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 |
|---|---|---|---|---|---|---|
WittVector.irreducible | Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean | theorem irreducible : Irreducible (p : 𝕎 k) | p : ℕ
hp✝ : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : CharP k p
hp : ¬IsUnit ↑p
a b : 𝕎 k
hab : ↑p = a * b
⊢ a ≠ 0 ∧ b ≠ 0 | rw [← mul_ne_zero_iff] | p : ℕ
hp✝ : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : CharP k p
hp : ¬IsUnit ↑p
a b : 𝕎 k
hab : ↑p = a * b
⊢ a * b ≠ 0 | bbcbb7f9941815f7 |
Multiset.lt_replicate_succ | Mathlib/Data/Multiset/Replicate.lean | theorem lt_replicate_succ {m : Multiset α} {x : α} {n : ℕ} :
m < replicate (n + 1) x ↔ m ≤ replicate n x | case mpr
α : Type u_1
m : Multiset α
x : α
n : ℕ
h : m ≤ replicate n x
⊢ ∃ a, a ::ₘ m ≤ replicate (n + 1) x | rw [replicate_succ] | case mpr
α : Type u_1
m : Multiset α
x : α
n : ℕ
h : m ≤ replicate n x
⊢ ∃ a, a ::ₘ m ≤ x ::ₘ replicate n x | c4daba5a64d21b93 |
Units.map_id | Mathlib/Algebra/Group/Units/Hom.lean | theorem map_id : map (MonoidHom.id M) = MonoidHom.id Mˣ | M : Type u
inst✝ : Monoid M
⊢ map (MonoidHom.id M) = MonoidHom.id Mˣ | ext | case h.a
M : Type u
inst✝ : Monoid M
x✝ : Mˣ
⊢ ↑((map (MonoidHom.id M)) x✝) = ↑((MonoidHom.id Mˣ) x✝) | 1b22afc8edfce4c1 |
exists_associated_pow_of_associated_pow_mul | Mathlib/RingTheory/PrincipalIdealDomain.lean | theorem exists_associated_pow_of_associated_pow_mul {a b c : R} (hab : IsCoprime a b) {k : ℕ}
(h : Associated (c ^ k) (a * b)) : ∃ d : R, Associated (d ^ k) a | R : Type u
inst✝² : CommRing R
inst✝¹ : IsBezout R
inst✝ : IsDomain R
a b c : R
hab : IsCoprime a b
k : ℕ
h : Associated (c ^ k) (a * b)
⊢ ∃ d, Associated (d ^ k) a | obtain ⟨u, hu⟩ := h.symm | case intro
R : Type u
inst✝² : CommRing R
inst✝¹ : IsBezout R
inst✝ : IsDomain R
a b c : R
hab : IsCoprime a b
k : ℕ
h : Associated (c ^ k) (a * b)
u : Rˣ
hu : a * b * ↑u = c ^ k
⊢ ∃ d, Associated (d ^ k) a | bc8e53465e1fd9a4 |
Lean.Order.PProd.chain.chain_snd | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Basic.lean | theorem PProd.chain.chain_snd [CCPO α] [CCPO β] {c : α ×' β → Prop} (hchain : chain c) :
chain (chain.snd c) | α : Sort u
β : Sort v
inst✝¹ : CCPO α
inst✝ : CCPO β
c : α ×' β → Prop
hchain : chain c
⊢ chain (snd c) | intro b₁ b₂ ⟨a₁, h₁⟩ ⟨a₂, h₂⟩ | α : Sort u
β : Sort v
inst✝¹ : CCPO α
inst✝ : CCPO β
c : α ×' β → Prop
hchain : chain c
b₁ b₂ : β
a₁ : α
h₁ : c ⟨a₁, b₁⟩
a₂ : α
h₂ : c ⟨a₂, b₂⟩
⊢ b₁ ⊑ b₂ ∨ b₂ ⊑ b₁ | 2afa3a24950d92b0 |
wellQuasiOrdered_iff_exists_monotone_subseq | Mathlib/Order/WellQuasiOrder.lean | theorem wellQuasiOrdered_iff_exists_monotone_subseq [IsPreorder α r] :
WellQuasiOrdered r ↔ ∀ f : ℕ → α, ∃ g : ℕ ↪o ℕ, ∀ m n : ℕ, m ≤ n → r (f (g m)) (f (g n)) | case mpr
α : Type u_1
r : α → α → Prop
inst✝ : IsPreorder α r
h : ∀ (f : ℕ → α), ∃ g, ∀ (m n : ℕ), m ≤ n → r (f (g m)) (f (g n))
f : ℕ → α
⊢ ∃ m n, m < n ∧ r (f m) (f n) | obtain ⟨g, gmon⟩ := h f | case mpr.intro
α : Type u_1
r : α → α → Prop
inst✝ : IsPreorder α r
h : ∀ (f : ℕ → α), ∃ g, ∀ (m n : ℕ), m ≤ n → r (f (g m)) (f (g n))
f : ℕ → α
g : ℕ ↪o ℕ
gmon : ∀ (m n : ℕ), m ≤ n → r (f (g m)) (f (g n))
⊢ ∃ m n, m < n ∧ r (f m) (f n) | 7dbc797b6874056a |
MulAction.mem_stabilizer_set' | Mathlib/Algebra/Pointwise/Stabilizer.lean | @[to_additive]
lemma mem_stabilizer_set' {s : Set α} (hs : s.Finite) :
a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s | case intro
G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
a : G
s : Finset α
⊢ a ∈ stabilizer G ↑s ↔ ∀ ⦃b : α⦄, b ∈ ↑s → a • b ∈ ↑s | classical simp [-mem_stabilizer_iff, mem_stabilizer_finset'] | no goals | e7c951482f224830 |
MeasureTheory.le_iInf₂_lintegral | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem le_iInf₂_lintegral {ι : Sort*} {ι' : ι → Sort*} (f : ∀ i, ι' i → α → ℝ≥0∞) :
∫⁻ a, ⨅ (i) (h : ι' i), f i h a ∂μ ≤ ⨅ (i) (h : ι' i), ∫⁻ a, f i h a ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Sort u_5
ι' : ι → Sort u_6
f : (i : ι) → ι' i → α → ℝ≥0∞
⊢ ∫⁻ (a : α), ⨅ i, ⨅ h, f i h a ∂μ ≤ ⨅ i, ⨅ h, ∫⁻ (a : α), f i h a ∂μ | convert (monotone_lintegral μ).map_iInf₂_le f with a | case h.e'_3.h.e'_4.h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Sort u_5
ι' : ι → Sort u_6
f : (i : ι) → ι' i → α → ℝ≥0∞
a : α
⊢ ⨅ i, ⨅ h, f i h a = (⨅ i, ⨅ j, f i j) a | 7d117a4e84758191 |
Finset.card_Icc_finset | Mathlib/Data/Finset/Interval.lean | theorem card_Icc_finset (h : s ⊆ t) : (Icc s t).card = 2 ^ (t.card - s.card) | α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
h : s ⊆ t
u : Finset α
hu : u ∈ ↑(t \ s).powerset
v : Finset α
hv : v ∈ ↑(t \ s).powerset
huv : s ⊔ u = s ⊔ v
⊢ u = v | rw [mem_coe, mem_powerset] at hu hv | α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
h : s ⊆ t
u : Finset α
hu : u ⊆ t \ s
v : Finset α
hv : v ⊆ t \ s
huv : s ⊔ u = s ⊔ v
⊢ u = v | 0e69e1d78dcfe219 |
Std.DHashMap.Internal.Raw₀.Const.isHashSelf_updateBucket_alter | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem isHashSelf_updateBucket_alter [BEq α] [EquivBEq α] [Hashable α] [LawfulHashable α]
{m : Raw₀ α (fun _ => β)} (h : Raw.WFImp m.1) {a : α} {f : Option β → Option β} :
IsHashSelf (updateBucket m.1.buckets m.2 a (AssocList.Const.alter a f)) | α : Type u
β : Type v
inst✝³ : BEq α
inst✝² : EquivBEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Raw₀ α fun x => β
h : Raw.WFImp m.val
a : α
f : Option β → Option β
l : AssocList α fun x => β
p : (_ : α) × β
hp : p ∈ (AssocList.Const.alter a f l).toList
⊢ containsKey p.fst l.toList = true ∨ hash p.fst = hash a | rw [AssocList.Const.toList_alter.mem_iff] at hp | α : Type u
β : Type v
inst✝³ : BEq α
inst✝² : EquivBEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Raw₀ α fun x => β
h : Raw.WFImp m.val
a : α
f : Option β → Option β
l : AssocList α fun x => β
p : (_ : α) × β
hp : p ∈ Const.alterKey a f l.toList
⊢ containsKey p.fst l.toList = true ∨ hash p.fst = hash a | 3a3a3bad02424c04 |
Finset.prod_filter_mul_prod_filter_not | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem prod_filter_mul_prod_filter_not
(s : Finset α) (p : α → Prop) [DecidablePred p] [∀ x, Decidable (¬p x)] (f : α → β) :
(∏ x ∈ s with p x, f x) * ∏ x ∈ s with ¬p x, f x = ∏ x ∈ s, f x | α : Type u_3
β : Type u_4
inst✝² : CommMonoid β
s : Finset α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : (x : α) → Decidable ¬p x
f : α → β
⊢ (∏ x ∈ filter (fun x => p x) s, f x) * ∏ x ∈ filter (fun x => ¬p x) s, f x = ∏ x ∈ s, f x | have := Classical.decEq α | α : Type u_3
β : Type u_4
inst✝² : CommMonoid β
s : Finset α
p : α → Prop
inst✝¹ : DecidablePred p
inst✝ : (x : α) → Decidable ¬p x
f : α → β
this : DecidableEq α
⊢ (∏ x ∈ filter (fun x => p x) s, f x) * ∏ x ∈ filter (fun x => ¬p x) s, f x = ∏ x ∈ s, f x | cd0e75d5f61fce7d |
IsDenseInducing.extend_Z_bilin_key | Mathlib/Topology/Algebra/UniformGroup/Basic.lean | theorem extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
∀ x ∈ U, ∀ x' ∈ U, ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
(fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W' | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
in... | rw [← nhds_prod_eq] at lim_sub_sub | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹² : TopologicalSpace α
inst✝¹¹ : AddCommGroup α
inst✝¹⁰ : IsTopologicalAddGroup α
inst✝⁹ : TopologicalSpace β
inst✝⁸ : AddCommGroup β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : AddCommGroup γ
inst✝⁵ : IsTopologicalAddGroup γ
inst✝⁴ : TopologicalSpace δ
in... | 965b7a7f03c398d2 |
ApproximatesLinearOn.norm_fderiv_sub_le | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem _root_.ApproximatesLinearOn.norm_fderiv_sub_le {A : E →L[ℝ] E} {δ : ℝ≥0}
(hf : ApproximatesLinearOn f A s δ) (hs : MeasurableSet s) (f' : E → E →L[ℝ] E)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : ∀ᵐ x ∂μ.restrict s, ‖f' x - A‖₊ ≤ δ | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, ... | simp only [add_zero, mul_zero] at this | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
A : E →L[ℝ] E
δ : ℝ≥0
hf : ApproximatesLinearOn f A s δ
hs : MeasurableSet s
f' : E → E →L[ℝ] E
hf' : ∀ x ∈ s, ... | 99ae4fc7a76b96c9 |
MeasureTheory.IsStoppingTime.measurableSet_gt' | Mathlib/Probability/Process/Stopping.lean | theorem measurableSet_gt' (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[hτ.measurableSpace] {ω | i < τ ω} | case h
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → ι
hτ : IsStoppingTime f τ
i : ι
ω : Ω
⊢ ω ∈ {ω | i < τ ω} ↔ ω ∈ {ω | τ ω ≤ i}ᶜ | simp | no goals | 5f50fd378320aa23 |
AlgebraicGeometry.universallyClosed_eq_universallySpecializing | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | lemma universallyClosed_eq_universallySpecializing :
@UniversallyClosed = (topologically @SpecializingMap).universally ⊓ @QuasiCompact | case a
⊢ (topologically @SpecializingMap ⊓ @QuasiCompact).universally ≤ (topologically @IsClosedMap).universally | exact universally_mono fun X Y f ⟨h₁, h₂⟩ ↦ (isClosedMap_iff_specializingMap _).mpr h₁ | no goals | 862e4b8e16a6249b |
RightDerivMeasurableAux.differentiable_set_subset_D | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem differentiable_set_subset_D :
{ x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } ⊆ D f K | case intro.intro
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
⊢ x ∈ ⋃ n... | obtain ⟨n, hn⟩ : ∃ n : ℕ, (1 / 2) ^ n < R :=
exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : ℝ) / 2 < 1) | case intro.intro.intro
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
e : ℕ
this : 0 < (1 / 2) ^ e
R : ℝ
R_pos : R > 0
hR : ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ((1 / 2) ^ e)
n :... | 02ae9336af5f07d4 |
mem_nhds_induced | Mathlib/Topology/Order.lean | theorem mem_nhds_induced [T : TopologicalSpace α] (f : β → α) (a : β) (s : Set β) :
s ∈ @nhds β (TopologicalSpace.induced f T) a ↔ ∃ u ∈ 𝓝 (f a), f ⁻¹' u ⊆ s | α : Type u
β : Type v
T : TopologicalSpace α
f : β → α
a : β
s : Set β
this : TopologicalSpace β := induced f T
⊢ s ∈ 𝓝 a ↔ ∃ u ∈ 𝓝 (f a), f ⁻¹' u ⊆ s | simp_rw [mem_nhds_iff, isOpen_induced_iff] | α : Type u
β : Type v
T : TopologicalSpace α
f : β → α
a : β
s : Set β
this : TopologicalSpace β := induced f T
⊢ (∃ t ⊆ s, (∃ t_1, IsOpen t_1 ∧ f ⁻¹' t_1 = t) ∧ a ∈ t) ↔ ∃ u, (∃ t ⊆ u, IsOpen t ∧ f a ∈ t) ∧ f ⁻¹' u ⊆ s | 64abb6c0366449b9 |
AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | /-- The main proof of the lower bound part of the Akra-Bazzi theorem. The factor
`1 + ε n` does not change the asymptotic order, but is needed for the induction step to go
through. -/
lemma smoothingFn_mul_asympBound_isBigO_T :
(fun (n : ℕ) => (1 + ε n) * asympBound g a b n) =O[atTop] T | case bc.h.h.a0
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
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_... | exact Or.inl ⟨R.g_nonneg j (by positivity), by positivity⟩ | no goals | 82bb51116c3f899e |
IsCyclotomicExtension.union_right | Mathlib/NumberTheory/Cyclotomic/Basic.lean | theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B | S T : Set ℕ+
A : Type u
B : Type v
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : Algebra A B
h : IsCyclotomicExtension (S ∪ T) A B
this : {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} = {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, b ^ ↑n = 1}
⊢ IsCyclotomicExtension T (↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1})) B | refine ⟨fun hn => ((isCyclotomicExtension_iff _ A _).1 h).1 (mem_union_right S hn), fun b => ?_⟩ | S T : Set ℕ+
A : Type u
B : Type v
inst✝² : CommRing A
inst✝¹ : CommRing B
inst✝ : Algebra A B
h : IsCyclotomicExtension (S ∪ T) A B
this : {b | ∃ n ∈ S ∪ T, b ^ ↑n = 1} = {b | ∃ n ∈ S, b ^ ↑n = 1} ∪ {b | ∃ n ∈ T, b ^ ↑n = 1}
b : B
⊢ b ∈ adjoin ↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1}) {b | ∃ n ∈ T, b ^ ↑n = 1} | e81df93a8ce1e666 |
Batteries.RBSet.findP?_insert_of_ne | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem findP?_insert_of_ne [@TransCmp α cmp] [IsStrictCut cmp cut]
(t : RBSet α cmp) (h : cut v ≠ .eq) : (t.insert v).findP? cut = t.findP? cut | α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
v : α
inst✝¹ : TransCmp cmp
inst✝ : IsStrictCut cmp cut
t : RBSet α cmp
h : cut v ≠ Ordering.eq
u : α
h' : cut u = Ordering.eq
⊢ t.find? v ≠ some u | exact mt (fun h => by rwa [IsCut.congr (cut := cut) (find?_some_eq_eq h)]) h | no goals | 885241d4721d2736 |
isJacobsonRing_localization | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isJacobsonRing_localization [H : IsJacobsonRing R] : IsJacobsonRing S | case pos
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
y : R
inst✝¹ : Algebra R S
inst✝ : Away y S
H : IsJacobsonRing R
P' : Ideal S
hP'✝ : P'.IsPrime
hP' : (Ideal.comap (algebraMap R S) P').IsPrime
hPM : Disjoint ↑(powers y) ↑(Ideal.comap (algebraMap R S) P')
hP : (Ideal.comap (algebraMap R S) P').... | exact J.mul_mem_left x h | no goals | 4266ea15949c3cae |
Polynomial.IsPrimitive.dvd_of_fraction_map_dvd_fraction_map | Mathlib/RingTheory/Polynomial/GaussLemma.lean | theorem IsPrimitive.dvd_of_fraction_map_dvd_fraction_map {p q : R[X]} (hp : p.IsPrimitive)
(hq : q.IsPrimitive) (h_dvd : p.map (algebraMap R K) ∣ q.map (algebraMap R K)) : p ∣ q | case intro.intro.mk
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
r : K[X]
hr : map (algebraMap R K) q = map (algebraMap R K) p * r
s : R
hs : map (algebra... | simp [s0, mem_nonZeroDivisors_iff_ne_zero] | no goals | e016a9b0390ddacb |
linearIndependent_sum | Mathlib/LinearAlgebra/LinearIndependent/Basic.lean | theorem linearIndependent_sum {v : ι ⊕ ι' → M} :
LinearIndependent R v ↔
LinearIndependent R (v ∘ Sum.inl) ∧
LinearIndependent R (v ∘ Sum.inr) ∧
Disjoint (Submodule.span R (range (v ∘ Sum.inl)))
(Submodule.span R (range (v ∘ Sum.inr))) | case refine_2.intro.intro.inl
ι : Type u'
ι' : Type u_1
R : Type u_2
M : Type u_4
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
v : ι ⊕ ι' → M
hl : ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • (v ∘ Sum.inl) i = 0 → ∀ i ∈ s, g i = 0
hr : ∀ (s : Finset ι') (g : ι' → R), ∑ i ∈ s, g i • (v ∘ Sum.inr) i = 0 → ∀... | exact hl _ _ A i (Finset.mem_preimage.2 hi) | no goals | 2518c65a54e13190 |
CategoryTheory.MorphismProperty.map_id_eq_isoClosure | Mathlib/CategoryTheory/MorphismProperty/Basic.lean | lemma map_id_eq_isoClosure (P : MorphismProperty C) :
P.map (𝟭 _) = P.isoClosure | case a
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
⊢ P ≤ P.isoClosure.inverseImage (𝟭 C) | intro X Y f hf | case a
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : P f
⊢ P.isoClosure.inverseImage (𝟭 C) f | be21352197170292 |
MeasureTheory.SimpleFunc.const_lintegral | Mathlib/MeasureTheory/Function/SimpleFunc.lean | theorem const_lintegral (c : ℝ≥0∞) : (const α c).lintegral μ = c * μ univ | case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
h✝ : Nonempty α
⊢ c * μ ((fun x => c) ⁻¹' {c}) = c * μ univ | rw [preimage_const_of_mem (mem_singleton c)] | no goals | 4222a64e67e68cc4 |
IsPrimePow.factorization_minFac_ne_zero | Mathlib/Data/Nat/Factorization/PrimePow.lean | lemma IsPrimePow.factorization_minFac_ne_zero {n : ℕ} (hn : IsPrimePow n) :
n.factorization n.minFac ≠ 0 | n : ℕ
hn : IsPrimePow n
⊢ ¬(¬Nat.Prime n.minFac ∨ ¬n.minFac ∣ n ∨ n = 0) | push_neg | n : ℕ
hn : IsPrimePow n
⊢ Nat.Prime n.minFac ∧ n.minFac ∣ n ∧ n ≠ 0 | f87a3b63f3edf769 |
HasDerivWithinAt.clm_apply | Mathlib/Analysis/Calculus/Deriv/Mul.lean | theorem HasDerivWithinAt.clm_apply (hc : HasDerivWithinAt c c' s x)
(hu : HasDerivWithinAt u u' s x) :
HasDerivWithinAt (fun y => (c y) (u y)) (c' (u x) + c x u') s x | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type v
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
G : Type u_2
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
c : 𝕜 → F →L[𝕜] G
c' : F →L[𝕜] G
u : 𝕜 → F
u' : F
hc : HasDerivWithinAt c c' s x
hu : HasDerivWithinAt u u' s x
thi... | rwa [add_apply, comp_apply, flip_apply, smulRight_apply, smulRight_apply, one_apply, one_smul,
one_smul, add_comm] at this | no goals | b2df66ae9b2ccd18 |
SupIrred.finset_sup_eq | Mathlib/Order/Irreducible.lean | theorem SupIrred.finset_sup_eq (ha : SupIrred a) (h : s.sup f = a) : ∃ i ∈ s, f i = a | case empty
ι : Type u_1
α : Type u_2
inst✝¹ : SemilatticeSup α
a : α
inst✝ : OrderBot α
s : Finset ι
f : ι → α
ha : SupIrred a
h : ∅.sup f = a
⊢ ∃ i ∈ ∅, f i = a | simpa [ha.ne_bot] using h.symm | no goals | 4acb7029d3da247f |
ConvexOn.continuousOn_tfae | Mathlib/Analysis/Convex/Continuous.lean | lemma ConvexOn.continuousOn_tfae (hC : IsOpen C) (hC' : C.Nonempty) (hf : ConvexOn ℝ C f) : TFAE [
LocallyLipschitzOn C f,
ContinuousOn f C,
∃ x₀ ∈ C, ContinuousAt f x₀,
∃ x₀ ∈ C, (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoun... | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
x₀ : E
hx₀ : x₀ ∈ C
h : ContinuousAt f x₀
⊢ f x₀ < f x₀ + 1 | simp | no goals | d7a82928a7cc2bf5 |
IsMulFreimanHom.comp | Mathlib/Combinatorics/Additive/FreimanHom.lean | @[to_additive] lemma IsMulFreimanHom.comp (hg : IsMulFreimanHom n B C g)
(hf : IsMulFreimanHom n A B f) : IsMulFreimanHom n A C (g ∘ f) where
mapsTo := hg.mapsTo.comp hf.mapsTo
map_prod_eq_map_prod s t hsA htA hs ht h | α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.... | rw [← map_map, ← map_map] | α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : CommMonoid α
inst✝¹ : CommMonoid β
inst✝ : CommMonoid γ
A : Set α
B : Set β
C : Set γ
f : α → β
g : β → γ
n : ℕ
hg : IsMulFreimanHom n B C g
hf : IsMulFreimanHom n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = n
ht : t.... | df817a6de52f5c2c |
Wbtw.collinear | Mathlib/Analysis/Convex/Between.lean | theorem Wbtw.collinear {x y z : P} (h : Wbtw R x y z) : Collinear R ({x, y, z} : Set P) | case inr.inl.intro.intro
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
x z : P
t : R
⊢ ∃ r, (lineMap x z) t = r • (z -ᵥ x) +ᵥ x | exact ⟨t, rfl⟩ | no goals | d43d75b5cd5fb4e5 |
Fin.inv_partialProd_mul_eq_contractNth | Mathlib/Algebra/BigOperators/Fin.lean | theorem inv_partialProd_mul_eq_contractNth {G : Type*} [Group G] (g : Fin (n + 1) → G)
(j : Fin (n + 1)) (k : Fin n) :
(partialProd g (j.succ.succAbove (Fin.castSucc k)))⁻¹ * partialProd g (j.succAbove k).succ =
j.contractNth (· * ·) g k | case inr.inr.h
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ j.succ ≤ k.castSucc.castSucc | rw [le_iff_val_le_val, val_succ] | case inr.inr.h
n : ℕ
G : Type u_3
inst✝ : Group G
g : Fin (n + 1) → G
j : Fin (n + 1)
k : Fin n
h : ↑j < ↑k
⊢ ↑j + 1 ≤ ↑k.castSucc.castSucc | f84a17ef32860b19 |
zero_lt_one_add_norm_sq' | Mathlib/Analysis/Normed/Group/Basic.lean | theorem zero_lt_one_add_norm_sq' (x : E) : 0 < 1 + ‖x‖ ^ 2 | E : Type u_5
inst✝ : SeminormedGroup E
x : E
⊢ 0 < 1 + ‖x‖ ^ 2 | positivity | no goals | b35ff64791d12e79 |
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable | Mathlib/MeasureTheory/Integral/Layercake.lean | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | ... | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0 ..s, g t... | convert measure_empty (μ := μ) | case h.e'_2.h.e'_6
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0
H2 : ∀ s > 0, 0 < ∫ (t... | 6098eac6b53bede3 |
MultilinearMap.map_sum_finset_aux | Mathlib/LinearAlgebra/Multilinear/Basic.lean | theorem map_sum_finset_aux [DecidableEq ι] [Fintype ι] {n : ℕ} (h : (∑ i, #(A i)) = n) :
(f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i => g i (r i) | R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
f : MultilinearMap R M₁ M₂
α : ι → Type u_1
g : (i : ι) → α i → M₁ i
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
this : (i : ι)... | congr with i | case h.e_6.h.h
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
f : MultilinearMap R M₁ M₂
α : ι → Type u_1
g : (i : ι) → α i → M₁ i
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι... | 981a316f387f732a |
MeasureTheory.tendsto_lintegral_filter_of_dominated_convergence | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem tendsto_lintegral_filter_of_dominated_convergence {ι} {l : Filter ι}
[l.IsCountablyGenerated] {F : ι → α → ℝ≥0∞} {f : α → ℝ≥0∞} (bound : α → ℝ≥0∞)
(hF_meas : ∀ᶠ n in l, Measurable (F n)) (h_bound : ∀ᶠ n in l, ∀ᵐ a ∂μ, F n a ≤ bound a)
(h_fin : ∫⁻ a, bound a ∂μ ≠ ∞) (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n =... | α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_5
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => ... | rcases h with ⟨k, h⟩ | case intro
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_5
l : Filter ι
inst✝ : l.IsCountablyGenerated
F : ι → α → ℝ≥0∞
f bound : α → ℝ≥0∞
hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)
h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a
h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤
h_lim : ∀ᵐ (a : α) ∂μ, Tendsto... | 879da8b2b29c96c4 |
Nat.primeFactorsList_prime | Mathlib/Data/Nat/Factors.lean | theorem primeFactorsList_prime {p : ℕ} (hp : Nat.Prime p) : p.primeFactorsList = [p] | p : ℕ
hp : Prime p
this✝ : p = p - 2 + 2
this : p.minFac = p
⊢ p.minFac :: (p / p.minFac).primeFactorsList = [p] | simp only [this, primeFactorsList, Nat.div_self (Nat.Prime.pos hp)] | no goals | 671d32de56a629cc |
Submodule.spanRank_toENat_eq_iInf_finset_card | Mathlib/Algebra/Module/SpanRank.lean | lemma spanRank_toENat_eq_iInf_finset_card (p : Submodule R M) :
p.spanRank.toENat =
⨅ (s : {s : Set M // s.Finite ∧ span R s = p}), (s.2.1.toFinset.card : ℕ∞) | case inr
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤
⊢ ⨅ s, ⨅ (_ : span R s = p), s.encard = ⨅ s, ↑⋯.toFinset.card | apply le_antisymm | case inr.a
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
p : Submodule R M
h2 : ⨅ s, ⨅ (_ : span R s = p), s.encard ≠ ⊤
⊢ ⨅ s, ⨅ (_ : span R s = p), s.encard ≤ ⨅ s, ↑⋯.toFinset.card
case inr.a
R : Type u_1
M : Type u
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module ... | f74636938a98e2b3 |
CategoryTheory.MorphismProperty.pullbacks_le | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | lemma pullbacks_le [P.IsStableUnderBaseChange] : P.pullbacks ≤ P | C : Type u
inst✝¹ : Category.{v, u} C
P : MorphismProperty C
inst✝ : P.IsStableUnderBaseChange
⊢ P.pullbacks ≤ P | rwa [← isStableUnderBaseChange_iff_pullbacks_le] | no goals | 15b19b1863dec296 |
MeasureTheory.UnifTight.aeeq | Mathlib/MeasureTheory/Function/UnifTight.lean | theorem aeeq (hf : UnifTight f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
UnifTight g p μ | case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
hfg : ∀ (n : ι), f n =ᶠ[ae μ] g n
ε : ℝ≥0
hε : 0 < ε
s : Set α
hμs : μ s ≠ ⊤
hfε : ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
⊢ ∃ s, μ s ≠ ⊤ ∧ ∀ (i... | refine ⟨s, hμs, fun n => (le_of_eq <| eLpNorm_congr_ae ?_).trans (hfε n)⟩ | case intro.intro
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ι → α → β
p : ℝ≥0∞
hf : UnifTight f p μ
hfg : ∀ (n : ι), f n =ᶠ[ae μ] g n
ε : ℝ≥0
hε : 0 < ε
s : Set α
hμs : μ s ≠ ⊤
hfε : ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε
n : ι
⊢ sᶜ.indicator ... | 453e4514d55787ac |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.denote_go_eq_divRec_q | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_go_eq_divRec_q (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | case hq
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.g... | intro idx hidx | case hq
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.g... | edb5936f3962e4a9 |
LieAlgebra.isEngelian_of_subsingleton | Mathlib/Algebra/Lie/Engel.lean | theorem LieAlgebra.isEngelian_of_subsingleton [Subsingleton L] : LieAlgebra.IsEngelian R L | case h
R : Type u₁
L : Type u₂
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : Subsingleton L
M : Type u_1
_i1 : AddCommGroup M
_i2 : Module R M
_i3 : LieRingModule L M
_i4 : LieModule R L M
_h : ∀ (x : L), IsNilpotent ((toEnd R L M) x)
⊢ lowerCentralSeries ℤ L M 1 = ⊥ | simp | no goals | e1bc69c104947d71 |
SimpleGraph.IsCycles.exists_cycle_toSubgraph_verts_eq_connectedComponentSupp | Mathlib/Combinatorics/SimpleGraph/Matching.lean | lemma IsCycles.exists_cycle_toSubgraph_verts_eq_connectedComponentSupp [Finite V]
{c : G.ConnectedComponent} (h : G.IsCycles) (hv : v ∈ c.supp)
(hn : (G.neighborSet v).Nonempty) :
∃ (p : G.Walk v v), p.IsCycle ∧ p.toSubgraph.verts = c.supp | case intro.intro.intro
V : Type u_1
G : SimpleGraph V
v : V
inst✝ : Finite V
c : G.ConnectedComponent
h : G.IsCycles
hv : v ∈ c.supp
w : V
hw : w ∈ G.neighborSet v
u : V
p : G.Walk u u
hp : p.IsCycle ∧ s(v, w) ∈ p.edges
⊢ ∃ p, p.IsCycle ∧ p.toSubgraph.verts = c.supp | have hvp : v ∈ p.support := SimpleGraph.Walk.fst_mem_support_of_mem_edges _ hp.2 | case intro.intro.intro
V : Type u_1
G : SimpleGraph V
v : V
inst✝ : Finite V
c : G.ConnectedComponent
h : G.IsCycles
hv : v ∈ c.supp
w : V
hw : w ∈ G.neighborSet v
u : V
p : G.Walk u u
hp : p.IsCycle ∧ s(v, w) ∈ p.edges
hvp : v ∈ p.support
⊢ ∃ p, p.IsCycle ∧ p.toSubgraph.verts = c.supp | 449969e7dbb35e13 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.deleteOne_preserves_strongAssignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem deleteOne_preserves_strongAssignmentsInvariant {n : Nat} (f : DefaultFormula n) (id : Nat) :
StrongAssignmentsInvariant f → StrongAssignmentsInvariant (deleteOne f id) | case inl.h
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
l_e... | simp only [Array.set!, Array.setIfInBounds] | case inl.h
n : Nat
f : DefaultFormula n
id : Nat
hsize : f.assignments.size = n
hsize' : (f.deleteOne id).assignments.size = n
i : PosFin n
b : Bool
i_in_bounds : i.val < f.assignments.size
c : DefaultClause n
heq : f.clauses[id]! = some c
l : Literal (PosFin n)
hl : c = { clause := [l], nodupkey := ⋯, nodup := ⋯ }
l_e... | 0ab20a88767ce69e |
IsLocalizedModule.mk_eq_mk' | Mathlib/Algebra/Module/LocalizedModule/Basic.lean | theorem mk_eq_mk' (s : S) (m : M) :
LocalizedModule.mk m s = mk' (LocalizedModule.mkLinearMap S M) m s | R : Type u_1
inst✝² : CommSemiring R
S : Submonoid R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : ↥S
m : M
⊢ LocalizedModule.mk m s = mk' (LocalizedModule.mkLinearMap S M) m s | rw [eq_comm, mk'_eq_iff, Submonoid.smul_def, LocalizedModule.smul'_mk, ← Submonoid.smul_def,
LocalizedModule.mk_cancel, LocalizedModule.mkLinearMap_apply] | no goals | a62df2ede51f1c8a |
CliffordAlgebra.forall_mul_self_eq_iff | Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | theorem forall_mul_self_eq_iff {A : Type*} [Ring A] [Algebra R A] (h2 : IsUnit (2 : A))
(f : M →ₗ[R] A) :
(∀ x, f x * f x = algebraMap _ _ (Q x)) ↔
(LinearMap.mul R A).compl₂ f ∘ₗ f + (LinearMap.mul R A).flip.compl₂ f ∘ₗ f =
Q.polarBilin.compr₂ (Algebra.linearMap R A) | R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_4
inst✝¹ : Ring A
inst✝ : Algebra R A
h2 : IsUnit 2
f : M →ₗ[R] A
x : M
h : ∀ (x y : M), f x * f y + f y * f x = (algebraMap R A) (QuadraticMap.polar (⇑Q) x y)
⊢ f x * f x = (algebraMap R A) (Q x) | apply h2.mul_left_cancel | R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_4
inst✝¹ : Ring A
inst✝ : Algebra R A
h2 : IsUnit 2
f : M →ₗ[R] A
x : M
h : ∀ (x y : M), f x * f y + f y * f x = (algebraMap R A) (QuadraticMap.polar (⇑Q) x y)
⊢ 2 * (f x * f x) = 2 * (algebraMap R... | a627073501885bf0 |
NumberField.Units.dirichletUnitTheorem.exists_unit | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | theorem exists_unit (w₁ : InfinitePlace K) :
∃ u : (𝓞 K)ˣ, ∀ w : InfinitePlace K, w ≠ w₁ → Real.log (w u) < 0 | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₁ : InfinitePlace K
⊢ ∃ x, minkowskiBound K 1 < ↑x * ↑(convexBodyLTFactor K) | exact ENNReal.exists_nat_mul_gt (ENNReal.coe_ne_zero.mpr (convexBodyLTFactor_ne_zero K))
(ne_of_lt (minkowskiBound_lt_top K 1)) | no goals | a5c11596a1c04e50 |
faaDiBruno_aux2 | Mathlib/Analysis/Calculus/ContDiff/FaaDiBruno.lean | /-- Composing a formal multilinear series with an ordered partition extended by adding a left point
to an already existing atom of index `i` corresponds to updating the `i`th block,
using `p (c.partSize i + 1)` instead of `p (c.partSize i)` there.
This is one of the terms that appears when differentiating in the Faa di... | case h.H
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : FormalMult... | congr | case h.H.h.e_6.h
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
m : ℕ
q : FormalMultilinearSeries 𝕜 F G
p : Fo... | 0043cf301ba04e46 |
CategoryTheory.isVanKampenColimit_of_isEmpty | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem isVanKampenColimit_of_isEmpty [HasStrictInitialObjects C] [IsEmpty J] {F : J ⥤ C}
(c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c | J : Type v'
inst✝³ : Category.{u', v'} J
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasStrictInitialObjects C
inst✝ : IsEmpty J
F : J ⥤ C
c : Cocone F
hc : IsColimit c
this : IsVanKampenColimit (asEmptyCocone c.pt)
⊢ IsVanKampenColimit (Cocone.whisker (equivalenceOfIsEmpty (Discrete PEmpty.{1}) J).functor c) | exact (this.precompose_isIso (Functor.uniqueFromEmpty
((equivalenceOfIsEmpty (Discrete PEmpty.{1}) J).functor ⋙ F)).hom).of_iso
(Cocones.ext (Iso.refl _) (by simp)) | no goals | eeac24dc087b2172 |
exists_squarefree_dvd_pow_of_ne_zero | Mathlib/Algebra/Squarefree/Basic.lean | lemma _root_.exists_squarefree_dvd_pow_of_ne_zero {x : R} (hx : x ≠ 0) :
∃ (y : R) (n : ℕ), Squarefree y ∧ y ∣ x ∧ x ∣ y ^ n | case pos
R : Type u_1
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : UniqueFactorizationMonoid R
z p : R
hz : z ≠ 0
hp : Irreducible p
ih : z ≠ 0 → ∃ y n, Squarefree y ∧ y ∣ z ∧ z ∣ y ^ n
hx : p * z ≠ 0
y : R
n : ℕ
hy : Squarefree y
hyx : y ∣ z
hy' : z ∣ y ^ n
hn : n > 0
hp' : p ∣ y
⊢ ∃ y n, Squarefree y ∧ y ∣ p * z ∧ p *... | exact ⟨y, n + 1, hy, dvd_mul_of_dvd_right hyx _,
mul_comm p z ▸ pow_succ y n ▸ mul_dvd_mul hy' hp'⟩ | no goals | 2f11f1533d70f050 |
Vector.attach_map | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Attach.lean | theorem attach_map {l : Vector α n} (f : α → β) :
(l.map f).attach = l.attach.map (fun ⟨x, h⟩ => ⟨f x, mem_map_of_mem f h⟩) | α : Type u_1
n : Nat
β : Type u_2
l : Vector α n
f : α → β
⊢ (map f l).attach =
map
(fun x =>
match x with
| ⟨x, h⟩ => ⟨f x, ⋯⟩)
l.attach | cases l | case mk
α : Type u_1
n : Nat
β : Type u_2
f : α → β
toArray✝ : Array α
size_toArray✝ : toArray✝.size = n
⊢ (map f { toArray := toArray✝, size_toArray := size_toArray✝ }).attach =
map
(fun x =>
match x with
| ⟨x, h⟩ => ⟨f x, ⋯⟩)
{ toArray := toArray✝, size_toArray := size_toArray✝ }.attac... | 136b979f9fc12efc |
conformalAt_iff_isConformalMap_fderiv | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | theorem conformalAt_iff_isConformalMap_fderiv {f : X → Y} {x : X} :
ConformalAt f x ↔ IsConformalMap (fderiv ℝ f x) | case neg
X : Type u_1
Y : Type u_2
inst✝³ : NormedAddCommGroup X
inst✝² : NormedAddCommGroup Y
inst✝¹ : NormedSpace ℝ X
inst✝ : NormedSpace ℝ Y
f : X → Y
x : X
H : IsConformalMap (fderiv ℝ f x)
h : ¬DifferentiableAt ℝ f x
⊢ ConformalAt f x | nontriviality X | X : Type u_1
Y : Type u_2
inst✝³ : NormedAddCommGroup X
inst✝² : NormedAddCommGroup Y
inst✝¹ : NormedSpace ℝ X
inst✝ : NormedSpace ℝ Y
f : X → Y
x : X
H : IsConformalMap (fderiv ℝ f x)
h : ¬DifferentiableAt ℝ f x
a✝ : Nontrivial X
⊢ ConformalAt f x | 7b61d25756689bca |
Nat.nth_lt_of_lt_count | Mathlib/Data/Nat/Nth.lean | theorem nth_lt_of_lt_count {n k : ℕ} (h : k < count p n) : nth p k < n | p : ℕ → Prop
inst✝ : DecidablePred p
n k : ℕ
h : k < count p n
⊢ nth p k < n | refine (count_monotone p).reflect_lt ?_ | p : ℕ → Prop
inst✝ : DecidablePred p
n k : ℕ
h : k < count p n
⊢ count p (nth p k) < count p n | 136d9032986f9438 |
ProbabilityTheory.evariance_mul | Mathlib/Probability/Variance.lean | theorem evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
evariance (fun ω => c * X ω) μ = ENNReal.ofReal (c ^ 2) * evariance X μ | case e_f.h
Ω : Type u_1
mΩ : MeasurableSpace Ω
c : ℝ
X : Ω → ℝ
μ : Measure Ω
ω : Ω
⊢ ‖c * X ω - ∫ (x : Ω), c * X x ∂μ‖ₑ ^ 2 = ENNReal.ofReal (c ^ 2) * ‖X ω - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 | rw [integral_mul_left, ← mul_sub, enorm_mul, mul_pow, ← enorm_pow,
Real.enorm_of_nonneg (sq_nonneg _)] | no goals | 87d7e9ed9d5e8329 |
HurwitzZeta.cosKernel_neg | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | @[simp]
lemma cosKernel_neg (a : UnitAddCircle) (x : ℝ) : cosKernel (-a) x = cosKernel a x | case H
x z✝ : ℝ
⊢ cosKernel (-↑z✝) x = cosKernel (↑z✝) x | simp [← QuotientAddGroup.mk_neg, ← ofReal_inj, cosKernel_def] | no goals | d86ba7455a7df237 |
Finset.affineCombinationLineMapWeights_apply_of_ne | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem affineCombinationLineMapWeights_apply_of_ne [DecidableEq ι] {i j t : ι} (hi : t ≠ i)
(hj : t ≠ j) (c : k) : affineCombinationLineMapWeights i j c t = 0 | k : Type u_1
inst✝¹ : Ring k
ι : Type u_4
inst✝ : DecidableEq ι
i j t : ι
hi : t ≠ i
hj : t ≠ j
c : k
⊢ affineCombinationLineMapWeights i j c t = 0 | simp [affineCombinationLineMapWeights, hi, hj] | no goals | 4ae27d481b521f8c |
Finset.subset_union_elim | Mathlib/Data/Finset/Basic.lean | theorem subset_union_elim {s : Finset α} {t₁ t₂ : Set α} (h : ↑s ⊆ t₁ ∪ t₂) :
∃ s₁ s₂ : Finset α, s₁ ∪ s₂ = s ∧ ↑s₁ ⊆ t₁ ∧ ↑s₂ ⊆ t₂ \ t₁ | case refine_3
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
t₁ t₂ : Set α
h : ↑s ⊆ t₁ ∪ t₂
x : α
⊢ x ∈ ↑(filter (fun x => x ∉ t₁) s) → x ∈ t₂ \ t₁ | simp only [not_not, coe_filter, Set.mem_setOf_eq, Set.mem_diff, and_imp] | case refine_3
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
t₁ t₂ : Set α
h : ↑s ⊆ t₁ ∪ t₂
x : α
⊢ x ∈ s → x ∉ t₁ → x ∈ t₂ ∧ x ∉ t₁ | 124422836b3c4eb9 |
MeasureTheory.lpMeasSubgroupToLpTrim_right_inv | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | theorem lpMeasSubgroupToLpTrim_right_inv (hm : m ≤ m0) :
Function.RightInverse (lpTrimToLpMeasSubgroup F p μ hm) (lpMeasSubgroupToLpTrim F p μ hm) | α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp F p (μ.trim hm))
⊢ lpMeasSubgroupToLpTrim F p μ hm (lpTrimToLpMeasSubgroup F p μ hm f) = f | ext1 | case h
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp F p (μ.trim hm))
⊢ ↑↑(lpMeasSubgroupToLpTrim F p μ hm (lpTrimToLpMeasSubgroup F p μ hm f)) =ᶠ[ae (μ.trim hm)] ↑↑f | 1f4f24d97edaec61 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastRotateLeft.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/RotateLeft.lean | theorem go_get_aux (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr),
(go input distance curr hcurr s).get idx (by omega)
=
s.get idx hidx | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
distance : Nat
input : aig.RefVec w
curr : Nat
hcurr : curr ≤ w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
h✝ : ¬curr < w
⊢ curr = w | omega | no goals | e1ea21e0c26513f1 |
IsDedekindDomain.HeightOneSpectrum.adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | lemma adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers (v : HeightOneSpectrum R)
(a : v.adicCompletion K) : ∃ b ∈ R⁰, a * b ∈ v.adicCompletionIntegers K | case neg.intro.intro
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
a : adicCompletion K v
d : ℤ
ha : 0 < d
hd : Valued.v a = ↑(ofAdd d)
ϖ : R
hϖ : v.intValuationDef ϖ = ↑(ofAdd (-1))
hϖ0 : ϖ ≠ 0
⊢ d + d ... | omega | no goals | 8327a4d56ea55b92 |
Set.diff_iInter | Mathlib/Data/Set/Lattice.lean | theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i | β : Type u_2
ι : Sort u_5
s : Set β
t : ι → Set β
⊢ s \ ⋂ i, t i = ⋃ i, s \ t i | rw [diff_eq, compl_iInter, inter_iUnion] | β : Type u_2
ι : Sort u_5
s : Set β
t : ι → Set β
⊢ ⋃ i, s ∩ (t i)ᶜ = ⋃ i, s \ t i | 2a5c736d07abb686 |
List.getElem?_eraseIdx_of_lt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Erase.lean | theorem getElem?_eraseIdx_of_lt (l : List α) (i : Nat) (j : Nat) (h : j < i) :
(l.eraseIdx i)[j]? = l[j]? | α : Type u_1
l : List α
i j : Nat
h : j < i
⊢ (l.eraseIdx i)[j]? = l[j]? | rw [getElem?_eraseIdx] | α : Type u_1
l : List α
i j : Nat
h : j < i
⊢ (if j < i then l[j]? else l[j + 1]?) = l[j]? | 01167548286c9ee6 |
PNat.XgcdType.reduce_a | Mathlib/Data/PNat/Xgcd.lean | theorem reduce_a {u : XgcdType} (h : u.r = 0) : u.reduce = u.finish | u : XgcdType
h : u.r = 0
⊢ u.reduce = u.finish | rw [reduce] | u : XgcdType
h : u.r = 0
⊢ (if x : u.r = 0 then u.finish else u.step.reduce.flip) = u.finish | 6004cbbfaf421ae5 |
BoxIntegral.norm_volume_sub_integral_face_upper_sub_lower_smul_le | Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean | theorem norm_volume_sub_integral_face_upper_sub_lower_smul_le {f : (Fin (n + 1) → ℝ) → E}
{f' : (Fin (n + 1) → ℝ) →L[ℝ] E} (hfc : ContinuousOn f (Box.Icc I)) {x : Fin (n + 1) → ℝ}
(hxI : x ∈ (Box.Icc I)) {a : E} {ε : ℝ} (h0 : 0 < ε)
(hε : ∀ y ∈ (Box.Icc I), ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖) {c : ℝ≥0}
... | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
hfc : ContinuousOn f (Box.Icc I)
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
hε : ∀ y ∈ Box.Icc I, ‖f y - a - ... | refine norm_integral_le_of_le_const (fun y hy => (this y hy).trans ?_) volume | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
i : Fin (n + 1)
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) →L[ℝ] E
hfc : ContinuousOn f (Box.Icc I)
x : Fin (n + 1) → ℝ
hxI : x ∈ Box.Icc I
a : E
ε : ℝ
h0 : 0 < ε
hε : ∀ y ∈ Box.Icc I, ‖f y - a - ... | 6dcd75375355bac1 |
Matrix.SpecialLinearGroup.SL2_inv_expl_det | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | theorem SL2_inv_expl_det (A : SL(2, R)) :
det ![![A.1 1 1, -A.1 0 1], ![-A.1 1 0, A.1 0 0]] = 1 | R : Type v
inst✝ : CommRing R
A : SL(2, R)
⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1 | have := A.2 | R : Type v
inst✝ : CommRing R
A : SL(2, R)
this : (↑A).det = 1
⊢ ↑A 0 0 * ↑A 1 1 - ↑A 0 1 * ↑A 1 0 = 1 | 54dbb0fb2273aa1b |
WeierstrassCurve.Projective.negY_of_Z_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma negY_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
W.negY P / P z = W.toAffine.negY (P x / P z) (P y / P z) | case a.a
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ W.negY P / P z + -W.a₃ * 1 - ((toAffine W).negY (P x / P z) (P y / P z) + -W.a₃ * (P z / P z)) = 0 | rw [negY, Affine.negY] | case a.a
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (-P y - W.a₁ * P x - W.a₃ * P z) / P z + -W.a₃ * 1 -
(-(P y / P z) - (toAffine W).a₁ * (P x / P z) - (toAffine W).a₃ + -W.a₃ * (P z / P z)) =
0 | bbf88f4f5524fe5a |
CategoryTheory.BinaryCofan.isPullback_initial_to_of_isVanKampen | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem BinaryCofan.isPullback_initial_to_of_isVanKampen [HasInitial C] {c : BinaryCofan X Y}
(h : IsVanKampenColimit c) : IsPullback (initial.to _) (initial.to _) c.inl c.inr | case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
inst✝ : HasInitial C
c : BinaryCofan X Y
h : IsVanKampenColimit c
⊢ mapPair (initial.to X) (𝟙 Y) ≫ c.ι =
(BinaryCofan.mk (initial.to Y) (𝟙 Y)).ι ≫ (Functor.const (Discrete WalkingPair)).map c.inr | ext ⟨⟨⟩⟩ <;> dsimp | case refine_1.w.h.mk.left
C : Type u
inst✝¹ : Category.{v, u} C
X Y : C
inst✝ : HasInitial C
c : BinaryCofan X Y
h : IsVanKampenColimit c
⊢ initial.to X ≫ c.inl = initial.to Y ≫ c.inr | 1d49a7be83228de8 |
AddMonoidWithOne.ext | Mathlib/Algebra/Ring/Ext.lean | theorem AddMonoidWithOne.ext ⦃inst₁ inst₂ : AddMonoidWithOne R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one : R)) :
inst₁ = inst₂ | case mk.mk.mk.mk
R : Type u
toAddMonoid✝¹ : AddMonoid R
toOne✝¹ : One R
natCast✝¹ : ℕ → R
natCast_zero✝¹ : NatCast.natCast 0 = 0
natCast_succ✝¹ : ∀ (n : ℕ), NatCast.natCast (n + 1) = NatCast.natCast n + 1
toAddMonoid✝ : AddMonoid R
toOne✝ : One R
natCast✝ : ℕ → R
natCast_zero✝ : NatCast.natCast 0 = 0
natCast_succ✝ : ∀ ... | congr | no goals | 8c33f554d55741a3 |
ENNReal.tendsto_atTop_zero_iff_lt_of_antitone | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem tendsto_atTop_zero_iff_lt_of_antitone {β : Type*} [Nonempty β] [SemilatticeSup β]
{f : β → ℝ≥0∞} (hf : Antitone f) :
Filter.Tendsto f Filter.atTop (𝓝 0) ↔ ∀ ε, 0 < ε → ∃ n : β, f n < ε | case neg
β : Type u_4
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → ℝ≥0∞
hf : Antitone f
h : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, f n ≤ ε
ε : ℝ≥0∞
hε : 0 < ε
n : β
hn : f n ≤ 1 ⊓ ε / 2
hε_top : ¬ε = ⊤
⊢ 1 < 2 | norm_num | no goals | 8491377f7bac7c56 |
Algebra.TensorProduct.includeLeft_map_center_le | Mathlib/Algebra/Central/TensorProduct.lean | lemma Algebra.TensorProduct.includeLeft_map_center_le :
(Subalgebra.center K B).map includeLeft ≤ Subalgebra.center K (B ⊗[K] C) | case intro.intro.add
K : Type u_1
B : Type u_2
C : Type u_3
inst✝⁴ : CommSemiring K
inst✝³ : Semiring B
inst✝² : Semiring C
inst✝¹ : Algebra K B
inst✝ : Algebra K C
b : B
hb0 : ∀ (b_1 : B), b_1 * b = b * b_1
x✝ y✝ : B ⊗[K] C
a✝¹ : x✝ * includeLeft b = includeLeft b * x✝
a✝ : y✝ * includeLeft b = includeLeft b * y✝
⊢ (x... | simp_all [add_mul, mul_add] | no goals | b5e2e4a224adead8 |
SetTheory.PGame.Impartial.exists_right_move_equiv_iff_fuzzy_zero | Mathlib/SetTheory/Game/Impartial.lean | theorem exists_right_move_equiv_iff_fuzzy_zero : (∃ j, G.moveRight j ≈ 0) ↔ G ‖ 0 | G : PGame
inst✝ : G.Impartial
hn : ∃ j, G.moveRight j ≤ 0
⊢ ∃ j, G.moveRight j ≈ 0 | obtain ⟨i, hi⟩ := hn | case intro
G : PGame
inst✝ : G.Impartial
i : G.RightMoves
hi : G.moveRight i ≤ 0
⊢ ∃ j, G.moveRight j ≈ 0 | dab6002e73196c9e |
LieAlgebra.hasCentralRadical_and_of_isIrreducible_of_isFaithful | Mathlib/Algebra/Lie/Semisimple/Lemmas.lean | lemma hasCentralRadical_and_of_isIrreducible_of_isFaithful :
HasCentralRadical k L ∧ (∀ x, x ∈ center k L ↔ toEnd k L M x ∈ k ∙ LinearMap.id) | 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 ... | replace hχ (x : L) (hx : x ∈ radical k L) : toEnd k _ M x = χ ⟨x, hx⟩ • LinearMap.id := by
ext m
have hm : ∀ (y : L) (hy : y ∈ radical k L), ⁅y, m⁆ = χ ⟨y, hy⟩ • m := by
simpa [N, weightSpaceOfIsLieTower, mem_weightSpace] using (hχ ▸ mem_top _ : m ∈ N)
simpa using hm x hx | 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 ... | 92e7bead74bccd0a |
CategoryTheory.MorphismProperty.universally_le | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | theorem universally_le (P : MorphismProperty C) : P.universally ≤ P | C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : P.universally f
⊢ f ≫ 𝟙 Y = 𝟙 X ≫ f | rw [Category.comp_id, Category.id_comp] | no goals | 155ec1c81c975a26 |
EuclideanGeometry.angle_lt_pi_div_two_of_angle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | theorem angle_lt_pi_div_two_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
(h0 : p₃ ≠ p₂) : ∠ p₂ p₃ p₁ < π / 2 | 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 : ∠ p₁ p₂ p₃ = π / 2
h0 : p₃ ≠ p₂
⊢ ∠ p₂ p₃ p₁ < π / 2 | rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0
h0 : p₃ ≠ p₂
⊢ ∠ p₂ p₃ p₁ < π / 2 | eb5d456a193260b4 |
IsUpperSet.div_right | Mathlib/Algebra/Order/UpperLower.lean | theorem IsUpperSet.div_right (hs : IsUpperSet s) : IsUpperSet (s / t) | α : Type u_1
inst✝ : OrderedCommGroup α
s t : Set α
hs : IsUpperSet s
⊢ IsUpperSet (s * t⁻¹) | exact hs.mul_right | no goals | 7bc864668d916ff9 |
tendsto_div_of_monotone_of_tendsto_div_floor_pow | Mathlib/Analysis/SpecificLimits/FloorPow.lean | theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
(c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
(hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) | u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
n : ℕ
⊢ 1 ≤ ↑⌊c k ^ n⌋₊ | simp only [Real.rpow_natCast, Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow₀ (cone k).le] | no goals | 39b6908bafcd74f7 |
LinearIndependent.iSupIndep_span_singleton | Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean | theorem LinearIndependent.iSupIndep_span_singleton (hv : LinearIndependent R v) :
iSupIndep fun i => R ∙ v i | ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hv : LinearIndependent R v
i : ι
m : M
hm : m ∈ span R {v i} ⊓ ⨆ j, ⨆ (_ : j ≠ i), span R {v j}
⊢ m ∈ ⊥ | simp only [mem_inf, mem_span_singleton, iSup_subtype'] at hm | ι : Type u'
R : Type u_2
M : Type u_4
v : ι → M
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
hv : LinearIndependent R v
i : ι
m : M
hm : (∃ a, a • v i = m) ∧ m ∈ ⨆ x, span R {v ↑x}
⊢ m ∈ ⊥ | 15c8499de3d7ad45 |
ZMod.subsingleton_iff | Mathlib/Data/ZMod/Basic.lean | lemma subsingleton_iff {n : ℕ} : Subsingleton (ZMod n) ↔ n = 1 | n : ℕ
⊢ Subsingleton (ZMod n) ↔ n = 1 | constructor | case mp
n : ℕ
⊢ Subsingleton (ZMod n) → n = 1
case mpr
n : ℕ
⊢ n = 1 → Subsingleton (ZMod n) | 3a46eadf39430283 |
IsAtom.le_iff | Mathlib/Order/Atoms.lean | theorem IsAtom.le_iff (h : IsAtom a) : x ≤ a ↔ x = ⊥ ∨ x = a | α : Type u_2
inst✝¹ : PartialOrder α
inst✝ : OrderBot α
a x : α
h : IsAtom a
⊢ x ≤ a ↔ x = ⊥ ∨ x = a | rw [le_iff_lt_or_eq, h.lt_iff] | no goals | 56462cd397d981f8 |
IsOpen.exists_subset_affineIndependent_span_eq_top | Mathlib/Analysis/Normed/Affine/AddTorsorBases.lean | theorem IsOpen.exists_subset_affineIndependent_span_eq_top {u : Set P} (hu : IsOpen u)
(hne : u.Nonempty) : ∃ s ⊆ u, AffineIndependent ℝ ((↑) : s → P) ∧ affineSpan ℝ s = ⊤ | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
u : Set P
hu : IsOpen u
hne : u.Nonempty
⊢ ∃ s ⊆ u, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤ | rcases hne with ⟨x, hx⟩ | case intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
u : Set P
hu : IsOpen u
x : P
hx : x ∈ u
⊢ ∃ s ⊆ u, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤ | 5ed0ed984803d56e |
Set.Icc_union_Icc' | Mathlib/Order/Interval/Set/Basic.lean | theorem Icc_union_Icc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Icc a b ∪ Icc c d = Icc (min a c) (max b d) | case pos
α : Type u_1
inst✝ : LinearOrder α
a b c d : α
h₁ : c ≤ b
h₂ : a ≤ d
x : α
hc : c ≤ x
hd : x ≤ d
⊢ a ≤ x ∧ x ≤ b ∨ c ≤ x ∧ x ≤ d ↔ (a ≤ x ∨ c ≤ x) ∧ (x ≤ b ∨ x ≤ d) | simp only [hc, hd, and_self, or_true] | no goals | 75dfe9f0a698fa98 |
BitVec.signExtend_eq_not_setWidth_not_of_msb_true | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem signExtend_eq_not_setWidth_not_of_msb_true {x : BitVec w} {v : Nat} (hmsb : x.msb = true) :
x.signExtend v = ~~~((~~~x).setWidth v) | case a.a
w : Nat
x : BitVec w
v : Nat
hmsb : x.msb = true
⊢ (2 ^ w - 1 - x.toNat) % 2 ^ v + 1 ≤ ?a.m✝ | apply Nat.succ_le_of_lt | case a.a.h
w : Nat
x : BitVec w
v : Nat
hmsb : x.msb = true
⊢ (2 ^ w - 1 - x.toNat) % 2 ^ v < ?a.m✝ | f7344d955b079af2 |
LocallyBoundedVariationOn.ae_differentiableAt | Mathlib/Analysis/BoundedVariation.lean | theorem ae_differentiableAt {f : ℝ → V} (h : LocallyBoundedVariationOn f univ) :
∀ᵐ x, DifferentiableAt ℝ f x | case h
V : Type u_3
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
h : LocallyBoundedVariationOn f univ
x : ℝ
hx : x ∈ univ → DifferentiableWithinAt ℝ f univ x
⊢ DifferentiableAt ℝ f x | rw [differentiableWithinAt_univ] at hx | case h
V : Type u_3
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
f : ℝ → V
h : LocallyBoundedVariationOn f univ
x : ℝ
hx : x ∈ univ → DifferentiableAt ℝ f x
⊢ DifferentiableAt ℝ f x | 33a41388c6885a3e |
NumberField.InfinitePlace.card_complex_embeddings | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem card_complex_embeddings :
card { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } = 2 * nrComplexPlaces K | K : Type u_2
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝ : { x // mkComplex x = ⟨w, hw⟩ }
φ : { φ // ¬ComplexEmbedding.IsReal φ }
hφ : mkComplex φ = ⟨w, hw⟩
⊢ mk ↑φ = w | rwa [Subtype.ext_iff] at hφ | no goals | 20c4978209cddac1 |
Matroid.cRk_mono | Mathlib/Data/Matroid/Rank/Cardinal.lean | theorem cRk_mono (M : Matroid α) : Monotone M.cRk | α : Type u
M : Matroid α
⊢ ∀ ⦃a b : Set α⦄, a ⊆ b → ∀ ⦃I : Set α⦄, M.IsBasis' I a → #↑I ≤ M.cRk b | intro X Y hXY I hIX | α : Type u
M : Matroid α
X Y : Set α
hXY : X ⊆ Y
I : Set α
hIX : M.IsBasis' I X
⊢ #↑I ≤ M.cRk Y | e3dc976a5390a380 |
UniqueFactorizationMonoid.radical_eq_of_associated | Mathlib/RingTheory/Radical.lean | theorem radical_eq_of_associated {a b : M} (h : Associated a b) : radical a = radical b | M : Type u_1
inst✝² : CancelCommMonoidWithZero M
inst✝¹ : NormalizationMonoid M
inst✝ : UniqueFactorizationMonoid M
a b : M
h : Associated a b
⊢ (primeFactors a).prod id = (primeFactors b).prod id | rw [h.primeFactors_eq] | no goals | 2fbd16dae46948ca |
Nat.Prime.factorization_pos_of_dvd | Mathlib/Data/Nat/Factorization/Defs.lean | theorem Prime.factorization_pos_of_dvd {n p : ℕ} (hp : p.Prime) (hn : n ≠ 0) (h : p ∣ n) :
0 < n.factorization p | n p : ℕ
hp : Prime p
hn : n ≠ 0
h : p ∣ n
⊢ 0 < n.factorization p | rwa [← primeFactorsList_count_eq, count_pos_iff, mem_primeFactorsList_iff_dvd hn hp] | no goals | ea65c5f724cd4b24 |
CochainComplex.mappingCone.inl_snd_assoc | Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | @[simp]
lemma inl_snd_assoc {K : CochainComplex C ℤ} {d e f : ℤ} (γ : Cochain G K d)
(he : 0 + d = e) (hf : -1 + e = f) :
(inl φ).comp ((snd φ).comp γ he) hf = 0 | C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
d e f : ℤ
γ : Cochain G K d
he : 0 + d = e
hf : -1 + e = f
⊢ (inl φ).comp ((snd φ).comp γ he) hf = 0 | obtain rfl : e = d := by omega | C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
e f : ℤ
hf : -1 + e = f
γ : Cochain G K e
he : 0 + e = e
⊢ (inl φ).comp ((snd φ).comp γ he) hf = 0 | 91c568454ffe6b85 |
Sbtw.dist_lt_max_dist | Mathlib/Analysis/Convex/StrictConvexBetween.lean | theorem Sbtw.dist_lt_max_dist (p : P) {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) :
dist p₂ p < max (dist p₁ p) (dist p₃ p) | case intro.inr.inl.intro
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p p₁ p₂ p₃ : P
hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p
h : p₂ = p₃
hp₂p₁ : p₂ ≠ p₁
hp₂p₃ : p₂ ≠ p₃
⊢ dist p₂ p < dist p₁ p ⊔ dist p₃ p | exact False.elim (hp₂p₃ h) | no goals | 9aa41ffc37e88f54 |
Order.exists_series_of_le_height | Mathlib/Order/KrullDimension.lean | /-- There exist a series ending in a element for any length up to the element’s height. -/
lemma exists_series_of_le_height (a : α) {n : ℕ} (h : n ≤ height a) :
∃ p : LTSeries α, p.last = a ∧ p.length = n | case h.left
α : Type u_1
inst✝ : Preorder α
a : α
n : ℕ
hne : Nonempty { p // RelSeries.last p = a }
m : ℕ
h : n ≤ m
ha : ⨆ x, ↑(↑x).length = ↑m
p : LTSeries α
hlast : RelSeries.last p = a
hlen : p.length = m
⊢ (RelSeries.drop p ⟨m - n, ⋯⟩).last = a | simp [hlast] | no goals | 0e15cb2199a1ee03 |
CochainComplex.HomComplex.Cochain.comp_smul | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | @[simp]
protected lemma comp_smul {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (k : R) (z₂ : Cochain G K n₂)
(h : n₁ + n₂ = n₁₂ ) : z₁.comp (k • z₂) h = k • (z₁.comp z₂ h) | case h
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
k : R
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp (k • z₂) h).v p q hpq = (k • z₁.comp z₂ h).v p q hpq | simp only [comp_v _ _ h p _ q rfl (by omega), smul_v, Linear.comp_smul] | no goals | e74ac95f7982b3df |
WeierstrassCurve.Jacobian.nonsingular_add | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma nonsingular_add {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q) :
W.Nonsingular <| W.add P Q | case pos
F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Nonsingular P
hQ : W.Nonsingular Q
hPz : ¬P z = 0
hQz : Q z = 0
⊢ W.Nonsingular (W.add P Q) | simpa only [add_of_Z_eq_zero_right hQ.left hPz hQz,
nonsingular_smul _ ((isUnit_X_of_Z_eq_zero hQ hQz).mul <| Ne.isUnit hPz).neg] | no goals | 756dd04fce6f14ae |
UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors | Mathlib/RingTheory/UniqueFactorizationDomain/NormalizedFactors.lean | theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a) : p ∣ a | case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a p : α
H : p ∈ normalizedFactors a
hcases : ¬a = 0
⊢ p ∣ a | exact dvd_trans (Multiset.dvd_prod H) (Associated.dvd (prod_normalizedFactors hcases)) | no goals | bc3d62cbe119c393 |
ax_grothendieck_of_definable | Mathlib/FieldTheory/AxGrothendieck.lean | theorem ax_grothendieck_of_definable [CompatibleRing K] {c : Set K}
(S : Set (ι → K)) (hS : c.Definable Language.ring S)
(ps : ι → MvPolynomial ι K) :
S.MapsTo (fun v i => eval v (ps i)) S →
S.InjOn (fun v i => eval v (ps i)) →
S.SurjOn (fun v i => eval v (ps i)) S | K : Type u_1
ι : Type u_2
inst✝³ : Field K
inst✝² : IsAlgClosed K
inst✝¹ : Finite ι
inst✝ : CompatibleRing K
c : Set K
S : Set (ι → K)
hS : c.Definable ring S
ps : ι → MvPolynomial ι K
this : Fintype ι := Fintype.ofFinite ι
⊢ Set.MapsTo (fun v i => (eval v) (ps i)) S S →
Set.InjOn (fun v i => (eval v) (ps i)) S → S... | let p : ℕ := ringChar K | K : Type u_1
ι : Type u_2
inst✝³ : Field K
inst✝² : IsAlgClosed K
inst✝¹ : Finite ι
inst✝ : CompatibleRing K
c : Set K
S : Set (ι → K)
hS : c.Definable ring S
ps : ι → MvPolynomial ι K
this : Fintype ι := Fintype.ofFinite ι
p : ℕ := ringChar K
⊢ Set.MapsTo (fun v i => (eval v) (ps i)) S S →
Set.InjOn (fun v i => (e... | 933d6add8d5d058a |
Surreal.Multiplication.ih1_neg_left | Mathlib/SetTheory/Surreal/Multiplication.lean | lemma ih1_neg_left : IH1 x y → IH1 (-x) y :=
fun h x₁ x₂ y' h₁ h₂ hy ↦ by
rw [isOption_neg] at h₁ h₂
exact P24_neg_left.2 (h h₂ h₁ hy)
| x y : PGame
h : IH1 x y
x₁ x₂ y' : PGame
h₁ : (-x₁).IsOption x
h₂ : (-x₂).IsOption x
hy : y' = y ∨ y'.IsOption y
⊢ P24 x₁ x₂ y' | exact P24_neg_left.2 (h h₂ h₁ hy) | no goals | cae1455068d0158a |
AffineBasis.exists_affineBasis_of_finiteDimensional | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | theorem exists_affineBasis_of_finiteDimensional [Fintype ι] [FiniteDimensional k V]
(h : Fintype.card ι = Module.finrank k V + 1) : Nonempty (AffineBasis ι k P) | case intro.intro
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : DivisionRing k
inst✝² : Module k V
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional k V
h : Fintype.card ι = Module.finrank k V + 1
s : Set P
b : AffineBasis (↑s) k P
hb : ⇑b = Subtype.val
⊢ Nonempty (... | lift s to Finset P using b.finite_set | case intro.intro.intro
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : DivisionRing k
inst✝² : Module k V
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional k V
h : Fintype.card ι = Module.finrank k V + 1
s : Finset P
b : AffineBasis (↑↑s) k P
hb : ⇑b = Subtype.val
⊢ ... | e510723a41bc6600 |
CategoryTheory.finrank_hom_simple_simple_eq_zero_iff | Mathlib/CategoryTheory/Preadditive/Schur.lean | theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
[FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X] [Simple Y] :
finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty (X ≅ Y) | C : Type u_1
inst✝⁹ : Category.{u_3, u_1} C
inst✝⁸ : Preadditive C
𝕜 : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : IsAlgClosed 𝕜
inst✝⁵ : Linear 𝕜 C
inst✝⁴ : HasKernels C
X Y : C
inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)
inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)
inst✝¹ : Simple X
inst✝ : Simple Y
this : finrank 𝕜 (X ⟶ Y) ≤ 1
⊢ fin... | omega | no goals | 36d01ca65cae1c3e |
geom_sum_ne_zero | Mathlib/Algebra/GeomSum.lean | theorem geom_sum_ne_zero [LinearOrderedRing α] (hx : x ≠ -1) (hn : n ≠ 0) :
∑ i ∈ range n, x ^ i ≠ 0 | case succ.succ
α : Type u
x : α
inst✝ : LinearOrderedRing α
hx : x ≠ -1
n : ℕ
hn : n + 1 + 1 ≠ 0
⊢ ∑ i ∈ range (n + 1 + 1), x ^ i ≠ 0 | rw [Ne, eq_neg_iff_add_eq_zero, ← Ne] at hx | case succ.succ
α : Type u
x : α
inst✝ : LinearOrderedRing α
hx : x + 1 ≠ 0
n : ℕ
hn : n + 1 + 1 ≠ 0
⊢ ∑ i ∈ range (n + 1 + 1), x ^ i ≠ 0 | 276116b8a692415e |
Batteries.UnionFind.setParentBump_rankD_lt | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | theorem setParentBump_rankD_lt {arr : Array UFNode} {x y : Fin arr.size}
(hroot : arr[x.1].rank < arr[y.1].rank ∨ arr[y.1].parent = y)
(H : arr[x.1].rank ≤ arr[y.1].rank) {i : Nat}
(rankD_lt : parentD arr i ≠ i → rankD arr i < rankD arr (parentD arr i))
(hP : parentD arr' i = if x.1 = i then y.1 else pa... | case isFalse.isFalse
arr' arr : Array UFNode
x y : Fin arr.size
hroot : arr[↑x].rank < arr[↑y].rank ∨ arr[↑y].parent = ↑y
H : arr[↑x].rank ≤ arr[↑y].rank
i : Nat
rankD_lt : ¬parentD arr i = i → rankD arr i < rankD arr (parentD arr i)
hP hR : True
h₁ : ¬↑x = i
h₂ : ¬(↑y = i ∧ arr[↑x].rank = arr[↑y].rank)
h : ¬parentD ar... | split <;> rename_i h₃ | case isFalse.isFalse.isTrue
arr' arr : Array UFNode
x y : Fin arr.size
hroot : arr[↑x].rank < arr[↑y].rank ∨ arr[↑y].parent = ↑y
H : arr[↑x].rank ≤ arr[↑y].rank
i : Nat
rankD_lt : ¬parentD arr i = i → rankD arr i < rankD arr (parentD arr i)
hP hR : True
h₁ : ¬↑x = i
h₂ : ¬(↑y = i ∧ arr[↑x].rank = arr[↑y].rank)
h : ¬par... | 9fe4c92bd6627be1 |
UV.le_of_mem_compression_of_not_mem | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem le_of_mem_compression_of_not_mem (h : a ∈ 𝓒 u v s) (ha : a ∉ s) : u ≤ a | case neg
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
ha : a ∉ s
b : α
hb : b ∈ s
h : ¬(Disjoint u b ∧ v ≤ b)
hba : b = a
⊢ u ≤ a | cases ne_of_mem_of_not_mem hb ha hba | no goals | 2a4bf00f4796d46c |
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