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 |
|---|---|---|---|---|---|---|
Metric.hausdorffDist_empty' | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | theorem hausdorffDist_empty' : hausdorffDist ∅ s = 0 | α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ hausdorffDist ∅ s = 0 | simp [hausdorffDist_comm] | no goals | f5c469b3da5e54f5 |
Nat.maxPowDiv.zero_base | Mathlib/Data/Nat/MaxPowDiv.lean | theorem zero_base {n : ℕ} : maxPowDiv 0 n = 0 | n : ℕ
⊢ maxPowDiv 0 n = 0 | dsimp [maxPowDiv] | n : ℕ
⊢ go 0 0 n = 0 | e2a28f08a422f4d5 |
List.zipIdx_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem zipIdx_eq_append_iff {l : List α} {k : Nat} :
zipIdx l k = l₁ ++ l₂ ↔
∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = zipIdx l₁' k ∧ l₂ = zipIdx l₂' (k + l₁'.length) | α : Type u_1
l₁ l₂ : List (α × Nat)
l : List α
k : Nat
⊢ l.zipIdx k = l₁ ++ l₂ ↔ ∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = l₁'.zipIdx k ∧ l₂ = l₂'.zipIdx (k + l₁'.length) | rw [zipIdx_eq_zip_range', zip_eq_append_iff] | α : Type u_1
l₁ l₂ : List (α × Nat)
l : List α
k : Nat
⊢ (∃ w x y z, w.length = y.length ∧ l = w ++ x ∧ range' k l.length = y ++ z ∧ l₁ = w.zip y ∧ l₂ = x.zip z) ↔
∃ l₁' l₂', l = l₁' ++ l₂' ∧ l₁ = l₁'.zipIdx k ∧ l₂ = l₂'.zipIdx (k + l₁'.length) | 16570ba8d162672d |
SetTheory.PGame.isOption_neg | Mathlib/SetTheory/Game/PGame.lean | theorem isOption_neg {x y : PGame} : IsOption x (-y) ↔ IsOption (-x) y | x y : PGame
⊢ ((∃ i, x = (-y).moveRight i) ∨ ∃ i, x = (-y).moveLeft i) ↔ (∃ i, -x = y.moveLeft i) ∨ ∃ i, -x = y.moveRight i | cases y | case mk
x : PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ ((∃ i, x = (-mk α✝ β✝ a✝¹ a✝).moveRight i) ∨ ∃ i, x = (-mk α✝ β✝ a✝¹ a✝).moveLeft i) ↔
(∃ i, -x = (mk α✝ β✝ a✝¹ a✝).moveLeft i) ∨ ∃ i, -x = (mk α✝ β✝ a✝¹ a✝).moveRight i | e3315ca605f5205c |
List.maximum_eq_coe_iff | Mathlib/Data/List/MinMax.lean | theorem maximum_eq_coe_iff : maximum l = m ↔ m ∈ l ∧ ∀ a ∈ l, a ≤ m | α : Type u_1
inst✝ : LinearOrder α
l : List α
m : α
⊢ l.maximum = ↑m ↔ m ∈ l ∧ ∀ (a : α), a ∈ l → a ≤ m | rw [maximum, ← WithBot.some_eq_coe, argmax_eq_some_iff] | α : Type u_1
inst✝ : LinearOrder α
l : List α
m : α
⊢ (m ∈ l ∧ (∀ (a : α), a ∈ l → id a ≤ id m) ∧ ∀ (a : α), a ∈ l → id m ≤ id a → idxOf m l ≤ idxOf a l) ↔
m ∈ l ∧ ∀ (a : α), a ∈ l → a ≤ m | c19313dd889e7fe8 |
UniqueFactorizationMonoid.normalize_normalized_factor | Mathlib/RingTheory/UniqueFactorizationDomain/NormalizedFactors.lean | theorem normalize_normalized_factor {a : α} :
∀ x : α, x ∈ normalizedFactors a → normalize x = x | α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
⊢ ∀ x ∈ Multiset.map (⇑normalize) (if h : a = 0 then 0 else Classical.choose ⋯), normalize x = x | split_ifs with h | case pos
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
h : a = 0
⊢ ∀ x ∈ Multiset.map (fun x => normalize x) 0, normalize x = x
case neg
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizatio... | 6feac4ff4b873185 |
continuous_gaugeRescale | Mathlib/Analysis/Convex/GaugeRescale.lean | theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
(ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
Continuous (gaugeRescale s t) | case inr
E : Type u_1
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℝ E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : T1Space E
s t : Set E
hs : Convex ℝ s
hs₀ : s ∈ 𝓝 0
ht : Convex ℝ t
ht₀ : t ∈ 𝓝 0
htb : IsVonNBounded ℝ t
hta : Absorbent ℝ t
x : E
hx : x ≠ 0
⊢ Continuous... | exact ((continuousAt_gauge hs hs₀).div (continuousAt_gauge ht ht₀)
((gauge_pos hta htb).2 hx).ne').smul continuousAt_id | no goals | 0c1e2ac317ba137d |
PowerBasis.minpolyGen_monic | Mathlib/RingTheory/PowerBasis.lean | theorem minpolyGen_monic (pb : PowerBasis A S) : Monic (minpolyGen pb) | S : Type u_2
inst✝² : Ring S
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra A S
pb : PowerBasis A S
a✝ : Nontrivial A
⊢ pb.minpolyGen.Monic | apply (monic_X_pow _).sub_of_left _ | S : Type u_2
inst✝² : Ring S
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra A S
pb : PowerBasis A S
a✝ : Nontrivial A
⊢ (∑ i : Fin pb.dim, C ((pb.basis.repr (pb.gen ^ pb.dim)) i) * X ^ ↑i).degree < (X ^ pb.dim).degree | 5f73e594e4d01056 |
MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le | Mathlib/MeasureTheory/Function/StronglyMeasurable/AEStronglyMeasurable.lean | theorem nullMeasurableSet_le [Preorder β] [OrderClosedTopology β] [PseudoMetrizableSpace β]
{f g : α → β} (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
NullMeasurableSet { a | f a ≤ g a } μ | α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : Preorder β
inst✝¹ : OrderClosedTopology β
inst✝ : PseudoMetrizableSpace β
f g : α → β
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
⊢ {a | AEStronglyMeasurable.mk f hf a ≤ AEStronglyMeasurable.mk g hg a} =ᶠ... | filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx | case h
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace β
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : Preorder β
inst✝¹ : OrderClosedTopology β
inst✝ : PseudoMetrizableSpace β
f g : α → β
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
x : α
hfx : f x = AEStronglyMeasurable.mk f hf x
hgx : g x = AEStro... | 4bd628dd7e9ae638 |
Vector.eq_mkVector_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem eq_mkVector_iff {a : α} {n} {l : Vector α n} :
l = mkVector n a ↔ ∀ (b) (_ : b ∈ l), b = a | α : Type u_1
a : α
n : Nat
l : Vector α n
⊢ l.toArray = (mkVector n a).toArray ↔ ∀ (b : α), b ∈ l → b = a | simpa using Array.eq_mkArray_iff (l := l.toArray) (n := n) | no goals | 4bca813e57f2444f |
normEDS_even_ofNat | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | lemma normEDS_even_ofNat (m : ℕ) : normEDS b c d (2 * (m + 3)) * b =
normEDS b c d (m + 2) ^ 2 * normEDS b c d (m + 3) * normEDS b c d (m + 5) -
normEDS b c d (m + 1) * normEDS b c d (m + 3) * normEDS b c d (m + 4) ^ 2 | R : Type u
inst✝ : CommRing R
b c d : R
m : ℕ
⊢ (preNormEDS' (b ^ 4) c d (2 * (m + 3)) * if Even (2 * (m + 3)) then b else 1) * b =
(preNormEDS' (b ^ 4) c d (m + 2) * if Even (m + 2) then b else 1) ^ 2 *
(preNormEDS' (b ^ 4) c d (m + 3) * if Even (m + 3) then b else 1) *
(preNormEDS' (b ^ 4) c d (... | simp only [preNormEDS'_even, if_pos <| even_two_mul _, Nat.even_add_one, ite_not] | R : Type u
inst✝ : CommRing R
b c d : R
m : ℕ
⊢ (preNormEDS' (b ^ 4) c d (m + 2) ^ 2 * preNormEDS' (b ^ 4) c d (m + 3) * preNormEDS' (b ^ 4) c d (m + 5) -
preNormEDS' (b ^ 4) c d (m + 1) * preNormEDS' (b ^ 4) c d (m + 3) * preNormEDS' (b ^ 4) c d (m + 4) ^ 2) *
b *
b =
(preNormEDS' (b ^ 4) c... | f90b888db7e274f6 |
HasFPowerSeriesWithinAt.comp | Mathlib/Analysis/Analytic/Composition.lean | theorem HasFPowerSeriesWithinAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
{p : FormalMultilinearSeries 𝕜 E F} {x : E} {t : Set F} {s : Set E}
(hg : HasFPowerSeriesWithinAt g q t (f x)) (hf : HasFPowerSeriesWithinAt f p s x)
(hs : Set.MapsTo f s t) : HasFPowerSeriesWithinAt (g ∘ f) (... | case intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q : FormalMultilinearSeries... | rcases q.comp_summable_nnreal p Hg.radius_pos Hf.radius_pos with ⟨r, r_pos : 0 < r, hr⟩ | case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
q : FormalMulti... | 919c288392263270 |
Submodule.basis_of_pid_aux | Mathlib/LinearAlgebra/FreeModule/PID.lean | theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
(M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
(∀ (c : R) (z : O), z... | case neg.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N <... | refine ⟨y', y'M, a, a_smul_y'.symm ▸ yN, ?_⟩ | case neg.intro.intro.intro
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N <... | 48ee91e21f4317c3 |
DirectedOn.pi | Mathlib/Order/Directed.lean | lemma pi {d : (i : ι) → Set (α i)} (hd : ∀ (i : ι), DirectedOn (r i) (d i)) :
DirectedOn (fun x y => ∀ i, r i (x i) (y i)) (Set.pi Set.univ d) | ι : Type u_1
α : ι → Type u_2
r : (i : ι) → α i → α i → Prop
d : (i : ι) → Set (α i)
hd : ∀ (i : ι), DirectedOn (r i) (d i)
a : (i : ι) → α i
ha : a ∈ Set.univ.pi d
b : (i : ι) → α i
hb : b ∈ Set.univ.pi d
⊢ ∃ z ∈ Set.univ.pi d, (fun x y => ∀ (i : ι), r i (x i) (y i)) a z ∧ (fun x y => ∀ (i : ι), r i (x i) (y i)) b z | choose f hfd haf hbf using fun i => hd i (a i) (ha i trivial) (b i) (hb i trivial) | ι : Type u_1
α : ι → Type u_2
r : (i : ι) → α i → α i → Prop
d : (i : ι) → Set (α i)
hd : ∀ (i : ι), DirectedOn (r i) (d i)
a : (i : ι) → α i
ha : a ∈ Set.univ.pi d
b : (i : ι) → α i
hb : b ∈ Set.univ.pi d
f : (i : ι) → α i
hfd : ∀ (i : ι), f i ∈ d i
haf : ∀ (i : ι), r i (a i) (f i)
hbf : ∀ (i : ι), r i (b i) (f i)
⊢ ∃... | 63e7fdecd35a8aa3 |
Std.DHashMap.Internal.IsHashSelf.updateAllBuckets | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Model.lean | theorem updateAllBuckets [BEq α] [Hashable α] [LawfulHashable α] {m : Array (AssocList α β)}
{f : AssocList α β → AssocList α δ} (hf : ∀ l p, p ∈ (f l).toList → containsKey p.1 l.toList)
(hm : IsHashSelf m) : IsHashSelf (updateAllBuckets m f) | α : Type u
β : α → Type v
δ : α → Type w
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Array (AssocList α β)
f : AssocList α β → AssocList α δ
hf : ∀ (l : AssocList α β) (p : (a : α) × δ a), p ∈ (f l).toList → containsKey p.fst l.toList = true
hm : IsHashSelf m
j : Nat
hj : j < (Array.map f m).size
h ... | rcases containsKey_eq_true_iff_exists_mem.1 (hf _ _ hp) with ⟨q, hq₁, hq₂⟩ | case intro.intro
α : Type u
β : α → Type v
δ : α → Type w
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulHashable α
m : Array (AssocList α β)
f : AssocList α β → AssocList α δ
hf : ∀ (l : AssocList α β) (p : (a : α) × δ a), p ∈ (f l).toList → containsKey p.fst l.toList = true
hm : IsHashSelf m
j : Nat
hj : j < (Array... | b66ffbf90867ba8f |
CategoryTheory.Limits.preservesBiproduct_of_epi_biproductComparison' | Mathlib/CategoryTheory/Preadditive/Biproducts.lean | /-- If the (coproduct-like) biproduct comparison for `F` and `f` is an epimorphism, then `F`
preserves the biproduct of `F` and `f`. For the converse, see `mapBiproduct`. -/
lemma preservesBiproduct_of_epi_biproductComparison' {f : J → C} [HasBiproduct f]
[HasBiproduct (F.obj ∘ f)] [Epi (biproductComparison' F ... | C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : Preadditive C
D : Type u'
inst✝⁶ : Category.{v', u'} D
inst✝⁵ : Preadditive D
F : C ⥤ D
inst✝⁴ : F.PreservesZeroMorphisms
J : Type
inst✝³ : Fintype J
f : J → C
inst✝² : HasBiproduct f
inst✝¹ : HasBiproduct (F.obj ∘ f)
inst✝ : Epi (F.biproductComparison' f)
⊢ Epi (F.splitEp... | simpa | no goals | e5350692752cdd00 |
Int.sq_add_sq_of_two_mul_sq_add_sq | Mathlib/NumberTheory/SumFourSquares.lean | theorem sq_add_sq_of_two_mul_sq_add_sq {m x y : ℤ} (h : 2 * m = x ^ 2 + y ^ 2) :
m = ((x - y) / 2) ^ 2 + ((x + y) / 2) ^ 2 :=
have : Even (x ^ 2 + y ^ 2) | m x y : ℤ
h : 2 * m = x ^ 2 + y ^ 2
this : Even (x ^ 2 + y ^ 2)
⊢ 2 * 2 ≠ 0 | decide | no goals | 8f6d51bb57e6acfe |
Filter.Tendsto.partialSups | Mathlib/Topology/Order/PartialSups.lean | protected lemma partialSups (hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))) :
Tendsto (partialSups f n) l (𝓝 (partialSups g n)) | L : Type u_1
inst✝² : SemilatticeSup L
inst✝¹ : TopologicalSpace L
inst✝ : ContinuousSup L
α : Type u_2
l : Filter α
f : ℕ → α → L
g : ℕ → L
n : ℕ
hf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))
⊢ ∀ i ∈ Finset.range (n + 1), Tendsto (f i) l (𝓝 (g i)) | simpa [Nat.lt_succ_iff] | no goals | 0e804d78aae53ace |
Ordinal.isNormal_iff_strictMono_and_continuous | Mathlib/SetTheory/Ordinal/Topology.lean | theorem isNormal_iff_strictMono_and_continuous (f : Ordinal.{u} → Ordinal.{u}) :
IsNormal f ↔ StrictMono f ∧ Continuous f | case refine_2.intro
f : Ordinal.{u} → Ordinal.{u}
h✝ : StrictMono f
h' : Continuous f
o : Ordinal.{u}
ho : o.IsLimit
a : Ordinal.{u}
h : ∀ b < o, f b ≤ a
⊢ o ∈ f ⁻¹' Set.Iic a | rw [mem_iff_iSup_of_isClosed (IsClosed.preimage h' (@isClosed_Iic _ _ _ _ a))] | case refine_2.intro
f : Ordinal.{u} → Ordinal.{u}
h✝ : StrictMono f
h' : Continuous f
o : Ordinal.{u}
ho : o.IsLimit
a : Ordinal.{u}
h : ∀ b < o, f b ≤ a
⊢ ∃ ι, ∃ (_ : Nonempty ι), ∃ f_1, (∀ (i : ι), f_1 i ∈ f ⁻¹' Set.Iic a) ∧ ⨆ i, f_1 i = o | a6bfa64dac95f188 |
IsFreeGroupoid.SpanningTree.endIsFree | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | /-- Given a free groupoid and an arborescence of its generating quiver, the vertex
group at the root is freely generated by loops coming from generating arrows
in the complement of the tree. -/
lemma endIsFree : IsFreeGroup (End (root' T)) :=
IsFreeGroup.ofUniqueLift ((wideSubquiverEquivSetTotal <| wideSubqui... | case intro.intro.refine_1
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Genera... | intros x y q | case intro.intro.refine_1
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Genera... | 4742e81f46ca4ae4 |
RingHom.finiteType_isStableUnderBaseChange | Mathlib/RingTheory/RingHom/FiniteType.lean | theorem finiteType_isStableUnderBaseChange : IsStableUnderBaseChange @FiniteType | case h₂
⊢ ∀ ⦃R S T : Type u_1⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
[inst_4 : Algebra R T], (algebraMap R T).FiniteType → Algebra.TensorProduct.includeLeftRingHom.FiniteType | introv h | case h₂
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : (algebraMap R T).FiniteType
⊢ Algebra.TensorProduct.includeLeftRingHom.FiniteType | 32090bc17de5e6dc |
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 | ... | 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... | exact disjoint_left.mpr (fun a ha ↦ by simpa using ha) | no goals | 6098eac6b53bede3 |
MeasureTheory.progMeasurable_of_tendsto' | Mathlib/Probability/Process/Adapted.lean | theorem progMeasurable_of_tendsto' {γ} [MeasurableSpace ι] [PseudoMetrizableSpace β]
(fltr : Filter γ) [fltr.NeBot] [fltr.IsCountablyGenerated] {U : γ → ι → Ω → β}
(h : ∀ l, ProgMeasurable f (U l)) (h_tendsto : Tendsto U fltr (𝓝 u)) : ProgMeasurable f u | Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : Preorder ι
u : ι → Ω → β
f : Filtration ι m
γ : Type u_4
inst✝³ : MeasurableSpace ι
inst✝² : PseudoMetrizableSpace β
fltr : Filter γ
inst✝¹ : fltr.NeBot
inst✝ : fltr.IsCountablyGenerated
U : γ → ι → Ω → β
h : ∀ (l : γ), Pr... | rw [tendsto_pi_nhds] at h_tendsto ⊢ | Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : Preorder ι
u : ι → Ω → β
f : Filtration ι m
γ : Type u_4
inst✝³ : MeasurableSpace ι
inst✝² : PseudoMetrizableSpace β
fltr : Filter γ
inst✝¹ : fltr.NeBot
inst✝ : fltr.IsCountablyGenerated
U : γ → ι → Ω → β
h : ∀ (l : γ), Pr... | 952758a0436228d6 |
not_differentiableWithinAt_of_deriv_tendsto_atBot_Iio | Mathlib/Analysis/Calculus/MeanValue.lean | theorem not_differentiableWithinAt_of_deriv_tendsto_atBot_Iio (f : ℝ → ℝ) {a : ℝ}
(hf : Tendsto (deriv f) (𝓝[<] a) atBot) : ¬ DifferentiableWithinAt ℝ f (Iio a) a | case intro.intro
f : ℝ → ℝ
a : ℝ
f' : ℝ → ℝ := f ∘ Neg.neg
b : ℝ
hb₁ : b < a
hb₂ : ∀ ⦃x : ℝ⦄, x ∈ Ioo b a → deriv f x ∈ Iic (-1)
x : ℝ
hx : -a < x ∧ x < -b
h₁ : -x ∈ Ioo b a
h₂ : deriv f (-x) ≤ -1
⊢ deriv f (-x) ≠ 0 | exact ne_of_lt (by linarith) | no goals | fe115a5e1bee6195 |
IsPrimitiveRoot.self_sub_one_pow_dvd_order | Mathlib/RingTheory/RootsOfUnity/Lemmas.lean | /-- If `μ` is a primitive `n`th root of unity in `R` and `k < n`, then `n` is divisible
by `(μ-1)^k` in `ℤ[μ] ⊆ R`. -/
lemma self_sub_one_pow_dvd_order {k n : ℕ} (hn : k < n) {μ : R} (hμ : IsPrimitiveRoot μ n) :
∃ z ∈ adjoin ℤ {μ}, n = z * (μ - 1) ^ k | case intro
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ z ∈ adjoin ℤ {μ}, μ ^ k - 1 = z * (μ - 1)
Z : ℕ → R := fun k => Classical.choose ⋯
Zdef : ∀ (k : ℕ), Z k ∈ adjoin ℤ {μ} ∧ μ ^ k - 1 = Z k * (μ - 1)
⊢ ∃ z ∈ adjoin ... | refine ⟨(-1) ^ (m + k) * (∏ j ∈ range k, Z (j + 1)) * ∏ j ∈ Ico k (m + k), (μ ^ (j + 1) - 1),
?_, ?_⟩ | case intro.refine_1
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
k n : ℕ
μ : R
m : ℕ
hn : k < m + k + 1
hμ : IsPrimitiveRoot μ (m + k + 1)
hdvd : ∀ (k : ℕ), ∃ z ∈ adjoin ℤ {μ}, μ ^ k - 1 = z * (μ - 1)
Z : ℕ → R := fun k => Classical.choose ⋯
Zdef : ∀ (k : ℕ), Z k ∈ adjoin ℤ {μ} ∧ μ ^ k - 1 = Z k * (μ - 1)
⊢ ((-1... | 08cc616e07efde3d |
Num.gcd_to_nat_aux | Mathlib/Data/Num/Lemmas.lean | theorem gcd_to_nat_aux :
∀ {n} {a b : Num}, a ≤ b → (a * b).natSize ≤ n → (gcdAux n a b : ℕ) = Nat.gcd a b
| 0, 0, _, _ab, _h => (Nat.gcd_zero_left _).symm
| 0, pos _, 0, ab, _h => (not_lt_of_ge ab).elim rfl
| 0, pos _, pos _, _ab, h => (not_lt_of_le h).elim <| PosNum.natSize_pos _
| Nat.succ _, 0, _, _ab, ... | case a
n : ℕ
a : PosNum
b : Num
ab : pos a ≤ b
h : ↑(pos a) * ↑b < 2 ^ n.succ
⊢ ↑(b % pos a) * ↑(pos a) < 2 ^ n | rw [mod_to_nat, mul_comm] | case a
n : ℕ
a : PosNum
b : Num
ab : pos a ≤ b
h : ↑(pos a) * ↑b < 2 ^ n.succ
⊢ ↑(pos a) * (↑b % ↑(pos a)) < 2 ^ n | 2a436717f8fc21a6 |
Finset.sup'_image₂_right | Mathlib/Data/Finset/NAry.lean | lemma sup'_image₂_right (g : γ → δ) (h : (image₂ f s t).Nonempty) :
sup' (image₂ f s t) h g =
sup' t h.of_image₂_right fun y ↦ sup' s h.of_image₂_left (g <| f · y) | α : Type u_1
β : Type u_3
γ : Type u_5
δ : Type u_7
inst✝¹ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
inst✝ : SemilatticeSup δ
g : γ → δ
h : (image₂ f s t).Nonempty
⊢ (image₂ f s t).sup' h g = t.sup' ⋯ fun y => s.sup' ⋯ fun x => g (f x y) | simp only [image₂, sup'_image, sup'_product_right] | α : Type u_1
β : Type u_3
γ : Type u_5
δ : Type u_7
inst✝¹ : DecidableEq γ
f : α → β → γ
s : Finset α
t : Finset β
inst✝ : SemilatticeSup δ
g : γ → δ
h : (image₂ f s t).Nonempty
⊢ (t.sup' ⋯ fun i' => s.sup' ⋯ fun i => (g ∘ uncurry f) (i, i')) = t.sup' ⋯ fun y => s.sup' ⋯ fun x => g (f x y) | 8e6b3cf6cd7a2f59 |
LipschitzOnWith.extend_finite_dimension | Mathlib/Analysis/Normed/Module/FiniteDimension.lean | theorem LipschitzOnWith.extend_finite_dimension {α : Type*} [PseudoMetricSpace α] {E' : Type*}
[NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E'] {s : Set α} {f : α → E'}
{K : ℝ≥0} (hf : LipschitzOnWith K f s) :
∃ g : α → E', LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s | α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s | let ι : Type _ := Basis.ofVectorSpaceIndex ℝ E' | α : Type u_1
inst✝³ : PseudoMetricSpace α
E' : Type u_2
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace ℝ E'
inst✝ : FiniteDimensional ℝ E'
s : Set α
f : α → E'
K : ℝ≥0
hf : LipschitzOnWith K f s
ι : Type u_2 := ↑(Basis.ofVectorSpaceIndex ℝ E')
⊢ ∃ g, LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s | ca32574ab8f0dfb2 |
Polynomial.leval_coe_eq_smeval | Mathlib/Algebra/Polynomial/Smeval.lean | theorem leval_coe_eq_smeval {R : Type*} [Semiring R] (r : R) :
⇑(leval r) = fun p => p.smeval r | R : Type u_3
inst✝ : Semiring R
r : R
x✝ : R[X]
⊢ (leval r) x✝ = x✝.smeval r | rw [leval_apply, smeval_def, eval_eq_sum] | R : Type u_3
inst✝ : Semiring R
r : R
x✝ : R[X]
⊢ (x✝.sum fun e a => a * r ^ e) = x✝.sum (smul_pow r) | e735235b3cd420bb |
MvPolynomial.coeff_C_mul | Mathlib/Algebra/MvPolynomial/Basic.lean | theorem coeff_C_mul (m) (a : R) (p : MvPolynomial σ R) : coeff m (C a * p) = a * coeff m p | R : Type u
σ : Type u_1
inst✝ : CommSemiring R
m : σ →₀ ℕ
a : R
p : MvPolynomial σ R
⊢ (sum p fun n b => (monomial (0 + n)) (0 * b)) = 0 | simp | no goals | a1985ef9ff455954 |
StieltjesFunction.length_Ioc | Mathlib/MeasureTheory/Measure/Stieltjes.lean | theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a) | f : StieltjesFunction
a b : ℝ
⊢ f.length (Ioc a b) = ofReal (↑f b - ↑f a) | refine
le_antisymm (iInf_le_of_le a <| iInf₂_le b Subset.rfl)
(le_iInf fun a' => le_iInf fun b' => le_iInf fun h => ENNReal.coe_le_coe.2 ?_) | f : StieltjesFunction
a b a' b' : ℝ
h : Ioc a b ⊆ Ioc a' b'
⊢ (↑f b - ↑f a).toNNReal ≤ (↑f b' - ↑f a').toNNReal | 580531cab106ec56 |
Set.Finite.encard_lt_top | Mathlib/Data/Set/Card.lean | theorem Finite.encard_lt_top (h : s.Finite) : s.encard < ⊤ | α : Type u_1
s : Set α
h : s.Finite
a : α
t : Set α
hat : a ∉ t
hs✝ : t.Finite
ht' : t.encard < ⊤
⊢ (insert a t).encard < ⊤ | rw [encard_insert_of_not_mem hat] | α : Type u_1
s : Set α
h : s.Finite
a : α
t : Set α
hat : a ∉ t
hs✝ : t.Finite
ht' : t.encard < ⊤
⊢ t.encard + 1 < ⊤ | 57fd237c16876548 |
isEmbedding_of_iSup_eq_top_of_preimage_subset_range | Mathlib/Topology/LocalAtTarget.lean | theorem isEmbedding_of_iSup_eq_top_of_preimage_subset_range
{X Y} [TopologicalSpace X] [TopologicalSpace Y]
(f : X → Y) (h : Continuous f) {ι : Type*}
(U : ι → Opens Y) (hU : Set.range f ⊆ (iSup U :))
(V : ι → Type*) [∀ i, TopologicalSpace (V i)]
(iV : ∀ i, V i → X) (hiV : ∀ i, Continuous (iV i)) (h... | case inr.refine_3
X : Type u_6
Y : Type u_7
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
h : Continuous f
ι : Type u_4
U : ι → Opens Y
hU : range f ⊆ ↑(iSup U)
V : ι → Type u_5
inst✝ : (i : ι) → TopologicalSpace (V i)
iV : (i : ι) → V i → X
hiV : ∀ (i : ι), Continuous (iV i)
hV : ∀ (i : ι), f ⁻¹' ↑... | exact hV | no goals | 3659690abb1e1161 |
mdifferentiableWithinAt_iff_source_of_mem_maximalAtlas | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | theorem mdifferentiableWithinAt_iff_source_of_mem_maximalAtlas
[IsManifold I 1 M] (he : e ∈ maximalAtlas I 1 M) (hx : x ∈ e.source) :
MDifferentiableWithinAt I I' f s x ↔
MDifferentiableWithinAt 𝓘(𝕜, E) I' (f ∘ (e.extend I).symm) ((e.extend I).symm ⁻¹' s ∩ range I)
(e.extend I x) | 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpa... | simp_rw [MDifferentiableWithinAt,
differentiableWithinAt_localInvariantProp.liftPropWithinAt_indep_chart_source he hx,
StructureGroupoid.liftPropWithinAt_self_source,
e.extend_symm_continuousWithinAt_comp_right_iff, differentiableWithinAtProp_self_source,
DifferentiableWithinAtProp, Function.comp, e.left_inv hx... | 𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpa... | d5fc77d4f090866e |
ContMDiffFiberwiseLinear.locality_aux₂ | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | theorem ContMDiffFiberwiseLinear.locality_aux₂
(n : WithTop ℕ∞) (e : PartialHomeomorph (B × F) (B × F)) (U : Set B) (hU : e.source = U ×ˢ univ)
(h : ∀ x ∈ U,
∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_hUu : u ⊆ U) (_hux : x ∈ u)
(hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (φ x : ... | 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorne... | have hU' : IsOpen U := by
rw [hUu']
apply isOpen_iUnion hu | 𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorne... | 1201be7e3e0cc0c8 |
LinearOrderedAddCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete | Mathlib/GroupTheory/ArchimedeanDensely.lean | lemma LinearOrderedAddCommGroup.wellFoundedOn_setOf_le_lt_iff_nonempty_discrete
{G : Type*} [LinearOrderedAddCommGroup G] [Nontrivial G] {g : G} :
Set.WellFoundedOn {x : G | g ≤ x} (· < ·) ↔ Nonempty (G ≃+o ℤ) | case pos
G : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : Archimedean G
⊢ Nonempty (G ≃+o ℤ) | rw [LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered] | case pos
G : Type u_2
inst✝¹ : LinearOrderedAddCommGroup G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : Archimedean G
⊢ ¬DenselyOrdered G | 59c60e31ffce1071 |
Multiset.gcd_ndinsert | Mathlib/Algebra/GCDMonoid/Multiset.lean | theorem gcd_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).gcd = GCDMonoid.gcd a s.gcd | α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizedGCDMonoid α
inst✝ : DecidableEq α
a : α
s : Multiset α
⊢ ∀ (a_1 : α), a_1 ∈ ndinsert a s ↔ a_1 ∈ a ::ₘ s | simp | no goals | 0e0612dac568d850 |
CategoryTheory.ComposableArrows.ext₀ | Mathlib/CategoryTheory/ComposableArrows.lean | lemma ext₀ {F G : ComposableArrows C 0} (h : F.obj' 0 = G.obj 0) : F = G :=
ext (fun i => match i with
| ⟨0, _⟩ => h) (fun i hi => by simp at hi)
| C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : ComposableArrows C 0
h : F.obj' 0 ⋯ = G.obj 0
i : ℕ
hi : i < 0
⊢ F.map' i (i + 1) ⋯ hi = eqToHom ⋯ ≫ G.map' i (i + 1) ⋯ hi ≫ eqToHom ⋯ | simp at hi | no goals | b3e0293b24ba6232 |
Matrix.smul_vec3 | Mathlib/Data/Matrix/Notation.lean | theorem smul_vec3 {R : Type*} [SMul R α] (x : R) (a₀ a₁ a₂ : α) :
x • ![a₀, a₁, a₂] = ![x • a₀, x • a₁, x • a₂] | α : Type u
R : Type u_1
inst✝ : SMul R α
x : R
a₀ a₁ a₂ : α
⊢ x • ![a₀, a₁, a₂] = ![x • a₀, x • a₁, x • a₂] | rw [smul_cons, smul_cons, smul_cons, smul_empty] | no goals | 881bdf72c76d4c88 |
Trivialization.Prod.left_inv | Mathlib/Topology/FiberBundle/Constructions.lean | theorem Prod.left_inv {x : TotalSpace (F₁ × F₂) (E₁ ×ᵇ E₂)}
(h : x ∈ π (F₁ × F₂) (E₁ ×ᵇ E₂) ⁻¹' (e₁.baseSet ∩ e₂.baseSet)) :
Prod.invFun' e₁ e₂ (Prod.toFun' e₁ e₂ x) = x | B : Type u_1
inst✝⁶ : TopologicalSpace B
F₁ : Type u_2
inst✝⁵ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝³ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivialization F₂ Total... | obtain ⟨x, v₁, v₂⟩ := x | case mk.mk
B : Type u_1
inst✝⁶ : TopologicalSpace B
F₁ : Type u_2
inst✝⁵ : TopologicalSpace F₁
E₁ : B → Type u_3
inst✝⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_4
inst✝³ : TopologicalSpace F₂
E₂ : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F₂ E₂)
e₁ : Trivialization F₁ TotalSpace.proj
e₂ : Trivializati... | f07d66bfab43f8d0 |
MeasureTheory.Measure.pi_caratheodory | Mathlib/MeasureTheory/Constructions/Pi.lean | theorem pi_caratheodory :
MeasurableSpace.pi ≤ (OuterMeasure.pi fun i => (μ i).toOuterMeasure).caratheodory | case intro.intro.hs
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ ∏ x : ι, (μ x).toOuterMeasure (eval x '' (t ∩ (fun b => b i) ⁻¹' s)) +
∏ x : ι, (μ x).toOuterMeasure (eval x '... | refine Finset.prod_add_prod_le' (Finset.mem_univ i) ?_ ?_ ?_ | case intro.intro.hs.refine_1
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
i : ι
s : Set (α i)
hs : MeasurableSet s
t : Set ((a : ι) → α a)
⊢ (μ i).toOuterMeasure (eval i '' (t ∩ (fun b => b i) ⁻¹' s)) +
(μ i).toOuterMeasure (eval i '' (t \ (f... | 564b11a9d820f16f |
Lean.Data.AC.Context.evalList_sort_congr | Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean | theorem Context.evalList_sort_congr
(ctx : Context α)
(h : Commutative ctx.op)
(h₂ : evalList α ctx a = evalList α ctx b)
(h₃ : a ≠ [])
(h₄ : b ≠ [])
: evalList α ctx (sort.loop a c) = evalList α ctx (sort.loop b c) | α : Sort u_1
a b c : List Nat
ctx : Context α
h : Commutative ctx.op
h₂ : evalList α ctx a = evalList α ctx b
h₃ : a ≠ []
h₄ : b ≠ []
⊢ evalList α ctx (sort.loop a c) = evalList α ctx (sort.loop b c) | induction c generalizing a b with
| nil => simp [sort.loop, h₂]
| cons c _ ih =>
simp [sort.loop]; apply ih; simp [evalList_insert ctx h, evalList]
cases a with
| nil => apply absurd h₃; simp
| cons a as =>
cases b with
| nil => apply absurd h₄; simp
| cons b bs => simp [evalList, h₂]
all_goals a... | no goals | 21bdef0510a8391b |
Complex.continuousWithinAt_log_of_re_neg_of_im_zero | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | theorem continuousWithinAt_log_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt log { z : ℂ | 0 ≤ z.im } z | case convert_1.intro
z : ℝ
hre : (↑z).re < 0
⊢ ‖↑z‖ ≠ 0 | simpa using hre.ne | no goals | fa596a77b673a65a |
Real.logDeriv_cos | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | theorem Real.logDeriv_cos : logDeriv (Real.cos) = -Real.tan | ⊢ logDeriv cos = -tan | ext | case h
x✝ : ℝ
⊢ logDeriv cos x✝ = (-tan) x✝ | d6feb2344ec29a8e |
CategoryTheory.presheafHom_isSheafFor | Mathlib/CategoryTheory/Sites/SheafHom.lean | lemma presheafHom_isSheafFor :
Presieve.IsSheafFor (presheafHom F G) S.arrows | case e_self.e_f
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
hg : S.arrows g
... | simp | no goals | 03e610e11f4d0fc5 |
subset_tangentCone_prod_right | Mathlib/Analysis/Calculus/TangentCone.lean | theorem subset_tangentCone_prod_right {t : Set F} {y : F} (hs : x ∈ closure s) :
LinearMap.inr 𝕜 E F '' tangentConeAt 𝕜 t y ⊆ tangentConeAt 𝕜 (s ×ˢ t) (x, y) | case intro.intro.intro.intro.intro.intro.refine_1
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
s : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ... | filter_upwards [hd] with n hn | case h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : E
s : Set E
t : Set F
y : F
hs : x ∈ closure s
w : F
c : ℕ → 𝕜
d : ℕ → F
hd : ∀ᶠ (n : ℕ) in atTop, y + d n ∈ t
hc : Tends... | 039d3fbcf94f6b57 |
MeasureTheory.StronglyMeasurable.norm_approxBounded_le | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} {c : ℝ} (hf : StronglyMeasurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
‖hf.approxBounded c n x‖ ≤ c | α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
⊢ ‖(1 ⊓ c / ‖(hf.approx n) x‖) • (hf.approx n) x‖ ≤ c | refine (norm_smul_le _ _).trans ?_ | α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
⊢ ‖1 ⊓ c / ‖(hf.approx n) x‖‖ * ‖(hf.approx n) x‖ ≤ c | c6dcf159a9df0ec3 |
ProbabilityTheory.Kernel.lintegral_parallelComp | Mathlib/Probability/Kernel/Composition/ParallelComp.lean | lemma lintegral_parallelComp (κ : Kernel α β) [IsSFiniteKernel κ]
(η : Kernel γ δ) [IsSFiniteKernel η]
(ac : α × γ) {g : β × δ → ℝ≥0∞} (hg : Measurable g) :
∫⁻ bd, g bd ∂(κ ∥ₖ η) ac = ∫⁻ b, ∫⁻ d, g (b, d) ∂η ac.2 ∂κ ac.1 | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel γ δ
inst✝ : IsSFiniteKernel η
ac : α × γ
g : β × δ → ℝ≥0∞
hg : Measurable g
⊢ ∫⁻ (b : β), ∫⁻ (c : δ), g (b, c) ∂(prodMkLeft... | simp | no goals | 73bc0f1b7ac795e9 |
Rat.add.aux | Mathlib/.lake/packages/batteries/Batteries/Data/Rat/Basic.lean | theorem add.aux (a b : Rat) {g ad bd} (hg : g = a.den.gcd b.den)
(had : ad = a.den / g) (hbd : bd = b.den / g) :
let den := ad * b.den; let num := a.num * bd + b.num * ad
num.natAbs.gcd g = num.natAbs.gcd den | a b : Rat
g ad bd : Nat
hg : g = a.den.gcd b.den
had : ad = a.den / g
hbd : bd = b.den / g
den : Nat := ad * b.den
num : Int := a.num * ↑bd + b.num * ↑ad
ae : ad * g = a.den
be : bd * g = b.den
hden : den = ad * bd * g
cop : ad.Coprime bd
d : Nat
this : d.gcd num.natAbs ∣ num.natAbs
⊢ d.gcd num.natAbs ∣ a.num.natAbs * ... | rw [← Int.ofNat_dvd, Int.dvd_natAbs] at this | a b : Rat
g ad bd : Nat
hg : g = a.den.gcd b.den
had : ad = a.den / g
hbd : bd = b.den / g
den : Nat := ad * b.den
num : Int := a.num * ↑bd + b.num * ↑ad
ae : ad * g = a.den
be : bd * g = b.den
hden : den = ad * bd * g
cop : ad.Coprime bd
d : Nat
this : ↑(d.gcd num.natAbs) ∣ num
⊢ d.gcd num.natAbs ∣ a.num.natAbs * bd ↔... | a66f7dc45eb48062 |
HNNExtension.NormalWord.prod_empty | Mathlib/GroupTheory/HNNExtension.lean | theorem prod_empty : (empty : NormalWord d).prod φ = 1 | G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
⊢ ReducedWord.prod φ empty.toReducedWord = 1 | simp [ReducedWord.prod] | no goals | 4fb909f145ac3bb8 |
ModuleCat.span_exact | Mathlib/Algebra/Category/ModuleCat/Free.lean | theorem span_exact {β : Type*} {u : ι ⊕ β → S.X₂} (huv : u ∘ Sum.inl = S.f ∘ v)
(hv : ⊤ ≤ span R (range v))
(hw : ⊤ ≤ span R (range (S.g ∘ u ∘ Sum.inr))) :
⊤ ≤ span R (range u) | case intro.intro.intro.a
ι : Type u_1
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
β : Type u_4
u : ι ⊕ β → ↑S.X₂
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
hv : ⊤ ≤ span R (range v)
hw : ⊤ ≤ span R (range (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr))
m : ↑S.X₂
a✝ : m ∈ ... | use cn.mapDomain (Sum.inl) | case h
ι : Type u_1
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
β : Type u_4
u : ι ⊕ β → ↑S.X₂
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
hv : ⊤ ≤ span R (range v)
hw : ⊤ ≤ span R (range (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr))
m : ↑S.X₂
a✝ : m ∈ ⊤
cm : β →₀ R
hm :... | 1696945178be2940 |
Seminorm.continuous_from_bounded | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem continuous_from_bounded {p : SeminormFamily 𝕝 E ι} {q : SeminormFamily 𝕝₂ F ι'}
{_ : TopologicalSpace E} (hp : WithSeminorms p) {_ : TopologicalSpace F} (hq : WithSeminorms q)
(f : E →ₛₗ[τ₁₂] F) (hf : Seminorm.IsBounded p q f) : Continuous f | 𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
ι : Type u_8
ι' : Type u_9
inst✝⁸ : AddCommGroup E
inst✝⁷ : NormedField 𝕝
inst✝⁶ : Module 𝕝 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : NormedField 𝕝₂
inst✝³ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝² : RingHomIsometric τ₁₂
inst✝¹ : Nonempty ι
inst✝ : Nonempty ι'
p : Seminor... | have : IsTopologicalAddGroup E := hp.topologicalAddGroup | 𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
ι : Type u_8
ι' : Type u_9
inst✝⁸ : AddCommGroup E
inst✝⁷ : NormedField 𝕝
inst✝⁶ : Module 𝕝 E
inst✝⁵ : AddCommGroup F
inst✝⁴ : NormedField 𝕝₂
inst✝³ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝² : RingHomIsometric τ₁₂
inst✝¹ : Nonempty ι
inst✝ : Nonempty ι'
p : Seminor... | 0044c0551bf1b760 |
Finset.inductive_claim_mul | Mathlib/Combinatorics/Additive/SmallTripling.lean | @[to_additive]
private lemma inductive_claim_mul (hm : 3 ≤ m)
(h : ∀ ε : Fin 3 → ℤ, (∀ i, |ε i| = 1) → #((finRange 3).map fun i ↦ A ^ ε i).prod ≤ k * #A)
(ε : Fin m → ℤ) (hε : ∀ i, |ε i| = 1) :
#((finRange m).map fun i ↦ A ^ ε i).prod ≤ k ^ (m - 2) * #A | G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i => A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
∀ (ε : Fin (m + 1) → ℤ),
(∀ (i : Fin (m + 1)), |ε i| = 1) →
↑(#(List.map (fun i => A ^ ε... | gcongr | case h₁
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i => A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
∀ (ε : Fin (m + 1) → ℤ),
(∀ (i : Fin (m + 1)), |ε i| = 1) →
↑(#(List.map (fun i ... | 75599e5c7cd8b516 |
List.find?_iota_eq_some | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Range.lean | theorem find?_iota_eq_some {n : Nat} {i : Nat} {p : Nat → Bool} :
(iota n).find? p = some i ↔ p i ∧ i ∈ iota n ∧ ∀ j, i < j → j ≤ n → !p j | case mpr.intro.intro.refine_1.right
n i : Nat
p : Nat → Bool
h✝ : p i = true
h : ∀ (j : Nat), i < j → j ≤ n → p j = false
i₁ : 1 ≤ i
i₂ : i < 1 + n
⊢ n - i + 1 = n - (i - 1) ∧ i = 1 + (i - 1) | omega | no goals | 75d9b906d97f27ae |
AlgebraicGeometry.exists_lift_of_germInjective | Mathlib/AlgebraicGeometry/SpreadingOut.lean | /--
Suppose `X` is a scheme, `x : X` such that the germ map at `x` is (locally) injective,
and `U` is a neighborhood of `x`.
Given a commutative diagram of `CommRingCat`
```
R ⟶ Γ(X, U)
↓ ↓
A ⟶ 𝒪_{X, x}
```
such that `R` is of finite type over `A`, we may lift `A ⟶ 𝒪_{X, x}` to some `A ⟶ Γ(X, V)`.
-/
lemma exists_... | X : Scheme
R A : CommRingCat
x : ↑↑X.toPresheafedSpace
inst✝ : X.IsGermInjectiveAt x
U : X.Opens
hxU : x ∈ U
φ : A ⟶ X.presheaf.stalk x
φRA : R ⟶ A
φRX : R ⟶ Γ(X, U)
hφRA : (CommRingCat.Hom.hom φRA).FiniteType
e : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU
⊢ ∃ V,
∃ (hxV : x ∈ V),
∃ φ', ∃ (i : V ≤ U), IsAffineOpe... | obtain ⟨V, hxV, iVU, hV⟩ := exists_lift_of_germInjective_aux hxU φ φRA φRX hφRA e | case intro.intro.intro
X : Scheme
R A : CommRingCat
x : ↑↑X.toPresheafedSpace
inst✝ : X.IsGermInjectiveAt x
U : X.Opens
hxU : x ∈ U
φ : A ⟶ X.presheaf.stalk x
φRA : R ⟶ A
φRX : R ⟶ Γ(X, U)
hφRA : (CommRingCat.Hom.hom φRA).FiniteType
e : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU
V : X.Opens
hxV : x ∈ V
iVU : V ≤ U
hV : (C... | 1c2551a743f9c2e9 |
ProbabilityTheory.defaultRatCDF_nonneg | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | lemma defaultRatCDF_nonneg (q : ℚ) : 0 ≤ defaultRatCDF q | q : ℚ
⊢ 0 ≤ if q < 0 then 0 else 1 | split_ifs | case pos
q : ℚ
h✝ : q < 0
⊢ 0 ≤ 0
case neg
q : ℚ
h✝ : ¬q < 0
⊢ 0 ≤ 1 | 4d211d4d56c218f3 |
cpow_eq_nhds' | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | theorem cpow_eq_nhds' {p : ℂ × ℂ} (hp_fst : p.fst ≠ 0) :
(fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) | p : ℂ × ℂ
hp_fst : p.1 ≠ 0
this : ∀ᶠ (x : ℂ × ℂ) in 𝓝 p, x.1 ≠ 0
x : ℂ × ℂ
hx : x.1 ≠ 0
⊢ x.1 ^ x.2 = cexp (Complex.log x.1 * x.2) | rw [cpow_def_of_ne_zero hx] | no goals | 502a83ba7126bfd4 |
Equiv.Perm.cycleFactorsFinset_mul_inv_mem_eq_sdiff | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem cycleFactorsFinset_mul_inv_mem_eq_sdiff [DecidableEq α] [Fintype α] {f g : Perm α}
(h : f ∈ cycleFactorsFinset g) : cycleFactorsFinset (g * f⁻¹) = cycleFactorsFinset g \ {f} | case refine_2
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ : Perm α
hσ : σ.IsCycle
f : Perm α
hf : f ∈ σ.cycleFactorsFinset
⊢ (σ * f⁻¹).cycleFactorsFinset = σ.cycleFactorsFinset \ {f} | simp only [cycleFactorsFinset_eq_singleton_self_iff.mpr hσ, mem_singleton] at hf ⊢ | case refine_2
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g f✝ σ : Perm α
hσ : σ.IsCycle
f : Perm α
hf : f = σ
⊢ (σ * f⁻¹).cycleFactorsFinset = {σ} \ {f} | 3d77fd0de031f082 |
SeminormFamily.filter_eq_iInf | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem filter_eq_iInf (p : SeminormFamily 𝕜 E ι) :
p.moduleFilterBasis.toFilterBasis.filter = ⨅ i, (𝓝 0).comap (p i) | case refine_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Nonempty ι
p : SeminormFamily 𝕜 E ι
s : Finset ι
r : ℝ
hr : 0 < r
hU : (s.sup p).ball 0 r ∈ p.basisSets
⊢ id ((s.sup p).ball 0 r) ∈ ⨅ i, comap (⇑(p i)) (𝓝 0) | rw [id, Seminorm.ball_finset_sup_eq_iInter _ _ _ hr, s.iInter_mem_sets] | case refine_2.intro.intro.intro
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Nonempty ι
p : SeminormFamily 𝕜 E ι
s : Finset ι
r : ℝ
hr : 0 < r
hU : (s.sup p).ball 0 r ∈ p.basisSets
⊢ ∀ i ∈ s, (p i).ball 0 r ∈ ⨅ i, comap (⇑(p i)) (𝓝 0) | 043221e62cb70beb |
Array.foldrM_start_stop | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem foldrM_start_stop {m} [Monad m] (l : Array α) (f : α → β → m β) (b) (start stop : Nat) :
l.foldrM f b start stop = (l.extract stop start).foldrM f b | case isFalse.isTrue
α : Type u_1
β : Type u_2
m : Type u_2 → Type u_3
inst✝ : Monad m
l : Array α
f : α → β → m β
b : β
start stop : Nat
this :
stop ≤ min start l.size →
foldrM.fold f l stop (min start l.size) ⋯ b = foldrM.fold f (l.extract stop start) 0 (min start l.size - stop) ⋯ b
h✝¹ : ¬start ≤ l.size
h✝ : st... | specialize this (by omega) | case isFalse.isTrue
α : Type u_1
β : Type u_2
m : Type u_2 → Type u_3
inst✝ : Monad m
l : Array α
f : α → β → m β
b : β
start stop : Nat
h✝¹ : ¬start ≤ l.size
h✝ : stop < l.size
h : min start l.size = l.size
this :
foldrM.fold f l stop (min start l.size) ⋯ b = foldrM.fold f (l.extract stop start) 0 (min start l.size ... | 3459c1e77bea9092 |
Std.DHashMap.Internal.List.containsKey_flatMap_eq_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem containsKey_flatMap_eq_false [BEq α] {γ : Type w} {l : List γ} {f : γ → List ((a : α) × β a)}
{a : α} (h : ∀ (i : Nat) (h : i < l.length), containsKey a (f l[i]) = false) :
containsKey a (l.flatMap f) = false | case refine_2
α : Type u
β : α → Type v
inst✝ : BEq α
γ : Type w
f : γ → List ((a : α) × β a)
a : α
g : γ
t : List γ
ih : (∀ (i : Nat) (h : i < t.length), containsKey a (f t[i]) = false) → containsKey a (List.flatMap f t) = false
h : ∀ (i : Nat) (h : i < (g :: t).length), containsKey a (f (g :: t)[i]) = false
i : Nat
h... | simpa using h (i + 1) (by simp only [List.length_cons]; omega) | no goals | f8a381b6bfd54777 |
Localization.mem_range_mapToFractionRing_iff | Mathlib/RingTheory/Localization/AsSubring.lean | theorem mem_range_mapToFractionRing_iff (B : Type*) [CommRing B] [Algebra A B] [IsLocalization S B]
(hS : S ≤ A⁰) (x : K) :
x ∈ (mapToFractionRing K S B hS).range ↔
∃ (a s : A) (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, hS hs⟩ :=
⟨by
rintro ⟨x, rfl⟩
obtain ⟨a, s, rfl⟩ := IsLocalization.mk'_sur... | A : Type u_1
K : Type u_2
inst✝⁶ : CommRing A
S : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsFractionRing A K
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
hS : S ≤ A⁰
x : K
⊢ (∃ a s, ∃ (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, ⋯⟩) → x ∈ (mapToFractionRing K S ... | rintro ⟨a, s, hs, rfl⟩ | case intro.intro.intro
A : Type u_1
K : Type u_2
inst✝⁶ : CommRing A
S : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsFractionRing A K
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
hS : S ≤ A⁰
a s : A
hs : s ∈ S
⊢ IsLocalization.mk' K a ⟨s, ⋯⟩ ∈ (mapToFractionRing K... | 68d824c2f8020338 |
MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem of_diff_eq_zero_of_symmDiff_eq_zero_positive (hu : MeasurableSet u) (hv : MeasurableSet v)
(hsu : 0 ≤[u] s) (hsv : 0 ≤[v] s) (hs : s (u ∆ v) = 0) : s (u \ v) = 0 ∧ s (v \ u) = 0 | α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
u v : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u ∆ v) = 0
a : ↑0 (u \ v) ≤ ↑s (u \ v)
⊢ ↑s (u \ v) = 0 ∧ ↑s (v \ u) = 0 | have b := hsv (hv.diff hu) diff_subset | α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
u v : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hsu : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ u → ↑0 j ≤ ↑s j
hsv : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ v → ↑0 j ≤ ↑s j
hs : ↑s (u ∆ v) = 0
a : ↑0 (u \ v) ≤ ↑s (u \ v)
b : ↑0 (v \ u) ≤ ↑s (v \ u)
⊢ ↑s (u \ v) ... | 0d8415720512f584 |
Order.IsSuccPrelimit.isMax | Mathlib/Order/SuccPred/Limit.lean | theorem IsSuccPrelimit.isMax (h : IsSuccPrelimit (succ a)) : IsMax a | α : Type u_1
a : α
inst✝¹ : Preorder α
inst✝ : SuccOrder α
h : IsSuccPrelimit (succ a)
H : ¬IsMax a
⊢ False | exact h a (covBy_succ_of_not_isMax H) | no goals | c596da599f52eec8 |
prod_mem_ideal_map_of_mem_conductor | Mathlib/NumberTheory/KummerDedekind.lean | theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
(hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) | case intro.intro.refine_3
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
p : R
z : S
hp : p ∈ comap (algebraMap R S) (conductor R x)
l : R →₀ S
H : l ∈ Finsupp.supported S S ↑I
H' : (l.sum fun i a => a • (algebraMap R S) i) = z
lem :
∀ {a : R},
a ∈ I →
... | exact lem ((Finsupp.mem_supported _ l).mp H hy) | no goals | 4b278fcdea19caab |
Equiv.Perm.pow_eq_on_of_mem_support | Mathlib/GroupTheory/Perm/Support.lean | theorem pow_eq_on_of_mem_support (h : ∀ x ∈ f.support ∩ g.support, f x = g x) (k : ℕ) :
∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x | case succ
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ (f ^ (k + 1)) x = (g ^ (k + 1)) x | rw [pow_succ, mul_apply, pow_succ, mul_apply, h _ hx, hk] | case succ.a
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f g : Perm α
h : ∀ x ∈ f.support ∩ g.support, f x = g x
k : ℕ
hk : ∀ x ∈ f.support ∩ g.support, (f ^ k) x = (g ^ k) x
x : α
hx : x ∈ f.support ∩ g.support
⊢ g x ∈ f.support ∩ g.support | 0bbbd941baa698a7 |
MeromorphicOn.isClopen_setOf_order_eq_top | Mathlib/Analysis/Meromorphic/Order.lean | theorem isClopen_setOf_order_eq_top : IsClopen { u : U | (hf u.1 u.2).order = ⊤ } | case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
t' : Set 𝕜
h₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y = 0
h₂t' : IsOpen t'
h₃t' : ↑z ∈ t'
w : ↑U
hw : w ∈ Subtype.val ⁻¹' t'
⊢ ∃ t, (∀ y ∈ t, ¬y = ↑w → ... | by_cases h₁w : w = z | case pos
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
t' : Set 𝕜
h₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y = 0
h₂t' : IsOpen t'
h₃t' : ↑z ∈ t'
w : ↑U
hw : w ∈ Subtype.val ⁻¹' t'
h₁w : w = z
⊢ ∃ t, (∀ y ∈... | 7349d8844a909f0d |
MeasureTheory.Measure.setIntegral_comp_smul | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ | E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.... | calc
∫ x in s, f (R • x) ∂μ
= ∫ x in e ⁻¹' (e.symm ⁻¹' s), f (e x) ∂μ := by simp [← preimage_comp]; rfl
_ = ∫ y in e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ := (setIntegral_map_equiv _ _ _).symm
_ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.symm ⁻¹' s, f y ∂μ := by
simp [map_addHaar_smul μ hR, integral_smul_measure, ENNReal.... | no goals | 74f76e652d1fafa7 |
Algebra.Generators.ofComp_kerCompPreimage | Mathlib/RingTheory/Generators.lean | lemma ofComp_kerCompPreimage (Q : Generators S T) (P : Generators R S) (x : Q.ker) :
(Q.ofComp P).toAlgHom (kerCompPreimage Q P x) = x | R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_2
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
⊢ (Q.ofComp P).toAlgHom ↑(Q.kerCompPreimage P x) = ∑ v ∈ support ↑x, (monomial v) (co... | rw [kerCompPreimage, map_finsupp_sum, Finsupp.sum] | R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
T : Type u_2
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
⊢ ∑ a ∈ (↑x).support,
(Q.ofComp P).toAlgHom ((rename Sum.inr) (P.σ (↑x a)) * (m... | 85aa8a3a9a7d4b4d |
Batteries.AssocList.toList_replace | Mathlib/.lake/packages/batteries/Batteries/Data/AssocList.lean | theorem toList_replace [BEq α] (a : α) (b : β) (l : AssocList α β) :
(replace a b l).toList =
l.toList.replaceF (bif ·.1 == a then some (a, b) else none) | case cons
α : Type u_1
β : Type u_2
inst✝ : BEq α
a : α
b : β
key✝ : α
value✝ : β
tail✝ : AssocList α β
tail_ih✝ : (replace a b tail✝).toList = List.replaceF (fun x => bif x.fst == a then some (a, b) else none) tail✝.toList
⊢ (match key✝ == a with
| true => cons a b tail✝
| false => cons key✝ value✝ (replac... | split <;> simp [*] | no goals | 99bfa7f9c29afc6d |
MvPolynomial.IsWeightedHomogeneous.weighted_total_degree | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | theorem weighted_total_degree [SemilatticeSup M] {w : σ → M} (hφ : IsWeightedHomogeneous w φ n)
(h : φ ≠ 0) : weightedTotalDegree' w φ = n | case a
R : Type u_1
M : Type u_2
inst✝² : CommSemiring R
σ : Type u_3
inst✝¹ : AddCommMonoid M
φ : MvPolynomial σ R
n : M
inst✝ : SemilatticeSup M
w : σ → M
hφ : IsWeightedHomogeneous w φ n
h : φ ≠ 0
⊢ ∀ (b : σ →₀ ℕ), coeff b φ ≠ 0 → (weight w) b ≤ n | exact fun d hd => le_of_eq (hφ hd) | no goals | 5d7899c9c0084c3b |
Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical | Mathlib/RingTheory/Polynomial/IrreducibleRing.lean | theorem Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical
{R S : Type*} [CommRing R] [(nilradical R).IsPrime] [CommRing S] [IsDomain S]
(φ : R →+* S) (f : R[X]) (hm : f.Monic) (hi : Irreducible (f.map φ)) : Irreducible f | case intro.intro
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : (nilradical R).IsPrime
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
f : R[X]
hm : f.Monic
R' : Type u_1 := R ⧸ nilradical R
ψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯
ι : R →+* R' := algebraMap R R'
hi✝ : Irreducible (Polynomial.m... | simpa only [coeff_map, coeff_C, hi, ite_false, ← RingHom.mem_ker,
show RingHom.ker ι = nilradical R from Ideal.mk_ker] using congr(coeff $(h.symm) i) | no goals | 765e7ca5277138ea |
PartENat.pos_iff_one_le | Mathlib/Data/Nat/PartENat.lean | theorem pos_iff_one_le {x : PartENat} : 0 < x ↔ 1 ≤ x :=
PartENat.casesOn x
(by simp only [le_top, natCast_lt_top, ← @Nat.cast_zero PartENat])
fun n => by
rw [← Nat.cast_zero, ← Nat.cast_one, PartENat.coe_lt_coe, PartENat.coe_le_coe]
rfl
| x : PartENat
⊢ 0 < ⊤ ↔ 1 ≤ ⊤ | simp only [le_top, natCast_lt_top, ← @Nat.cast_zero PartENat] | no goals | 1ba41efab5b17963 |
Real.logb_le_logb_of_le | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | theorem logb_le_logb_of_le (h : 0 < x) (hxy : x ≤ y) : logb b x ≤ logb b y :=
(logb_le_logb hb h (by linarith)).mpr hxy
| b x y : ℝ
hb : 1 < b
h : 0 < x
hxy : x ≤ y
⊢ 0 < y | linarith | no goals | 5c7cb77e0e709c8a |
Polynomial.natDegree_le_of_dvd | Mathlib/Algebra/Polynomial/Degree/Domain.lean | lemma natDegree_le_of_dvd (h1 : p ∣ q) (h2 : q ≠ 0) : p.natDegree ≤ q.natDegree | case intro
R : Type u
inst✝¹ : Semiring R
inst✝ : NoZeroDivisors R
p q : R[X]
h2 : p ≠ 0 ∧ q ≠ 0
⊢ p.natDegree ≤ p.natDegree + q.natDegree | exact Nat.le_add_right _ _ | no goals | 2a42eadd2621d181 |
FintypeCat.Skeleton.is_skeletal | Mathlib/CategoryTheory/FintypeCat.lean | theorem is_skeletal : Skeletal Skeleton.{u} := fun X Y ⟨h⟩ =>
ext _ _ <|
Fin.equiv_iff_eq.mp <|
Nonempty.intro <|
{ toFun := fun x => (h.hom ⟨x⟩).down
invFun := fun x => (h.inv ⟨x⟩).down
left_inv | X Y : Skeleton
x✝ : IsIsomorphic X Y
h : X ≅ Y
a : Fin X.len
⊢ (h.inv { down := (fun x => (h.hom { down := x }).down) a }).down = a | rw [ULift.up_down] | X Y : Skeleton
x✝ : IsIsomorphic X Y
h : X ≅ Y
a : Fin X.len
⊢ (h.inv (h.hom { down := a })).down = a | 8277ccba905521be |
Metric.uniformity_edist_aux | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | theorem Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } =
⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } | case refine_2.intro
α : Type u_3
d : α → α → ℝ≥0
ε : ℝ≥0
hε : ↑ε > 0
⊢ {p | ↑(d p.1 p.2) < ↑ε} ∈ 𝓟 {p | ↑(d p.1 p.2) < ↑ε} | exact fun _ => ENNReal.coe_lt_coe.1 | no goals | ab44719fcc963446 |
CategoryTheory.FintypeCat.Action.isConnected_of_transitive | Mathlib/CategoryTheory/Galois/Examples.lean | theorem Action.isConnected_of_transitive (X : FintypeCat) [MulAction G X]
[MulAction.IsPretransitive G X] [h : Nonempty X] :
IsConnected (Action.FintypeCat.ofMulAction G X) where
notInitial := not_initial_of_inhabited (Action.forget _ _) h.some
noTrivialComponent Y i hm hni | case refine_2
G : Type u
inst✝² : Group G
X : FintypeCat
inst✝¹ : MulAction G X.carrier
inst✝ : MulAction.IsPretransitive G X.carrier
h : Nonempty X.carrier
Y : Action FintypeCat G
i : Y ⟶ Action.FintypeCat.ofMulAction G X
hm : Mono i
hni : IsInitial Y → False
y : Y.V.carrier
x' : (Action.FintypeCat.ofMulAction G X).V.... | obtain ⟨σ, hσ⟩ := MulAction.exists_smul_eq G x x' | case refine_2.intro
G : Type u
inst✝² : Group G
X : FintypeCat
inst✝¹ : MulAction G X.carrier
inst✝ : MulAction.IsPretransitive G X.carrier
h : Nonempty X.carrier
Y : Action FintypeCat G
i : Y ⟶ Action.FintypeCat.ofMulAction G X
hm : Mono i
hni : IsInitial Y → False
y : Y.V.carrier
x' : (Action.FintypeCat.ofMulAction G... | 598f0ed4a0644baa |
eVariationOn.add_le_union | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | theorem add_le_union (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) | α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈... | rcases hi with hi | hi | case inl
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ... | 4878120f0018b659 |
Int.self_le_zpow_clog | Mathlib/Data/Int/Log.lean | theorem self_le_zpow_clog {b : ℕ} (hb : 1 < b) (r : R) : r ≤ (b : R) ^ clog b r | case inr
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
⊢ r ≤ ↑b ^ clog b r | rw [← neg_log_inv_eq_clog, zpow_neg, le_inv_comm₀ hr (zpow_pos ..)] | case inr
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
⊢ ↑b ^ log b r⁻¹ ≤ r⁻¹
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ
hb : 1 < b
r : R
hr : 0 < r
⊢ 0 < ↑b | 0b96d1798dc9477a |
ENNReal.essSup_liminf_le | Mathlib/MeasureTheory/Function/EssSup.lean | theorem essSup_liminf_le {ι} [Countable ι] [LinearOrder ι] (f : ι → α → ℝ≥0∞) :
essSup (fun x => atTop.liminf fun n => f n x) μ ≤
atTop.liminf fun n => essSup (fun x => f n x) μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_3
inst✝¹ : Countable ι
inst✝ : LinearOrder ι
f : ι → α → ℝ≥0∞
⊢ essSup (fun x => liminf (fun n => f n x) atTop) μ ≤ liminf (fun n => essSup (fun x => f n x) μ) atTop | simp_rw [essSup] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
ι : Type u_3
inst✝¹ : Countable ι
inst✝ : LinearOrder ι
f : ι → α → ℝ≥0∞
⊢ limsup (fun x => liminf (fun n => f n x) atTop) (ae μ) ≤ liminf (fun n => limsup (fun x => f n x) (ae μ)) atTop | 0ddf986c8d3565cd |
HasFDerivAt.abs_of_neg | Mathlib/Analysis/Calculus/Deriv/Abs.lean | theorem HasFDerivAt.abs_of_neg (hf : HasFDerivAt f f' x)
(h₀ : f x < 0) : HasFDerivAt (fun x ↦ |f x|) (-f') x | case h.e'_12
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : E →L[ℝ] ℝ
x : E
hf : HasFDerivAt f f' x
h₀ : f x < 0
⊢ -f' = (ContinuousLinearMap.smulRight 1 (-1)).comp f' | ext y | case h.e'_12.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
f' : E →L[ℝ] ℝ
x : E
hf : HasFDerivAt f f' x
h₀ : f x < 0
y : E
⊢ (-f') y = ((ContinuousLinearMap.smulRight 1 (-1)).comp f') y | efef028e1cd91933 |
UnitAddCircle.mem_approxAddOrderOf_iff | Mathlib/NumberTheory/WellApproximable.lean | theorem mem_approxAddOrderOf_iff {δ : ℝ} {x : UnitAddCircle} {n : ℕ} (hn : 0 < n) :
x ∈ approxAddOrderOf UnitAddCircle n δ ↔ ∃ m < n, gcd m n = 1 ∧ ‖x - ↑((m : ℝ) / n)‖ < δ | case mp.intro.intro.intro.intro.intro
δ : ℝ
x : UnitAddCircle
n : ℕ
hn : 0 < n
m : ℕ
hm₁ : m < n
hm₂ : m.gcd n = 1
hx : ‖x - ↑(↑m / ↑n)‖ < δ
⊢ ∃ m < n, gcd m n = 1 ∧ ‖x - ↑(↑m / ↑n)‖ < δ | exact ⟨m, hm₁, hm₂, hx⟩ | no goals | 6c2242c810a53b2d |
Lean.Data.AC.Context.evalList_sort_cons | Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean | theorem Context.evalList_sort_cons
(ctx : Context α)
(h : Commutative ctx.op)
(x : Nat)
(xs : List Nat)
: evalList α ctx (sort (x :: xs)) = evalList α ctx (x :: sort xs) | case cons
α : Sort u_1
ctx : Context α
h : Commutative ctx.op
z : Nat
zs : List Nat
ih : ∀ (x : Nat) (ys : List Nat), evalList α ctx (sort.loop (insert x ys) zs) = evalList α ctx (x :: sort.loop ys zs)
x : Nat
ys : List Nat
⊢ evalList α ctx (sort.loop (z :: insert x ys) zs) = evalList α ctx (x :: sort.loop ys (z :: zs)... | simp [sort.loop, ←ih] | case cons
α : Sort u_1
ctx : Context α
h : Commutative ctx.op
z : Nat
zs : List Nat
ih : ∀ (x : Nat) (ys : List Nat), evalList α ctx (sort.loop (insert x ys) zs) = evalList α ctx (x :: sort.loop ys zs)
x : Nat
ys : List Nat
⊢ evalList α ctx (sort.loop (z :: insert x ys) zs) = evalList α ctx (sort.loop (insert x (insert... | c430d06d071ec3a0 |
Equiv.Perm.cycleType_extendDomain | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem cycleType_extendDomain {β : Type*} [Fintype β] [DecidableEq β] {p : β → Prop}
[DecidablePred p] (f : α ≃ Subtype p) {g : Perm α} :
cycleType (g.extendDomain f) = cycleType g | case base_cycles
α : Type u_1
inst✝⁴ : Fintype α
inst✝³ : DecidableEq α
β : Type u_2
inst✝² : Fintype β
inst✝¹ : DecidableEq β
p : β → Prop
inst✝ : DecidablePred p
f : α ≃ Subtype p
σ : Perm α
hσ : σ.IsCycle
⊢ (σ.extendDomain f).cycleType = σ.cycleType | rw [(hσ.extendDomain f).cycleType, hσ.cycleType, card_support_extend_domain] | no goals | a5e15d1fef0edcde |
ArithmeticFunction.cardDistinctFactors_eq_cardFactors_iff_squarefree | Mathlib/NumberTheory/ArithmeticFunction.lean | theorem cardDistinctFactors_eq_cardFactors_iff_squarefree {n : ℕ} (h0 : n ≠ 0) :
ω n = Ω n ↔ Squarefree n | n : ℕ
h0 : n ≠ 0
⊢ n.primeFactorsList.dedup.length = Ω n ↔ n.primeFactorsList.Nodup | constructor <;> intro h | case mp
n : ℕ
h0 : n ≠ 0
h : n.primeFactorsList.dedup.length = Ω n
⊢ n.primeFactorsList.Nodup
case mpr
n : ℕ
h0 : n ≠ 0
h : n.primeFactorsList.Nodup
⊢ n.primeFactorsList.dedup.length = Ω n | dd911b39979b8cd2 |
ZetaAsymptotics.term_of_lt | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | lemma term_of_lt {n : ℕ} (hn : 0 < n) {s : ℝ} (hs : 1 < s) :
term n s = 1 / (s - 1) * (1 / n ^ (s - 1) - 1 / (n + 1) ^ (s - 1))
- n / s * (1 / n ^ s - 1 / (n + 1) ^ s) | case hg.hf
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ IntervalIntegrable (fun x => x ^ (-(s + 1))) volume (↑n) (↑n + 1) | refine intervalIntegral.intervalIntegrable_rpow (Or.inr <| not_mem_uIcc_of_lt ?_ ?_) | case hg.hf.refine_1
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n
case hg.hf.refine_2
n : ℕ
hn : 0 < n
s : ℝ
hs : 1 < s
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
⊢ 0 < ↑n + 1 | edd6f36d5371f792 |
MeasureTheory.SimpleFunc.setToSimpleFunc_zero' | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToSimpleFunc_zero' {T : Set α → E →L[ℝ] F'}
(h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) (f : α →ₛ E) (hf : Integrable f μ) :
setToSimpleFunc T f = 0 | α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
⊢ ∑ x ∈ f.range,... | refine sum_eq_zero fun x _ => ?_ | α : Type u_1
E : Type u_2
F' : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F'
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f... | 7e4d817c9fd3a3f2 |
WeierstrassCurve.Affine.CoordinateRing.mk_ψ₂_sq | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | lemma Affine.CoordinateRing.mk_ψ₂_sq : mk W W.ψ₂ ^ 2 = mk W (C W.Ψ₂Sq) | R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ (mk W) W.ψ₂ ^ 2 = (mk W) (C W.Ψ₂Sq) | rw [C_Ψ₂Sq, map_sub, map_mul, AdjoinRoot.mk_self, mul_zero, sub_zero, map_pow] | no goals | 0b1725304b5dbd94 |
Int.lt_sub_right_of_add_lt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem lt_sub_right_of_add_lt {a b c : Int} (h : a + b < c) : a < c - b | a b c : Int
h✝ : a + b < c
h : a + b + -b < c + -b
⊢ a < c - b | rwa [Int.add_neg_cancel_right] at h | no goals | 31c652de4f40c5d3 |
SimplexCategoryGenRel.factor_δ_σ | Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/EpiMono.lean | private lemma factor_δ_σ {n : ℕ} (i : Fin (n + 1)) (i' : Fin (n + 2)) :
∃ (z : SimplexCategoryGenRel) (e : mk n ⟶ z) (m : z ⟶ mk n)
(_ : P_σ e) (_ : P_δ m), δ i' ≫ σ i = e ≫ m | i : Fin (0 + 1)
i' : Fin (0 + 2)
⊢ δ i' ≫ σ i = 𝟙 (mk 0) ≫ 𝟙 (mk 0) | simp [switch_δ_σ₀] | no goals | 6f8eb316d6efa99c |
PartialEquiv.trans_transEquiv | Mathlib/Logic/Equiv/PartialEquiv.lean | theorem trans_transEquiv (e : PartialEquiv α β) (e' : PartialEquiv β γ) (f'' : γ ≃ δ) :
(e.trans e').transEquiv f'' = e.trans (e'.transEquiv f'') | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
e' : PartialEquiv β γ
f'' : γ ≃ δ
⊢ (e.trans e').transEquiv f'' = e.trans (e'.transEquiv f'') | simp only [transEquiv_eq_trans, trans_assoc, Equiv.trans_toPartialEquiv] | no goals | 9e064f6cca787340 |
Matrix.PosDef.add_posSemidef | Mathlib/LinearAlgebra/Matrix/PosDef.lean | protected lemma add_posSemidef [AddLeftMono R]
{A : Matrix m m R} {B : Matrix m m R}
(hA : A.PosDef) (hB : B.PosSemidef) : (A + B).PosDef :=
⟨hA.isHermitian.add hB.isHermitian, fun x hx => by
rw [add_mulVec, dotProduct_add]
exact add_pos_of_pos_of_nonneg (hA.2 x hx) (hB.2 x)⟩
| m : Type u_1
R : Type u_3
inst✝⁴ : Fintype m
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : AddLeftMono R
A B : Matrix m m R
hA : A.PosDef
hB : B.PosSemidef
x : m → R
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ (A + B) *ᵥ x | rw [add_mulVec, dotProduct_add] | m : Type u_1
R : Type u_3
inst✝⁴ : Fintype m
inst✝³ : CommRing R
inst✝² : PartialOrder R
inst✝¹ : StarRing R
inst✝ : AddLeftMono R
A B : Matrix m m R
hA : A.PosDef
hB : B.PosSemidef
x : m → R
hx : x ≠ 0
⊢ 0 < star x ⬝ᵥ A *ᵥ x + star x ⬝ᵥ B *ᵥ x | 414045ca030d9893 |
bernoulli'PowerSeries_mul_exp_sub_one | Mathlib/NumberTheory/Bernoulli.lean | theorem bernoulli'PowerSeries_mul_exp_sub_one :
bernoulli'PowerSeries A * (exp A - 1) = X * exp A | case h.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ (coeff A (n + 1, 0).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernoulli' n / ↑n !)) *
(coeff A (n + 1, 0).2) (exp A - 1) +
∑ p ∈ antidiagonal n,
(coeff A (p.1, p.2 + 1).1) (PowerSeries.mk fun n => (algebraMap ℚ A) (bernou... | suffices (∑ p ∈ antidiagonal n,
bernoulli' p.1 / p.1! * ((p.2 + 1) * p.2! : ℚ)⁻¹) = (n ! : ℚ)⁻¹ by
simpa [map_sum, Nat.factorial] using congr_arg (algebraMap ℚ A) this | case h.succ
A : Type u_1
inst✝¹ : CommRing A
inst✝ : Algebra ℚ A
n : ℕ
⊢ ∑ p ∈ antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹ | 42e6710e61d9a85a |
Finset.prod_range_induction | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem prod_range_induction (f s : ℕ → β) (base : s 0 = 1)
(step : ∀ n, s (n + 1) = s n * f n) (n : ℕ) :
∏ k ∈ Finset.range n, f k = s n | case zero
β : Type u_4
inst✝ : CommMonoid β
f s : ℕ → β
base : s 0 = 1
step : ∀ (n : ℕ), s (n + 1) = s n * f n
⊢ ∏ k ∈ range 0, f k = s 0 | rw [Finset.prod_range_zero, base] | no goals | 0d8711ebae4259f7 |
ProbabilityTheory.hasFiniteIntegral_prod_mk_left | Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean | theorem hasFiniteIntegral_prod_mk_left (a : α) {s : Set (β × γ)} (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
HasFiniteIntegral (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) | case h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
s : Set (β × γ)
h2s : ((κ ⊗ₖ η) a) s ≠ ⊤
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
b : β
hb : (η (a, b)) ... | rw [ofReal_toReal hb.ne] | case h
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
s : Set (β × γ)
h2s : ((κ ⊗ₖ η) a) s ≠ ⊤
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
b : β
hb : (η (a, b)) ... | 18121c7d8b5b8137 |
exists_associated_pow_of_mul_eq_pow | Mathlib/Algebra/GCDMonoid/Basic.lean | theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
{k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a | case h.intro.intro.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ... | use Units.mkOfMulEqOne _ _ h' | case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd b a)
k : ℕ
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₂ * d₁
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
h0₂ : d₂ ^ k ≠ 0
b' : α
h : d₂ ^ k * ... | 4887d5e0abe68a27 |
Filter.tendsto_atTop_add_left_of_le' | Mathlib/Order/Filter/AtTopBot/Group.lean | theorem tendsto_atTop_add_left_of_le' (C : β) (hf : ∀ᶠ x in l, C ≤ f x) (hg : Tendsto g l atTop) :
Tendsto (fun x => f x + g x) l atTop :=
.atTop_of_isBoundedUnder_le_add (f := -f) ⟨-C, by simpa⟩ (by simpa)
| α : Type u_1
β : Type u_2
inst✝ : OrderedAddCommGroup β
l : Filter α
f g : α → β
C : β
hf : ∀ᶠ (x : α) in l, C ≤ f x
hg : Tendsto g l atTop
⊢ ∀ᶠ (x : β) in map (-f) l, (fun x1 x2 => x1 ≤ x2) x (-C) | simpa | no goals | b721368f278ac4e5 |
Fin.natAdd_eq_addNat | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean | theorem natAdd_eq_addNat (n : Nat) (i : Fin n) : Fin.natAdd n i = i.addNat n | n : Nat
i : Fin n
⊢ natAdd n i = i.addNat n | ext | case h
n : Nat
i : Fin n
⊢ ↑(natAdd n i) = ↑(i.addNat n) | f79a7fce64f01a77 |
WittVector.RecursionBase.solution_spec' | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | theorem solution_spec' {a₁ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (a₂ : 𝕎 k) :
solution p a₁ a₂ ^ p * a₁.coeff 0 = solution p a₁ a₂ * a₂.coeff 0 | p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : IsAlgClosed k
a₁ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
a₂ : 𝕎 k
this : solution p a₁ a₂ ^ (p - 1) = a₂.coeff 0 / a₁.coeff 0
⊢ solution p a₁ a₂ ^ p * a₁.coeff 0 = solution p a₁ a₂ * a₂.coeff 0 | obtain ⟨q, hq⟩ := Nat.exists_eq_succ_of_ne_zero hp.out.ne_zero | case intro
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : IsAlgClosed k
a₁ : 𝕎 k
ha₁ : a₁.coeff 0 ≠ 0
a₂ : 𝕎 k
this : solution p a₁ a₂ ^ (p - 1) = a₂.coeff 0 / a₁.coeff 0
q : ℕ
hq : p = q.succ
⊢ solution p a₁ a₂ ^ p * a₁.coeff 0 = solution p a₁ a₂ * a₂.coeff 0 | 76b3a3217e505467 |
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