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
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|---|---|---|---|---|---|---|
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 a < ↑δ ∧ a < a'
|
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 x ∈ Ico a (a + p)
⊢ ∃ u,
(∃ l u_1, x ∈ Ioo l u_1 ∧ ∀ x ∈ Ioo l u_1, x ∈ u) ∧ ∀ (x_1 : 𝕜), x_1 ∈ u ∧ x_1 ∈ Ici x → x_1 ∈ toIcoMod hp a ⁻¹' s
|
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 : toIcoMod hp a x ∈ Ico a (a + p)
⊢ l < x - d ∧ x - d < a + p ∧ x - d < u
case intro.intro.intro.refine_2
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 x ∈ Ico a (a + p)
y : 𝕜
⊢ (l < y - d ∧ y - d < a + p ∧ y - d < u) ∧ y ∈ Ici x → y ∈ toIcoMod hp a ⁻¹' s
|
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 = f₂
|
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
x : S
⊢ f₁ x = f₂ 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 : Triangle C
hT : T ∈ distinguishedTriangles
n₀ : ℤ
⊢ (F.shift n₀).map T.mor₁ ≫ (F.shift n₀).map T.mor₂ = 0
|
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, 0 < δ →
∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior s) :
¬Tendsto (fun r ↦ volume (closure s ∩ closedBall x r) / volume (closedBall x r)) (𝓝[>] 0)
(𝓝 0)
|
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) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ volume (closedBall (f (ε n) ⋯) (ε n / 4)) / volume (closedBall x (ε n)) ≤
volume (closure s ∩ closedBall x (ε n)) / volume (closedBall x (ε n))
|
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)) (𝓝[>] 0) (𝓝 0)
ε : ℕ → ℝ
hε' : ∀ (n : ℕ), 0 < ε n
hε₀ : Tendsto ε atTop (𝓝[>] 0)
n : ℕ
⊢ closedBall (f (ε n) ⋯) (ε n / 4) ⊆ closure s ∩ closedBall x (ε n)
|
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.linearCombination R v) g
hc : IsChain r ∅
hfsupp : f ∈ Finsupp.supported R R (⋃ I ∈ ∅, ↑I)
hgsupp : g ∈ Finsupp.supported R R (⋃ I ∈ ∅, ↑I)
⊢ g = 0
|
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
obtain ⟨z, hz, rfl⟩ | hz := hz
· rw [Sym.mem_cons] at ha
obtain rfl | ha := ha
· simp
· exact mem_of_mem_of_mem_sym ha hz
· rw [mem_cons]
right
exact mem_of_mem_of_mem_sym ha 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).support ⊆ s
⊢ ((of R).symm (x • y)).coeff a = ∑ ij ∈ VAddAntidiagonal ⋯ hs a, x.coeff ij.1 • ((of R).symm y).coeff ij.2
|
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).support ⊆ s
⊢ ∑ ij ∈ VAddAntidiagonal ⋯ ⋯ a, x.coeff ij.1 • ((of R).symm y).coeff ij.2 =
∑ ij ∈ VAddAntidiagonal ⋯ hs a, x.coeff ij.1 • ((of R).symm y).coeff ij.2
|
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 < μ K
|
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)
inst✝ : Module.Free A B
x : B
⊢ (intNorm A B) x = (norm A) x
|
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)
inst✝ : Module.Free A B
x : B
this : IsIntegralClosure B A (FractionRing B)
⊢ (intNorm A B) x = (norm A) x
|
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 (f n) x
|
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 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
k : ℕ
IH : ↑k ≤ N → (iteratedFDeriv 𝕜 k fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x
hk : ↑(k + 1) ≤ N
h'k : ↑k < N
A : Summable fun n => iteratedFDeriv 𝕜 k (f n) 0
n : α
x : E
⊢ ‖fderiv 𝕜 (fun x => iteratedFDeriv 𝕜 k (f n) x) x‖ ≤ v (k + 1) n
|
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 i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
⊢ ∫⁻ (a : α), ∏ i ∈ insert i₀ s, f i a ^ p i ∂μ ≤ ∏ i ∈ insert i₀ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
|
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 i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
⊢ ∫⁻ (a : α), ∏ i ∈ insert i₀ s, f i a ^ p i ∂μ ≤ ∏ i ∈ insert i₀ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
|
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 : deriv f b * (y - w) < f y - f w
this : deriv f a * (y - w) < deriv f b * (y - w)
⊢ deriv f a * (y - x) < f y - f x
|
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 atTop, ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑n)}
|
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 - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑i_1)}
|
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) ∧ bsup.{u, u} o f = a,
∃ (ι : Type u), Nonempty ι ∧ ∃ f : ι → Ordinal, (∀ i, f i ∈ s) ∧ ⨆ i, f i = a]
|
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 ∧ BddAbove t ∧ sSup t = a) →
∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ o.bsup f = a
⊢ [a ∈ closure s, a ∈ closure (s ∩ Iic a), (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t ⊆ s, t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ o.bsup f = a,
∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ ⨆ i, f i = a].TFAE
|
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 ∧ BddAbove t ∧ sSup t = a) →
∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ o.bsup f = a
tfae_5_to_6 :
(∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ o.bsup f = a) →
∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ ⨆ i, f i = a
⊢ [a ∈ closure s, a ∈ closure (s ∩ Iic a), (s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t ⊆ s, t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ o.bsup f = a,
∃ ι, Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ ⨆ i, f i = a].TFAE
|
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✝⁵ : ν.IsMulLeftInvariant
inst✝⁴ : Countable ↥Γ
inst✝³ : ν.IsMulRightInvariant
inst✝² : SigmaFinite ν
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : SigmaFinite μ
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 (QuotientGroup.mk ⁻¹' V ∩ s)
neTopV : μ V ≠ ⊤
U : Set (G ⧸ Γ)
a✝ : MeasurableSet U
meas_π : Measurable QuotientGroup.mk
μ' : Measure (G ⧸ Γ) := map QuotientGroup.mk (ν.restrict s)
has_fund : HasFundamentalDomain (↥Γ.op) G ν
i : QuotientMeasureEqMeasurePreimage ν μ'
this✝ : μ'.IsMulLeftInvariant
this : SigmaFinite μ'
⊢ ν (QuotientGroup.mk ⁻¹' V ∩ s) ≠ ⊤
|
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) ≤ _ by simp; exact Nat.pred_le _
have a1m : (⟨a, 1⟩ * ⟨a, -1⟩ : ℤ√(d a1)) = 1 := isPell_norm.1 (isPell_one a1)
if ha : (⟨↑a, 1⟩ : ℤ√(d a1)) ≤ b then
let ⟨m, e⟩ :=
eq_pell_lem n (b * ⟨a, -1⟩) (by rw [← a1m]; exact mul_le_mul_of_nonneg_right ha am1p)
(isPell_mul hp (isPell_star.1 (isPell_one a1)))
(by
have t := mul_le_mul_of_nonneg_right h am1p
rwa [pellZd_succ, mul_assoc, a1m, mul_one] at t)
⟨m + 1, by
rw [show b = b * ⟨a, -1⟩ * ⟨a, 1⟩ by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp,
pellZd_succ, e]⟩
else
suffices ¬1 < b from ⟨0, show b = 1 from (Or.resolve_left (lt_or_eq_of_le h1) this).symm⟩
fun h1l => by
obtain ⟨x, y⟩ := b
exact by
have bm : (_ * ⟨_, _⟩ : ℤ√d a1) = 1 := Pell.isPell_norm.1 hp
have y0l : (0 : ℤ√d a1) < ⟨x - x, y - -y⟩ :=
sub_lt_sub h1l fun hn : (1 : ℤ√d a1) ≤ ⟨x, -y⟩ => by
have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)
rw [bm, mul_one] at t
exact h1l t
have yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩ :=
show (⟨x, y⟩ - ⟨x, -y⟩ : ℤ√d a1) < ⟨a, 1⟩ - ⟨a, -1⟩ from
sub_lt_sub ha fun hn : (⟨x, -y⟩ : ℤ√d a1) ≤ ⟨a, -1⟩ => by
have t := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p
rw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t
exact ha t
simp only [sub_self, sub_neg_eq_add] at y0l; simp only [Zsqrtd.neg_re, add_neg_cancel,
Zsqrtd.neg_im, neg_neg] at yl2
exact
match y, y0l, (yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩) with
| 0, y0l, _ => y0l (le_refl 0)
| (y + 1 : ℕ), _, yl2 =>
yl2
(Zsqrtd.le_of_le_le (by simp [sub_eq_add_neg])
(let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)
add_le_add t t))
| Int.negSucc _, y0l, _ => y0l trivial
|
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
hn : 1 ≤ n
hx : x ∈ s
A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0)
this : ↑↑0 < n
e_8✝ : SeminormedAddCommGroup.toAddCommGroup = ContinuousMultilinearMap.instAddCommGroup
he✝ : NormedSpace.toModule = ContinuousMultilinearMap.instModule
e_10✝ : UniformSpace.toTopologicalSpace = ContinuousMultilinearMap.instTopologicalSpace
y : E
v : Fin 0 → E
⊢ (((↑{ toLinearEquiv := (continuousMultilinearCurryFin0 𝕜 E F).symm.toLinearEquiv, continuous_toFun := ⋯,
continuous_invFun := ⋯ }).comp
((continuousMultilinearCurryFin1 𝕜 E F) (p x 1)))
y)
v =
((p x (Nat.succ 0)).curryLeft y) v
|
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
hn : 1 ≤ n
hx : x ∈ s
A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0)
this : ↑↑0 < n
e_8✝ : SeminormedAddCommGroup.toAddCommGroup = ContinuousMultilinearMap.instAddCommGroup
he✝ : NormedSpace.toModule = ContinuousMultilinearMap.instModule
e_10✝ : UniformSpace.toTopologicalSpace = ContinuousMultilinearMap.instTopologicalSpace
y : E
v : Fin 0 → E
⊢ (p x 1) (snoc 0 y) = (p x 1) (cons y v)
|
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_isNoetherian_of_isJacobsonRing
[IsNoetherianRing R] [IsJacobsonRing R] (x : PrimeSpectrum R) :
List.TFAE [IsOpen {x}, IsClopen {x}, IsClosed {x} ∧ StableUnderGeneralization {x}]
|
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≥0 := ⟨ε, ⋯⟩
δpos : 0 < δ
t : ℕ → Set E
A : ℕ → E →L[ℝ] E
t_disj : Pairwise (Disjoint on t)
t_meas : ∀ (n : ℕ), MeasurableSet (t n)
t_cover : s ⊆ ⋃ n, t n
ht : ∀ (n : ℕ), ApproximatesLinearOn f (A n) (s ∩ t n) δ
right✝ : s.Nonempty → ∀ (n : ℕ), ∃ y ∈ s, A n = f' y
g : E → E →L[ℝ] E
g_meas : Measurable g
hg : ∀ (n : ℕ), ∀ x ∈ t n, g x = A n
H : ∀ᵐ (x : E) ∂sum fun n => μ.restrict (s ∩ t n), dist (g x) (f' x) ≤ ε
this : μ.restrict s ≤ sum fun n => μ.restrict (s ∩ t n)
⊢ ∀ᵐ (x : E) ∂μ.restrict s, dist (g x) (f' x) ≤ ε
|
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 x ∂μ) atTop (𝓝 (g 0))
|
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 : ∫ (x : F), 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
I : Integrable φ μ
u : Set F
u_open : IsOpen u
hu : 0 ∈ u
ε : ℝ
εpos : ε > 0
⊢ ∀ᶠ (n : ℝ) in atTop, ∀ x ∈ uᶜ, dist (0 x) (n ^ finrank ℝ F * φ (n • x)) < ε
|
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 : F), 0 ≤ φ x
h'φ : ∫ (x : F), 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
I : Integrable φ μ
u : Set F
u_open : IsOpen u
hu : 0 ∈ u
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : δ > 0
h'u : ball 0 δ ⊆ u
⊢ ∀ᶠ (n : ℝ) in atTop, ∀ x ∈ uᶜ, dist (0 x) (n ^ finrank ℝ F * φ (n • x)) < ε
|
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 fun m n => 1 ≤ ‖f m - f n‖
⊢ False
|
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 fun m n => 1 ≤ ‖f m - f n‖
rRpos : 0 < r / R
⊢ False
|
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.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z - soln
q : ℤ_[p]
hq :
Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2
this✝¹ : (Polynomial.eval soln (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this✝ : Polynomial.eval soln (Polynomial.derivative F) + q * h = 0
this : Polynomial.eval soln (Polynomial.derivative F) = -q * h
⊢ ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
|
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.comp P).Ring
e : ((Q.comp P).vars →₀ ℕ) ≃+ (Q.vars →₀ ℕ) × (P.vars →₀ ℕ) := Finsupp.sumFinsuppAddEquivProdFinsupp
i : Q.vars →₀ ℕ
this :
∀ (x : (Q.comp P).Ring),
(Function.support fun a => if a.1 = i then (aeval P.val) ((monomial a.2) (coeff (e.symm a) x)) else 0) ⊆
↑(Finset.map (↑e).toEmbedding (support x))
⊢ ∑ x_1 ∈ Finset.filter (fun x => x.1 = i) (Finset.map (↑e).toEmbedding (support x)),
(aeval P.val) ((monomial x_1.2) (coeff (e.symm x_1) x)) =
coeff i ((Q.ofComp P).toAlgHom x)
|
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.comp P).Ring
e : ((Q.comp P).vars →₀ ℕ) ≃+ (Q.vars →₀ ℕ) × (P.vars →₀ ℕ) := Finsupp.sumFinsuppAddEquivProdFinsupp
i : Q.vars →₀ ℕ
this :
∀ (x : (Q.comp P).Ring),
(Function.support fun a => if a.1 = i then (aeval P.val) ((monomial a.2) (coeff (e.symm a) x)) else 0) ⊆
↑(Finset.map (↑e).toEmbedding (support x))
⊢ (∑ᶠ (i_1 : (Q.vars →₀ ℕ) × (P.vars →₀ ℕ)),
if i_1.1 = i then (aeval P.val) ((monomial i_1.2) (coeff (e.symm i_1) x)) else 0) =
coeff i ((Q.ofComp P).toAlgHom x)
|
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 M) x
|
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⟩)
· exact Or.inl ⟨toLeftMovesAdd (Sum.inl i'), add_le_add_right' ih⟩
· refine Or.inr ⟨toRightMovesAdd (Sum.inl j), ?_⟩
convert add_le_add_right' jh
apply add_moveRight_inl
· exact Or.inl ⟨@toLeftMovesAdd _ ⟨_, _, _, _⟩ (Sum.inr i), add_le_add_right' h⟩
· rw [le_def] at h
rcases h.right i with (⟨i, ih⟩ | ⟨j', jh⟩)
· refine Or.inl ⟨toLeftMovesAdd (Sum.inl i), ?_⟩
convert add_le_add_right' ih
apply add_moveLeft_inl
· exact Or.inr ⟨toRightMovesAdd (Sum.inl j'), add_le_add_right' jh⟩
· exact
Or.inr ⟨@toRightMovesAdd _ ⟨_, _, _, _⟩ (Sum.inr i), add_le_add_right' h⟩
termination_by x y z => (x, y, z)
|
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).moveLeft i).moveRight j ≤ mk yl yr yL yR) ∧
∀ (j : (mk yl yr yL yR).RightMoves),
(∃ i, mk xl xr xL xR ≤ ((mk yl yr yL yR).moveRight j).moveLeft i) ∨
∃ j', (mk xl xr xL xR).moveRight j' ≤ (mk yl yr yL yR).moveRight j
i : yr
⊢ (∃ i_1, mk xl xr xL xR + mk zl zr zL zR ≤ ((mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i)).moveLeft i_1) ∨
∃ j', (mk xl xr xL xR + mk zl zr zL zR).moveRight j' ≤ (mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i)
|
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 xR).moveLeft i).moveRight j ≤ mk yl yr yL yR) ∧
∀ (j : (mk yl yr yL yR).RightMoves),
(∃ i, mk xl xr xL xR ≤ ((mk yl yr yL yR).moveRight j).moveLeft i) ∨
∃ j', (mk xl xr xL xR).moveRight j' ≤ (mk yl yr yL yR).moveRight j
i✝ : yr
i : ((mk yl yr yL yR).moveRight i✝).LeftMoves
ih : mk xl xr xL xR ≤ ((mk yl yr yL yR).moveRight i✝).moveLeft i
⊢ (∃ i, mk xl xr xL xR + mk zl zr zL zR ≤ ((mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i✝)).moveLeft i) ∨
∃ j', (mk xl xr xL xR + mk zl zr zL zR).moveRight j' ≤ (mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i✝)
case refine_2.inl.inr.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 xR).moveLeft i).moveRight j ≤ mk yl yr yL yR) ∧
∀ (j : (mk yl yr yL yR).RightMoves),
(∃ i, mk xl xr xL xR ≤ ((mk yl yr yL yR).moveRight j).moveLeft i) ∨
∃ j', (mk xl xr xL xR).moveRight j' ≤ (mk yl yr yL yR).moveRight j
i : yr
j' : (mk xl xr xL xR).RightMoves
jh : (mk xl xr xL xR).moveRight j' ≤ (mk yl yr yL yR).moveRight i
⊢ (∃ i_1, mk xl xr xL xR + mk zl zr zL zR ≤ ((mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i)).moveLeft i_1) ∨
∃ j', (mk xl xr xL xR + mk zl zr zL zR).moveRight j' ≤ (mk yl yr yL yR + mk zl zr zL zR).moveRight (Sum.inl i)
|
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).Countable
⊢ ∃ V D, V.Countable ∧ Perfect D ∧ C = V ∪ D
|
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 ∩ C).Countable
⊢ IsClosed D
case intro.intro.intro.refine_2
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).Countable
⊢ Preperfect D
case intro.intro.intro.refine_3
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).Countable
⊢ C = V ∩ C ∪ D
|
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 ∈ t, μ (closedBall (c a) (3 * r a)) ≤ C * μ (B a))
(ht : ∀ a ∈ t, (interior (B a)).Nonempty) (h't : ∀ a ∈ t, IsClosed (B a))
(hf : ∀ x ∈ s, ∀ ε > (0 : ℝ), ∃ a ∈ t, r a ≤ ε ∧ c a = x) :
∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
|
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 a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
⊢ ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
|
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 a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
⊢ ∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
|
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 ++ r }.get i) a;
foldrAux f a { data := l ++ [].reverse ++ r } i { byteIdx := utf8Len l }
else a) =
List.foldr f a [].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 (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
(h'iφ : ∀ᶠ i in l, AEStronglyMeasurable (φ i) (μ.restrict s))
(hmg : IntegrableOn g s μ) (hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)) :
Tendsto (fun i : ι ↦ ∫ x in s, φ i x • g x ∂μ) l (𝓝 a)
|
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 : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
h't : μ t ≠ ⊤
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i => ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
h : α → E := g - t.indicator fun x => a
A : Tendsto (fun i => ∫ (x : α) in s, φ i x • h x ∂μ + (∫ (x : α) in t, φ i x ∂μ) • a) l (𝓝 a)
i : ι
hi : IntegrableOn (fun x => φ i x • g x) s μ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < 1
⊢ ∫ (x : α) in s, φ i x • h x ∂μ + (∫ (x : α) in t, φ i x ∂μ) • a = ∫ (x : α) in s, φ i x • g x ∂μ
|
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 : MeasurableSet t
hts : t ⊆ s
h'ts : t ∈ 𝓝[s] x₀
h't : μ t ≠ ⊤
hnφ : ∀ᶠ (i : ι) in l, ∀ x ∈ s, 0 ≤ φ i x
hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u)
hiφ : Tendsto (fun i => ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)
h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) (μ.restrict s)
hmg : IntegrableOn g s μ
hcg : Tendsto g (𝓝[s] x₀) (𝓝 a)
h : α → E := g - t.indicator fun x => a
A : Tendsto (fun i => ∫ (x : α) in s, φ i x • h x ∂μ + (∫ (x : α) in t, φ i x ∂μ) • a) l (𝓝 a)
i : ι
hi : IntegrableOn (fun x => φ i x • g x) s μ
h'i : ‖∫ (x : α) in t, φ i x ∂μ - 1‖ < 1
⊢ ∫ (x : α) in s, φ i x • g x - t.indicator (fun a_1 => φ i a_1 • a) x ∂μ + (∫ (x : α) in t, φ i x ∂μ) • a =
∫ (x : α) in s, φ i x • g x ∂μ
|
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 fun m ↦ sInf_le (isCoatom_map_of_ker_le surj (le_iInf_iff.mp le m) m.2)
|
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.Surjective ⇑f
le : LinearMap.ker f ≤ ⨅ a, ↑a
⊢ jacobson R₂ M₂ ≤ ⨅ i, map f ↑i
|
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 < l₃
|
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 (fun x1 x2 => x1 < x2) as bs
ih : ∀ {l₁ : List α}, l₁ ≤ as → l₁ < bs
l₁ : List α
h₁ : a :: l₁ ≤ a :: as
w₄ : ¬a < a
⊢ a :: l₁ < a :: bs
|
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 ▷ P.X) ≫ P.actLeft =
R.X ◁ coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ≫
R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft
|
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 R.X P.X).hom ≫ R.X ◁ P.actLeft ≫ P.actLeft =
R.X ◁ coequalizer.π (R.mul ▷ P.X) ((α_ R.X R.X P.X).hom ≫ R.X ◁ P.actLeft) ≫
R.X ◁ coequalizer.desc P.actLeft ⋯ ≫ P.actLeft
|
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 := (evalEquiv 𝕜 E).symm φ
i : ι
⊢ ∃ y ∈ span 𝕜 (Set.range L), φ y = (L i) φs
|
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_le_nhds _ (le_principal_iff.2 hi) with ⟨x, hxi, hlx⟩
exact ⟨x, mem_iUnion₂.2 ⟨i, his, hxi⟩, hlx⟩
|
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)).toRingHom x✝ = f.toRingHom x✝
|
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⟩
rcases dvd with ⟨k2, pr⟩
have k2_nonzero : k2 ≠ 0 := fun k2_zero => by simp [k2_zero] at pr
rw [Nat.mul_div_cancel_left k (Nat.pos_of_ne_zero a_split), pr,
Nat.mul_div_cancel_left k2 (Nat.pos_of_ne_zero c_split), Nat.mul_comm ((c + 1) * k2) k, ←
Nat.mul_assoc k (c + 1) k2, Nat.mul_div_cancel _ (Nat.pos_of_ne_zero k2_nonzero),
Nat.mul_div_cancel _ (Nat.pos_of_ne_zero c_split)]
|
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 : AddCommGroup M₂
inst✝² : Module R M₂
inst✝¹ : LieRingModule L₂ M₂
f : L →ₗ⁅R⁆ L₂
g : M →ₗ[R] M₂
hfg : ∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆
inst✝ : LieModule R L₂ M₂
hf_surj : Surjective ⇑f
hg_surj : Surjective ⇑g
k : ℕ
ih : ↑(lowerCentralSeries R L₂ M₂ k) ≤ Submodule.map g ↑(lowerCentralSeries R L M k)
n : M
hn : n ∈ lowerCentralSeries R L M k
y : L
⊢ ⁅f y, g n⁆ ∈ ⇑g '' {m | ∃ x, ∃ n ∈ lowerCentralSeries R L M k, ⁅x, n⁆ = m}
|
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 p]
{a : D} (ha : a ∉ k) (hinsep : ∀ x : D, IsSeparable k x → x ∈ k):
∃ m, 1 ≤ m ∧ ∀ n, p ^ m ≤ n → (ad k D a)^[n] = 0
|
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 × M) := map (Prod.map f g) (cocompact (α × β))
l_def : l = map (Prod.map f g) (cocompact (α × β))
⊢ Tendsto (fun i => f i.1 * g i.2) (cocompact (α × β)) (𝓝 0)
|
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 × M) := map (Prod.map f g) (cocompact (α × β))
l_def : l = map (Prod.map f g) (cocompact (α × β))
K : Set (M × M) := insert 0 (range f) ×ˢ insert 0 (range g)
⊢ Tendsto (fun i => f i.1 * g i.2) (cocompact (α × β)) (𝓝 0)
|
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 ∩ closure s ⊆ {x | (fun x => f x ≤ f a) x}
|
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 : X
hxU : x ∈ U
hxs : x ∈ closure s
⊢ x ∈ {x | (fun x => f x ≤ f a) 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 i₂, p₂]) :
Sbtw R (t.points i₁) p p₁
|
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)
h₁' : p ∈ affineSpan R {t.points i₁, p₁}
h₂' : p ∈ affineSpan R {t.points i₂, p₂}
this : DecidableRel fun x1 x2 => x1 < x2 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hp : p ∈ affineSpan R (Set.range t.points)
h₁i : p₁ ∈ ⇑(lineMap (t.points i₂) (t.points i₃)) '' Set.Ioo 0 1
⊢ Sbtw R (t.points i₁) p p₁
|
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)
h₁' : p ∈ affineSpan R {t.points i₁, p₁}
h₂' : p ∈ affineSpan R {t.points i₂, p₂}
this : DecidableRel fun x1 x2 => x1 < x2 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
hp : p ∈ affineSpan R (Set.range t.points)
h₁i : p₁ ∈ ⇑(lineMap (t.points i₂) (t.points i₃)) '' Set.Ioo 0 1
h₂i : p₂ ∈ ⇑(lineMap (t.points i₁) (t.points i₃)) '' Set.Ioo 0 1
⊢ Sbtw R (t.points i₁) p p₁
|
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
this : (i : ι) → DecidableEq (α i) := fun i => Classical.decEq (α i)
n : ℕ
IH :
∀ m < n,
∀ (A : (i : ι) → Finset (α i)),
∑ i : ι, #(A i) = m → (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i => g i (r i)
A : (i : ι) → Finset (α i)
h : ∑ i : ι, #(A i) = n
Ai_empty : ∀ (i : ι), A i ≠ ∅
i₀ : ι
hi₀ : 1 < #(A i₀)
j₁ j₂ : α i₀
left✝ : j₁ ∈ A i₀
hj₂ : j₂ ∈ A i₀
right✝ : j₁ ≠ j₂
B : (a : ι) → Finset (α a) := update A i₀ (A i₀ \ {j₂})
C : (a : ι) → Finset (α a) := update A i₀ {j₂}
B_subset_A : ∀ (i : ι), B i ⊆ A i
C_subset_A : ∀ (i : ι), C i ⊆ A i
A_eq_BC : (fun i => ∑ j ∈ A i, g i j) = update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j + ∑ j ∈ C i₀, g i₀ j)
Beq : update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j) = fun i => ∑ j ∈ B i, g i j
Ceq : update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ C i₀, g i₀ j) = fun i => ∑ j ∈ C i, g i j
Brec : (f fun i => ∑ j ∈ B i, g i j) = ∑ r ∈ piFinset B, f fun i => g i (r i)
Crec : (f fun i => ∑ j ∈ C i, g i j) = ∑ r ∈ piFinset C, f fun i => g i (r i)
D : Disjoint (piFinset B) (piFinset C)
r : (a : ι) → α a
hr : r ∈ piFinset A
hri₀ : ¬r i₀ = j₂
⊢ r ∈ piFinset B
|
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
this : (i : ι) → DecidableEq (α i) := fun i => Classical.decEq (α i)
n : ℕ
IH :
∀ m < n,
∀ (A : (i : ι) → Finset (α i)),
∑ i : ι, #(A i) = m → (f fun i => ∑ j ∈ A i, g i j) = ∑ r ∈ piFinset A, f fun i => g i (r i)
A : (i : ι) → Finset (α i)
h : ∑ i : ι, #(A i) = n
Ai_empty : ∀ (i : ι), A i ≠ ∅
i₀ : ι
hi₀ : 1 < #(A i₀)
j₁ j₂ : α i₀
left✝ : j₁ ∈ A i₀
hj₂ : j₂ ∈ A i₀
right✝ : j₁ ≠ j₂
B : (a : ι) → Finset (α a) := update A i₀ (A i₀ \ {j₂})
C : (a : ι) → Finset (α a) := update A i₀ {j₂}
B_subset_A : ∀ (i : ι), B i ⊆ A i
C_subset_A : ∀ (i : ι), C i ⊆ A i
A_eq_BC : (fun i => ∑ j ∈ A i, g i j) = update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j + ∑ j ∈ C i₀, g i₀ j)
Beq : update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j) = fun i => ∑ j ∈ B i, g i j
Ceq : update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ C i₀, g i₀ j) = fun i => ∑ j ∈ C i, g i j
Brec : (f fun i => ∑ j ∈ B i, g i j) = ∑ r ∈ piFinset B, f fun i => g i (r i)
Crec : (f fun i => ∑ j ∈ C i, g i j) = ∑ r ∈ piFinset C, f fun i => g i (r i)
D : Disjoint (piFinset B) (piFinset C)
r : (a : ι) → α a
hr : r ∈ piFinset A
hri₀ : ¬r i₀ = j₂
i : ι
⊢ r i ∈ B i
|
981a316f387f732a
|
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