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 |
|---|---|---|---|---|---|---|
StieltjesFunction.outer_Ioc | Mathlib/MeasureTheory/Measure/Stieltjes.lean | theorem outer_Ioc (a b : ℝ) : f.outer (Ioc a b) = ofReal (f b - f a) | f : StieltjesFunction
a b : ℝ
s : ℕ → Set ℝ
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ⊤
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a
⊢ ∃ a', ↑f a' - ↑f a < ↑δ ∧ a < a' | have B : f a - f a < δ := by rwa [sub_self, NNReal.coe_pos, ← ENNReal.coe_pos] | f : StieltjesFunction
a b : ℝ
s : ℕ → Set ℝ
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ⊤
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
A : ContinuousWithinAt (fun r => ↑f r - ↑f a) (Ioi a) a
B : ↑f a - ↑f a < ↑δ
⊢ ∃ a', ↑f a' - ↑f ... | c3b0d539cbbc47aa |
continuous_right_toIcoMod | Mathlib/Topology/Instances/AddCircle.lean | theorem continuous_right_toIcoMod : ContinuousWithinAt (toIcoMod hp a) (Ici x) x | case intro.intro.intro
𝕜 : Type u_1
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
s : Set 𝕜
this : Nontrivial 𝕜
l u : 𝕜
hxI : toIcoMod hp a x ∈ Ioo l u
hIs : Ioo l u ⊆ s
d : 𝕜 := toIcoDiv hp a x • p
hd : toIcoMod hp a ... | refine ⟨_, ⟨l + d, min (a + p) u + d, ?_, fun x => id⟩, fun y => ?_⟩ <;>
simp_rw [← sub_mem_Ioo_iff_left, mem_Ioo, lt_min_iff] | case intro.intro.intro.refine_1
𝕜 : Type u_1
inst✝³ : LinearOrderedAddCommGroup 𝕜
inst✝² : Archimedean 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p : 𝕜
hp : 0 < p
a x : 𝕜
s : Set 𝕜
this : Nontrivial 𝕜
l u : 𝕜
hxI : toIcoMod hp a x ∈ Ioo l u
hIs : Ioo l u ⊆ s
d : 𝕜 := toIcoDiv hp a x • p
hd : toIco... | d6bbe1e27d3a7548 |
Sum.isConnected_iff | Mathlib/Topology/Connected/Clopen.lean | theorem Sum.isConnected_iff [TopologicalSpace β] {s : Set (α ⊕ β)} :
IsConnected s ↔
(∃ t, IsConnected t ∧ s = Sum.inl '' t) ∨ ∃ t, IsConnected t ∧ s = Sum.inr '' t | case refine_2.inr.intro.intro
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
t : Set β
ht : IsConnected t
⊢ IsConnected (inr '' t) | exact ht.image _ continuous_inr.continuousOn | no goals | a8bbf511dc872e45 |
eq_of_powMul_faithful | Mathlib/Analysis/Normed/Ring/IsPowMulFaithful.lean | theorem eq_of_powMul_faithful (f₁ : AlgebraNorm R S) (hf₁_pm : IsPowMul f₁) (f₂ : AlgebraNorm R S)
(hf₂_pm : IsPowMul f₂)
(h_eq : ∀ y : S, ∃ (C₁ C₂ : ℝ) (_ : 0 < C₁) (_ : 0 < C₂),
∀ x : Algebra.adjoin R {y}, f₁ x.val ≤ C₁ * f₂ x.val ∧ f₂ x.val ≤ C₂ * f₁ x.val) :
f₁ = f₂ | R : Type u_1
S : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
f₁ : AlgebraNorm R S
hf₁_pm : IsPowMul ⇑f₁
f₂ : AlgebraNorm R S
hf₂_pm : IsPowMul ⇑f₂
h_eq :
∀ (y : S),
∃ C₁ C₂, ∃ (_ : 0 < C₁) (_ : 0 < C₂), ∀ (x : ↥(Algebra.adjoin R {y})), f₁ ↑x ≤ C₁ * f₂ ↑x ∧ f₂ ↑x ≤ C₂ * f₁ ↑x
⊢ f₁ = ... | ext x | case a
R : Type u_1
S : Type u_2
inst✝² : NormedCommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
f₁ : AlgebraNorm R S
hf₁_pm : IsPowMul ⇑f₁
f₂ : AlgebraNorm R S
hf₂_pm : IsPowMul ⇑f₂
h_eq :
∀ (y : S),
∃ C₁ C₂, ∃ (_ : 0 < C₁) (_ : 0 < C₂), ∀ (x : ↥(Algebra.adjoin R {y})), f₁ ↑x ≤ C₁ * f₂ ↑x ∧ f₂ ↑x ≤ C₂ * f₁ ↑x
... | f86a56fc3f1b670d |
MulAction.IsTrivialBlock.image | Mathlib/GroupTheory/GroupAction/Blocks.lean | theorem IsTrivialBlock.image {φ : M → N} {f : α →ₑ[φ] β}
(hf : Function.Surjective f) {B : Set α} (hB : IsTrivialBlock B) :
IsTrivialBlock (f '' B) | case inr
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = univ
⊢ IsTrivialBlock (⇑f '' B) | apply Or.intro_right | case inr.h
M : Type u_3
α : Type u_4
N : Type u_5
β : Type u_6
inst✝³ : Monoid M
inst✝² : MulAction M α
inst✝¹ : Monoid N
inst✝ : MulAction N β
φ : M → N
f : α →ₑ[φ] β
hf : Function.Surjective ⇑f
B : Set α
hB : B = univ
⊢ ⇑f '' B = univ | 83c940b50d5b79be |
Real.tendsto_harmonic_sub_log | Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | lemma tendsto_harmonic_sub_log :
Tendsto (fun n : ℕ ↦ harmonic n - log n) atTop (𝓝 eulerMascheroniConstant) | case h
n : ℕ
hn : n ≠ 0
⊢ eulerMascheroniSeq' n = ↑(harmonic n) - log ↑n | simp_rw [eulerMascheroniSeq', hn, if_false] | no goals | 7c3b53a7104901b8 |
Std.Sat.AIG.RefVec.get_append | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVec.lean | theorem get_append (lhs : RefVec aig lw) (rhs : RefVec aig rw) (idx : Nat)
(hidx : idx < lw + rw) :
(lhs.append rhs).get idx hidx
=
if h : idx < lw then
lhs.get idx h
else
rhs.get (idx - lw) (by omega) | case isTrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
lw rw : Nat
lhs : aig.RefVec lw
rhs : aig.RefVec rw
idx : Nat
hidx : idx < lw + rw
h✝ : idx < lw
⊢ (lhs.refs ++ rhs.refs)[idx] = lhs.refs[idx] | rw [Array.getElem_append_left] | no goals | b668eb71e597aaa0 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastReplicate.aux1 | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Replicate.lean | theorem aux1 {a b c : Nat} (h : b < a * c) : 0 < a | case neg
a b c : Nat
h : b < a * c
h✝ : ¬a = 0
⊢ 0 < a | omega | no goals | 740fe05bb7fedf9b |
CategoryTheory.Functor.homologySequence_comp | Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean | @[reassoc]
lemma homologySequence_comp :
(F.shift n₀).map T.mor₁ ≫ (F.shift n₀).map T.mor₂ = 0 | C : Type u_1
A : Type u_3
inst✝⁹ : Category.{u_5, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{u_4, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
T : Tri... | rw [← Functor.map_comp, comp_distTriang_mor_zero₁₂ _ hT, Functor.map_zero] | no goals | 369f9bf05fcdb78d |
Function.support_div | Mathlib/Algebra/GroupWithZero/Indicator.lean | @[simp] lemma support_div (f g : ι → G₀) : support (fun a ↦ f a / g a) = support f ∩ support g | ι : Type u_1
G₀ : Type u_3
inst✝ : GroupWithZero G₀
f g : ι → G₀
⊢ (support fun a => f a / g a) = support f ∩ support g | simp [div_eq_mul_inv] | no goals | 07752fa867abf796 |
Filter.prod_map_left | Mathlib/Order/Filter/Prod.lean | theorem prod_map_left (f : α → β) (F : Filter α) (G : Filter γ) :
map f F ×ˢ G = map (Prod.map f id) (F ×ˢ G) | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
F : Filter α
G : Filter γ
⊢ map f F ×ˢ G = map (Prod.map f id) (F ×ˢ G) | rw [← prod_map_map_eq', map_id] | no goals | 677f419e95dcad77 |
aux₀ | Mathlib/MeasureTheory/Order/UpperLower.lean | /-- If we can fit a small ball inside a set `s` intersected with any neighborhood of `x`, then the
density of `s` near `x` is not `0`.
Along with `aux₁`, this proves that `x` is a Lebesgue point of `s`. This will be used to prove that
the frontier of an order-connected set is null. -/
private lemma aux₀
(h : ∀ δ, ... | ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0... | gcongr | case h.h
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
x : ι → ℝ
f : (δ : ℝ) → 0 < δ → ι → ℝ
hf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ
hf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s
H : Tendsto (fun r => volume (closure s ∩ closedBall x r) / volume (closedBall x r))... | 0d32daafddddd57d |
Order.PFilter.sInf_gc | Mathlib/Order/PFilter.lean | theorem sInf_gc :
GaloisConnection (fun x => toDual (principal x)) fun F => sInf (ofDual F : PFilter P) :=
fun x F => by simp only [le_sInf_iff, SetLike.mem_coe, toDual_le, SetLike.le_def, mem_principal]
| P : Type u_1
inst✝ : CompleteSemilatticeInf P
x : P
F : (PFilter P)ᵒᵈ
⊢ (fun x => toDual (principal x)) x ≤ F ↔ x ≤ (fun F => sInf ↑(ofDual F)) F | simp only [le_sInf_iff, SetLike.mem_coe, toDual_le, SetLike.le_def, mem_principal] | no goals | 5d08432b704928b0 |
Primrec.vector_ofFn | Mathlib/Computability/Primrec.lean | theorem vector_ofFn {n} {f : Fin n → α → σ} (hf : ∀ i, Primrec (f i)) :
Primrec fun a => List.Vector.ofFn fun i => f i a :=
vector_toList_iff.1 <| by simp [list_ofFn hf]
| α : Type u_1
σ : Type u_3
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
n : ℕ
f : Fin n → α → σ
hf : ∀ (i : Fin n), Primrec (f i)
⊢ Primrec fun a => (List.Vector.ofFn fun i => f i a).toList | simp [list_ofFn hf] | no goals | 638f102a9470be59 |
Behrend.roth_lower_bound_explicit | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
(N : ℝ) * exp (-4 * √(log N)) < rothNumberNat N | N : ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
hn₂ : 2 < n
this : (2 * dValue N - 1) ^ n ≤ N
⊢ rexp (-4 * √(log ↑N)) ≤ (↑N ^ (2 / ↑n) * (rexp (↑n - 2) * ↑n))⁻¹ | rw [mul_inv, mul_inv, ← exp_neg, ← rpow_neg (cast_nonneg _), neg_sub, ← div_eq_mul_inv] | N : ℕ
hN : 4096 ≤ N
n : ℕ := nValue N
hn : 0 < ↑n
hd : 0 < dValue N
hN₀ : 0 < ↑N
hn₂ : 2 < n
this : (2 * dValue N - 1) ^ n ≤ N
⊢ rexp (-4 * √(log ↑N)) ≤ ↑N ^ (-(2 / ↑n)) * (rexp (2 - ↑n) / ↑n) | 9f846f1b063b879f |
exists_maximal_linearIndepOn' | Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean | theorem exists_maximal_linearIndepOn' (v : ι → M) :
∃ s : Set ι, (LinearIndepOn R v s) ∧ ∀ t : Set ι, s ⊆ t → (LinearIndepOn R v t) → s = t | ι : Type u'
R : Type u_2
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
v : ι → M
indep : Set ι → Prop := fun s => LinearIndepOn R v s
X : Type (max 0 u') := { I // indep I }
r : X → X → Prop := fun I J => ↑I ⊆ ↑J
f g : ι →₀ R
hsum : (Finsupp.linearCombination R v) f = (Finsupp.linearCombi... | simpa using hgsupp | no goals | 31de1692d3396ae5 |
List.mem_of_mem_of_mem_sym | Mathlib/Data/List/Sym.lean | theorem mem_of_mem_of_mem_sym {n : ℕ} {xs : List α} {a : α} {z : Sym α n}
(ha : a ∈ z) (hz : z ∈ xs.sym n) : a ∈ xs :=
match n, xs with
| 0, xs => by
cases Sym.eq_nil_of_card_zero z
simp at ha
| n + 1, [] => by simp [List.sym] at hz
| n + 1, x :: xs => by
rw [List.sym, mem_append, mem_map] at hz... | case inl.intro.intro.inr
α : Type u_1
n✝ : ℕ
xs✝ : List α
a : α
n : ℕ
x : α
xs : List α
z : Sym α n
hz : z ∈ List.sym n (x :: xs)
ha : a ∈ z
⊢ a ∈ x :: xs | exact mem_of_mem_of_mem_sym ha hz | no goals | 05f3e79165190c49 |
HahnModule.coeff_smul_right | Mathlib/RingTheory/HahnSeries/Multiplication.lean | theorem coeff_smul_right [SMulZeroClass R V] {x : HahnSeries Γ R} {y : HahnModule Γ' R V} {a : Γ'}
{s : Set Γ'} (hs : s.IsPWO) (hys : ((of R).symm y).support ⊆ s) :
((of R).symm <| x • y).coeff a =
∑ ij ∈ VAddAntidiagonal x.isPWO_support hs a,
x.coeff ij.fst • ((of R).symm y).coeff ij.snd | Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Zero R
inst✝ : SMulZeroClass R V
x : HahnSeries Γ R
y : HahnModule Γ' R V
a : Γ'
s : Set Γ'
hs : s.IsPWO
hys : ((of R).symm y).supp... | rw [coeff_smul] | Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : Zero R
inst✝ : SMulZeroClass R V
x : HahnSeries Γ R
y : HahnModule Γ' R V
a : Γ'
s : Set Γ'
hs : s.IsPWO
hys : ((of R).symm y).supp... | c0ba122707806e17 |
EuclideanGeometry.continuousAt_oangle | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | theorem continuousAt_oangle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∡ y.1 y.2.1 y.2.2) x | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
x : P × P × P
hx12 : x.1 ≠ x.2.1
hx32 : x.2.2 ≠ x.2.1
⊢ ContinuousAt (fun y => ∡ y.1 y.2.1 y.2.2) x | unfold oangle | V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
x : P × P × P
hx12 : x.1 ≠ x.2.1
hx32 : x.2.2 ≠ x.2.1
⊢ ContinuousAt (fun y => o.oangle (y.1 -ᵥ y.2.1) (y.2.2 -ᵥ y.2.1)) x | 11c77fc630e9dc97 |
Rat.cast_sub_of_ne_zero | Mathlib/Data/Rat/Cast/Defs.lean | @[norm_cast] lemma cast_sub_of_ne_zero (hp : (p.den : α) ≠ 0) (hq : (q.den : α) ≠ 0) :
↑(p - q) = (p - q : α) | α : Type u_3
inst✝ : DivisionRing α
p q : ℚ
hp : ↑p.den ≠ 0
hq : ↑q.den ≠ 0
⊢ ↑(p - q) = ↑p - ↑q | simp [sub_eq_add_neg, cast_add_of_ne_zero, hp, hq] | no goals | 4aced8ac9767b150 |
List.dedup_append | Mathlib/Data/List/Dedup.lean | theorem dedup_append (l₁ l₂ : List α) : dedup (l₁ ++ l₂) = l₁ ∪ dedup l₂ | α : Type u_1
inst✝ : DecidableEq α
l₁ l₂ : List α
⊢ (l₁ ++ l₂).dedup = l₁ ∪ l₂.dedup | induction' l₁ with a l₁ IH | case nil
α : Type u_1
inst✝ : DecidableEq α
l₂ : List α
⊢ ([] ++ l₂).dedup = [] ∪ l₂.dedup
case cons
α : Type u_1
inst✝ : DecidableEq α
l₂ : List α
a : α
l₁ : List α
IH : (l₁ ++ l₂).dedup = l₁ ∪ l₂.dedup
⊢ (a :: l₁ ++ l₂).dedup = a :: l₁ ∪ l₂.dedup | b11f71f7a5212231 |
ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime | Mathlib/NumberTheory/SumTwoSquares.lean | theorem ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime {n x y : ℤ} (h : n = x ^ 2 + y ^ 2)
(hc : IsCoprime x y) : IsSquare (-1 : ZMod n.natAbs) | n x y : ℤ
h : n = x ^ 2 + y ^ 2
hc : IsCoprime x y
u v : ℤ
huv : u * x + v * n = 1
⊢ u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v) | linear_combination -u ^ 2 * h + (n * v - u * x - 1) * huv | no goals | 1f943d046c7310cd |
MeasureTheory.Measure.InnerRegularWRT.of_sigmaFinite | Mathlib/MeasureTheory/Measure/Regular.lean | /-- Given a σ-finite measure, any measurable set can be approximated from inside by a measurable
set of finite measure. -/
lemma of_sigmaFinite [SigmaFinite μ] :
InnerRegularWRT μ (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) (fun s ↦ MeasurableSet s) | case intro
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
B : ℕ → Set α := spanningSets μ
hr : r < ⨆ n, μ (s ∩ B n)
hBU : ⋃ n, s ∩ B n = s
this : μ s = ⨆ n, μ (s ∩ B n)
n : ℕ
hn : r < μ (s ∩ B n)
⊢ ∃ K ⊆ s, (fun s => MeasurableSet s ∧ μ s ≠ ⊤) K ∧ r <... | refine ⟨s ∩ B n, inter_subset_left, ⟨hs.inter (measurableSet_spanningSets μ n), ?_⟩, hn⟩ | case intro
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
B : ℕ → Set α := spanningSets μ
hr : r < ⨆ n, μ (s ∩ B n)
hBU : ⋃ n, s ∩ B n = s
this : μ s = ⨆ n, μ (s ∩ B n)
n : ℕ
hn : r < μ (s ∩ B n)
⊢ μ (s ∩ B n) ≠ ⊤ | b786574f15e6f5b1 |
Algebra.intNorm_eq_norm | Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean | lemma Algebra.intNorm_eq_norm [Module.Free A B] : Algebra.intNorm A B = Algebra.norm A | case h
A : Type u_1
B : Type u_4
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra A B
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsDomain B
inst✝⁴ : IsIntegrallyClosed B
inst✝³ : Module.Finite A B
inst✝² : NoZeroSMulDivisors A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
ins... | haveI : IsIntegralClosure B A (FractionRing B) :=
IsIntegralClosure.of_isIntegrallyClosed _ _ _ | case h
A : Type u_1
B : Type u_4
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra A B
inst✝⁷ : IsIntegrallyClosed A
inst✝⁶ : IsDomain A
inst✝⁵ : IsDomain B
inst✝⁴ : IsIntegrallyClosed B
inst✝³ : Module.Finite A B
inst✝² : NoZeroSMulDivisors A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
ins... | c8310f9857a3230d |
leftCoset_assoc | Mathlib/GroupTheory/Coset/Basic.lean | theorem leftCoset_assoc (s : Set α) (a b : α) : a • (b • s) = (a * b) • s | α : Type u_1
inst✝ : Semigroup α
s : Set α
a b : α
⊢ a • b • s = (a * b) • s | simp [← image_smul, (image_comp _ _ _).symm, Function.comp, mul_assoc] | no goals | 14599ccdec7663b9 |
Set.Nontrivial.image_of_injOn | Mathlib/Data/Set/Image.lean | theorem Nontrivial.image_of_injOn (hs : s.Nontrivial) (hf : s.InjOn f) :
(f '' s).Nontrivial | α : Type u_1
β : Type u_2
s : Set α
f : α → β
hs : s.Nontrivial
hf : InjOn f s
⊢ (f '' s).Nontrivial | obtain ⟨x, hx, y, hy, hxy⟩ := hs | case intro.intro.intro.intro
α : Type u_1
β : Type u_2
s : Set α
f : α → β
hf : InjOn f s
x : α
hx : x ∈ s
y : α
hy : y ∈ s
hxy : x ≠ y
⊢ (f '' s).Nontrivial | d15cb0cda6972799 |
Order.height_le_krullDim | Mathlib/Order/KrullDimension.lean | lemma height_le_krullDim (a : α) : height a ≤ krullDim α | α : Type u_1
inst✝ : Preorder α
a : α
this : Nonempty α
⊢ height a ≤ ⨆ p, ↑p.length | exact height_le fun p _ ↦ le_iSup_of_le p le_rfl | no goals | 2d71222255195130 |
CategoryTheory.Limits.Types.coequalizer_preimage_image_eq_of_preimage_eq | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | theorem coequalizer_preimage_image_eq_of_preimage_eq (π : Y ⟶ Z) (e : f ≫ π = g ≫ π)
(h : IsColimit (Cofork.ofπ π e)) (U : Set Y) (H : f ⁻¹' U = g ⁻¹' U) : π ⁻¹' (π '' U) = U | X Y Z : Type u
f g : X ⟶ Y
π : Y ⟶ Z
e : f ≫ π = g ≫ π
h : IsColimit (Cofork.ofπ π e)
U : Set Y
H : f ⁻¹' U = g ⁻¹' U
lem : ∀ (x y : Y), Function.Coequalizer.Rel f g x y → (x ∈ U ↔ y ∈ U)
eqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U
⊢ π ⁻¹' (π '' U) = U | ext | case h
X Y Z : Type u
f g : X ⟶ Y
π : Y ⟶ Z
e : f ≫ π = g ≫ π
h : IsColimit (Cofork.ofπ π e)
U : Set Y
H : f ⁻¹' U = g ⁻¹' U
lem : ∀ (x y : Y), Function.Coequalizer.Rel f g x y → (x ∈ U ↔ y ∈ U)
eqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U
x✝ : Y
⊢ x✝ ∈ π ⁻¹' (π '' U) ↔ x✝ ∈ U | 4cf815c00d783414 |
iteratedFDeriv_tsum | Mathlib/Analysis/Calculus/SmoothSeries.lean | theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
(hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
(h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
(hk : (k : ℕ∞) ≤ N) :
(iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k ... | α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 ... | simpa only [iteratedFDeriv_succ_eq_comp_left, LinearIsometryEquiv.norm_map, comp_apply]
using h'f k.succ n x hk | no goals | 1428749b8bc916e3 |
ENNReal.lintegral_prod_norm_pow_le | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem lintegral_prod_norm_pow_le {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
(s : Finset ι) {f : ι → α → ℝ≥0∞} (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
{p : ι → ℝ} (hp : ∑ i ∈ s, p i = 1) (h2p : ∀ i ∈ s, 0 ≤ p i) :
∫⁻ a, ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ a, f i a ∂μ) ^ p i | case insert.inr
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
(∀ i ∈ s, AEMeasurable (f i) μ) →
∀ {p : ι → ℝ},
∑ i ∈ s, p i = 1 →
(∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p ... | have h2pi₀ : 1 - p i₀ ≠ 0 := by
rwa [sub_ne_zero, ne_comm] | case insert.inr
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
(∀ i ∈ s, AEMeasurable (f i) μ) →
∀ {p : ι → ℝ},
∑ i ∈ s, p i = 1 →
(∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p ... | c6067aa6ebb0db88 |
Array.map_flatten | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem map_flatten (f : α → β) (L : Array (Array α)) :
(flatten L).map f = (map (map f) L).flatten | α : Type u_1
β : Type u_2
f : α → β
L : Array (Array α)
⊢ map f L.flatten = (map (map f) L).flatten | induction L using array₂_induction with
| of xss =>
simp only [flatten_toArray_map, List.map_toArray, List.map_flatten, List.map_map,
Function.comp_def]
rw [← Function.comp_def, ← List.map_map, flatten_toArray_map] | no goals | 9b8ec0763d48eb33 |
StrictMonoOn.exists_deriv_lt_slope | Mathlib/Analysis/Convex/Deriv.lean | theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | case neg.intro.intro.intro.intro.intro.intro.intro.intro.intro
x y : ℝ
f : ℝ → ℝ
hf : ContinuousOn f (Icc x y)
hxy : x < y
hf'_mono : StrictMonoOn (deriv f) (Ioo x y)
w : ℝ
hw : deriv f w = 0
hxw : x < w
hwy : w < y
a : ℝ
hxa : x < a
haw : a < w
b : ℝ
hwb : w < b
hby : b < y
ha : deriv f a * (w - x) < f w - f x
hb : de... | linarith | no goals | 824891cae2affaba |
setOf_liouvilleWith_subset_aux | Mathlib/NumberTheory/Transcendental/Liouville/Measure.lean | theorem setOf_liouvilleWith_subset_aux :
{ x : ℝ | ∃ p > 2, LiouvilleWith p x } ⊆
⋃ m : ℤ, (· + (m : ℝ)) ⁻¹' ⋃ n > (0 : ℕ),
{ x : ℝ | ∃ᶠ b : ℕ in atTop, ∃ a ∈ Finset.Icc (0 : ℤ) b,
|x - (a : ℤ) / b| < 1 / (b : ℝ) ^ (2 + 1 / n : ℝ) } | x p : ℝ
hp : p > 2
hxp : LiouvilleWith p x
n : ℕ
hn : 2 + 1 / (↑n + 1) < p
this :
∀ (y : ℝ),
LiouvilleWith p y →
y ∈ Ico 0 1 → ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |y - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))
⊢ x ∈
⋃ m,
(fun x => x + ↑m) ⁻¹'
⋃ n, ⋃ (_ : n > 0), {x | ∃ᶠ (b : ℕ) in atT... | simp only [mem_iUnion, mem_preimage] | x p : ℝ
hp : p > 2
hxp : LiouvilleWith p x
n : ℕ
hn : 2 + 1 / (↑n + 1) < p
this :
∀ (y : ℝ),
LiouvilleWith p y →
y ∈ Ico 0 1 → ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |y - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))
⊢ ∃ i i_1,
∃ (_ : i_1 > 0), x + ↑i ∈ {x | ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |x ... | 0ab923a0bda46abb |
MeasureTheory.FiniteMeasure.self_eq_mass_mul_normalize | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | theorem self_eq_mass_mul_normalize (s : Set Ω) : μ s = μ.mass * μ.normalize s | Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
s : Set Ω
⊢ μ s = μ.mass * μ.normalize s | obtain rfl | h := eq_or_ne μ 0 | case inl
Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
s : Set Ω
⊢ 0 s = mass 0 * (normalize 0) s
case inr
Ω : Type u_1
inst✝ : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
s : Set Ω
h : μ ≠ 0
⊢ μ s = μ.mass * μ.normalize s | c8d2346619166e24 |
Ordinal.mem_closure_tfae | Mathlib/SetTheory/Ordinal/Topology.lean | theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
TFAE [a ∈ closure s,
a ∈ closure (s ∩ Iic a),
(s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
(∀ x hx, f x hx ∈ s) ∧ b... | a : Ordinal.{u}
s : Set Ordinal.{u}
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a → ∃ t ⊆ s, t.Nonempty ∧ BddAbove t ∧ sSup t = a
tfae_4_to_5 :
(∃ t ⊆ s, t.Nonempty ∧ B... | tfae_have 5 → 6 := by
rintro ⟨o, h₀, f, hfs, rfl⟩
exact ⟨_, toType_nonempty_iff_ne_zero.2 h₀, familyOfBFamily o f, fun _ => hfs _ _, rfl⟩ | a : Ordinal.{u}
s : Set Ordinal.{u}
tfae_1_to_2 : a ∈ closure s → a ∈ closure (s ∩ Iic a)
tfae_2_to_3 : a ∈ closure (s ∩ Iic a) → (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a
tfae_3_to_4 : (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a → ∃ t ⊆ s, t.Nonempty ∧ BddAbove t ∧ sSup t = a
tfae_4_to_5 :
(∃ t ⊆ s, t.Nonempty ∧ B... | f361ec1995f661dc |
MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | theorem MeasureTheory.Measure.IsMulLeftInvariant.quotientMeasureEqMeasurePreimage_of_set {s : Set G}
(fund_dom_s : IsFundamentalDomain Γ.op s ν) {V : Set (G ⧸ Γ)}
(meas_V : MeasurableSet V) (neZeroV : μ V ≠ 0) (hV : μ V = ν (π ⁻¹' V ∩ s))
(neTopV : μ V ≠ ⊤) : QuotientMeasureEqMeasurePreimage ν μ | case h.convert_3
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
ν : Measure G
inst✝... | exact trans hV.symm neTopV | no goals | a1958c2e021a6ca0 |
Pell.eq_pell_lem | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem eq_pell_lem : ∀ (n) (b : ℤ√(d a1)), 1 ≤ b → IsPell b →
b ≤ pellZd a1 n → ∃ n, b = pellZd a1 n
| 0, _ => fun h1 _ hl => ⟨0, @Zsqrtd.le_antisymm _ (dnsq a1) _ _ hl h1⟩
| n + 1, b => fun h1 hp h =>
have a1p : (0 : ℤ√(d a1)) ≤ ⟨a, 1⟩ := trivial
have am1p : (0 : ℤ√(d a1)) ≤ ⟨a, -1⟩ := show (_ : Nat) ... | a : ℕ
a1 : 1 < a
n : ℕ
b : ℤ√↑(Pell.d a1)
h1 : 1 ≤ b
hp : IsPell b
h : b ≤ pellZd a1 (n + 1)
a1p : 0 ≤ { re := ↑a, im := 1 }
am1p : 0 ≤ { re := ↑a, im := -1 }
a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1
ha : { re := ↑a, im := 1 } ≤ b
m : ℕ
e : b * { re := ↑a, im := -1 } = pellZd a1 m
⊢ b = pellZd a1 (m + 1... | rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,
pellZd_succ, e] | no goals | f5b138704daee74a |
MeasureTheory.Measure.rnDeriv_lt_top | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem rnDeriv_lt_top (μ ν : Measure α) [SigmaFinite μ] : ∀ᵐ x ∂ν, μ.rnDeriv ν x < ∞ | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
n : ℕ
⊢ ∀ᵐ (x : α) ∂ν.restrict (spanningSets μ n), μ.rnDeriv ν x < ⊤ | apply ae_lt_top (measurable_rnDeriv _ _) | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
n : ℕ
⊢ ∫⁻ (x : α) in spanningSets μ n, μ.rnDeriv ν x ∂ν ≠ ⊤ | 1320a499e197f7dd |
HasFTaylorSeriesUpToOn.hasFDerivWithinAt | Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean | theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
(hx : x ∈ s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x | case h.e'_12.h.h.h.H
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : WithTop ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpToOn n f p s
h... | change (p x 1) (snoc 0 y) = (p x 1) (cons y v) | case h.e'_12.h.h.h.H
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : WithTop ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
h : HasFTaylorSeriesUpToOn n f p s
h... | 08ce69563124364e |
AkraBazziRecurrence.T_pos | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | @[aesop safe apply] lemma T_pos (n : ℕ) : 0 < T n | case ind.inr
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
n : ℕ
h_ind : ∀ m < n, 0 < T m
hn : R.n₀ ≤ n
⊢ 0 < ∑ i : α, a i * T (r i n) + g ↑n | have := R.g_nonneg | case ind.inr
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
n : ℕ
h_ind : ∀ m < n, 0 < T m
hn : R.n₀ ≤ n
this : ∀ x ≥ 0, 0 ≤ g x
⊢ 0 < ∑ i : α, a i * T (r i n) + g ↑n | 5c29d3478f140bd0 |
PrimeSpectrum.isOpen_singleton_tfae_of_isNoetherian_of_isJacobsonRing | Mathlib/RingTheory/Spectrum/Prime/Jacobson.lean | /--
If `R` is both noetherian and jacobson, then the following are equivalent for `x : Spec R`:
1. `{x}` is open (i.e. `x` is an isolated point)
2. `{x}` is clopen
3. `{x}` is both closed and stable under generalization
(i.e. `x` is both a minimal prime and a maximal ideal)
-/
lemma isOpen_singleton_tfae_of_isNoether... | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
tfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}
h₁ : IsMax x
h₂ : StableUnderGeneralization {x}
⊢ IsOpen {x} | suffices {x} = (⋃ p ∈ { p : PrimeSpectrum R | IsMin p ∧ p ≠ x }, closure {p})ᶜ by
rw [this, isOpen_compl_iff]
refine Set.Finite.isClosed_biUnion ?_ (fun _ _ ↦ isClosed_closure)
exact (finite_setOf_isMin R).subset fun x h ↦ h.1 | R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsNoetherianRing R
inst✝ : IsJacobsonRing R
x : PrimeSpectrum R
tfae_1_to_2 : IsOpen {x} → IsClopen {x}
tfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}
h₁ : IsMax x
h₂ : StableUnderGeneralization {x}
⊢ {x} = (⋃ p ∈ {p | IsMin p ∧ p ≠ x}, closure {p})ᶜ | 2ced83ded0998596 |
IsCompact.isLindelof | Mathlib/Topology/Compactness/Lindelof.lean | theorem IsCompact.isLindelof (hs : IsCompact s) :
IsLindelof s | X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
⊢ IsLindelof s | tauto | no goals | 2825f02c9b3a2dbb |
Sigma.nhds_eq | Mathlib/Topology/Constructions.lean | theorem Sigma.nhds_eq (x : Sigma σ) : 𝓝 x = Filter.map (Sigma.mk x.1) (𝓝 x.2) | case mk
ι : Type u_5
σ : ι → Type u_7
inst✝ : (i : ι) → TopologicalSpace (σ i)
fst✝ : ι
snd✝ : σ fst✝
⊢ 𝓝 ⟨fst✝, snd✝⟩ = Filter.map (mk ⟨fst✝, snd✝⟩.fst) (𝓝 ⟨fst✝, snd✝⟩.snd) | apply Sigma.nhds_mk | no goals | daa20c54c5de9dcb |
Set.union_pi_inter | Mathlib/Data/Set/Prod.lean | theorem union_pi_inter
(ht₁ : ∀ i ∉ s₁, t₁ i = univ) (ht₂ : ∀ i ∉ s₂, t₂ i = univ) :
(s₁ ∪ s₂).pi (fun i ↦ t₁ i ∩ t₂ i) = s₁.pi t₁ ∩ s₂.pi t₂ | case h
ι : Type u_1
α : ι → Type u_2
s₁ s₂ : Set ι
t₁ t₂ : (i : ι) → Set (α i)
ht₁ : ∀ i ∉ s₁, t₁ i = univ
ht₂ : ∀ i ∉ s₂, t₂ i = univ
x : (i : ι) → α i
⊢ (∀ (i : ι), i ∈ s₁ ∨ i ∈ s₂ → x i ∈ t₁ i ∧ x i ∈ t₂ i) ↔ (∀ i ∈ s₁, x i ∈ t₁ i) ∧ ∀ i ∈ s₂, x i ∈ t₂ i | refine ⟨fun h ↦ ⟨fun i his₁ ↦ (h i (Or.inl his₁)).1, fun i his₂ ↦ (h i (Or.inr his₂)).2⟩,
fun h i hi ↦ ?_⟩ | case h
ι : Type u_1
α : ι → Type u_2
s₁ s₂ : Set ι
t₁ t₂ : (i : ι) → Set (α i)
ht₁ : ∀ i ∉ s₁, t₁ i = univ
ht₂ : ∀ i ∉ s₂, t₂ i = univ
x : (i : ι) → α i
h : (∀ i ∈ s₁, x i ∈ t₁ i) ∧ ∀ i ∈ s₂, x i ∈ t₂ i
i : ι
hi : i ∈ s₁ ∨ i ∈ s₂
⊢ x i ∈ t₁ i ∧ x i ∈ t₂ i | c857e90e721d68e9 |
integral_cpow | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_cpow {r : ℂ} (h : -1 < r.re ∨ r ≠ -1 ∧ (0 : ℝ) ∉ [[a, b]]) :
(∫ x : ℝ in a..b, (x : ℂ) ^ r) = ((b : ℂ) ^ (r + 1) - (a : ℂ) ^ (r + 1)) / (r + 1) | a b : ℝ
r : ℂ
h : -1 < r.re ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ r + 1 ≠ 0 | rcases h with h | h | case inl
a b : ℝ
r : ℂ
h : -1 < r.re
⊢ r + 1 ≠ 0
case inr
a b : ℝ
r : ℂ
h : r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ r + 1 ≠ 0 | 6570340d40c90ca3 |
SimpleGraph.Walk.append_nil | Mathlib/Combinatorics/SimpleGraph/Walk.lean | theorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p | V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
⊢ p.append nil = p | induction p with
| nil => rw [nil_append]
| cons _ _ ih => rw [cons_append, ih] | no goals | ca6be4bee7478dfc |
MeasureTheory.aemeasurable_fderivWithin | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem aemeasurable_fderivWithin (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) : AEMeasurable f' (μ.restrict s) | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
ε : ℝ
εpos : ε > 0
δ : ℝ≥... | exact ae_mono this H | no goals | ee4b6aa7dbac5ef5 |
Submodule.isClosed_or_dense_of_isCoatom | Mathlib/Topology/Algebra/Module/Basic.lean | theorem Submodule.isClosed_or_dense_of_isCoatom (s : Submodule R M) (hs : IsCoatom s) :
IsClosed (s : Set M) ∨ Dense (s : Set M) | R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : ContinuousConstSMul R M
inst✝ : ContinuousAdd M
s : Submodule R M
hs : IsCoatom s
⊢ IsClosed ↑s ∨ Dense ↑s | refine (hs.le_iff.mp s.le_topologicalClosure).symm.imp ?_ dense_iff_topologicalClosure_eq_top.mpr | R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : ContinuousConstSMul R M
inst✝ : ContinuousAdd M
s : Submodule R M
hs : IsCoatom s
⊢ s.topologicalClosure = s → IsClosed ↑s | 4b422b561aabb861 |
tendsto_integral_comp_smul_smul_of_integrable | Mathlib/MeasureTheory/Integral/PeakFunction.lean | theorem tendsto_integral_comp_smul_smul_of_integrable
{φ : F → ℝ} (hφ : ∀ x, 0 ≤ φ x) (h'φ : ∫ x, φ x ∂μ = 1)
(h : Tendsto (fun x ↦ ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0))
{g : F → E} (hg : Integrable g μ) (h'g : ContinuousAt g 0) :
Tendsto (fun (c : ℝ) ↦ ∫ x, (c ^ (finrank ℝ F) * φ (c • x)) • g ... | E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x : F), 0 ≤ φ x
h'φ ... | obtain ⟨δ, δpos, h'u⟩ : ∃ δ > 0, ball 0 δ ⊆ u := Metric.isOpen_iff.1 u_open _ hu | case intro.intro
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : CompleteSpace E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
μ : Measure F
inst✝ : μ.IsAddHaarMeasure
φ : F → ℝ
hφ : ∀ (x :... | 06046c21dacd4c17 |
MeasurableEmbedding.eLpNorm_map_measure | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | theorem _root_.MeasurableEmbedding.eLpNorm_map_measure {g : β → F} (hf : MeasurableEmbedding f) :
eLpNorm g p (Measure.map f μ) = eLpNorm (g ∘ f) p μ | case pos
α : Type u_1
F : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup F
β : Type u_6
mβ : MeasurableSpace β
f : α → β
g : β → F
hf : MeasurableEmbedding f
hp_zero : ¬p = 0
hp : p = ⊤
⊢ eLpNormEssSup g (Measure.map f μ) = eLpNormEssSup (g ∘ f) μ | exact hf.essSup_map_measure | no goals | cb6bd843dd6b53b1 |
InformationTheory.klDiv_eq_top_iff | Mathlib/InformationTheory/KullbackLeibler/Basic.lean | lemma klDiv_eq_top_iff : klDiv μ ν = ∞ ↔ μ ≪ ν → ¬ Integrable (llr μ ν) μ | case mp
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h : klDiv μ ν = ⊤
⊢ μ ≪ ν → ¬Integrable (llr μ ν) μ | contrapose! h | case mp
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
h : μ ≪ ν ∧ Integrable (llr μ ν) μ
⊢ klDiv μ ν ≠ ⊤ | ea0b9b82484805cd |
FiniteDimensional.of_isCompact_closedBall₀ | Mathlib/Analysis/Normed/Module/FiniteDimension.lean | theorem FiniteDimensional.of_isCompact_closedBall₀ {r : ℝ} (rpos : 0 < r)
(h : IsCompact (Metric.closedBall (0 : E) r)) : FiniteDimensional 𝕜 E | case intro.intro.intro.intro
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
r : ℝ
rpos : 0 < r
h : IsCompact (closedBall 0 r)
hfin : ¬FiniteDimensional 𝕜 E
R : ℝ
f : ℕ → E
Rgt : 1 < R
fle : ∀ (n : ℕ), ‖f n‖ ≤ R
lef : Pairwise ... | have rRpos : 0 < r / R := div_pos rpos (zero_lt_one.trans Rgt) | case intro.intro.intro.intro
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
r : ℝ
rpos : 0 < r
h : IsCompact (closedBall 0 r)
hfin : ¬FiniteDimensional 𝕜 E
R : ℝ
f : ℕ → E
Rgt : 1 < R
fle : ∀ (n : ℕ), ‖f n‖ ≤ R
lef : Pairwise ... | c91708e1e8291456 |
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))
y : ↑↑(pullback (D.f i j) (D.f i k))
⊢ (ConcreteCategory.hom (inv (D.t' k i j) ≫ pullback.fst (D.f k i) (D.f k j)).base) y =
(ConcreteCategory.hom (pullback.snd (D.f i j) (D.f i k) ≫ D.t i k).base) y | congr 3 | case h.e_a.e_a.e_self
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))
y : ↑↑(pullback (D.f i j) (D.f i k))
⊢ inv (D.t' k i j) ≫ pullback.fst (D.f k i) (D.f k j) = pullback.snd (D.f i j) (D.f i k) ≫ D.t i k | e08cb0129b520e6d |
soln_unique | Mathlib/NumberTheory/Padics/Hensel.lean | theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
(hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
have soln_dist : ‖z - soln‖ < ‖F.derivative.eval a‖ :=
calc
‖z - soln‖ = ‖z - a + (a - soln)‖ | p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
z : ℤ_[p]
hev : Polynomial.eval z F = 0
hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
soln_dist : ‖z - soln‖ < ‖Polynomial.ev... | apply soln_dist | no goals | 9538cf328c481f11 |
Finset.sup'_const | Mathlib/Data/Finset/Lattice/Fold.lean | theorem sup'_const (a : α) : s.sup' H (fun _ => a) = a | case a
α : Type u_2
β : Type u_3
inst✝ : SemilatticeSup α
s : Finset β
H : s.Nonempty
a : α
⊢ a ≤ s.sup' H fun x => a | apply le_sup' (fun _ => a) H.choose_spec | no goals | 19a233cc520bfbb7 |
FirstOrder.Language.Substructure.fg_iff_structure_fg | Mathlib/ModelTheory/FinitelyGenerated.lean | theorem Substructure.fg_iff_structure_fg (S : L.Substructure M) : S.FG ↔ Structure.FG L S | case refine_2
L : Language
M : Type u_1
inst✝ : L.Structure M
S : L.Substructure M
h : ⊤.FG
⊢ S.FG | have h := h.map S.subtype.toHom | case refine_2
L : Language
M : Type u_1
inst✝ : L.Structure M
S : L.Substructure M
h✝ : ⊤.FG
h : (map S.subtype.toHom ⊤).FG
⊢ S.FG | 5f0b8ca0a2c04578 |
Algebra.Generators.map_toComp_ker | Mathlib/RingTheory/Generators.lean | lemma map_toComp_ker (Q : Generators S T) (P : Generators R S) :
P.ker.map (Q.toComp P).toAlgHom = RingHom.ker (Q.ofComp P).toAlgHom | case a.convert_4.a
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
this✝ : DecidableEq (Q.vars →₀ ℕ) := Classical.decEq (Q.vars →₀ ℕ)
x : (Q.com... | rw [Finset.sum_filter, ← finsum_eq_sum_of_support_subset _ (this x)] | case a.convert_4.a
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
this✝ : DecidableEq (Q.vars →₀ ℕ) := Classical.decEq (Q.vars →₀ ℕ)
x : (Q.com... | 5cc741e4fe814640 |
List.mapM'_eq_mapM | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem mapM'_eq_mapM [Monad m] [LawfulMonad m] (f : α → m β) (l : List α) :
mapM' f l = mapM f l | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
l✝ : List α
a : α
l : List α
acc : List β
⊢ mapM.loop f (a :: l) acc = do
let __do_lift ← mapM' f (a :: l)
pure (acc.reverse ++ __do_lift) | simp [go l, mapM.loop, mapM'] | no goals | d8bbb1f21029aff1 |
MulActionHom.comp_inverse' | Mathlib/GroupTheory/GroupAction/Hom.lean | theorem comp_inverse' {f : X →ₑ[φ] Y} {g : Y → X}
{k₁ : Function.LeftInverse φ' φ} {k₂ : Function.RightInverse φ' φ}
{h₁ : Function.LeftInverse g f} {h₂ : Function.RightInverse g f} :
(inverse' f g k₂ h₁ h₂).comp f (κ := CompTriple.comp_inv k₁)
= MulActionHom.id M | M : Type u_2
N : Type u_3
φ : M → N
X : Type u_5
inst✝¹ : SMul M X
Y : Type u_6
inst✝ : SMul N Y
φ' : N → M
f : X →ₑ[φ] Y
g : Y → X
k₁ : Function.LeftInverse φ' φ
k₂ : Function.RightInverse φ' φ
h₁ : Function.LeftInverse g ⇑f
h₂ : Function.RightInverse g ⇑f
x : X
⊢ ((f.inverse' g k₂ h₁ h₂).comp f) x = (MulActionHom.id ... | simp only [comp_apply, inverse_apply, id_apply] | M : Type u_2
N : Type u_3
φ : M → N
X : Type u_5
inst✝¹ : SMul M X
Y : Type u_6
inst✝ : SMul N Y
φ' : N → M
f : X →ₑ[φ] Y
g : Y → X
k₁ : Function.LeftInverse φ' φ
k₂ : Function.RightInverse φ' φ
h₁ : Function.LeftInverse g ⇑f
h₂ : Function.RightInverse g ⇑f
x : X
⊢ (f.inverse' g k₂ h₁ h₂) (f x) = x | 2b5eb2627520d5c3 |
SetTheory.PGame.add_le_add_right' | Mathlib/SetTheory/Game/PGame.lean | theorem add_le_add_right' : ∀ {x y z : PGame}, x ≤ y → x + z ≤ y + z
| mk xl xr xL xR, mk yl yr yL yR, mk zl zr zL zR => fun h => by
refine le_def.2 ⟨fun i => ?_, fun i => ?_⟩ <;> obtain i | i := i
· rw [le_def] at h
obtain ⟨h_left, h_right⟩ := h
rcases h_left i with (⟨i', ih⟩ | ⟨j, jh⟩)
· e... | case refine_2.inl
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
h :
(∀ (i : (mk xl xr xL xR).LeftMoves),
(∃ i', (mk xl xr xL xR).moveLeft i ≤ (mk yl yr yL yR).moveLeft i') ∨
∃ j, ((mk xl xr xL xR).moveLe... | rcases h.right i with (⟨i, ih⟩ | ⟨j', jh⟩) | case refine_2.inl.inl.intro
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
h :
(∀ (i : (mk xl xr xL xR).LeftMoves),
(∃ i', (mk xl xr xL xR).moveLeft i ≤ (mk yl yr yL yR).moveLeft i') ∨
∃ j, ((mk xl xr xL ... | 21f7612be5d148f4 |
FractionalIdeal.mul_inv_cancel_of_le_one | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem mul_inv_cancel_of_le_one [h : IsDedekindDomain A] {I : Ideal A} (hI0 : I ≠ ⊥)
(hI : (I * (I : FractionalIdeal A⁰ K)⁻¹)⁻¹ ≤ 1) : I * (I : FractionalIdeal A⁰ K)⁻¹ = 1 | case pos
A : Type u_2
K : Type u_3
inst✝³ : CommRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hJ : ↑⊥ = ↑I * (↑I)⁻¹
⊢ I = ⊥ | rw [eq_bot_iff, ← coeIdeal_le_coeIdeal K, hJ] | case pos
A : Type u_2
K : Type u_3
inst✝³ : CommRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : IsDedekindDomain A
I : Ideal A
hI0 : I ≠ ⊥
hI : (↑I * (↑I)⁻¹)⁻¹ ≤ 1
hJ : ↑⊥ = ↑I * (↑I)⁻¹
⊢ ↑I ≤ ↑I * (↑I)⁻¹ | 59c657b0d051c4ed |
exists_countable_union_perfect_of_isClosed | Mathlib/Topology/Perfect.lean | theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
(hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D | case intro.intro.intro
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : b.Countable
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ (U ∩ C).Countable}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : (V ∩ C).Count... | refine ⟨V ∩ C, D, Vct, ⟨?_, ?_⟩, ?_⟩ | case intro.intro.intro.refine_1
α : Type u_1
inst✝¹ : TopologicalSpace α
C : Set α
inst✝ : SecondCountableTopology α
hclosed : IsClosed C
b : Set (Set α)
bct : b.Countable
left✝ : ∅ ∉ b
bbasis : IsTopologicalBasis b
v : Set (Set α) := {U | U ∈ b ∧ (U ∩ C).Countable}
V : Set α := ⋃ U ∈ v, U
D : Set α := C \ V
Vct : (V ∩... | f45aad82d45e9e73 |
Int.lt_floor_iff | Mathlib/Algebra/Order/Floor.lean | theorem lt_floor_iff : z < ⌊a⌋ ↔ z + 1 ≤ a | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a : α
⊢ z < ⌊a⌋ ↔ ↑z + 1 ≤ a | rw [← add_one_le_iff, le_floor] | α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a : α
⊢ ↑(z + 1) ≤ a ↔ ↑z + 1 ≤ a | 956616311ceb77ab |
Hindman.exists_FP_of_large | Mathlib/Combinatorics/Hindman.lean | theorem exists_FP_of_large {M} [Semigroup M] (U : Ultrafilter M) (U_idem : U * U = U) (s₀ : Set M)
(sU : s₀ ∈ U) : ∃ a, FP a ⊆ s₀ | M : Type u_1
inst✝ : Semigroup M
U : Ultrafilter M
U_idem : U * U = U
s₀ : Set M
sU : s₀ ∈ U
exists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty
⊢ ∃ a, FP a ⊆ s₀ | let elem : { s // s ∈ U } → M := fun p => (exists_elem p.property).some | M : Type u_1
inst✝ : Semigroup M
U : Ultrafilter M
U_idem : U * U = U
s₀ : Set M
sU : s₀ ∈ U
exists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty
elem : { s // s ∈ U } → M := fun p => ⋯.some
⊢ ∃ a, FP a ⊆ s₀ | 8a116562f2fd2b3f |
Vitali.exists_disjoint_covering_ae | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈... | case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r... | have ut : u ⊆ t := fun a hau => (ut' hau).1 | case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r... | 4466d2fc3e80cb3f |
String.foldrAux_of_valid | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem foldrAux_of_valid (f : Char → α → α) (l m r a) :
foldrAux f a ⟨l ++ m ++ r⟩ ⟨utf8Len l + utf8Len m⟩ ⟨utf8Len l⟩ = m.foldr f a | case nil
α : Type u_1
f : Char → α → α
l m r : List Char
a : α
⊢ foldrAux f a { data := l ++ [].reverse ++ r } { byteIdx := utf8Len l + utf8Len [].reverse } { byteIdx := utf8Len l } =
List.foldr f a [].reverse | unfold foldrAux | case nil
α : Type u_1
f : Char → α → α
l m r : List Char
a : α
⊢ (if h : { byteIdx := utf8Len l } < { byteIdx := utf8Len l + utf8Len [].reverse } then
let_fun this := ⋯;
let i := { data := l ++ [].reverse ++ r }.prev { byteIdx := utf8Len l + utf8Len [].reverse };
let a := f ({ data := l ++ [].reverse ... | 5d4fc670d1273631 |
tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto | Mathlib/MeasureTheory/Integral/PeakFunction.lean | theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
(h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (f... | case h
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : MeasurableSe... | simp only [h, Pi.sub_apply, smul_sub, ← indicator_smul_apply] | case h
α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
s : Set α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
hs : MeasurableSet s
t : Set α
ht : MeasurableSe... | fcf9a03d4108054c |
Module.map_jacobson_of_ker_le | Mathlib/RingTheory/Jacobson/Radical.lean | theorem map_jacobson_of_ker_le (surj : Function.Surjective f)
(le : LinearMap.ker f ≤ jacobson R M) :
map f (jacobson R M) = jacobson R₂ M₂ :=
le_antisymm (map_jacobson_le f) <| by
rw [jacobson, sInf_eq_iInf'] at le
conv_rhs => rw [jacobson, sInf_eq_iInf', map_iInf_of_ker_le surj le]
exact le_iInf... | R : Type u_1
R₂ : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝⁸ : Ring R
inst✝⁷ : Ring R₂
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝² : RingHomSurjective τ₁₂
F : Type u_5
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F τ₁₂ M M₂
f : F
surj : Function... | exact le_iInf fun m ↦ sInf_le (isCoatom_map_of_ker_le surj (le_iInf_iff.mp le m) m.2) | no goals | 12c07be335000e33 |
List.lt_of_le_of_lt | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean | theorem lt_of_le_of_lt [DecidableEq α] [LT α] [DecidableLT α]
[i₀ : Std.Irrefl (· < · : α → α → Prop)]
[i₁ : Std.Asymm (· < · : α → α → Prop)]
[i₂ : Std.Antisymm (¬ · < · : α → α → Prop)]
[i₃ : Trans (¬ · < · : α → α → Prop) (¬ · < ·) (¬ · < ·)]
{l₁ l₂ l₃ : List α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ < l₃) : l₁... | case pos
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : LT α
inst✝ : DecidableLT α
i₀ : Std.Irrefl fun x1 x2 => x1 < x2
i₁ : Std.Asymm fun x1 x2 => x1 < x2
i₂ : Std.Antisymm fun x1 x2 => ¬x1 < x2
i₃ : Trans (fun x1 x2 => ¬x1 < x2) (fun x1 x2 => ¬x1 < x2) fun x1 x2 => ¬x1 < x2
l₂ l₃ : List α
a : α
as bs : List α
w₃ : Lex ... | exact Lex.cons (ih (le_of_cons_le_cons h₁)) | no goals | 1ae657684e05c692 |
Bimod.LeftUnitorBimod.hom_left_act_hom' | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem hom_left_act_hom' :
((regular R).tensorBimod P).actLeft ≫ hom P = (R.X ◁ hom P) ≫ P.actLeft | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ ((α_ R.X R.X P.X).inv ≫ R.mul ▷... | slice_lhs 2 3 => rw [left_assoc] | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
⊢ (α_ R.X R.X P.X).inv ≫ (α_ R.X ... | 1a8d528bed358917 |
abs_eq_iff_mul_self_eq | Mathlib/Algebra/Order/Ring/Abs.lean | lemma abs_eq_iff_mul_self_eq : |a| = |b| ↔ a * a = b * b | α : Type u_1
inst✝ : LinearOrderedRing α
a b : α
⊢ |a| = |b| ↔ |a| * |a| = |b| * |b| | exact (mul_self_inj (abs_nonneg a) (abs_nonneg b)).symm | no goals | 708a5883b44c61f2 |
Equiv.Perm.mem_cycleFactorsFinset_support_le | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem mem_cycleFactorsFinset_support_le {p f : Perm α} (h : p ∈ cycleFactorsFinset f) :
p.support ≤ f.support | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
p f : Perm α
h : p.IsCycle ∧ ∀ a ∈ p.support, p a = f a
x : α
hx : x ∈ p.support
⊢ x ∈ f.support | rwa [mem_support, ← h.right x hx, ← mem_support] | no goals | 203063167af9775b |
List.isInfix_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Sublist.lean | theorem isInfix_iff : l₁ <:+: l₂ ↔
∃ k, l₁.length + k ≤ l₂.length ∧ ∀ i (h : i < l₁.length), l₂[i + k]? = some l₁[i] | case mpr
α✝ : Type u_1
l₁ l₂ : List α✝
⊢ (∃ k, l₁.length + k ≤ l₂.length ∧ ∀ (i : Nat) (h : i < l₁.length), l₂[i + k]? = some l₁[i]) → l₁ <:+: l₂ | rintro ⟨k, le, w⟩ | case mpr.intro.intro
α✝ : Type u_1
l₁ l₂ : List α✝
k : Nat
le : l₁.length + k ≤ l₂.length
w : ∀ (i : Nat) (h : i < l₁.length), l₂[i + k]? = some l₁[i]
⊢ l₁ <:+: l₂ | 8c836fc6d0b14b98 |
CochainComplex.mappingCone.inr_descCochain | Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | @[simp]
lemma inr_descCochain :
(Cochain.ofHom (inr φ)).comp (descCochain φ α β h) (zero_add n) = β | C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
⊢ (Cochain.ofHom (inr φ)).comp (descCochain φ α β h) ⋯ = β | simp [descCochain] | no goals | c8fce629a3043f89 |
FiniteDimensional.mem_span_of_iInf_ker_le_ker | Mathlib/LinearAlgebra/Dual.lean | theorem _root_.FiniteDimensional.mem_span_of_iInf_ker_le_ker [FiniteDimensional 𝕜 E]
{L : ι → E →ₗ[𝕜] 𝕜} {K : E →ₗ[𝕜] 𝕜}
(h : ⨅ i, LinearMap.ker (L i) ≤ ker K) : K ∈ span 𝕜 (range L) | ι : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝³ : Field 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
L : ι → E →ₗ[𝕜] 𝕜
K : E →ₗ[𝕜] 𝕜
h : ⨅ i, ker (L i) ≤ ker K
hK : K ∉ span 𝕜 (Set.range L)
φ : Dual 𝕜 (E →ₗ[𝕜] 𝕜)
φne : φ K ≠ 0
hφ : map φ (span 𝕜 (Set.range L)) = ⊥
φs : E := (ev... | exact ⟨L i, Submodule.subset_span ⟨i, rfl⟩, (apply_evalEquiv_symm_apply 𝕜 E _ φ).symm⟩ | no goals | 25255e000b48b4e5 |
Set.Finite.isCompact_biUnion | Mathlib/Topology/Compactness/Compact.lean | theorem Set.Finite.isCompact_biUnion {s : Set ι} {f : ι → Set X} (hs : s.Finite)
(hf : ∀ i ∈ s, IsCompact (f i)) : IsCompact (⋃ i ∈ s, f i) :=
isCompact_iff_ultrafilter_le_nhds'.2 fun l hl => by
rw [Ultrafilter.finite_biUnion_mem_iff hs] at hl
rcases hl with ⟨i, his, hi⟩
rcases (hf i his).ultrafilter_... | case intro.intro
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set ι
f : ι → Set X
hs : s.Finite
hf : ∀ i ∈ s, IsCompact (f i)
l : Ultrafilter X
i : ι
his : i ∈ s
hi : f i ∈ l
⊢ ∃ x ∈ ⋃ i ∈ s, f i, ↑l ≤ 𝓝 x | rcases (hf i his).ultrafilter_le_nhds _ (le_principal_iff.2 hi) with ⟨x, hxi, hlx⟩ | case intro.intro.intro.intro
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
s : Set ι
f : ι → Set X
hs : s.Finite
hf : ∀ i ∈ s, IsCompact (f i)
l : Ultrafilter X
i : ι
his : i ∈ s
hi : f i ∈ l
x : X
hxi : x ∈ f i
hlx : ↑l ≤ 𝓝 x
⊢ ∃ x ∈ ⋃ i ∈ s, f i, ↑l ≤ 𝓝 x | 314e5eedf1f80172 |
RatFunc.intDegree_add_le | Mathlib/FieldTheory/RatFunc/Degree.lean | theorem intDegree_add_le {x y : RatFunc K} (hy : y ≠ 0) (hxy : x + y ≠ 0) :
intDegree (x + y) ≤ max (intDegree x) (intDegree y) | case pos
K : Type u
inst✝ : Field K
x y : RatFunc K
hy : y ≠ 0
hx : x = 0
hxy : ¬y = 0
⊢ (x + y).intDegree ≤ x.intDegree ⊔ y.intDegree | simp [hx, hxy] | no goals | df522905dbc696e6 |
AnalyticAt.order_eq_zero_iff | Mathlib/Analysis/Analytic/Order.lean | /-- The order of an analytic function `f` at `z₀` is zero iff `f` does not vanish at `z₀`. -/
lemma order_eq_zero_iff (hf : AnalyticAt 𝕜 f z₀) :
hf.order = 0 ↔ f z₀ ≠ 0 | case mp
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
g : 𝕜 → E
left✝¹ : AnalyticAt 𝕜 g z₀
left✝ : g z₀ ≠ 0
hg : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ 0 • g z
⊢ f z₀ ≠ 0 | simpa [hg.self_of_nhds] | no goals | f89c330d4911bc56 |
Algebra.Extension.Hom.comp_id | Mathlib/RingTheory/Extension.lean | @[simp]
lemma Hom.comp_id (f : Hom P P') : f.comp (Hom.id P) = f | R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
P : Extension R S
R' : Type u_1
S' : Type u_2
inst✝⁴ : CommRing R'
inst✝³ : CommRing S'
inst✝² : Algebra R' S'
P' : Extension R' S'
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
f : P.Hom P'
⊢ f.comp (Hom.id P) = f | ext | case toRingHom.a
R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
P : Extension R S
R' : Type u_1
S' : Type u_2
inst✝⁴ : CommRing R'
inst✝³ : CommRing S'
inst✝² : Algebra R' S'
P' : Extension R' S'
inst✝¹ : Algebra R R'
inst✝ : Algebra S S'
f : P.Hom P'
x✝ : P.Ring
⊢ (f.comp (Hom.id P)... | 576140ccab98f1be |
CategoryTheory.Limits.isIso_of_source_target_iso_zero | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | theorem isIso_of_source_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) (j : Y ≅ 0) :
IsIso f | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X Y : C
f : X ⟶ Y
i : X ≅ 0
j : Y ≅ 0
⊢ IsIso f | rw [zero_of_source_iso_zero f i] | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
X Y : C
f : X ⟶ Y
i : X ≅ 0
j : Y ≅ 0
⊢ IsIso 0 | a5aa73cbe5291376 |
Filter.le_comap_top | Mathlib/Order/Filter/Map.lean | theorem le_comap_top (f : α → β) (l : Filter α) : l ≤ comap f ⊤ | α : Type u_1
β : Type u_2
f : α → β
l : Filter α
⊢ l ≤ ⊤ | exact le_top | no goals | f174d317256c6beb |
ContDiffAt.eventually | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem ContDiffAt.eventually (h : ContDiffAt 𝕜 n f x) (h' : n ≠ ∞) :
∀ᶠ y in 𝓝 x, ContDiffAt 𝕜 n f y | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : WithTop ℕ∞
h : ContDiffAt 𝕜 n f x
h' : n ≠ ∞
⊢ ∀ᶠ (y : E) in 𝓝 x, ContDiffAt 𝕜 n f y | simpa [nhdsWithin_univ] using ContDiffWithinAt.eventually h h' | no goals | b27b5906a80f601a |
IsNilpotent.exp_of_nilpotent_is_unit | Mathlib/RingTheory/Nilpotent/Exp.lean | theorem exp_of_nilpotent_is_unit {a : A} (h : IsNilpotent a) : IsUnit (exp a) | A : Type u_1
inst✝¹ : Ring A
inst✝ : Algebra ℚ A
a : A
h : IsNilpotent a
h₁ : Commute a (-a)
h₂ : IsNilpotent (-a)
h₃ : 1 = exp a * exp (-a)
⊢ exp (-a) * exp a = 1 | rw [← exp_add_of_commute h₁.symm h₂ h, neg_add_cancel a, exp_zero_eq_one] | no goals | ecb00174aeba1950 |
linearEquiv_det_rotation | Mathlib/Analysis/Complex/Isometry.lean | theorem linearEquiv_det_rotation (a : Circle) : LinearEquiv.det (rotation a).toLinearEquiv = 1 | a : Circle
⊢ LinearEquiv.det (rotation a).toLinearEquiv = 1 | rw [← Units.eq_iff, LinearEquiv.coe_det, det_rotation, Units.val_one] | no goals | 28ee5d0195ffd788 |
CategoryTheory.ShortComplex.HasRightHomology.hasKernel | Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | lemma hasKernel [S.HasRightHomology] [HasCokernel S.f] :
HasKernel (cokernel.desc S.f S.g S.zero) | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
S : ShortComplex C
inst✝¹ : S.HasRightHomology
inst✝ : HasCokernel S.f
h : S.RightHomologyData := S.rightHomologyData
⊢ HasKernel (cokernel.desc S.f S.g ⋯) | haveI : HasLimit (parallelPair h.g' 0) := ⟨⟨⟨_, h.hι'⟩⟩⟩ | C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
S : ShortComplex C
inst✝¹ : S.HasRightHomology
inst✝ : HasCokernel S.f
h : S.RightHomologyData := S.rightHomologyData
this : HasLimit (parallelPair h.g' 0)
⊢ HasKernel (cokernel.desc S.f S.g ⋯) | 0960e28a5330c4e9 |
Int.bmod_zero | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem bmod_zero : Int.bmod 0 m = 0 | m : Nat
⊢ (if 0 < (↑m + 1) / 2 then 0 else 0 - ↑m) = 0 | simp only [Int.zero_sub, ite_eq_left_iff, Int.neg_eq_zero] | m : Nat
⊢ ¬0 < (↑m + 1) / 2 → ↑m = 0 | aafdd0ab2162fca6 |
Nat.div_div_div_eq_div | Mathlib/Data/Nat/Init.lean | @[simp] lemma div_div_div_eq_div (dvd : b ∣ a) (dvd2 : a ∣ c) : c / (a / b) / b = c / a :=
match a, b, c with
| 0, _, _ => by simp
| a + 1, 0, _ => by simp at dvd
| a + 1, c + 1, _ => by
have a_split : a + 1 ≠ 0 := succ_ne_zero a
have c_split : c + 1 ≠ 0 := succ_ne_zero c
rcases dvd2 with ⟨k, rfl⟩
... | a✝ b c a x✝ : ℕ
dvd : 0 ∣ a + 1
dvd2 : a + 1 ∣ x✝
⊢ x✝ / ((a + 1) / 0) / 0 = x✝ / (a + 1) | simp at dvd | no goals | eab95b5d7fe19e6b |
Function.Surjective.lieModule_lcs_map_eq | Mathlib/Algebra/Lie/Nilpotent.lean | theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
(lowerCentralSeries R L M k : Submodule R M).map g = lowerCentralSeries R L₂ M₂ k | case succ.intro.intro.intro.intro
R : Type u
L : Type v
M : Type w
inst✝¹² : CommRing R
inst✝¹¹ : LieRing L
inst✝¹⁰ : LieAlgebra R L
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
inst✝⁷ : LieRingModule L M
inst✝⁶ : LieModule R L M
L₂ : Type u_1
M₂ : Type u_2
inst✝⁵ : LieRing L₂
inst✝⁴ : LieAlgebra R L₂
inst✝³ : AddCommGr... | exact ⟨⁅y, n⁆, ⟨y, n, hn, rfl⟩, (hfg y n).symm⟩ | no goals | 027902f382a3a0a8 |
Metric.closure_eq_iInter_cthickening | Mathlib/Topology/MetricSpace/Thickening.lean | theorem closure_eq_iInter_cthickening (E : Set α) :
closure E = ⋂ (δ : ℝ) (_ : 0 < δ), cthickening δ E | α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
⊢ closure E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E | rw [← cthickening_zero] | α : Type u
inst✝ : PseudoEMetricSpace α
E : Set α
⊢ cthickening 0 E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E | c1e605eb0b1be19a |
gauge_eq_zero | Mathlib/Analysis/Convex/Gauge.lean | theorem gauge_eq_zero (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
gauge s x = 0 ↔ x = 0 | E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : T1Space E
hs : Absorbent ℝ s
hb : Bornology.IsVonNBounded ℝ s
h₀ : gauge s x = 0
hne : x ≠ 0
⊢ False | have : {x}ᶜ ∈ comap (gauge s) (𝓝 0) :=
comap_gauge_nhds_zero_le hs hb (isOpen_compl_singleton.mem_nhds hne.symm) | E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : T1Space E
hs : Absorbent ℝ s
hb : Bornology.IsVonNBounded ℝ s
h₀ : gauge s x = 0
hne : x ≠ 0
this : {x}ᶜ ∈ comap (gauge s) (𝓝 0)
⊢ False | 51b46a70cefabb44 |
MeasureTheory.stoppedProcess_eq_of_mem_finset | Mathlib/Probability/Process/Stopping.lean | theorem stoppedProcess_eq_of_mem_finset [LinearOrder ι] [AddCommMonoid E] {s : Finset ι} (n : ι)
(hbdd : ∀ ω, τ ω < n → τ ω ∈ s) : stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) +
∑ i ∈ s.filter (· < n), Set.indicator {ω | τ ω = i} (u i) | case h.inl
Ω : Type u_1
ι : Type u_3
τ : Ω → ι
E : Type u_4
u : ι → Ω → E
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid E
s : Finset ι
n : ι
hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s
ω : Ω
h : n ≤ τ ω
m : ι
hm : m ∈ Finset.filter (fun x => x < n) s
⊢ ω ∉ {ω | τ ω = m} | rw [Finset.mem_filter] at hm | case h.inl
Ω : Type u_1
ι : Type u_3
τ : Ω → ι
E : Type u_4
u : ι → Ω → E
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid E
s : Finset ι
n : ι
hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s
ω : Ω
h : n ≤ τ ω
m : ι
hm : m ∈ s ∧ m < n
⊢ ω ∉ {ω | τ ω = m} | c67823aca57b6859 |
JacobsonNoether.exist_pow_eq_zero_of_le | Mathlib/FieldTheory/JacobsonNoether.lean | /-- If `D` is a purely inseparable extension of `k` of characteristic `p`,
then for every element `a` of `D \ k`, there exists a natural number `m`
greater than 0 such that `(a * x - x * a) ^ n = 0` (as linear maps) for
every `n` greater than `(p ^ m)`. -/
lemma exist_pow_eq_zero_of_le (p : ℕ) [hchar : ExpChar D ... | case h
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
x : D
⊢ (⇑((ad (↥k) D) a))^[p ^ m] x = 0 x | rw [ad_eq_lmul_left_sub_lmul_right, ← pow_apply, Pi.sub_apply,
sub_pow_expChar_pow_of_commute p m (commute_mulLeft_right a a), sub_apply,
pow_mulLeft, mulLeft_apply, pow_mulRight, mulRight_apply, Pi.zero_apply,
Subring.mem_center_iff.1 hm.2 x] | case h
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
p : ℕ
hchar : ExpChar D p
a : D
ha : a ∉ k
hinsep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
m : ℕ
hm : 1 ≤ m ∧ a ^ p ^ m ∈ k
n : ℕ
hn : p ^ m ≤ n
x : D
⊢ a ^ p ^ m * x - a ^ p ^ m * x = 0 | 6921e21d57128a1f |
compl_mul_closure_one_eq | Mathlib/Topology/Algebra/Group/Basic.lean | @[to_additive]
lemma compl_mul_closure_one_eq {t : Set G} (ht : t * (closure {1} : Set G) = t) :
tᶜ * (closure {1} : Set G) = tᶜ | G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
t : Set G
ht : t * closure {1} = t
⊢ tᶜ * closure {1} = tᶜ | refine Subset.antisymm ?_ (subset_mul_closure_one tᶜ) | G : Type w
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
t : Set G
ht : t * closure {1} = t
⊢ tᶜ * closure {1} ⊆ tᶜ | 8ee113e952415294 |
AffineSubspace.isPreconnected_setOf_sSameSide | Mathlib/Analysis/Convex/Side.lean | theorem isPreconnected_setOf_sSameSide (s : AffineSubspace ℝ P) (x : P) :
IsPreconnected { y | s.SSameSide x y } | case inl
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x : P
h : s = ⊥
⊢ IsPreconnected {y | False} | exact isPreconnected_empty | no goals | b6c6fe063db720d1 |
tendsto_mul_cocompact_nhds_zero | Mathlib/Topology/Algebra/Monoid.lean | theorem tendsto_mul_cocompact_nhds_zero [TopologicalSpace α] [TopologicalSpace β]
{f : α → M} {g : β → M} (f_cont : Continuous f) (g_cont : Continuous g)
(hf : Tendsto f (cocompact α) (𝓝 0)) (hg : Tendsto g (cocompact β) (𝓝 0)) :
Tendsto (fun i : α × β ↦ f i.1 * g i.2) (cocompact (α × β)) (𝓝 0) | M : Type u_3
α : Type u_6
β : Type u_7
inst✝⁴ : TopologicalSpace M
inst✝³ : MulZeroClass M
inst✝² : ContinuousMul M
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → M
g : β → M
f_cont : Continuous f
g_cont : Continuous g
hf : Tendsto f (cocompact α) (𝓝 0)
hg : Tendsto g (cocompact β) (𝓝 0)
l : Filter (M... | set K : Set (M × M) := (insert 0 (range f)) ×ˢ (insert 0 (range g)) | M : Type u_3
α : Type u_6
β : Type u_7
inst✝⁴ : TopologicalSpace M
inst✝³ : MulZeroClass M
inst✝² : ContinuousMul M
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → M
g : β → M
f_cont : Continuous f
g_cont : Continuous g
hf : Tendsto f (cocompact α) (𝓝 0)
hg : Tendsto g (cocompact β) (𝓝 0)
l : Filter (M... | a631c0c35ab89c7f |
IsLocalMaxOn.closure | Mathlib/Topology/Order/ExtrClosure.lean | theorem IsLocalMaxOn.closure (h : IsLocalMaxOn f s a) (hc : ContinuousOn f (closure s)) :
IsLocalMaxOn f (closure s) a | case intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : Preorder Y
inst✝ : OrderClosedTopology Y
f : X → Y
s : Set X
a : X
h : IsLocalMaxOn f s a
hc : ContinuousOn f (closure s)
U : Set X
Uo : IsOpen U
aU : a ∈ U
hU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}
⊢ U ∩ c... | rintro x ⟨hxU, hxs⟩ | case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : Preorder Y
inst✝ : OrderClosedTopology Y
f : X → Y
s : Set X
a : X
h : IsLocalMaxOn f s a
hc : ContinuousOn f (closure s)
U : Set X
Uo : IsOpen U
aU : a ∈ U
hU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}
x... | 4734a3091cd8637d |
Matrix.detp_mul | Mathlib/LinearAlgebra/Matrix/SemiringInverse.lean | theorem detp_mul :
detp 1 (A * B) + (detp 1 A * detp (-1) B + detp (-1) A * detp 1 B) =
detp (-1) (A * B) + (detp 1 A * detp 1 B + detp (-1) A * detp (-1) B) | case h
n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
s t : ℤˣ
σ : Perm n
hσ : σ ∈ ofSign s
τ : Perm n
⊢ τ ∈ ofSign (t * s) ↔ τ ∈ Finset.map (mulRightEmbedding σ) (ofSign t) | simp_rw [mem_map, mulRightEmbedding_apply, ← eq_mul_inv_iff_mul_eq, exists_eq_right,
mem_ofSign, _root_.map_mul, _root_.map_inv, mul_inv_eq_iff_eq_mul, mem_ofSign.mp hσ] | no goals | 1c7cf392f1553616 |
sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair | Mathlib/Analysis/Convex/Between.lean | theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
{t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
(h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
(h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i... | R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)
h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃)
h... | have h₂i := h₂.mem_image_Ioo | R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
p₁ p₂ p : P
h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)
h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃)
h... | 96830884953f795d |
List.zip_map' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean | theorem zip_map' (f : α → β) (g : α → γ) :
∀ l : List α, zip (l.map f) (l.map g) = l.map fun a => (f a, g a)
| [] => rfl
| a :: l => by simp only [map, zip_cons_cons, zip_map']
| α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β
g : α → γ
a : α
l : List α
⊢ (map f (a :: l)).zip (map g (a :: l)) = map (fun a => (f a, g a)) (a :: l) | simp only [map, zip_cons_cons, zip_map'] | no goals | 2e482eb52d78edf3 |
MultilinearMap.map_sum_finset_aux | Mathlib/LinearAlgebra/Multilinear/Basic.lean | theorem map_sum_finset_aux [DecidableEq ι] [Fintype ι] {n : ℕ} (h : (∑ i, #(A i)) = n) :
(f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i => g i (r i) | case neg.h
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
f : MultilinearMap R M₁ M₂
α : ι → Type u_1
g : (i : ι) → α i → M₁ i
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
thi... | refine mem_piFinset.2 fun i => ?_ | case neg.h
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁶ : Semiring R
inst✝⁵ : (i : ι) → AddCommMonoid (M₁ i)
inst✝⁴ : AddCommMonoid M₂
inst✝³ : (i : ι) → Module R (M₁ i)
inst✝² : Module R M₂
f : MultilinearMap R M₁ M₂
α : ι → Type u_1
g : (i : ι) → α i → M₁ i
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
thi... | 981a316f387f732a |
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