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 |
|---|---|---|---|---|---|---|
Algebra.isOpen_smoothLocus | Mathlib/RingTheory/Smooth/Locus.lean | lemma isOpen_smoothLocus [FinitePresentation R A] : IsOpen (smoothLocus R A) | case intro.intro.intro.intro.intro.intro
R A : Type u
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule (Submonoid.powers f) (Ω[A⁄R]))
⊢ ∃ t... | let Af := Localization.Away f | case intro.intro.intro.intro.intro.intro
R A : Type u
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
f : A
hxf : x ∈ ↑(basicOpen f)
hf : Module.Projective (Localization.Away f) (LocalizedModule (Submonoid.powers f) (Ω[A⁄R]))
Af : ... | f811a0293eb7511f |
not_irreducible_pow | Mathlib/Algebra/Prime/Lemmas.lean | theorem not_irreducible_pow {M} [Monoid M] {x : M} {n : ℕ} (hn : n ≠ 1) :
¬ Irreducible (x ^ n) | case zero
M : Type u_3
inst✝ : Monoid M
x : M
hn : 0 ≠ 1
⊢ ¬Irreducible (x ^ 0) | simp | no goals | b9fd06b49708f472 |
crossProduct_ne_zero_iff_linearIndependent | Mathlib/LinearAlgebra/CrossProduct.lean | lemma crossProduct_ne_zero_iff_linearIndependent {F : Type*} [Field F] {v w : Fin 3 → F} :
crossProduct v w ≠ 0 ↔ LinearIndependent F ![v, w] | F : Type u_2
inst✝ : Field F
v w : Fin 3 → F
hv : ¬v = 0
hv' : v = ![v 0, v 1, v 2]
⊢ w = ![w 0, w 1, w 2] | simp [← List.ofFn_inj] | no goals | 7afc9aa3161c3f50 |
lp.norm_const_smul_le | Mathlib/Analysis/Normed/Lp/lpSpace.lean | theorem norm_const_smul_le (hp : p ≠ 0) (c : 𝕜) (f : lp E p) : ‖c • f‖ ≤ ‖c‖ * ‖f‖ | case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), BoundedSMul 𝕜 (E i)
hp✝ : p ≠ 0
c : 𝕜
f : ↥(lp E p)
hp : 0 < p.toReal
inst : NNNorm ↥(lp E p) := { nnnorm := fun f => ⟨‖f‖, ⋯⟩ }
⊢ ... | have coe_nnnorm : ∀ f : lp E p, ↑‖f‖₊ = ‖f‖ := fun _ => rfl | case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), BoundedSMul 𝕜 (E i)
hp✝ : p ≠ 0
c : 𝕜
f : ↥(lp E p)
hp : 0 < p.toReal
inst : NNNorm ↥(lp E p) := { nnnorm := fun f => ⟨‖f‖, ⋯⟩ }
co... | 226c34467852710f |
emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso | Mathlib/RingTheory/ChainOfDivisors.lean | theorem emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso {m p : Associates M}
{n : Associates N} (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) :
emultiplicity p m ≤ emultiplicity (↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩)) n | case neg
M : Type u_1
inst✝³ : CancelCommMonoidWithZero M
N : Type u_2
inst✝² : CancelCommMonoidWithZero N
inst✝¹ : UniqueFactorizationMonoid N
inst✝ : UniqueFactorizationMonoid M
m p : Associates M
n : Associates N
hp : p ∈ normalizedFactors m
d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)
hn : ¬n = 0
hm : ¬m = 0
⊢ ↑(d ⟨p, ⋯⟩) ^ mu... | apply pow_image_of_prime_by_factor_orderIso_dvd hn hp d (pow_multiplicity_dvd ..) | no goals | 61c7ac6c10a4255e |
BoxIntegral.integrable_of_bounded_and_ae_continuousWithinAt | Mathlib/Analysis/BoxIntegral/Basic.lean | theorem integrable_of_bounded_and_ae_continuousWithinAt [CompleteSpace E] {I : Box ι} {f : ℝⁿ → E}
(hb : ∃ C : ℝ, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C) (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ]
(hc : ∀ᵐ x ∂(μ.restrict (Box.Icc I)), ContinuousWithinAt f (Box.Icc I) x) :
Integrable I l f μ.toBoxAdditive.toSMul | ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε... | refine le_of_eq (measure_biUnion (countable_toSet _) ?_ (fun J _ ↦ J.measurableSet_coe)).symm | ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε... | 603a1c16b71ac0cf |
Int.sub_eq_zero_of_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean | theorem sub_eq_zero_of_eq {a b : Int} (h : a = b) : a - b = 0 | a b : Int
h : a = b
⊢ a - b = 0 | rw [h, Int.sub_self] | no goals | f67aa3f6c837d54d |
LieModule.trace_toEnd_eq_zero_of_mem_lcs | Mathlib/Algebra/Lie/TraceForm.lean | lemma trace_toEnd_eq_zero_of_mem_lcs
{k : ℕ} {x : L} (hk : 1 ≤ k) (hx : x ∈ lowerCentralSeries R L L k) :
trace R _ (toEnd R L M x) = 0 | R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ lowerCentralSeries R L L 1
⊢ x ∈ Submodule.span R {m | ∃ u v, ⁅u, v⁆ = m} | rw [lowerCentralSeries_succ, ← LieSubmodule.mem_toSubmodule,
LieSubmodule.lieIdeal_oper_eq_linear_span'] at hx | R : Type u_1
L : Type u_3
M : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
k : ℕ
x : L
hk : 1 ≤ k
hx : x ∈ Submodule.span R {x | ∃ x_1 ∈ ⊤, ∃ n ∈ lowerCentralSeries R L L 0, ⁅x_1, n⁆ = x}
⊢ x ∈ Subm... | 93c4c71234cad1df |
MeasureTheory.Measure.haar.is_left_invariant_chaar | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem is_left_invariant_chaar {K₀ : PositiveCompacts G} (g : G) (K : Compacts G) :
chaar K₀ (K.map _ <| continuous_mul_left g) = chaar K₀ K | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) ⋯ K) - f K
this : Continuous eval
⊢ closure (prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}) ⊆ eval ⁻¹' {0} | rw [IsClosed.closure_subset_iff] | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) ⋯ K) - f K
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' {0}
G : Type... | 6ade7b3c444e172e |
Affine.Triangle.dist_orthocenter_reflection_circumcenter | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem dist_orthocenter_reflection_circumcenter (t : Triangle ℝ P) {i₁ i₂ : Fin 3} (h : i₁ ≠ i₂) :
dist t.orthocenter (reflection (affineSpan ℝ (t.points '' {i₁, i₂})) t.circumcenter) =
t.circumradius | 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
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
⊢ -(∑ x : Fin (0 + 2 + 1),
(∑ x_1 : Fin (0 + 2 + 1),
((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) *
... | have hu : ({i₁, i₂} : Finset (Fin 3)) ⊆ univ := subset_univ _ | 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
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
hu : {i₁, i₂} ⊆ univ
⊢ -(∑ x : Fin (0 + 2 + 1),
(∑ x_1 : Fin (0 + 2 + 1),
((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ th... | cf2667ffc8b63c24 |
Rel.interedges_mono | Mathlib/Combinatorics/SimpleGraph/Density.lean | theorem interedges_mono (hs : s₂ ⊆ s₁) (ht : t₂ ⊆ t₁) : interedges r s₂ t₂ ⊆ interedges r s₁ t₁ :=
fun x ↦ by
simp_rw [mem_interedges_iff]
exact fun h ↦ ⟨hs h.1, ht h.2.1, h.2.2⟩
| α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
x : α × β
⊢ x ∈ interedges r s₂ t₂ → x ∈ interedges r s₁ t₁ | simp_rw [mem_interedges_iff] | α : Type u_4
β : Type u_5
r : α → β → Prop
inst✝ : (a : α) → DecidablePred (r a)
s₁ s₂ : Finset α
t₁ t₂ : Finset β
hs : s₂ ⊆ s₁
ht : t₂ ⊆ t₁
x : α × β
⊢ x.1 ∈ s₂ ∧ x.2 ∈ t₂ ∧ r x.1 x.2 → x.1 ∈ s₁ ∧ x.2 ∈ t₁ ∧ r x.1 x.2 | 24cb9d9ef47a802b |
Pell.IsFundamental.eq_pow_of_nonneg | Mathlib/NumberTheory/Pell.lean | theorem eq_pow_of_nonneg {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 0 < a.x)
(hay : 0 ≤ a.y) : ∃ n : ℕ, a = a₁ ^ n | case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 = a.y
⊢ a = a₁ ^ 0 | simp only [pow_zero] | case intro.h.inl
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
x : ℕ
ih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n
a : Solution₁ d
hay : 0 ≤ a.y
hax' : ↑x = a.x
hax : 0 < ↑x
hy : 0 = a.y
⊢ a = 1 | e2cabc4b76df1e43 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.readyForRupAdd_insert | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem readyForRupAdd_insert {n : Nat} (f : DefaultFormula n) (c : DefaultClause n) :
ReadyForRupAdd f → ReadyForRupAdd (insert f c) | case pos
n : Nat
f : DefaultFormula n
c : DefaultClause n
f_readyForRupAdd : f.ReadyForRupAdd
x✝ : Option (Literal (PosFin n))
l : PosFin n
hc : c.isUnit = some (l, false)
hsize : (f.assignments.modify l.val addNegAssignment).size = n
i : PosFin n
b : Bool
hb : hasAssignment b (f.assignments.modify l.val addNegAssignme... | assumption | no goals | c1096ba3fc377c54 |
Fin.sizeOf | Mathlib/.lake/packages/lean4/src/lean/Init/SizeOfLemmas.lean | theorem Fin.sizeOf (a : Fin n) : sizeOf a = a.val + 1 | n : Nat
a : Fin n
⊢ sizeOf a = ↑a + 1 | cases a | case mk
n val✝ : Nat
isLt✝ : val✝ < n
⊢ sizeOf ⟨val✝, isLt✝⟩ = ↑⟨val✝, isLt✝⟩ + 1 | 79ad8ea852aac7e2 |
LinearMap.toSpanSingleton_homothety | Mathlib/Analysis/Normed/Module/Span.lean | theorem toSpanSingleton_homothety (x : E) (c : 𝕜) :
‖LinearMap.toSpanSingleton 𝕜 E x c‖ = ‖x‖ * ‖c‖ | 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : BoundedSMul 𝕜 E
x : E
c : 𝕜
⊢ ‖(toSpanSingleton 𝕜 E x) c‖ = ‖x‖ * ‖c‖ | rw [mul_comm] | 𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedDivisionRing 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : BoundedSMul 𝕜 E
x : E
c : 𝕜
⊢ ‖(toSpanSingleton 𝕜 E x) c‖ = ‖c‖ * ‖x‖ | ddfe2b87fc17adc4 |
List.zip_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean | theorem zip_append :
∀ {l₁ r₁ : List α} {l₂ r₂ : List β} (_h : length l₁ = length l₂),
zip (l₁ ++ r₁) (l₂ ++ r₂) = zip l₁ l₂ ++ zip r₁ r₂
| [], r₁, l₂, r₂, h => by simp only [eq_nil_of_length_eq_zero h.symm]; rfl
| l₁, r₁, [], r₂, h => by simp only [eq_nil_of_length_eq_zero h]; rfl
| a :: l₁, r₁, b :: l... | α : Type u_1
β : Type u_2
a : α
l₁ r₁ : List α
b : β
l₂ r₂ : List β
h : (a :: l₁).length = (b :: l₂).length
⊢ (a :: l₁ ++ r₁).zip (b :: l₂ ++ r₂) = (a :: l₁).zip (b :: l₂) ++ r₁.zip r₂ | simp only [cons_append, zip_cons_cons, zip_append (Nat.succ.inj h)] | no goals | cb39c7e8a1d8f244 |
Lean.Omega.IntList.mul_distrib_left | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/IntList.lean | theorem mul_distrib_left (xs ys zs : IntList) : (xs + ys) * zs = xs * zs + ys * zs | case cons.cons.nil
x : Int
xs : List Int
ih₁ : ∀ (ys zs : IntList), (xs + ys) * zs = xs * zs + ys * zs
head✝ : Int
tail✝ : List Int
⊢ (x :: xs + head✝ :: tail✝) * [] = (x :: xs) * [] + (head✝ :: tail✝) * [] | simp | no goals | ad435ae640ae9032 |
Algebra.FormallyUnramified.isSeparable | Mathlib/RingTheory/Unramified/Field.lean | theorem isSeparable : Algebra.IsSeparable K L | K : Type u_1
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FormallyUnramified K L
inst✝ : EssFiniteType K L
this✝¹ : Module.Finite K L
this✝ : FormallyUnramified (↥(separableClosure K L)) L
this : EssFiniteType (↥(separableClosure K L)) L
⊢ separableClosure K L = ⊤ | ext | case h
K : Type u_1
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : FormallyUnramified K L
inst✝ : EssFiniteType K L
this✝¹ : Module.Finite K L
this✝ : FormallyUnramified (↥(separableClosure K L)) L
this : EssFiniteType (↥(separableClosure K L)) L
x✝ : L
⊢ x✝ ∈ separableClosure K L ↔ x✝ ∈ ⊤ | 098767b1d96a2eff |
Std.DHashMap.Internal.Raw₀.expand.go_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem expand.go_eq [BEq α] [Hashable α] [PartialEquivBEq α] (source : Array (AssocList α β))
(target : {d : Array (AssocList α β) // 0 < d.size}) : expand.go 0 source target =
(toListModel source).foldl (fun acc p => reinsertAux hash acc p.1 p.2) target | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
i : PartialEquivBEq α
source : Nat
target : Array (AssocList α β)
target✝ : { d // 0 < d.size }
hi : source < target.size
es : AssocList α β := target[source]
newSource : Array (AssocList α β) := target.set source AssocList.nil hi
newTarget : { d // 0 < d.size... | rw [List.drop_eq_getElem_cons hi, List.flatMap_cons, List.foldl_append,
List.drop_set_of_lt _ _ (by omega), Array.getElem_toList] | no goals | c4dec2d2e93626cd |
LightCondensed.isLocallySurjective_iff_locallySurjective_on_lightProfinite | Mathlib/Condensed/Light/Epi.lean | lemma isLocallySurjective_iff_locallySurjective_on_lightProfinite : IsLocallySurjective f ↔
∀ (S : LightProfinite) (y : ToType (Y.val.obj ⟨S⟩)),
(∃ (S' : LightProfinite) (φ : S' ⟶ S) (_ : Function.Surjective φ)
(x : ToType (X.val.obj ⟨S'⟩)),
f.val.app ⟨S'⟩ x = Y.val.map ⟨φ⟩ y) | A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
inst✝ : PreservesFiniteProducts (CategoryTheory.forget A)
X Y : LightCondensed A
f : X ⟶ Y
⊢ IsLocallySurjective f ↔
∀ (S : LightProfinite) (y : ToType (Y... | rw [coherentTopology.isLocallySurjective_iff,
regularTopology.isLocallySurjective_iff] | A : Type u'
inst✝³ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
inst✝ : PreservesFiniteProducts (CategoryTheory.forget A)
X Y : LightCondensed A
f : X ⟶ Y
⊢ (∀ (X_1 : LightProfinite) (y : ToType (Y.val.obj (Opposite.op X_1... | 0a92c944fa2c53b6 |
PowerSeries.invOneSubPow_val_one_eq_invUnitSub_one | Mathlib/RingTheory/PowerSeries/WellKnown.lean | theorem invOneSubPow_val_one_eq_invUnitSub_one :
(invOneSubPow S 1).val = invUnitsSub (1 : Sˣ) | S : Type u_1
inst✝ : CommRing S
⊢ ↑(invOneSubPow S 1) = invUnitsSub 1 | simp [invOneSubPow, invUnitsSub] | no goals | f51ab1b0b25494bc |
Nat.factorization_prod | Mathlib/Data/Nat/Factorization/Defs.lean | theorem factorization_prod {α : Type*} {S : Finset α} {g : α → ℕ} (hS : ∀ x ∈ S, g x ≠ 0) :
(S.prod g).factorization = S.sum fun x => (g x).factorization | case h.refine_2
α : Type u_1
S : Finset α
g : α → ℕ
hS : ∀ x ∈ S, g x ≠ 0
p : ℕ
⊢ ∀ {a : α} {s : Finset α},
a ∈ S →
s ⊆ S →
a ∉ s →
(s.prod g).factorization p = (∑ x ∈ s, (g x).factorization) p →
((insert a s).prod g).factorization p = (∑ x ∈ insert a s, (g x).factorization) p | intro x T hxS hTS hxT IH | case h.refine_2
α : Type u_1
S : Finset α
g : α → ℕ
hS : ∀ x ∈ S, g x ≠ 0
p : ℕ
x : α
T : Finset α
hxS : x ∈ S
hTS : T ⊆ S
hxT : x ∉ T
IH : (T.prod g).factorization p = (∑ x ∈ T, (g x).factorization) p
⊢ ((insert x T).prod g).factorization p = (∑ x ∈ insert x T, (g x).factorization) p | 0f85b322fea385d6 |
Ordnode.Valid'.map_aux | Mathlib/Data/Ordmap/Ordset.lean | theorem Valid'.map_aux {β} [Preorder β] {f : α → β} (f_strict_mono : StrictMono f) {t a₁ a₂}
(h : Valid' a₁ t a₂) :
Valid' (Option.map f a₁) (map f t) (Option.map f a₂) ∧ (map f t).size = t.size | case node.intro.intro.bal.right.left
α : Type u_1
inst✝¹ : Preorder α
β : Type u_2
inst✝ : Preorder β
f : α → β
f_strict_mono : StrictMono f
size✝ : ℕ
l✝ : Ordnode α
x✝ : α
r✝ : Ordnode α
a₁ : WithBot α
a₂ : WithTop α
h : Valid' a₁ (Ordnode.node size✝ l✝ x✝ r✝) a₂
t_l_valid : Valid' (Option.map f a₁) (map f l✝) (Option... | exact t_l_valid.bal | no goals | ff36f5e3d5092df5 |
CliffordAlgebra.even_induction | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | theorem even_induction {motive : ∀ x, x ∈ evenOdd Q 0 → Prop}
(algebraMap : ∀ r : R, motive (algebraMap _ _ r) (SetLike.algebraMap_mem_graded _ _))
(add : ∀ x y hx hy, motive x hx → motive y hy → motive (x + y) (Submodule.add_mem _ hx hy))
(ι_mul_ι_mul :
∀ m₁ m₂ x hx,
motive x hx →
m... | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q 0 → Prop
algebraMap : ∀ (r : R), motive ((_root_.algebraMap R (CliffordAlgebra Q)) r) ⋯
add :
∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 0) (hy : y ∈ eve... | obtain ⟨r, rfl⟩ := Submodule.mem_one.mp h | case intro
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
motive : (x : CliffordAlgebra Q) → x ∈ evenOdd Q 0 → Prop
algebraMap : ∀ (r : R), motive ((_root_.algebraMap R (CliffordAlgebra Q)) r) ⋯
add :
∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 0) (h... | 24e125e8024e9761 |
Polynomial.Chebyshev.S_add_two | Mathlib/RingTheory/Polynomial/Chebyshev.lean | theorem S_add_two : ∀ n, S R (n + 2) = X * S R (n + 1) - S R n
| (k : ℕ) => S.eq_3 R k
| -(k + 1 : ℕ) => by linear_combination (norm := (simp [Int.negSucc_eq]; ring_nf)) S.eq_4 R k
| R : Type u_1
inst✝ : CommRing R
k : ℕ
⊢ S R (-↑(k + 1) + 2) = X * S R (-↑(k + 1) + 1) - S R (-↑(k + 1)) | linear_combination (norm := (simp [Int.negSucc_eq]; ring_nf)) S.eq_4 R k | no goals | 440be322469157c0 |
FormalMultilinearSeries.radius_shift | Mathlib/Analysis/Analytic/Basic.lean | theorem radius_shift (p : FormalMultilinearSeries 𝕜 E F) : p.shift.radius = p.radius | case h.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
C : ℝ
⊢ (∀ (n : ℕ), ‖p (n + 1)‖ * ↑r ^ n ≤ C) → ∃ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ... | intro h | case h.h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
C : ℝ
h : ∀ (n : ℕ), ‖p (n + 1)‖ * ↑r ^ n ≤ C
⊢ ∃ (_ : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ... | b15148c409de31ac |
String.firstDiffPos_loop_eq | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem firstDiffPos_loop_eq (l₁ l₂ r₁ r₂ stop p)
(hl₁ : p = utf8Len l₁) (hl₂ : p = utf8Len l₂)
(hstop : stop = min (utf8Len l₁ + utf8Len r₁) (utf8Len l₂ + utf8Len r₂)) :
firstDiffPos.loop ⟨l₁ ++ r₁⟩ ⟨l₂ ++ r₂⟩ ⟨stop⟩ ⟨p⟩ =
⟨p + utf8Len (List.takeWhile₂ (· = ·) r₁ r₂).1⟩ | case hnc
l₁ l₂ r₁ r₂ : List Char
stop p : Nat
hl₁ : p = utf8Len l₁
hl₂ : p = utf8Len l₂
hstop : stop = min (utf8Len l₁ + utf8Len r₁) (utf8Len l₂ + utf8Len r₂)
x✝¹ x✝ : List Char
h : ∀ (a : Char) (as : List Char) (b : Char) (bs : List Char), r₁ = a :: as → r₂ = b :: bs → False
⊢ 0 < utf8Len r₁ → ¬0 < utf8Len r₂ | intro h₁ h₂ | case hnc
l₁ l₂ r₁ r₂ : List Char
stop p : Nat
hl₁ : p = utf8Len l₁
hl₂ : p = utf8Len l₂
hstop : stop = min (utf8Len l₁ + utf8Len r₁) (utf8Len l₂ + utf8Len r₂)
x✝¹ x✝ : List Char
h : ∀ (a : Char) (as : List Char) (b : Char) (bs : List Char), r₁ = a :: as → r₂ = b :: bs → False
h₁ : 0 < utf8Len r₁
h₂ : 0 < utf8Len r₂
⊢ F... | 5881499f2c507cc8 |
Std.Sat.AIG.mkOrCached_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedGatesLemmas.lean | theorem mkOrCached_decl_eq idx (aig : AIG α) (input : BinaryInput aig) {h : idx < aig.decls.size}
{h2} :
(aig.mkOrCached input).aig.decls[idx]'h2 = aig.decls[idx] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkOrCached input).aig.decls.size
⊢ (aig.mkGateCached (input.asGateInput true true)).aig.decls[idx] = aig.decls[idx] | rw [AIG.LawfulOperator.decl_eq (f := mkGateCached)] | case h2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
idx : Nat
aig : AIG α
input : aig.BinaryInput
h : idx < aig.decls.size
h2 : idx < (aig.mkOrCached input).aig.decls.size
⊢ idx < (aig.mkGateCached (input.asGateInput true true)).aig.decls.size | 2825f1c9a489a1bb |
Equiv.Perm.signAux_eq_signAux2 | Mathlib/GroupTheory/Perm/Sign.lean | theorem signAux_eq_signAux2 {n : ℕ} :
∀ (l : List α) (f : Perm α) (e : α ≃ Fin n) (_h : ∀ x, f x ≠ x → x ∈ l),
signAux ((e.symm.trans f).trans e) = signAux2 l f
| [], f, e, h => by
have : f = 1 := Equiv.ext fun y => Classical.not_not.1 (mt (h y) (List.not_mem_nil _))
rw [this, one_def, Equiv.trans_r... | α : Type u
inst✝ : DecidableEq α
n : ℕ
f : Perm α
e : α ≃ Fin n
h : ∀ (x : α), f x ≠ x → x ∈ []
this : f = 1
⊢ signAux ((e.symm.trans f).trans e) = signAux2 [] f | rw [this, one_def, Equiv.trans_refl, Equiv.symm_trans_self, ← one_def, signAux_one, signAux2] | no goals | f94966561d754325 |
Nat.dvd_of_pow_dvd | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem dvd_of_pow_dvd {p k m : Nat} (hk : 1 ≤ k) (hpk : p ^ k ∣ m) : p ∣ m | p k m : Nat
hk : 1 ≤ k
hpk : p ^ k ∣ m
⊢ p ^ 1 ∣ m | exact pow_dvd_of_le_of_pow_dvd hk hpk | no goals | e8dc4bde1893e501 |
Function.Periodic.const_smul | Mathlib/Algebra/Ring/Periodic.lean | theorem Periodic.const_smul [AddMonoid α] [Group γ] [DistribMulAction γ α]
(h : Periodic f c) (a : γ) : Periodic (fun x => f (a • x)) (a⁻¹ • c) := fun x => by
simpa only [smul_add, smul_inv_smul] using h (a • x)
| α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
c : α
inst✝² : AddMonoid α
inst✝¹ : Group γ
inst✝ : DistribMulAction γ α
h : Periodic f c
a : γ
x : α
⊢ (fun x => f (a • x)) (x + a⁻¹ • c) = (fun x => f (a • x)) x | simpa only [smul_add, smul_inv_smul] using h (a • x) | no goals | dac44d0f1d21a425 |
Nat.succ_div | Mathlib/Data/Nat/Init.lean | lemma succ_div : ∀ a b : ℕ, (a + 1) / b = a / b + if b ∣ a + 1 then 1 else 0
| a, 0 => by simp
| 0, 1 => by simp
| 0, b + 2 => by
have hb2 : b + 2 > 1 | case pos
a b : ℕ
hb_eq_a : ¬b = a + 1
hb_le_a1 : b ≤ a + 1
hb_le_a : b ≤ a
h₁ : 0 < b + 1 ∧ b + 1 ≤ a + 1 + 1
⊢ (if 0 < b + 1 ∧ b + 1 ≤ a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =
(if 0 < b + 1 ∧ b + 1 ≤ a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 ∣ a + 1 + 1 then 1 else 0 | have h₂ : 0 < b + 1 ∧ b + 1 ≤ a + 1 := ⟨succ_pos _, Nat.add_le_add_iff_right.2 hb_le_a⟩ | case pos
a b : ℕ
hb_eq_a : ¬b = a + 1
hb_le_a1 : b ≤ a + 1
hb_le_a : b ≤ a
h₁ : 0 < b + 1 ∧ b + 1 ≤ a + 1 + 1
h₂ : 0 < b + 1 ∧ b + 1 ≤ a + 1
⊢ (if 0 < b + 1 ∧ b + 1 ≤ a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =
(if 0 < b + 1 ∧ b + 1 ≤ a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 ∣ ... | 77c7480efd928acc |
Complex.Gammaℝ_ne_zero_of_re_pos | Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean | lemma Gammaℝ_ne_zero_of_re_pos {s : ℂ} (hs : 0 < re s) : Gammaℝ s ≠ 0 | s : ℂ
hs : 0 < s.re
⊢ s.Gammaℝ ≠ 0 | apply mul_ne_zero | case ha
s : ℂ
hs : 0 < s.re
⊢ ↑π ^ (-s / 2) ≠ 0
case hb
s : ℂ
hs : 0 < s.re
⊢ Gamma (s / 2) ≠ 0 | c9782cf89a013ea8 |
EReal.continuousAt_add_bot_bot | Mathlib/Topology/Instances/EReal/Lemmas.lean | theorem continuousAt_add_bot_bot : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, ⊥) | r : ℝ
x✝ : EReal × EReal
h : x✝.1 < ↑0 ∧ x✝.2 < ↑r
⊢ x✝.1 + x✝.2 < ↑r | simpa only [coe_zero, zero_add] using add_lt_add h.1 h.2 | no goals | 025b907b97b97f54 |
LinearMap.det_conj | Mathlib/LinearAlgebra/Determinant.lean | theorem det_conj {N : Type*} [AddCommGroup N] [Module A N] (f : M →ₗ[A] M) (e : M ≃ₗ[A] N) :
LinearMap.det ((e : M →ₗ[A] N) ∘ₗ f ∘ₗ (e.symm : N →ₗ[A] M)) = LinearMap.det f | case neg
M : Type u_2
inst✝⁴ : AddCommGroup M
A : Type u_5
inst✝³ : CommRing A
inst✝² : Module A M
N : Type u_7
inst✝¹ : AddCommGroup N
inst✝ : Module A N
f : M →ₗ[A] M
e : M ≃ₗ[A] N
H : ¬∃ s, Nonempty (Basis { x // x ∈ s } A M)
⊢ LinearMap.det (↑e ∘ₗ f ∘ₗ ↑e.symm) = LinearMap.det f | have H' : ¬∃ t : Finset N, Nonempty (Basis t A N) := by
contrapose! H
rcases H with ⟨s, ⟨b⟩⟩
exact ⟨_, ⟨(b.map e.symm).reindexFinsetRange⟩⟩ | case neg
M : Type u_2
inst✝⁴ : AddCommGroup M
A : Type u_5
inst✝³ : CommRing A
inst✝² : Module A M
N : Type u_7
inst✝¹ : AddCommGroup N
inst✝ : Module A N
f : M →ₗ[A] M
e : M ≃ₗ[A] N
H : ¬∃ s, Nonempty (Basis { x // x ∈ s } A M)
H' : ¬∃ t, Nonempty (Basis { x // x ∈ t } A N)
⊢ LinearMap.det (↑e ∘ₗ f ∘ₗ ↑e.symm) = Linea... | 97fcc4f0d8a18a2c |
CategoryTheory.epi_iff_surjective_up_to_refinements | Mathlib/CategoryTheory/Abelian/Refinements.lean | lemma epi_iff_surjective_up_to_refinements (f : X ⟶ Y) :
Epi f ↔ ∀ ⦃A : C⦄ (y : A ⟶ Y),
∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (x : A' ⟶ X), π ≫ y = x ≫ f | case mp
C : Type u_2
inst✝¹ : Category.{u_1, u_2} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
a✝ : Epi f
A : C
a : A ⟶ Y
⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ a = x ≫ f | exact ⟨pullback a f, pullback.fst a f, inferInstance, pullback.snd a f, pullback.condition⟩ | no goals | 8dab997ceac1b3da |
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving | Mathlib/CategoryTheory/Sites/Over.lean | lemma over_map_compatiblePreserving {X Y : C} (f : X ⟶ Y) :
CompatiblePreserving (J.over Y) (Over.map f) where
compatible {F Z _ x hx Y₁ Y₂ W f₁ f₂ g₁ g₂ hg₁ hg₂ h} | case h.e'_3.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over X
x✝ : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) x✝
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
... | congr 1 | case h.e'_3.h.e_a.e_f
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : Sheaf (J.over Y) (Type u_1)
Z : Over X
x✝ : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) x✝
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj... | b30448b3f4083fa8 |
Path.delayReflRight_zero | Mathlib/Topology/Homotopy/HSpaces.lean | theorem delayReflRight_zero (γ : Path x y) : delayReflRight 0 γ = γ.trans (Path.refl y) | case pos
X : Type u
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
t : ↑I
h : ↑t ≤ 1 / 2
⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩ | apply congr_arg γ | case pos
X : Type u
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
t : ↑I
h : ↑t ≤ 1 / 2
⊢ qRight (t, 0) = ⟨2 * ↑t, ⋯⟩ | ece5c946a8b6b2c4 |
LinearPMap.mem_domain_iff | Mathlib/LinearAlgebra/LinearPMap.lean | theorem mem_domain_iff {f : E →ₗ.[R] F} {x : E} : x ∈ f.domain ↔ ∃ y : F, (x, y) ∈ f.graph | R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
x : E
⊢ x ∈ f.domain ↔ ∃ y, (x, y) ∈ f.graph | constructor <;> intro h | case mp
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
x : E
h : x ∈ f.domain
⊢ ∃ y, (x, y) ∈ f.graph
case mpr
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3... | 8da0e3a2043b5c33 |
MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_eq_inner | Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner {u : E → F'}
(hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
{p p' : ℝ≥0} (hp : 1 ≤ p) (hn : 0 < finrank ℝ E)
(hp' : (p' : ℝ)⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹) :
eLpNorm u p' μ ≤ eLpNormLESNormFDerivOfEqInnerConst μ p * eLpNorm (fderiv ℝ u) p μ | E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1... | rwa [NNReal.coe_inv, sub_pos,
inv_lt_inv₀ _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this | E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F' : Type u_5
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : CompleteSpace F'
u : E → F'
hu : ContDiff ℝ 1... | 728bbe5fb5258103 |
WeakDual.CharacterSpace.ext_ker | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | theorem ext_ker {φ ψ : characterSpace 𝕜 A} (h : RingHom.ker φ = RingHom.ker ψ) : φ = ψ | case h
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
x : A
⊢ φ x = ψ x | have : x - algebraMap 𝕜 A (ψ x) ∈ RingHom.ker φ := by
simpa only [h, RingHom.mem_ker, map_sub, AlgHomClass.commutes] using sub_self (ψ x) | case h
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
x : A
this : x - (al... | 536b9d00b0364064 |
Lean.Omega.Constraint.scale_sat | Mathlib/.lake/packages/lean4/src/lean/Init/Omega/Constraint.lean | theorem scale_sat {c : Constraint} (k) (w : c.sat t) : (scale k c).sat (k * t) | case isFalse.mk.none.some.isFalse
t k : Int
h✝ : ¬k = 0
u : Int
w : t ≤ u
h : k ≤ 0
⊢ k * u ≤ k * t | exact Int.mul_le_mul_of_nonpos_left h w | no goals | 498e5968042038b0 |
DFinsupp.support_update_ne_zero | Mathlib/Data/DFinsupp/Defs.lean | theorem support_update_ne_zero (f : Π₀ i, β i) (i : ι) {b : β i} (h : b ≠ 0) :
support (f.update i b) = insert i f.support | case h.inl
ι : Type u
β : ι → Type v
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Zero (β i)
inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)
f : Π₀ (i : ι), β i
i : ι
b : β i
h : b ≠ 0
⊢ i ∈ (f.update i b).support ↔ i ∈ insert i f.support | simp [h] | no goals | c7d5d2c6b8773acd |
Nat.getLast_digit_ne_zero | Mathlib/Data/Nat/Digits.lean | theorem getLast_digit_ne_zero (b : ℕ) {m : ℕ} (hm : m ≠ 0) :
(digits b m).getLast (digits_ne_nil_iff_ne_zero.mpr hm) ≠ 0 | case zero.zero
hm : 0 ≠ 0
⊢ (digits 0 0).getLast ⋯ ≠ 0 | cases hm rfl | no goals | f80a47e31809cc02 |
Subgroup.smul_mem_of_mem_closure_of_mem | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | theorem smul_mem_of_mem_closure_of_mem {X : Type*} [MulAction G X] {s : Set G} {t : Set X}
(hs : ∀ g ∈ s, g⁻¹ ∈ s) (hst : ∀ᵉ (g ∈ s) (x ∈ t), g • x ∈ t) {g : G}
(hg : g ∈ Subgroup.closure s) {x : X} (hx : x ∈ t) : g • x ∈ t | case inv_mem
G : Type u_2
inst✝¹ : Group G
X : Type u_5
inst✝ : MulAction G X
s : Set G
t : Set X
hs : ∀ g ∈ s, g⁻¹ ∈ s
hst : ∀ g ∈ s, ∀ x ∈ t, g • x ∈ t
g g' : G
hg' : g' ∈ s
x : X
hx : x ∈ t
⊢ g'⁻¹ • x ∈ t | exact hst g'⁻¹ (hs g' hg') x hx | no goals | 4eab67206dcd6372 |
orthonormal_fourier | Mathlib/Analysis/Fourier/AddCircle.lean | theorem orthonormal_fourier : Orthonormal ℂ (@fourierLp T _ 2 _) | T : ℝ
hT : Fact (0 < T)
i j : ℤ
h : ¬i = j
⊢ -i + j ≠ 0 | rw [add_comm] | T : ℝ
hT : Fact (0 < T)
i j : ℤ
h : ¬i = j
⊢ j + -i ≠ 0 | 1362cf6350008e93 |
AlgebraicGeometry.PresheafedSpace.GlueData.snd_invApp_t_app' | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | theorem snd_invApp_t_app' (i j k : D.J) (U : Opens (pullback (D.f i j) (D.f i k)).carrier) :
∃ eq,
(π₂⁻¹ i, j, k) U ≫ (D.t k i).c.app _ ≫ (D.V (k, i)).presheaf.map (eqToHom eq) =
(D.t' k i j).c.app _ ≫ (π₁⁻¹ k, j, i) (unop _) | case h
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(pullback (D.f i j) (D.f i k))
⊢ (D.t k i).c.app (op ((opensFunctor (pullback.snd (D.f i j) (D.f i k))).obj U)) ≫
(D.V (k, i)).presheaf.map (eqToHom ⋯) =
(((pullback.snd (D.f i j) (D.f i k)).c.app (op ((opensFunctor (pullback.s... | rw [← comp_c_app, congr_app (D.t_fac k i j), comp_c_app] | case h
C : Type u
inst✝ : Category.{v, u} C
D : GlueData C
i j k : D.J
U : Opens ↑↑(pullback (D.f i j) (D.f i k))
⊢ (D.t k i).c.app (op ((opensFunctor (pullback.snd (D.f i j) (D.f i k))).obj U)) ≫
(D.V (k, i)).presheaf.map (eqToHom ⋯) =
((((D.t k i).c.app (op ((opensFunctor (pullback.snd (D.f i j) (D.f i k)))... | e08cb0129b520e6d |
Equiv.Perm.mem_list_cycles_iff | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem mem_list_cycles_iff {α : Type*} [Finite α] {l : List (Perm α)}
(h1 : ∀ σ : Perm α, σ ∈ l → σ.IsCycle) (h2 : l.Pairwise Disjoint) {σ : Perm α} :
σ ∈ l ↔ σ.IsCycle ∧ ∀ a, σ a ≠ a → σ a = l.prod a | case h.e'_5
α : Type u_4
inst✝ : Finite α
l : List (Perm α)
h1 : ∀ σ ∈ l, σ.IsCycle
h2 : List.Pairwise Disjoint l
σ : Perm α
h3 : σ.IsCycle
val✝ : Fintype α
h : ∀ (a : α), σ a ≠ a → σ a = l.prod a
hσl : σ.support ⊆ l.prod.support
a : α
ha : a ∈ σ.support
τ : Perm α
hτ : τ ∈ l
hτa : a ∈ τ.support
hτl : ∀ x ∈ τ.support, ... | refine h3.eq_on_support_inter_nonempty_congr (h1 _ hτ) key ?_ ha | case h.e'_5
α : Type u_4
inst✝ : Finite α
l : List (Perm α)
h1 : ∀ σ ∈ l, σ.IsCycle
h2 : List.Pairwise Disjoint l
σ : Perm α
h3 : σ.IsCycle
val✝ : Fintype α
h : ∀ (a : α), σ a ≠ a → σ a = l.prod a
hσl : σ.support ⊆ l.prod.support
a : α
ha : a ∈ σ.support
τ : Perm α
hτ : τ ∈ l
hτa : a ∈ τ.support
hτl : ∀ x ∈ τ.support, ... | 0cc516b4ece5c647 |
Polynomial.natDegree_mod_lt | Mathlib/Algebra/Polynomial/FieldDivision.lean | lemma natDegree_mod_lt [Field k] (p : k[X]) {q : k[X]} (hq : q.natDegree ≠ 0) :
(p % q).natDegree < q.natDegree | k : Type y
inst✝ : Field k
p q : k[X]
hq : q.natDegree ≠ 0
⊢ (p % q).natDegree < q.natDegree | have hq' : q.leadingCoeff ≠ 0 := by
rw [leadingCoeff_ne_zero]
contrapose! hq
simp [hq] | k : Type y
inst✝ : Field k
p q : k[X]
hq : q.natDegree ≠ 0
hq' : q.leadingCoeff ≠ 0
⊢ (p % q).natDegree < q.natDegree | c1edaa831c1aaa7d |
List.countP_erase | Mathlib/Data/List/Count.lean | lemma countP_erase (p : α → Bool) (l : List α) (a : α) :
countP p (l.erase a) = countP p l - if a ∈ l ∧ p a then 1 else 0 | α : Type u_1
inst✝ : DecidableEq α
p : α → Bool
l : List α
a : α
⊢ countP p (l.erase a) = countP p l - if a ∈ l ∧ p a = true then 1 else 0 | rw [countP_eq_length_filter, countP_eq_length_filter, ← erase_filter, length_erase] | α : Type u_1
inst✝ : DecidableEq α
p : α → Bool
l : List α
a : α
⊢ (if a ∈ filter p l then (filter p l).length - 1 else (filter p l).length) =
(filter p l).length - if a ∈ l ∧ p a = true then 1 else 0 | 3c5560675ae847f4 |
Submodule.mem_iSup_iff_exists_finsupp | Mathlib/LinearAlgebra/DFinsupp.lean | lemma mem_iSup_iff_exists_finsupp (p : ι → Submodule R N) (x : N) :
x ∈ iSup p ↔ ∃ (f : ι →₀ N), (∀ i, f i ∈ p i) ∧ (f.sum fun _i xi ↦ xi) = x | case intro.intro.refine_2
ι : Type u_1
R : Type u_2
N : Type u_5
inst✝² : Semiring R
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
p : ι → Submodule R N
f : ι →₀ N
hf : ∀ (i : ι), f i ∈ p i
i : ι
hi : i ∈ f.support
⊢ (fun x xi => ↑xi) i ((DFinsupp.mk f.support fun i => ⟨f ↑i, ⋯⟩) i) = (fun _i xi => xi) i (f i) | simp [Finsupp.mem_support_iff.mp hi] | no goals | 1c0c258395c7890b |
Int.Cooper.resolve_left_dvd₁ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Cooper.lean | theorem resolve_left_dvd₁ (a c d p x : Int) (h₁ : p ≤ a * x) :
a ∣ resolve_left a c d p x + p | a c d p x : Int
h₁ : p ≤ a * x
k' : Nat
w : a * x = p + ↑k'
⊢ ↑k' + p = a * x | rw [w, Int.add_comm] | no goals | 4717869eaf119b13 |
FDRep.average_char_eq_finrank_invariants | Mathlib/RepresentationTheory/Character.lean | theorem average_char_eq_finrank_invariants (V : FDRep k G) :
⅟ (Fintype.card G : k) • ∑ g : G, V.character g = finrank k (invariants V.ρ) | k : Type u
inst✝³ : Field k
G : Type u
inst✝² : Group G
inst✝¹ : Fintype G
inst✝ : Invertible ↑(Fintype.card G)
V : FDRep k G
⊢ ⅟↑(Fintype.card G) • ∑ g : G, V.character g = (trace k ↑V.V) (averageMap V.ρ) | simp [character, GroupAlgebra.average, _root_.map_sum] | no goals | 905d0d1df2842315 |
List.forIn'_congr | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem forIn'_congr [Monad m] {as bs : List α} (w : as = bs)
{b b' : β} (hb : b = b')
{f : (a' : α) → a' ∈ as → β → m (ForInStep β)}
{g : (a' : α) → a' ∈ bs → β → m (ForInStep β)}
(h : ∀ a m b, f a (by simpa [w] using m) b = g a m b) :
forIn' as b f = forIn' bs b' g | case cons.cons.intro.e_a.h.yield
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m
b✝ b' : β
hb : b✝ = b'
a : α
as : List α
f : (a' : α) → a' ∈ a :: as → β → m (ForInStep β)
ih :
∀ {as_1 : List α} (w : as_1 = as) {b b' : β},
b = b' →
∀ {f : (a' : α) → a' ∈ as_1 → β → m (ForInStep β)} {g : (a... | intro a m b | case cons.cons.intro.e_a.h.yield
m✝ : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝ : Monad m✝
b✝¹ b' : β
hb : b✝¹ = b'
a✝ : α
as : List α
f : (a' : α) → a' ∈ a✝ :: as → β → m✝ (ForInStep β)
ih :
∀ {as_1 : List α} (w : as_1 = as) {b b' : β},
b = b' →
∀ {f : (a' : α) → a' ∈ as_1 → β → m✝ (ForInStep β)}... | b21e9d91722268c4 |
List.head?_zipWith | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean | theorem head?_zipWith {f : α → β → γ} :
(List.zipWith f as bs).head? = match as.head?, bs.head? with
| some a, some b => some (f a b) | _, _ => none | α : Type u_1
β : Type u_2
γ : Type u_3
as : List α
bs : List β
f : α → β → γ
⊢ (zipWith f as bs).head? =
match as.head?, bs.head? with
| some a, some b => some (f a b)
| x, x_1 => none | simp [head?_eq_getElem?, getElem?_zipWith] | no goals | 7f88082bfe80a9fd |
CategoryTheory.extensiveTopology.mem_sieves_iff_contains_colimit_cofan | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveTopology.lean | lemma extensiveTopology.mem_sieves_iff_contains_colimit_cofan {X : C} (S : Sieve X) :
S ∈ (extensiveTopology C) X ↔
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
Nonempty (IsColimit (Cofan.mk X π)) ∧ (∀ a : α, (S.arrows) (π a))) | case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : FinitaryPreExtensive C
X : C
S : Sieve X
⊢ (∃ α, ∃ (_ : Finite α), ∃ Y π, Nonempty (IsColimit (Cofan.mk X π)) ∧ ∀ (a : α), S.arrows (π a)) →
S ∈ (extensiveTopology C) X | intro ⟨α, _, Y, π, h, h'⟩ | case mpr
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : FinitaryPreExtensive C
X : C
S : Sieve X
α : Type
w✝ : Finite α
Y : α → C
π : (a : α) → Y a ⟶ X
h : Nonempty (IsColimit (Cofan.mk X π))
h' : ∀ (a : α), S.arrows (π a)
⊢ S ∈ (extensiveTopology C) X | 3464c4e8e1c1caf4 |
AffineSubspace.setOf_sSameSide_eq_image2 | Mathlib/Analysis/Convex/Side.lean | theorem setOf_sSameSide_eq_image2 {s : AffineSubspace R P} {x p : P} (hx : x ∉ s) (hp : p ∈ s) :
{ y | s.SSameSide x y } = Set.image2 (fun (t : R) q => t • (x -ᵥ p) +ᵥ q) (Set.Ioi 0) s | case h.mp
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
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ s.SSameSide x y → ∃ a, 0 < a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y | rw [sSameSide_iff_exists_left hp] | case h.mp
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
s : AffineSubspace R P
x p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ (x ∉ s ∧ y ∉ s ∧ ∃ p₂ ∈ s, SameRay R (x -ᵥ p) (y -ᵥ p₂)) → ∃ a, 0 < a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y | 96d8b7db8dee1b03 |
Ergodic.zero_measure | Mathlib/Dynamics/Ergodic/Ergodic.lean | theorem zero_measure {f : α → α} (hf : Measurable f) : @Ergodic α m f 0 where
measurable := hf
map_eq | α : Type u_1
m : MeasurableSpace α
f : α → α
hf : Measurable f
⊢ Measure.map f 0 = 0 | simp | no goals | a746010c0e268f32 |
piiUnionInter_singleton | Mathlib/MeasureTheory/PiSystem.lean | theorem piiUnionInter_singleton (π : ι → Set (Set α)) (i : ι) :
piiUnionInter π {i} = π i ∪ {univ} | case pos
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
i : ι
t : Finset ι
f : ι → Set α
hfπ : ∀ x ∈ t, f x ∈ π x
hti : ∀ y ∈ t, y = i
hi : i ∈ t
ht_eq_i : t = {i}
⊢ f i ∈ π i ∨ f i ∈ {univ} | exact Or.inl (hfπ i hi) | no goals | d8b76da0d2d10fc7 |
CategoryTheory.InjectiveResolution.ofCocomplex_exactAt_succ | Mathlib/CategoryTheory/Abelian/Injective/Resolution.lean | lemma ofCocomplex_exactAt_succ (n : ℕ) :
(ofCocomplex Z).ExactAt (n + 1) | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
Z : C
n : ℕ
⊢ (ShortComplex.mk
(CochainComplex.mkAux (under Z) (syzygies (Injective.ι Z)) (syzygies (d (Injective.ι Z))) (d (Injective.ι Z))
(d (d (Injective.ι Z))) ⋯ (fun S => ⟨syzygies S.g, ⟨d S.g, ⋯⟩⟩) 0).f
(Coc... | apply exact_f_d ((CochainComplex.mkAux _ _ _
(d (Injective.ι Z)) (d (d (Injective.ι Z))) _ _ 0).f) | no goals | 7abb2c8c6ef1cb12 |
dist_integral_mulExpNegMulSq_comp_le | Mathlib/Analysis/SpecialFunctions/MulExpNegMulSqIntegral.lean | theorem dist_integral_mulExpNegMulSq_comp_le (f : E →ᵇ ℝ)
{A : Subalgebra ℝ C(E, ℝ)} (hA : A.SeparatesPoints)
(hbound : ∀ g ∈ A, ∃ C, ∀ x y : E, dist (g x) (g y) ≤ C)
(heq : ∀ g ∈ A, ∫ x, (g : E → ℝ) x ∂P = ∫ x, (g : E → ℝ) x ∂P') (hε : 0 < ε) :
|∫ x, mulExpNegMulSq ε (f x) ∂P - ∫ x, mulExpNegMulSq ε (f... | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalge... | have line3 : |∫ x in K, mulExpNegMulSq ε (g x) ∂P
- ∫ x, mulExpNegMulSq ε (g x) ∂P| < sqrt ε := by
rw [abs_sub_comm]
exact (abs_integral_sub_setIntegral_mulExpNegMulSq_comp_lt
g (IsClosed.measurableSet hKcl) hε hKPbound) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalge... | 6c7688b02378f81c |
CategoryTheory.Subgroupoid.isNormal_map | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | theorem isNormal_map (hφ : Function.Injective φ.obj) (hφ' : im φ hφ = ⊤) (Sn : S.IsNormal) :
(map φ hφ S).IsNormal :=
{ wide := fun d => by
obtain ⟨c, rfl⟩ := obj_surjective_of_im_eq_top φ hφ hφ' d
change Map.Arrows φ hφ S _ _ (𝟙 _); rw [← Functor.map_id]
constructor; exact Sn.wide c
conj :... | case intro.intro.intro.intro.intro.intro.refl.intro
C : Type u
inst✝¹ : Groupoid C
S : Subgroupoid C
D : Type u_1
inst✝ : Groupoid D
φ : C ⥤ D
hφ : Function.Injective φ.obj
hφ' : im φ hφ = ⊤
Sn : S.IsNormal
c : C
γ : c ⟶ c
γS : γ ∈ S.arrows c c
cd' : φ.obj c = φ.obj c
c' : C
g : φ.obj c ⟶ φ.obj c'
⊢ Groupoid.inv g ≫ (e... | have : g ∈ (im φ hφ).arrows (φ.obj c) (φ.obj c') := by rw [hφ']; trivial | case intro.intro.intro.intro.intro.intro.refl.intro
C : Type u
inst✝¹ : Groupoid C
S : Subgroupoid C
D : Type u_1
inst✝ : Groupoid D
φ : C ⥤ D
hφ : Function.Injective φ.obj
hφ' : im φ hφ = ⊤
Sn : S.IsNormal
c : C
γ : c ⟶ c
γS : γ ∈ S.arrows c c
cd' : φ.obj c = φ.obj c
c' : C
g : φ.obj c ⟶ φ.obj c'
this : g ∈ (im φ hφ).... | 38dea252c0a3b817 |
uniformCauchySeqOn_ball_of_deriv | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | theorem uniformCauchySeqOn_ball_of_deriv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
(hf : ∀ n : ι, ∀ y : 𝕜, y ∈ Metric.ball x r → HasDerivAt (f n) (f' n y) y)
(hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) | ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f f' : ι → 𝕜 → G
x : 𝕜
inst✝ : IsRCLikeNormedField 𝕜
r : ℝ
hf'✝ : UniformCauchySeqOnFilter f' l (𝓟 (Metric.ball x r))
hfg : Cauchy (map (fun n => f n x) l)
hf : ∀ (n : ι),... | exact uniformCauchySeqOn_ball_of_fderiv hf' hf hfg | no goals | 38780b170b1d275d |
Complex.mul_cpow_ofReal_nonneg | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | theorem mul_cpow_ofReal_nonneg {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (r : ℂ) :
((a : ℂ) * (b : ℂ)) ^ r = (a : ℂ) ^ r * (b : ℂ) ^ r | case inr.inr.inr
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
r : ℂ
hr : r ≠ 0
ha' : 0 < a
hb' : 0 < b
⊢ (↑a * ↑b) ^ r = ↑a ^ r * ↑b ^ r | have ha'' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha'.ne' | case inr.inr.inr
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
r : ℂ
hr : r ≠ 0
ha' : 0 < a
hb' : 0 < b
ha'' : ↑a ≠ 0
⊢ (↑a * ↑b) ^ r = ↑a ^ r * ↑b ^ r | 385cc79a0a30c890 |
SatisfiesM_EStateM_eq | Mathlib/.lake/packages/batteries/Batteries/Classes/SatisfiesM.lean | theorem SatisfiesM_EStateM_eq :
SatisfiesM (m := EStateM ε σ) p x ↔ ∀ s a s', x.run s = .ok a s' → p a | case mpr.refine_2.h
ε σ α✝ : Type u_1
p : α✝ → Prop
x : EStateM ε σ α✝
w : ∀ (s : σ) (a : α✝) (s' : σ), x.run s = EStateM.Result.ok a s' → p a
s : σ
⊢ (Subtype.val <$> fun s =>
match q : x.run s with
| EStateM.Result.ok a s' => EStateM.Result.ok ⟨a, ⋯⟩ s'
| EStateM.Result.error e s' => ESt... | rw [EStateM.run_map, EStateM.run] | case mpr.refine_2.h
ε σ α✝ : Type u_1
p : α✝ → Prop
x : EStateM ε σ α✝
w : ∀ (s : σ) (a : α✝) (s' : σ), x.run s = EStateM.Result.ok a s' → p a
s : σ
⊢ EStateM.Result.map Subtype.val
(match q : x.run s with
| EStateM.Result.ok a s' => EStateM.Result.ok ⟨a, ⋯⟩ s'
| EStateM.Result.error e s' => EStateM.R... | 6fb1fa46dc09b1a6 |
MeasureTheory.SignedMeasure.toSignedMeasure_toJordanDecomposition | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem toSignedMeasure_toJordanDecomposition (s : SignedMeasure α) :
s.toJordanDecomposition.toSignedMeasure = s | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : 0 ≤[i] s
hi₃ : s ≤[iᶜ] 0
hμ : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂
hν : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃
⊢ s.toJordanDecomposition.... | simp only [JordanDecomposition.toSignedMeasure, hμ, hν] | case intro.intro.intro.intro.intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : 0 ≤[i] s
hi₃ : s ≤[iᶜ] 0
hμ : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂
hν : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃
⊢ (s.toMeasureOfZeroLE i h... | eb0fd2c3a693cf1a |
Subgroup.leftCoset_cover_filter_FiniteIndex_aux | Mathlib/GroupTheory/CosetCover.lean | theorem leftCoset_cover_filter_FiniteIndex_aux
[DecidablePred (FiniteIndex : Subgroup G → Prop)] :
(⋃ k ∈ s.filter (fun i => (H i).FiniteIndex), g k • (H k : Set G) = Set.univ) ∧
(1 ≤ ∑ i ∈ s, ((H i).index : ℚ)⁻¹) ∧
(∑ i ∈ s, ((H i).index : ℚ)⁻¹ = 1 → Set.PairwiseDisjoint
(s.filter (fun i =>... | case refine_2
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ ... | rw [one_mul, mul_assoc, inv_mul_cancel₀ (Nat.cast_ne_zero.mpr hD.finiteIndex), mul_one,
Nat.cast_le] | case refine_2
G : Type u_1
inst✝¹ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ ... | 6c7a447596ed19f6 |
Finset.prod_add_prod_le' | Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean | /-- If `g, h ≤ f` and `g i + h i ≤ f i`, then the product of `f` over `s` is at least the
sum of the products of `g` and `h`. This is the version for `CanonicallyOrderedAdd`.
-/
lemma prod_add_prod_le' (hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j)
(hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) : ((∏... | ι : Type u_1
R : Type u_2
inst✝² : CommSemiring R
inst✝¹ : PartialOrder R
inst✝ : CanonicallyOrderedAdd R
f g h : ι → R
s : Finset ι
i : ι
hi : i ∈ s
h2i : g i + h i ≤ f i
hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j
hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j
⊢ g i * ∏ i ∈ s \ {i}, g i + h i * ∏ i ∈ s \ {i}, h i ≤ g i * ∏ i ∈ s \ {i}, f i +... | gcongr with j hj j hj <;> simp_all | no goals | b1ae00d51a4733f1 |
EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | theorem mul_dist_le_mul_dist_add_mul_dist (a b c d : P) :
dist a c * dist b d ≤ dist a b * dist c d + dist b c * dist a d | case inr.inr
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
a b c d : P
hb : b ≠ a
hc : c ≠ a
⊢ dist a c * dist b d ≤ dist a b * dist c d + dist b c * dist a d | rcases eq_or_ne d a with (rfl | hd) | case inr.inr.inl
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
b c d : P
hb : b ≠ d
hc : c ≠ d
⊢ dist d c * dist b d ≤ dist d b * dist c d + dist b c * dist d d
case inr.inr.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommG... | d7abc1e0712fcb18 |
Batteries.TransCmp.compareOfLessAndEq | Mathlib/.lake/packages/batteries/Batteries/Classes/Order.lean | theorem TransCmp.compareOfLessAndEq
[LT α] [DecidableRel (LT.lt (α := α))] [DecidableEq α]
(lt_irrefl : ∀ x : α, ¬x < x)
(lt_trans : ∀ {x y z : α}, x < y → y < z → x < z)
(lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y) :
TransCmp (α := α) (compareOfLessAndEq · ·) | case isFalse.isTrue
α : Type u_1
inst✝² : LT α
inst✝¹ : DecidableRel LT.lt
inst✝ : DecidableEq α
lt_irrefl : ∀ (x : α), ¬x < x
lt_trans : ∀ {x y z : α}, x < y → y < z → x < z
lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y
x : α
h✝ : ¬x < x
⊢ Ordering.eq.swap = if x < x then Ordering.lt else if x = x then Ordering.e... | rw [if_neg ‹_›, if_pos rfl] | case isFalse.isTrue
α : Type u_1
inst✝² : LT α
inst✝¹ : DecidableRel LT.lt
inst✝ : DecidableEq α
lt_irrefl : ∀ (x : α), ¬x < x
lt_trans : ∀ {x y z : α}, x < y → y < z → x < z
lt_antisymm : ∀ {x y : α}, ¬x < y → ¬y < x → x = y
x : α
h✝ : ¬x < x
⊢ Ordering.eq.swap = Ordering.eq | 265051c307a250c5 |
Std.DHashMap.Raw.getD_insert | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getD_insert [LawfulBEq α] (h : m.WF) {k a : α} {fallback : β a} {v : β k} :
(m.insert k v).getD a fallback =
if h : k == a then cast (congrArg β (eq_of_beq h)) v else m.getD a fallback | α : Type u
β : α → Type v
m : Raw α β
inst✝² : BEq α
inst✝¹ : Hashable α
inst✝ : LawfulBEq α
h : m.WF
k a : α
fallback : β a
v : β k
⊢ (m.insert k v).getD a fallback = if h : (k == a) = true then cast ⋯ v else m.getD a fallback | simp_to_raw using Raw₀.getD_insert | no goals | 3fd9207edc078058 |
Set.uIcc_injective_right | Mathlib/Order/Interval/Set/UnorderedInterval.lean | lemma uIcc_injective_right (a : α) : Injective fun b => uIcc b a := fun b c h => by
rw [Set.ext_iff] at h
exact eq_of_mem_uIcc_of_mem_uIcc ((h _).1 left_mem_uIcc) ((h _).2 left_mem_uIcc)
| α : Type u_1
inst✝ : DistribLattice α
a b c : α
h : (fun b => [[b, a]]) b = (fun b => [[b, a]]) c
⊢ b = c | rw [Set.ext_iff] at h | α : Type u_1
inst✝ : DistribLattice α
a b c : α
h : ∀ (x : α), x ∈ (fun b => [[b, a]]) b ↔ x ∈ (fun b => [[b, a]]) c
⊢ b = c | e3501e8407613320 |
Nat.testBit_bitwise | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean | theorem testBit_bitwise (of_false_false : f false false = false) (x y i : Nat) :
(bitwise f x y).testBit i = f (x.testBit i) (y.testBit i) | f : Bool → Bool → Bool
of_false_false : f false false = false
i : Nat
hyp : ∀ (m : Nat), m < i → ∀ (x y : Nat), (bitwise f x y).testBit m = f (x.testBit m) (y.testBit m)
x y : Nat
x_zero : x = 0
⊢ (if x = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then x else... | cases p : f false true <;>
cases yi : testBit y i <;>
simp [x_zero, p, yi, of_false_false] | no goals | 19fb8270ffef0c8a |
ShrinkingLemma.PartialRefinement.exists_gt | Mathlib/Topology/ShrinkingLemma.lean | theorem exists_gt [NormalSpace X] (v : PartialRefinement u s ⊤) (hs : IsClosed s)
(i : ι) (hi : i ∉ v.carrier) :
∃ v' : PartialRefinement u s ⊤, v < v' | ι : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝ : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier
⊢ s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ ⊆ v.toFun i | simp only [subset_def, mem_inter_iff, mem_iInter, and_imp] | ι : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝ : NormalSpace X
v : PartialRefinement u s ⊤
hs : IsClosed s
i : ι
hi : i ∉ v.carrier
⊢ ∀ x ∈ s, (∀ (i_1 : ι), i_1 ≠ i → x ∈ (v.toFun i_1)ᶜ) → x ∈ v.toFun i | f07fb14ff2f0d64d |
CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le | Mathlib/Order/CompactlyGenerated/Basic.lean | theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
IsCompactElement k ↔
∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x | case h.left
α : Type u_2
inst✝ : CompleteLattice α
k : α
hk : ∀ (s : Set α), s.Nonempty → DirectedOn (fun x1 x2 => x1 ≤ x2) s → k ≤ sSup s → ∃ x ∈ s, k ≤ x
s : Set α
hsup : k ≤ sSup s
S : Set α := {x | ∃ t, ↑t ⊆ s ∧ x = t.sup id}
x : α
c : Finset α
hc : ↑c ⊆ s ∧ x = c.sup id
y : α
d : Finset α
hd : ↑d ⊆ s ∧ y = d.sup i... | simp only [hc.left, hd.left, Set.union_subset_iff, Finset.coe_union, and_self_iff] | no goals | 445bf3d7fb93c3e6 |
HasFPowerSeriesOnBall.tendstoUniformlyOn | Mathlib/Analysis/Analytic/Basic.lean | theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
(h : (r' : ℝ≥0∞) < r) :
TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
(Metric.ball (0 : E) r') | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
r' : ℝ≥0
hf : HasFPowerSeriesOnBall f p x r
h : ↑r' < r
⊢ TendstoUniforml... | rw [← hasFPowerSeriesWithinOnBall_univ] at hf | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
r : ℝ≥0∞
r' : ℝ≥0
hf : HasFPowerSeriesWithinOnBall f p univ x r
h : ↑r' < r
⊢ Tend... | 903b769ac55d34fe |
DFinsupp.lex_fibration | Mathlib/Data/DFinsupp/WellFounded.lean | theorem lex_fibration [∀ (i) (s : Set ι), Decidable (i ∈ s)] :
Fibration (InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd) (DFinsupp.Lex r s)
fun x => piecewise x.2.1 x.2.2 x.1 | case neg
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
hi : ¬r i i
⊢ s i (... | assumption | no goals | ce70f659976abe0e |
MeasureTheory.MemLp.induction | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | theorem MemLp.induction [_i : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) (P : (α → E) → Prop)
(h_ind : ∀ (c : E) ⦃s⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun _ => c))
(h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ →
P f → P g → P (f + g))
(h_closed : IsClosed { f ... | case h.e'_1.h
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ⊤
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → ... | simp [const] | no goals | bf4a6fb28ab310ad |
BumpCovering.exists_isSubordinate | Mathlib/Topology/PartitionOfUnity.lean | theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X)
(ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, f.IsSubordinate U | case intro.intro.intro.intro
ι : Type u
X : Type v
inst✝² : TopologicalSpace X
s : Set X
inst✝¹ : NormalSpace X
inst✝ : ParacompactSpace X
hs : IsClosed s
U : ι → Set X
ho : ∀ (i : ι), IsOpen (U i)
hU : s ⊆ ⋃ i, U i
V : ι → Set X
hVo : ∀ (i : ι), IsOpen (V i)
hsV : s ⊆ ⋃ i, V i
hVf : LocallyFinite V
hVU : ∀ (i : ι), V ... | rcases exists_isSubordinate_of_locallyFinite hs V hVo hVf hsV with ⟨f, hf⟩ | case intro.intro.intro.intro.intro
ι : Type u
X : Type v
inst✝² : TopologicalSpace X
s : Set X
inst✝¹ : NormalSpace X
inst✝ : ParacompactSpace X
hs : IsClosed s
U : ι → Set X
ho : ∀ (i : ι), IsOpen (U i)
hU : s ⊆ ⋃ i, U i
V : ι → Set X
hVo : ∀ (i : ι), IsOpen (V i)
hsV : s ⊆ ⋃ i, V i
hVf : LocallyFinite V
hVU : ∀ (i : ... | 2201cd5e2b224650 |
isPurelyInseparable_of_finSepDegree_eq_one | Mathlib/FieldTheory/PurelyInseparable/Basic.lean | theorem isPurelyInseparable_of_finSepDegree_eq_one
(hdeg : finSepDegree F E = 1) : IsPurelyInseparable F E | case pos
F : Type u
E : Type v
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
hdeg : finSepDegree F E = 1
H : Algebra.IsAlgebraic F E
x : E
hsep : IsSeparable F x
this✝ : Algebra.IsAlgebraic (↥F⟮x⟯) E
this : F⟮x⟯ = ⊥ ∧ finSepDegree (↥F⟮x⟯) E = 1
⊢ x ∈ (algebraMap F E).range | simpa only [this.1] using mem_adjoin_simple_self F x | no goals | 3a40061a1043c350 |
WfDvdMonoid.of_setOf_isPrincipal_wellFoundedOn_gt | Mathlib/RingTheory/UniqueFactorizationDomain/Ideal.lean | /-- The ascending chain condition on principal ideals in a domain is sufficient to prove that
the domain is `WfDvdMonoid`. -/
lemma WfDvdMonoid.of_setOf_isPrincipal_wellFoundedOn_gt [CommSemiring α] [IsDomain α]
(h : {I : Ideal α | I.IsPrincipal}.WellFoundedOn (· > ·)) :
WfDvdMonoid α | case wf
α : Type u_1
inst✝¹ : CommSemiring α
inst✝ : IsDomain α
h : {I | Submodule.IsPrincipal I}.WellFoundedOn fun x1 x2 => x1 > x2
this : WellFounded fun x1 x2 => x1 > x2
⊢ WellFounded DvdNotUnit | convert InvImage.wf (fun a => ⟨Ideal.span ({a} : Set α), _, rfl⟩) this | case h.e'_2
α : Type u_1
inst✝¹ : CommSemiring α
inst✝ : IsDomain α
h : {I | Submodule.IsPrincipal I}.WellFoundedOn fun x1 x2 => x1 > x2
this : WellFounded fun x1 x2 => x1 > x2
⊢ DvdNotUnit = InvImage (fun x1 x2 => x1 > x2) fun a => ⟨Ideal.span {a}, ⋯⟩ | 1d2e74032df7b812 |
UV.compress_idem | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | theorem compress_idem (u v a : α) : compress u v (compress u v a) = compress u v a | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
u v a : α
⊢ compress u v (compress u v a) = compress u v a | unfold compress | α : Type u_1
inst✝² : GeneralizedBooleanAlgebra α
inst✝¹ : DecidableRel Disjoint
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
u v a : α
⊢ (if
Disjoint u (if Disjoint u a ∧ v ≤ a then (a ⊔ u) \ v else a) ∧
v ≤ if Disjoint u a ∧ v ≤ a then (a ⊔ u) \ v else a then
((if Disjoint u a ∧ v ≤ a then (a ⊔ u... | 8839cd8a05af7046 |
IsNonarchimedean.finset_image_add_of_nonempty | Mathlib/Data/Real/IsNonarchimedean.lean | theorem finset_image_add_of_nonempty {F α β : Type*} [AddCommGroup α] [FunLike F α ℝ]
[AddGroupSeminormClass F α ℝ] [Nonempty β] {f : F} (hna : IsNonarchimedean f)
(g : β → α) {t : Finset β} (ht : t.Nonempty) :
∃ b : β, (b ∈ t) ∧ f (t.sum g) ≤ f (g b) | case intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : AddCommGroup α
inst✝² : FunLike F α ℝ
inst✝¹ : AddGroupSeminormClass F α ℝ
inst✝ : Nonempty β
f : F
hna : IsNonarchimedean ⇑f
g : β → α
t : Finset β
ht : t.Nonempty
b : β
hbt : t.Nonempty → b ∈ t
hbf : f (t.sum g) ≤ f (g b)
⊢ ∃ b ∈ t, f (t.sum g) ≤ f (g b... | exact ⟨b, hbt ht, hbf⟩ | no goals | c760e0a9e25a1e82 |
Set.Iio_subset_Iio_iff | Mathlib/Order/Interval/Set/Basic.lean | theorem Iio_subset_Iio_iff : Iio a ⊆ Iio b ↔ a ≤ b | α : Type u_1
inst✝ : LinearOrder α
a b : α
⊢ Iio a ⊆ Iio b ↔ a ≤ b | refine ⟨fun h => ?_, fun h => Iio_subset_Iio h⟩ | α : Type u_1
inst✝ : LinearOrder α
a b : α
h : Iio a ⊆ Iio b
⊢ a ≤ b | 3af4c0dba423e29b |
Turing.PartrecToTM2.head_stack_ok | Mathlib/Computability/TMToPartrec.lean | theorem head_stack_ok {q s L₁ L₂ L₃} :
Reaches₁ (TM2.step tr)
⟨some (head stack q), s, K'.elim (trList L₁) [] [] (trList L₂ ++ Γ'.consₗ :: L₃)⟩
⟨some q, none, K'.elim (trList (L₂.headI :: L₁)) [] [] L₃⟩ | case cons
q : Λ'
s : Option Γ'
L₁ : List ℕ
L₃ : List Γ'
a : ℕ
L₂ : List ℕ
⊢ Reaches₁ (TM2.step tr)
{ l := some (head stack q), var := s, stk := elim (trList L₁) [] [] (trList (a :: L₂) ++ Γ'.consₗ :: L₃) }
{ l := some q, var := none, stk := elim (trList ((a :: L₂).headI :: L₁)) [] [] L₃ } | refine
TransGen.trans
(move_ok (by decide)
(splitAtPred_eq _ _ (trNat a) (some Γ'.cons) (trList L₂ ++ Γ'.consₗ :: L₃)
(trNat_natEnd _) ⟨rfl, by simp⟩))
(TransGen.head rfl (TransGen.head rfl ?_)) | case cons
q : Λ'
s : Option Γ'
L₁ : List ℕ
L₃ : List Γ'
a : ℕ
L₂ : List ℕ
⊢ TransGen (fun a b => b ∈ TM2.step tr a)
(TM2.stepAux
(tr
((fun x =>
Λ'.read fun s =>
(if s = some Γ'.consₗ then id else Λ'.clear (fun x => decide (x = Γ'.consₗ)) stack) (unrev q))
(some Γ'.c... | 5cadd8176b1827d5 |
TopCat.GlueData.ι_eq_iff_rel | Mathlib/Topology/Gluing.lean | theorem ι_eq_iff_rel (i j : D.J) (x : D.U i) (y : D.U j) :
𝖣.ι i x = 𝖣.ι j y ↔ D.Rel ⟨i, x⟩ ⟨j, y⟩ | case mp
D : GlueData
i j : D.J
x : ↑(D.U i)
y : ↑(D.U j)
h :
(ConcreteCategory.hom (Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram)) x =
(ConcreteCategory.hom (Sigma.ι D.diagram.right j ≫ Multicoequalizer.sigmaπ D.diagram)) y
⊢ D.Rel (((sigmaIsoSigma D.U).inv ≫ (sigmaIsoSigma D.U).hom) ⟨i, x⟩) ⟨j, ... | rw [←
show _ = Sigma.mk j y from ConcreteCategory.congr_hom (sigmaIsoSigma.{_, u} D.U).inv_hom_id _] | case mp
D : GlueData
i j : D.J
x : ↑(D.U i)
y : ↑(D.U j)
h :
(ConcreteCategory.hom (Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram)) x =
(ConcreteCategory.hom (Sigma.ι D.diagram.right j ≫ Multicoequalizer.sigmaπ D.diagram)) y
⊢ D.Rel (((sigmaIsoSigma D.U).inv ≫ (sigmaIsoSigma D.U).hom) ⟨i, x⟩)
... | e0a110dda282edcf |
Filter.HasBasis.sup' | Mathlib/Order/Filter/Bases.lean | theorem HasBasis.sup' (hl : l.HasBasis p s) (hl' : l'.HasBasis p' s') :
(l ⊔ l').HasBasis (fun i : PProd ι ι' => p i.1 ∧ p' i.2) fun i => s i.1 ∪ s' i.2 :=
⟨by
intro t
simp_rw [mem_sup, hl.mem_iff, hl'.mem_iff, PProd.exists, union_subset_iff,
← exists_and_right, ← exists_and_left]
simp only [an... | α : Type u_1
ι : Sort u_4
ι' : Sort u_5
l l' : Filter α
p : ι → Prop
s : ι → Set α
p' : ι' → Prop
s' : ι' → Set α
hl : l.HasBasis p s
hl' : l'.HasBasis p' s'
t : Set α
⊢ (∃ x x_1, (p x ∧ s x ⊆ t) ∧ p' x_1 ∧ s' x_1 ⊆ t) ↔ ∃ a b, (p a ∧ p' b) ∧ s a ⊆ t ∧ s' b ⊆ t | simp only [and_assoc, and_left_comm] | no goals | 00fa2602b09511a8 |
Cardinal.mk_Ioi_real | Mathlib/Data/Real/Cardinality.lean | theorem mk_Ioi_real (a : ℝ) : #(Ioi a) = 𝔠 | a : ℝ
h : #↑(Ioi a) < 𝔠
⊢ #↑Set.univ < 𝔠 | have hu : Iio a ∪ {a} ∪ Ioi a = Set.univ := by
convert @Iic_union_Ioi ℝ _ _
exact Iio_union_right | a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑Set.univ < 𝔠 | 564ae2d1d94f837f |
Std.DHashMap.Raw.getKey?_eq_some_getKey! | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getKey?_eq_some_getKey! [EquivBEq α] [LawfulHashable α] [Inhabited α] (h : m.WF) {a : α} :
a ∈ m → m.getKey? a = some (m.getKey! a) | α : Type u
β : α → Type v
m : Raw α β
inst✝⁴ : BEq α
inst✝³ : Hashable α
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited α
h : m.WF
a : α
⊢ a ∈ m → m.getKey? a = some (m.getKey! a) | simpa [mem_iff_contains] using getKey?_eq_some_getKey!_of_contains h | no goals | 986b5940dcf7795b |
Multiset.Icc_eq_zero_iff | Mathlib/Order/Interval/Multiset.lean | theorem Icc_eq_zero_iff : Icc a b = 0 ↔ ¬a ≤ b | α : Type u_1
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
a b : α
⊢ Icc a b = 0 ↔ ¬a ≤ b | rw [Icc, Finset.val_eq_zero, Finset.Icc_eq_empty_iff] | no goals | f7346d2b96938a0d |
writtenInExtChartAt_comp | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | theorem writtenInExtChartAt_comp (h : ContinuousWithinAt f s x) :
{y | writtenInExtChartAt I I'' x (g ∘ f) y =
(writtenInExtChartAt I' I'' (f x) g ∘ writtenInExtChartAt I I' x f) y} ∈
𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x) x | 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : Norme... | apply
@Filter.mem_of_superset _ _ (f ∘ (extChartAt I x).symm ⁻¹' (extChartAt I' (f x)).source) _
(extChartAt_preimage_mem_nhdsWithin
(h.preimage_mem_nhdsWithin (extChartAt_source_mem_nhds _))) | 𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : Norme... | 4e42d7572fbdc0d2 |
List.replace_eq_replaceTR | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Impl.lean | theorem replace_eq_replaceTR : @List.replace = @replaceTR | ⊢ @replace = @replaceTR | funext α _ l b c | case h.h.h.h.h
α : Type u_1
x✝ : BEq α
l : List α
b c : α
⊢ l.replace b c = l.replaceTR b c | a82f1469e1b7fcc8 |
BddAbove.continuous_convolution_right_of_integrable | Mathlib/Analysis/Convolution.lean | theorem _root_.BddAbove.continuous_convolution_right_of_integrable
[FirstCountableTopology G] [SecondCountableTopologyEither G E']
(hbg : BddAbove (range fun x => ‖g x‖)) (hf : Integrable f μ) (hg : Continuous g) :
Continuous (f ⋆[L, μ] g) | case h
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →... | filter_upwards with t | case h.h
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E'... | 7cef45832d5d6aeb |
SetTheory.PGame.numeric_def | Mathlib/SetTheory/Surreal/Basic.lean | theorem numeric_def {x : PGame} :
Numeric x ↔
(∀ i j, x.moveLeft i < x.moveRight j) ∧
(∀ i, Numeric (x.moveLeft i)) ∧ ∀ j, Numeric (x.moveRight j) | case mk
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
⊢ (mk α✝ β✝ a✝¹ a✝).Numeric ↔
(∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).RightMoves),
(mk α✝ β✝ a✝¹ a✝).moveLeft i < (mk α✝ β✝ a✝¹ a✝).moveRight j) ∧
(∀ (i : (mk α✝ β✝ a✝¹ a✝).LeftMoves), ((mk α✝ β✝ a✝¹ a✝).moveLeft i).Numeric)... | rfl | no goals | d26425902c048a37 |
ProbabilityTheory.IndepFun.integral_mul_of_integrable | Mathlib/Probability/Integration.lean | theorem IndepFun.integral_mul_of_integrable (hXY : IndepFun X Y μ) (hX : Integrable X μ)
(hY : Integrable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : Integrable X μ
hY : Integrable Y μ
pos : ℝ → ℝ := fun x => x ⊔ 0
neg : ℝ → ℝ := fun x => -x ⊔ 0
posm : Measurable pos
negm : Measurable neg
Xp : Ω → ℝ := pos ∘ X
Xm : Ω → ℝ := neg ∘ X
Yp : Ω → ℝ := pos ∘ Y
Ym : Ω → ℝ := neg ∘ Y
hXpm... | have hm3 : AEMeasurable Ym μ := hY.1.aemeasurable.neg.max aemeasurable_const | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : Integrable X μ
hY : Integrable Y μ
pos : ℝ → ℝ := fun x => x ⊔ 0
neg : ℝ → ℝ := fun x => -x ⊔ 0
posm : Measurable pos
negm : Measurable neg
Xp : Ω → ℝ := pos ∘ X
Xm : Ω → ℝ := neg ∘ X
Yp : Ω → ℝ := pos ∘ Y
Ym : Ω → ℝ := neg ∘ Y
hXpm... | 1836e7e522e593a2 |
Finsupp.single_of_single_apply | Mathlib/Data/Finsupp/Single.lean | theorem single_of_single_apply (a a' : α) (b : M) :
single a ((single a' b) a) = single a' (single a' b) a | case neg
α : Type u_1
M : Type u_5
inst✝ : Zero M
a a' : α
b : M
a✝ : α
h : ¬a' = a
⊢ (single a 0) a✝ = 0 a✝ | rw [zero_apply, single_apply, ite_self] | no goals | 33af47e1a4958c4f |
Nat.Primrec'.sqrt | Mathlib/Computability/Primrec.lean | theorem sqrt : @Primrec' 1 fun v => v.head.sqrt | H : ∀ (n : ℕ), n.sqrt = Nat.rec 0 (fun x y => if x.succ < y.succ * y.succ then y else y.succ) n
v : List.Vector ℕ (1 + 2)
x : ℕ
⊢ ℕ | have y := v.tail.head | H : ∀ (n : ℕ), n.sqrt = Nat.rec 0 (fun x y => if x.succ < y.succ * y.succ then y else y.succ) n
v : List.Vector ℕ (1 + 2)
x y : ℕ
⊢ ℕ | 99d6f1beec25882a |
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
⊢ finrank 𝕜 (X ⟶ Y) = 0 ↔ IsEmpty ... | rw [← not_nonempty_iff, ← not_congr (finrank_hom_simple_simple_eq_one_iff 𝕜 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
⊢ finrank 𝕜 (X ⟶ Y) = 0 ↔ ¬finrank... | 36d01ca65cae1c3e |
Besicovitch.exist_disjoint_covering_families | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exist_disjoint_covering_families {N : ℕ} {τ : ℝ} (hτ : 1 < τ)
(hN : IsEmpty (SatelliteConfig α N τ)) (q : BallPackage β α) :
∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) ∧
range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, ball (q.c j) (q.r j) | case hs
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i => ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : ... | exact (p.color_lt (hk.trans jy_lt) hN).ne' | no goals | 416e48a035cdf3d5 |
MulAction.smul_bijective_of_is_unit | Mathlib/GroupTheory/GroupAction/Pointwise.lean | theorem MulAction.smul_bijective_of_is_unit
{M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
Function.Bijective (fun (a : α) ↦ m • a) | case h.left
M : Type u_1
inst✝¹ : Monoid M
α : Type u_2
inst✝ : MulAction M α
m : Mˣ
x : α
⊢ (fun a => m⁻¹ • a) ((fun a => ↑m • a) x) = x | simp [← Units.smul_def] | no goals | 91b4009f7a529152 |
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