nonEmpty : insert x xs ≠ [] | case cons
x✝ x : Nat
xs : List Nat
tail_ih✝ : insert x✝ xs ≠ []
⊢ insert x✝ (x :: xs) ≠ [] | simp [insert] | case cons
x✝ x : Nat
xs : List Nat
tail_ih✝ : insert x✝ xs ≠ []
⊢ ¬(if x✝ < x then x✝ :: x :: xs else x :: insert x✝ xs) = [] | 67572a68a4fc090d |
inseparable_iff_clusterPt_uniformity | Mathlib/Topology/UniformSpace/Separation.lean | theorem inseparable_iff_clusterPt_uniformity {x y : α} :
Inseparable x y ↔ ClusterPt (x, y) (𝓤 α) | α : Type u
inst✝ : UniformSpace α
x y : α
h : ClusterPt (x, y) (𝓤 α)
⊢ Inseparable x y | simp_rw [uniformity_hasBasis_closed.inseparable_iff_uniformity, isClosed_iff_clusterPt] | α : Type u
inst✝ : UniformSpace α
x y : α
h : ClusterPt (x, y) (𝓤 α)
⊢ ∀ (i : Set (α × α)), (i ∈ 𝓤 α ∧ ∀ (a : α × α), ClusterPt a (𝓟 i) → a ∈ i) → (x, y) ∈ id i | 2c0355318e9e169d |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.clear_insert_inductive_case | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem clear_insert_inductive_case {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(units : Array (Literal (PosFin n))) (units_nodup : ∀ i : Fin units.size, ∀ j : Fin units.size, i ≠ j → units[i] ≠ units[j])
(idx : Fin units.size) (assignments : Array Assignment)
(ih : ClearI... | case right.right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignmen... | simp only [clearUnit, Array.get_eq_getElem] | case right.right.right.right.left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignmen... | c90be518eb885cb1 |
Ideal.radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot | Mathlib/RingTheory/Jacobson/Ideal.lean | theorem radical_eq_jacobson_iff_radical_quotient_eq_jacobson_bot :
I.radical = I.jacobson ↔ radical (⊥ : Ideal (R ⧸ I)) = jacobson ⊥ | case mpr
R : Type u
inst✝ : CommRing R
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : ⊥.radical = ⊥.jacobson
⊢ I.radical = I.jacobson | have := congr_arg (comap (Ideal.Quotient.mk I)) h | case mpr
R : Type u
inst✝ : CommRing R
I : Ideal R
hf : Function.Surjective ⇑(Quotient.mk I)
h : ⊥.radical = ⊥.jacobson
this : comap (Quotient.mk I) ⊥.radical = comap (Quotient.mk I) ⊥.jacobson
⊢ I.radical = I.jacobson | c60dc84e4b282f7d |
HasFDerivAtFilter.iterate | Mathlib/Analysis/Calculus/FDeriv/Comp.lean | theorem HasFDerivAtFilter.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : ℕ) :
HasFDerivAtFilter f^[n] (f' ^ n) x L | case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
L : Filter E
f : E → E
f' : E →L[𝕜] E
hf : HasFDerivAtFilter f f' x L
hL : Tendsto f L L
hx : f x = x
n : ℕ
ihn : HasFDerivAtFilter f^[n] (f' ^ n) x L
⊢ HasFDerivAtFilter (f^[n] ∘ f) (f... | rw [← hx] at ihn | case succ
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : E
L : Filter E
f : E → E
f' : E →L[𝕜] E
hf : HasFDerivAtFilter f f' x L
hL : Tendsto f L L
hx : f x = x
n : ℕ
ihn : HasFDerivAtFilter f^[n] (f' ^ n) (f x) L
⊢ HasFDerivAtFilter (f^[n] ∘ f... | 1115050efc37856a |
Ideal.isPrimary_finset_inf | Mathlib/RingTheory/Ideal/IsPrimary.lean | lemma isPrimary_finset_inf {ι} {s : Finset ι} {f : ι → Ideal R} {i : ι} (hi : i ∈ s)
(hs : ∀ ⦃y⦄, y ∈ s → (f y).IsPrimary)
(hs' : ∀ ⦃y⦄, y ∈ s → (f y).radical = (f i).radical) :
IsPrimary (s.inf f) | case insert.inl
R : Type u_1
inst✝ : CommSemiring R
ι : Type u_2
f : ι → Ideal R
a i : ι
ha : a ∉ ∅
IH :
∀ {i : ι},
i ∈ ∅ →
(∀ ⦃y : ι⦄, y ∈ ∅ → (f y).IsPrimary) → (∀ ⦃y : ι⦄, y ∈ ∅ → (f y).radical = (f i).radical) → (∅.inf f).IsPrimary
hs : ∀ ⦃y : ι⦄, y ∈ insert a ∅ → (f y).IsPrimary
hs' : ∀ ⦃y : ι⦄, y ∈ in... | simpa [hi] using hs | no goals | 50431db21ffc3646 |
Set.exists_subset_encard_eq | Mathlib/Data/Set/Card.lean | theorem exists_subset_encard_eq {k : ℕ∞} (hk : k ≤ s.encard) : ∃ t, t ⊆ s ∧ t.encard = k | α : Type u_1
s : Set α
k : ℕ∞
n : ℕ
IH : ↑n ≤ s.encard → ∃ t ⊆ s, t.encard = ↑n
hle : ↑n.succ ≤ s.encard
⊢ ↑n ≤ ↑n.succ | simp | no goals | 05b71bd015f46ce9 |
EuclideanGeometry.oangle_right_eq_arccos_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem oangle_right_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₂ p₃ p₁ = Real.arccos (dist p₃ p₂ / dist p₁ p₃) | 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
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
h : ∡ p₁ p₂ p₃ = ↑(π / 2)
hs : (∡ p₂ p₃ p₁).sign = 1
⊢ ∡ p₂ p₃ p₁ = ↑(Real.arccos (dist p₃ p₂ / dist p₁ p₃)) | rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arccos_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)] | no goals | 04d931b2b5c6dacc |
List.MergeSort.Internal.mergeTR_go_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Impl.lean | theorem mergeTR_go_eq : mergeTR.go le l₁ l₂ acc = acc.reverse ++ merge l₁ l₂ le | case cons
α✝ : Type u_1
le : α✝ → α✝ → Bool
x : α✝
l₁ : List α✝
ih₁ : ∀ {l₂ acc : List α✝}, mergeTR.go le l₁ l₂ acc = acc.reverse ++ l₁.merge l₂ le
l₂ acc : List α✝
⊢ mergeTR.go le (x :: l₁) l₂ acc = acc.reverse ++ (x :: l₁).merge l₂ le | induction l₂ generalizing acc with
| nil => simp [mergeTR.go, merge, reverseAux_eq]
| cons y l₂ ih₂ =>
simp [mergeTR.go, merge]
split <;> simp [ih₁, ih₂] | no goals | 1b0fc125369876cf |
Int.normalize_of_nonpos | Mathlib/Algebra/GCDMonoid/Nat.lean | theorem normalize_of_nonpos {z : ℤ} (h : z ≤ 0) : normalize z = -z | case inr
z : ℤ
h✝ : z ≤ 0
h : z < 0
⊢ normalize z = -z | rw [normalize_apply, normUnit_eq, if_neg (not_le_of_gt h), Units.val_neg, Units.val_one,
mul_neg_one] | no goals | 0217b9405fd44be1 |
List.filterMap_eq_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem filterMap_eq_cons_iff {l} {b} {bs} :
filterMap f l = b :: bs ↔
∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ (∀ x, x ∈ l₁ → f x = none) ∧ f a = some b ∧
filterMap f l₂ = bs | case mp.cons.none
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → Option α✝
b : α✝
bs : List α✝
a : α✝¹
l : List α✝¹
h : f a = none
w : filterMap f l = b :: bs
ih : ∃ l₁ a l₂, l = l₁ ++ a :: l₂ ∧ (∀ (x : α✝¹), x ∈ l₁ → f x = none) ∧ f a = some b ∧ filterMap f l₂ = bs
⊢ ∃ l₁ a_1 l₂, a :: l = l₁ ++ a_1 :: l₂ ∧ (∀ (x : α✝¹), x ∈ l... | obtain ⟨l₁, a', l₂, rfl, w₁, w₂, w₃⟩ := ih | case mp.cons.none.intro.intro.intro.intro.intro.intro
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → Option α✝
b : α✝
bs : List α✝
a : α✝¹
h : f a = none
l₁ : List α✝¹
a' : α✝¹
l₂ : List α✝¹
w : filterMap f (l₁ ++ a' :: l₂) = b :: bs
w₁ : ∀ (x : α✝¹), x ∈ l₁ → f x = none
w₂ : f a' = some b
w₃ : filterMap f l₂ = bs
⊢ ∃ l₁_1 a_1... | a63f5fdfcdb83803 |
SimplicialObject.Splitting.decomposition_id | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | theorem decomposition_id (Δ : SimplexCategoryᵒᵖ) :
𝟙 (X.obj Δ) = ∑ A : IndexSet Δ, s.πSummand A ≫ (s.cofan Δ).inj A | case h.h₀
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
X : SimplicialObject C
s : Splitting X
inst✝ : Preadditive C
Δ : SimplexCategoryᵒᵖ
A B : IndexSet Δ
a✝ : B ∈ Finset.univ
h₂ : B ≠ A
⊢ (s.cofan Δ).inj A ≫ s.πSummand B ≫ (s.cofan Δ).inj B = 0 | rw [s.cofan_inj_πSummand_eq_zero_assoc _ _ h₂, zero_comp] | no goals | aefa811ca143ddcf |
GenContFract.convs_stable_of_terminated | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | theorem convs_stable_of_terminated (n_le_m : n ≤ m) (terminatedAt_n : g.TerminatedAt n) :
g.convs m = g.convs n | K : Type u_1
g : GenContFract K
n m : ℕ
inst✝ : DivisionRing K
n_le_m : n ≤ m
terminatedAt_n : g.TerminatedAt n
⊢ g.convs m = g.convs n | simp only [convs, dens_stable_of_terminated n_le_m terminatedAt_n,
nums_stable_of_terminated n_le_m terminatedAt_n] | no goals | f497afa64f2b8d89 |
ProbabilityTheory.condExpKernel_apply_eq_condDistrib | Mathlib/Probability/Kernel/Condexp.lean | lemma condExpKernel_apply_eq_condDistrib [Nonempty Ω] {ω : Ω} :
condExpKernel μ m ω = @condDistrib Ω Ω Ω mΩ _ _ mΩ (m ⊓ mΩ) id id μ _ (id ω) | Ω : Type u_1
m mΩ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : Nonempty Ω
ω : Ω
⊢ (condExpKernel μ m) ω = (condDistrib id id μ) (id ω) | simp [condExpKernel_eq, Kernel.comap_apply] | no goals | 23e1e74c8ca3cb5b |
WeierstrassCurve.exists_variableChange_of_char_two_of_j_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_two_of_j_eq_zero
[E.IsCharTwoJEqZeroNF] [E'.IsCharTwoJEqZeroNF] :
∃ C : VariableChange F, E.variableChange C = E' | case h.a₁
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 2
inst✝¹ : E.IsCharTwoJEqZeroNF
inst✝ : E'.IsCharTwoJEqZeroNF
ha₃ : E.a₃ ≠ 0
ha₃' : E'.a₃ ≠ 0
this : NeZero 3
u : F
hu : u ^ 3 = E.a₃ / E'.a₃
s : F
hs : 1 * s ^ 4 + E.a₃... | ring1 | no goals | 596c78b46310dbf8 |
jacobiSym.mod_right' | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | theorem mod_right' (a : ℕ) {b : ℕ} (hb : Odd b) : J(a | b) = J(a | b % (4 * a)) | case inr.intro.intro.intro.e_a.zero
a b : ℕ
hb : Odd b
ha₀ : a ≠ 0
hb' : Odd (b % (4 * a))
a' : ℕ
ha₁' : ¬2 ∣ a'
ha₁ : Odd a'
ha₂ : a = 2 ^ 0 * a'
⊢ χ₈ ↑b ^ 0 = χ₈ ↑(b % (4 * a)) ^ 0 | rfl | no goals | 3b374c961388648d |
isLocalHom_of_le_jacobson_bot | Mathlib/RingTheory/Henselian.lean | theorem isLocalHom_of_le_jacobson_bot {R : Type*} [CommRing R] (I : Ideal R)
(h : I ≤ Ideal.jacobson ⊥) : IsLocalHom (Ideal.Quotient.mk I) | case map_nonunit.intro.mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R
h✝ : I ≤ ⊥.jacobson
a : R
h : IsUnit ((Ideal.Quotient.mk I) a)
y : R
h1 : (Ideal.Quotient.mk ⊥.jacobson) a * (Ideal.Quotient.mk ⊥.jacobson) y = 1
h2 : (Ideal.Quotient.mk ⊥.jacobson) y * (Ideal.Quotient.mk ⊥.jacobson) a = 1
⊢ IsUnit a | rw [← (Ideal.Quotient.mk _).map_mul, ← (Ideal.Quotient.mk _).map_one, Ideal.Quotient.eq,
Ideal.mem_jacobson_bot] at h1 h2 | case map_nonunit.intro.mk.intro
R : Type u_1
inst✝ : CommRing R
I : Ideal R
h✝ : I ≤ ⊥.jacobson
a : R
h : IsUnit ((Ideal.Quotient.mk I) a)
y : R
h1 : ∀ (y_1 : R), IsUnit ((a * y - 1) * y_1 + 1)
h2 : ∀ (y_1 : R), IsUnit ((y * a - 1) * y_1 + 1)
⊢ IsUnit a | a961c7da2da42bc3 |
Filter.EventuallyEq.fderivWithin_eq | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | theorem Filter.EventuallyEq.fderivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f f₁ : E → F
x : E
s : Set E
hs : f₁ =ᶠ[𝓝[s] x] f
hx : f₁ x = f x
⊢ fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x | simp only [fderivWithin, DifferentiableWithinAt, hs.hasFDerivWithinAt_iff hx] | no goals | 59b54b8ec0db62c8 |
Int.clog_of_right_le_zero | Mathlib/Data/Int/Log.lean | theorem clog_of_right_le_zero (b : ℕ) {r : R} (hr : r ≤ 0) : clog b r = 0 | case inr
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
r : R
hr : r ≤ 0
hb : 1 < b
⊢ ⌊r⁻¹⌋₊ ≤ 1 | exact Nat.floor_le_one_of_le_one ((inv_nonpos.2 hr).trans zero_le_one) | no goals | 8bce8832680ba9f2 |
Fin.succ_cast_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean | theorem succ_cast_eq {n' : Nat} (i : Fin n) (h : n = n') :
(i.cast h).succ = i.succ.cast (by rw [h]) := rfl
| n n' : Nat
i : Fin n
h : n = n'
⊢ n + 1 = n' + 1 | rw [h] | no goals | bfea8d2a92f9243e |
Polynomial.exists_partition_polynomial_aux | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | theorem exists_partition_polynomial_aux (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : Fq[X]} (hb : b ≠ 0)
(A : Fin n → Fq[X]) : ∃ t : Fin n → Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊),
∀ i₀ i₁ : Fin n, t i₀ = t i₁ ↔
(cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε | case intro.intro.intro.refine_2.refine_1
Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
ε : ℝ
hε : 0 < ε
b : Fq[X]
hb : b ≠ 0
hbε : 0 < cardPowDegree b • ε
n : ℕ
ih :
∀ (A : Fin n → Fq[X]),
∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε
A : Fin (n + 1) → Fq[... | rw [Fin.cons_succ, Fin.cons_zero, ← not_le] at approx | case intro.intro.intro.refine_2.refine_1
Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Field Fq
ε : ℝ
hε : 0 < ε
b : Fq[X]
hb : b ≠ 0
hbε : 0 < cardPowDegree b • ε
n : ℕ
ih :
∀ (A : Fin n → Fq[X]),
∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε
A : Fin (n + 1) → Fq[... | d529c9a5494d9218 |
SimpleGraph.colorable_iff_exists_bdd_nat_coloring | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem colorable_iff_exists_bdd_nat_coloring (n : ℕ) :
G.Colorable n ↔ ∃ C : G.Coloring ℕ, ∀ v, C v < n | case mpr.intro.refine_1
V : Type u
G : SimpleGraph V
n : ℕ
C : G.Coloring ℕ
Cf : ∀ (v : V), C v < n
⊢ V → Fin n | exact fun v => ⟨C v, Cf v⟩ | no goals | d1efc174a6faab8a |
isEmpty_sigma | Mathlib/Logic/IsEmpty.lean | theorem isEmpty_sigma {α} {E : α → Type*} : IsEmpty (Sigma E) ↔ ∀ a, IsEmpty (E a) | α : Type u_5
E : α → Type u_4
⊢ IsEmpty (Sigma E) ↔ ∀ (a : α), IsEmpty (E a) | simp only [← not_nonempty_iff, nonempty_sigma, not_exists] | no goals | 3dc476767ef74be6 |
Set.Icc_top | Mathlib/Order/Interval/Set/Basic.lean | theorem Icc_top : Icc a ⊤ = Ici a | α : Type u_1
inst✝¹ : Preorder α
inst✝ : OrderTop α
a : α
⊢ Icc a ⊤ = Ici a | simp [← Ici_inter_Iic] | no goals | d75f3877112ae8bf |
ENNReal.tsum_fiberwise | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem tsum_fiberwise (f : β → ℝ≥0∞) (g : β → γ) :
∑' x, ∑' b : g ⁻¹' {x}, f b = ∑' i, f i | β : Type u_2
γ : Type u_3
f : β → ℝ≥0∞
g : β → γ
⊢ ∑' (x : γ) (b : ↑(g ⁻¹' {x})), f ↑b = ∑' (i : β), f i | apply HasSum.tsum_eq | case ha
β : Type u_2
γ : Type u_3
f : β → ℝ≥0∞
g : β → γ
⊢ HasSum (fun b => ∑' (b_1 : ↑(g ⁻¹' {b})), f ↑b_1) (∑' (i : β), f i) | 1774e3c0853c01ba |
IsZGroup.of_injective | Mathlib/GroupTheory/SpecificGroups/ZGroup.lean | theorem of_injective [hG' : IsZGroup G'] (hf : Function.Injective f) : IsZGroup G | G : Type u_1
G' : Type u_2
inst✝¹ : Group G
inst✝ : Group G'
f : G →* G'
hG' : ∀ (p : ℕ), Nat.Prime p → ∀ (P : Sylow p G'), IsCyclic ↥↑P
hf : Function.Injective ⇑f
⊢ ∀ (p : ℕ), Nat.Prime p → ∀ (P : Sylow p G), IsCyclic ↥↑P | intro p hp P | G : Type u_1
G' : Type u_2
inst✝¹ : Group G
inst✝ : Group G'
f : G →* G'
hG' : ∀ (p : ℕ), Nat.Prime p → ∀ (P : Sylow p G'), IsCyclic ↥↑P
hf : Function.Injective ⇑f
p : ℕ
hp : Nat.Prime p
P : Sylow p G
⊢ IsCyclic ↥↑P | c14359612b1097e9 |
Finset.aestronglyMeasurable_prod | Mathlib/MeasureTheory/Function/StronglyMeasurable/AEStronglyMeasurable.lean | theorem _root_.Finset.aestronglyMeasurable_prod {ι : Type*} {f : ι → α → M} (s : Finset ι)
(hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ) :
AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ | α : Type u_1
m₀ : MeasurableSpace α
μ : Measure α
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
ι : Type u_6
f : ι → α → M
s : Finset ι
hf : ∀ i ∈ s, AEStronglyMeasurable (f i) μ
⊢ AEStronglyMeasurable (fun a => ∏ i ∈ s, f i a) μ | simpa only [← Finset.prod_apply] using s.aestronglyMeasurable_prod' hf | no goals | 9e433ccebe899aad |
Quaternion.normSq_add | Mathlib/Algebra/Quaternion.lean | theorem normSq_add (a b : ℍ[R]) : normSq (a + b) = normSq a + normSq b + 2 * (a * star b).re :=
calc
normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b) | R : Type u_3
inst✝ : CommRing R
a b : ℍ[R]
⊢ normSq (a + b) = normSq a + (a * star b).re + ((b * star a).re + normSq b) | simp_rw [normSq_def, star_add, add_mul, mul_add, add_re] | no goals | f3df2da162650457 |
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