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
|
|---|---|---|---|---|---|---|
Array.array₃_induction
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem array₃_induction (P : Array (Array (Array α)) → Prop)
(of : ∀ (xss : List (List (List α))), P ((xss.map (fun xs => xs.map List.toArray)).map List.toArray).toArray)
(ass : Array (Array (Array α))) : P ass
|
α : Type u_1
P : Array (Array (Array α)) → Prop
of :
∀ (xss : List (List (List α))), P (List.map List.toArray (List.map (fun xs => List.map List.toArray xs) xss)).toArray
ass : Array (Array (Array α))
⊢ P ass
|
specialize of ((ass.toList.map toList).map (fun as => as.map toList))
|
α : Type u_1
P : Array (Array (Array α)) → Prop
ass : Array (Array (Array α))
of :
P
(List.map List.toArray
(List.map (fun xs => List.map List.toArray xs)
(List.map (fun as => List.map toList as) (List.map toList ass.toList)))).toArray
⊢ P ass
|
633f2b5cbd3730e2
|
Polynomial.induction_on
|
Mathlib/Algebra/Polynomial/Basic.lean
|
theorem induction_on {M : R[X] → Prop} (p : R[X]) (h_C : ∀ a, M (C a))
(h_add : ∀ p q, M p → M q → M (p + q))
(h_monomial : ∀ (n : ℕ) (a : R), M (C a * X ^ n) → M (C a * X ^ (n + 1))) : M p
|
case empty
R : Type u
inst✝ : Semiring R
M : R[X] → Prop
p : R[X]
h_C : ∀ (a : R), M (C a)
h_add : ∀ (p q : R[X]), M p → M q → M (p + q)
h_monomial : ∀ (n : ℕ) (a : R), M (C a * X ^ n) → M (C a * X ^ (n + 1))
A : ∀ {n : ℕ} {a : R}, M (C a * X ^ n)
⊢ M (∑ n ∈ ∅, C (p.coeff n) * X ^ n)
|
convert h_C 0
|
case h.e'_1
R : Type u
inst✝ : Semiring R
M : R[X] → Prop
p : R[X]
h_C : ∀ (a : R), M (C a)
h_add : ∀ (p q : R[X]), M p → M q → M (p + q)
h_monomial : ∀ (n : ℕ) (a : R), M (C a * X ^ n) → M (C a * X ^ (n + 1))
A : ∀ {n : ℕ} {a : R}, M (C a * X ^ n)
⊢ ∑ n ∈ ∅, C (p.coeff n) * X ^ n = C 0
|
9b29050224f0e20b
|
Batteries.BinomialHeap.Imp.Heap.realSize_deleteMin
|
Mathlib/.lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean
|
theorem Heap.realSize_deleteMin {s : Heap α} (eq : s.deleteMin le = some (a, s')) :
s.realSize = s'.realSize + 1
|
α : Type u_1
le : α → α → Bool
r : Nat
a : α
c : HeapNode α
s : Heap α
before : Heap α → Heap α
val : α
node : HeapNode α
next : Heap α
m : Nat
ih₁ : ∀ (s : Heap α), ({ before := before, val := val, node := node, next := next }.before s).realSize = m + s.realSize
ih₂ :
c.realSize + s.realSize + 1 =
m + { before := before, val := val, node := node, next := next }.node.realSize +
{ before := before, val := val, node := node, next := next }.next.realSize +
1
⊢ (cons r a c s).realSize =
(merge le { before := before, val := val, node := node, next := next }.node.toHeap
({ before := before, val := val, node := node, next := next }.before
{ before := before, val := val, node := node, next := next }.next)).realSize +
1
|
dsimp only at ih₁ ih₂
|
α : Type u_1
le : α → α → Bool
r : Nat
a : α
c : HeapNode α
s : Heap α
before : Heap α → Heap α
val : α
node : HeapNode α
next : Heap α
m : Nat
ih₁ : ∀ (s : Heap α), (before s).realSize = m + s.realSize
ih₂ : c.realSize + s.realSize + 1 = m + node.realSize + next.realSize + 1
⊢ (cons r a c s).realSize =
(merge le { before := before, val := val, node := node, next := next }.node.toHeap
({ before := before, val := val, node := node, next := next }.before
{ before := before, val := val, node := node, next := next }.next)).realSize +
1
|
36aad300d7e604ad
|
Set.Ioo_inter_Iio
|
Mathlib/Order/Interval/Set/Basic.lean
|
theorem Ioo_inter_Iio : Ioo a b ∩ Iio c = Ioo a (min b c)
|
α : Type u_1
inst✝ : LinearOrder α
a b c : α
⊢ Ioo a b ∩ Iio c = Ioo a (b ⊓ c)
|
ext
|
case h
α : Type u_1
inst✝ : LinearOrder α
a b c x✝ : α
⊢ x✝ ∈ Ioo a b ∩ Iio c ↔ x✝ ∈ Ioo a (b ⊓ c)
|
78bf76235a3f6efb
|
Submodule.basis_of_pid_aux
|
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
(M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
(∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0) ∧
∀ (n') (bN' : Basis (Fin n') R N'),
∃ bN : Basis (Fin (n' + 1)) R N,
∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
∀ as : Fin n' → R,
(∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)) →
∃ as' : Fin (n' + 1) → R,
∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O)
|
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
ϕ : ↥M →ₗ[R] R := this.choose
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N
a : R := generator (ϕ.submoduleImage N)
a_mem : a ∈ ϕ.submoduleImage N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ ⟨y, ⋯⟩ = a
_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0
c : ι → R
hc : ∀ (i : ι), (b'M.coord i) ⟨y, ⋯⟩ = a * c i
val✝ : Fintype ι
y' : O := ∑ i : ι, c i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : ⟨y', y'M⟩ = ∑ i : ι, c i • b'M i
i : ι
x✝ : i ∈ Finset.univ
⊢ a • c i • b'M i = (b'M.repr ⟨y, ⋯⟩) i • b'M i
|
rw [← mul_smul, ← hc]
|
ι : Type u_1
R : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : Finite ι
O : Type u_4
inst✝¹ : AddCommGroup O
inst✝ : Module R O
M N : Submodule R O
b'M : Basis ι R ↥M
N_bot : N ≠ ⊥
N_le_M : N ≤ M
this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N
ϕ : ↥M →ₗ[R] R := this.choose
ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N
a : R := generator (ϕ.submoduleImage N)
a_mem : a ∈ ϕ.submoduleImage N
a_zero : ¬a = 0
y : O
yN : y ∈ N
ϕy_eq : ϕ ⟨y, ⋯⟩ = a
_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0
c : ι → R
hc : ∀ (i : ι), (b'M.coord i) ⟨y, ⋯⟩ = a * c i
val✝ : Fintype ι
y' : O := ∑ i : ι, c i • ↑(b'M i)
y'M : y' ∈ M
mk_y' : ⟨y', y'M⟩ = ∑ i : ι, c i • b'M i
i : ι
x✝ : i ∈ Finset.univ
⊢ (b'M.coord i) ⟨y, ⋯⟩ • b'M i = (b'M.repr ⟨y, ⋯⟩) i • b'M i
|
48ee91e21f4317c3
|
Pell.IsFundamental.mul_inv_x_lt_x
|
Mathlib/NumberTheory/Pell.lean
|
theorem mul_inv_x_lt_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d} (hax : 1 < a.x)
(hay : 0 < a.y) : (a * a₁⁻¹).x < a.x
|
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a.x * a₁.x < d * (a.y * a₁.y) + a.x
|
refine (mul_lt_mul_left h.2.1).mp ?_
|
d : ℤ
a₁ : Solution₁ d
h : IsFundamental a₁
a : Solution₁ d
hax : 1 < a.x
hay : 0 < a.y
⊢ a₁.y * (a.x * a₁.x) < a₁.y * (d * (a.y * a₁.y) + a.x)
|
9b261d48a46b4210
|
Equiv.Perm.fin_5_not_solvable
|
Mathlib/GroupTheory/Solvable.lean
|
theorem Equiv.Perm.fin_5_not_solvable : ¬IsSolvable (Equiv.Perm (Fin 5))
|
case succ
x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }
y : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }
z : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_inv := ⋯, right_inv := ⋯ }
key : x = z * ⁅x, y * x * y⁻¹⁆ * z⁻¹
n : ℕ
ih : x ∈ derivedSeries (Perm (Fin 5)) n
⊢ x ∈ derivedSeries (Perm (Fin 5)) (n + 1)
|
rw [key, (derivedSeries_normal _ _).mem_comm_iff, inv_mul_cancel_left]
|
case succ
x : Perm (Fin 5) := { toFun := ![1, 2, 0, 3, 4], invFun := ![2, 0, 1, 3, 4], left_inv := ⋯, right_inv := ⋯ }
y : Perm (Fin 5) := { toFun := ![3, 4, 2, 0, 1], invFun := ![3, 4, 2, 0, 1], left_inv := ⋯, right_inv := ⋯ }
z : Perm (Fin 5) := { toFun := ![0, 3, 2, 1, 4], invFun := ![0, 3, 2, 1, 4], left_inv := ⋯, right_inv := ⋯ }
key : x = z * ⁅x, y * x * y⁻¹⁆ * z⁻¹
n : ℕ
ih : x ∈ derivedSeries (Perm (Fin 5)) n
⊢ ⁅x, y * x * y⁻¹⁆ ∈ derivedSeries (Perm (Fin 5)) (n + 1)
|
2cf02f2116026673
|
Int.le_bmod
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
|
theorem le_bmod {x : Int} {m : Nat} (h : 0 < m) : - (m/2) ≤ Int.bmod x m
|
case isFalse
x : Int
m : Nat
h : 0 < m
v : ↑m % 2 = 0 ∨ ↑m % 2 = 1
w : (↑m + 1) / 2 ≤ x % ↑m
⊢ -(↑m / 2) ≤ (↑m + 1) / 2 - ↑m
|
rw [← ediv_add_emod m 2]
|
case isFalse
x : Int
m : Nat
h : 0 < m
v : ↑m % 2 = 0 ∨ ↑m % 2 = 1
w : (↑m + 1) / 2 ≤ x % ↑m
⊢ -((2 * (↑m / 2) + ↑m % 2) / 2) ≤ (2 * (↑m / 2) + ↑m % 2 + 1) / 2 - (2 * (↑m / 2) + ↑m % 2)
|
c8cfe301519f3ee0
|
Set.Iic.succ_eq_of_not_isMax
|
Mathlib/Order/Interval/Set/SuccOrder.lean
|
lemma Iic.succ_eq_of_not_isMax
[SuccOrder J] {j : J} {i : Set.Iic j} (hi : ¬ IsMax i) :
Order.succ i = ⟨Order.succ i.1, by
rw [← coe_succ_of_not_isMax hi]
apply Subtype.coe_prop⟩
|
case a
J : Type u_1
inst✝¹ : PartialOrder J
inst✝ : SuccOrder J
j : J
i : ↑(Iic j)
hi : ¬IsMax i
⊢ ↑(Order.succ i) = ↑⟨Order.succ ↑i, ⋯⟩
|
simp only [coe_succ_of_not_isMax hi]
|
no goals
|
aa6f4dc6ce5e335e
|
Pell.Solution₁.exists_nontrivial_of_not_isSquare
|
Mathlib/NumberTheory/Pell.lean
|
theorem exists_nontrivial_of_not_isSquare (h₀ : 0 < d) (hd : ¬IsSquare d) :
∃ a : Solution₁ d, a ≠ 1 ∧ a ≠ -1
|
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
⊢ ∃ a, a ≠ 1 ∧ a ≠ -1
|
obtain ⟨x, y, prop, hy⟩ := exists_of_not_isSquare h₀ hd
|
case intro.intro.intro
d : ℤ
h₀ : 0 < d
hd : ¬IsSquare d
x y : ℤ
prop : x ^ 2 - d * y ^ 2 = 1
hy : y ≠ 0
⊢ ∃ a, a ≠ 1 ∧ a ≠ -1
|
f47e8ea11b2d637b
|
IsUltrametricDist.exists_norm_multiset_prod_le
|
Mathlib/Analysis/Normed/Group/Ultra.lean
|
theorem exists_norm_multiset_prod_le (s : Multiset ι) [Nonempty ι] {f : ι → M} :
∃ i : ι, (s ≠ 0 → i ∈ s) ∧ ‖(s.map f).prod‖ ≤ ‖f i‖
|
case neg.inr.refine_1
M✝ : Type u_1
ι : Type u_2
inst✝² : SeminormedCommGroup M✝
inst✝¹ : IsUltrametricDist M✝
inst✝ : Nonempty ι
f : ι → M✝
inhabited_h : Inhabited ι
a : ι
t : Multiset ι
M : ι
hMs : t ≠ 0 → M ∈ t
hM : ‖(Multiset.map f t).prod‖ ≤ ‖f M‖
hMa : ‖f a‖ < ‖f M‖
ht : t ≠ 0
⊢ a ::ₘ t ≠ 0 → M ∈ a ::ₘ t
|
simp [hMs ht]
|
no goals
|
76b69fa578fb2276
|
CategoryTheory.Functor.OfSequence.map_comp
|
Mathlib/CategoryTheory/Functor/OfSequence.lean
|
@[reassoc]
lemma map_comp (i j k : ℕ) (hij : i ≤ j) (hjk : j ≤ k) :
map f i k (hij.trans hjk) = map f i j hij ≫ map f j k hjk
|
C : Type u_1
inst✝ : Category.{u_2, u_1} C
j : ℕ
hj :
∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)) (k : ℕ) (hij : 0 ≤ j) (hjk : j ≤ k),
map f 0 k ⋯ = map f 0 j hij ≫ map f j k hjk
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
k : ℕ
hij : 0 ≤ j + 1
hjk : j + 1 ≤ k + 1 + 1
⊢ 0 ≤ j
|
omega
|
no goals
|
68dd418237e48daa
|
isLocalStructomorphOn_contDiffGroupoid_iff
|
Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean
|
theorem isLocalStructomorphOn_contDiffGroupoid_iff (f : PartialHomeomorph M M') :
LiftPropOn (contDiffGroupoid n I).IsLocalStructomorphWithinAt f f.source ↔
ContMDiffOn I I n f f.source ∧ ContMDiffOn I I n f.symm f.target
|
case h.e'_12
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
H✝ : Type u_3
inst✝⁵ : TopologicalSpace H✝
I : ModelWithCorners 𝕜 E H✝
M : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H✝ M
n : WithTop ℕ∞
inst✝² : IsManifold I n M
M' : Type u_5
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H✝ M'
IsM' : IsManifold I n M'
f : PartialHomeomorph M M'
h₁ : ContMDiffOn I I n (↑f) f.source
h₂ : ContMDiffOn I I n (↑f.symm) f.target
x : M
hx : x ∈ f.source
c : PartialHomeomorph M H✝ := chartAt H✝ x
c' : PartialHomeomorph M' H✝ := chartAt H✝ (↑f x)
hx' : ↑c x ∈ ↑c.symm ⁻¹' f.source
y : E
hy :
(↑I.symm y ∈ c'.target ∧ ↑c'.symm (↑I.symm y) ∈ f.target ∧ ↑f.symm (↑c'.symm (↑I.symm y)) ∈ c.source) ∧ y ∈ range ↑I
H :
ContinuousWithinAt (↑f.symm) (f.symm ≫ₕ c).source (↑(extChartAt I (↑f x)).symm y) ∧
ContDiffWithinAt 𝕜 n (↑(extChartAt I x) ∘ ↑f.symm ∘ ↑(extChartAt I (↑f x)).symm)
(↑(extChartAt I (↑f x)).symm ⁻¹' (f.symm ≫ₕ c).source ∩ range ↑I)
(↑(extChartAt I (↑f x)) (↑(extChartAt I (↑f x)).symm y))
hy' : ↑(extChartAt I (↑f x)).symm y ∈ c'.source
hy'' : ↑f.symm (↑(extChartAt I (↑f x)).symm y) ∈ c.source
⊢ y = ↑(extChartAt I (↑f x)) (↑(extChartAt I (↑f x)).symm y)
|
simp only [c', hy, mfld_simps]
|
no goals
|
52ce8090c899c270
|
Finset.card_le_card_shatterer
|
Mathlib/Combinatorics/SetFamily/Shatter.lean
|
/-- Pajor's variant of the **Sauer-Shelah lemma**. -/
lemma card_le_card_shatterer (𝒜 : Finset (Finset α)) : #𝒜 ≤ #𝒜.shatterer
|
case neg.intro.intro
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ : Finset (Finset α)
a : α
𝒜 : Finset (Finset α)
ih₀ : #(nonMemberSubfamily a 𝒜) ≤ #(nonMemberSubfamily a 𝒜).shatterer
ih₁ : #(memberSubfamily a 𝒜) ≤ #(memberSubfamily a 𝒜).shatterer
ℬ : Finset (Finset α) := image (insert a) ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
s t : Finset α
ht : t.erase a ⊆ s
hs : (memberSubfamily a 𝒜).Shatters s ∧ (nonMemberSubfamily a 𝒜).Shatters s
ha : a ∉ t
u : Finset α
hu : u ∈ 𝒜 ∧ a ∉ u
hsu : s ∩ u = t.erase a
⊢ ∃ u ∈ 𝒜, insert a s ∩ u = t
|
refine ⟨_, hu.1, ?_⟩
|
case neg.intro.intro
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ : Finset (Finset α)
a : α
𝒜 : Finset (Finset α)
ih₀ : #(nonMemberSubfamily a 𝒜) ≤ #(nonMemberSubfamily a 𝒜).shatterer
ih₁ : #(memberSubfamily a 𝒜) ≤ #(memberSubfamily a 𝒜).shatterer
ℬ : Finset (Finset α) := image (insert a) ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
s t : Finset α
ht : t.erase a ⊆ s
hs : (memberSubfamily a 𝒜).Shatters s ∧ (nonMemberSubfamily a 𝒜).Shatters s
ha : a ∉ t
u : Finset α
hu : u ∈ 𝒜 ∧ a ∉ u
hsu : s ∩ u = t.erase a
⊢ insert a s ∩ u = t
|
7b2e4a2398648e6f
|
contMDiffOn_isOpenEmbedding_symm
|
Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
|
/-- If the `ChartedSpace` structure on a manifold `M` is given by an open embedding `e : M → H`,
then the inverse of `e` is `C^n`. -/
lemma contMDiffOn_isOpenEmbedding_symm [Nonempty M] :
haveI := h.singletonChartedSpace; ContMDiffOn I I
n (IsOpenEmbedding.toPartialHomeomorph e h).symm (range e)
|
case right
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
H : Type u_3
inst✝² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹ : TopologicalSpace M
e : M → H
h : IsOpenEmbedding e
n : WithTop ℕ∞
inst✝ : Nonempty M
this✝ : IsManifold I ω M
z✝ : H
hz✝ : M
z : E
hz :
z ∈
(extChartAt I z✝).target ∩
↑(extChartAt I z✝).symm ⁻¹'
(range e ∩ ↑(IsOpenEmbedding.toPartialHomeomorph e h).symm ⁻¹' (extChartAt I hz✝).source)
this : ↑I.symm z ∈ range e
⊢ z ∈ range ↑I
|
exact mem_of_subset_of_mem (extChartAt_target_subset_range _) hz.1
|
no goals
|
9da88db66ba876e6
|
MeasureTheory.withDensity_inv_same_le
|
Mathlib/MeasureTheory/Measure/WithDensity.lean
|
lemma withDensity_inv_same_le {μ : Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
(μ.withDensity f).withDensity f⁻¹ ≤ μ
|
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
this : (f * fun x => (f x)⁻¹) ≤ᶠ[ae μ] 1
⊢ μ.withDensity 1 ≤ μ
|
rw [withDensity_one]
|
no goals
|
d6fc55f433791664
|
Ideal.ramificationIdx_tower
|
Mathlib/NumberTheory/RamificationInertia/Basic.lean
|
theorem ramificationIdx_tower [IsDedekindDomain S] [IsDedekindDomain T] {f : R →+* S} {g : S →+* T}
{p : Ideal R} {P : Ideal S} {Q : Ideal T} [hpm : P.IsPrime] [hqm : Q.IsPrime]
(hg0 : map g P ≠ ⊥) (hfg : map (g.comp f) p ≠ ⊥) (hg : map g P ≤ Q) :
ramificationIdx (g.comp f) p Q = ramificationIdx f p P * ramificationIdx g P Q
|
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
f : R →+* S
g : S →+* T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map g P ≠ ⊥
hfg : map (g.comp f) p ≠ ⊥
hg : map g P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
⊢ ramificationIdx (g.comp f) p Q = ramificationIdx f p P * ramificationIdx g P Q
|
have hq0 : Q ≠ ⊥ := ne_bot_of_le_ne_bot hg0 hg
|
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : IsDedekindDomain S
inst✝ : IsDedekindDomain T
f : R →+* S
g : S →+* T
p : Ideal R
P : Ideal S
Q : Ideal T
hpm : P.IsPrime
hqm : Q.IsPrime
hg0 : map g P ≠ ⊥
hfg : map (g.comp f) p ≠ ⊥
hg : map g P ≤ Q
hf0 : map f p ≠ ⊥
hp0 : P ≠ ⊥
hq0 : Q ≠ ⊥
⊢ ramificationIdx (g.comp f) p Q = ramificationIdx f p P * ramificationIdx g P Q
|
57208cf4c3229e08
|
spectrum.isUnit_one_sub_smul_of_lt_inv_radius
|
Mathlib/Analysis/Normed/Algebra/Spectrum.lean
|
theorem isUnit_one_sub_smul_of_lt_inv_radius {a : A} {z : 𝕜} (h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹) :
IsUnit (1 - z • a)
|
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
hu : IsUnit (u⁻¹ • 1 - a)
⊢ IsUnit (1 - z • a)
|
rwa [IsUnit.smul_sub_iff_sub_inv_smul, inv_inv u] at hu
|
no goals
|
7c24ad5e48f2ddf5
|
WittVector.IsPoly₂.ext
|
Mathlib/RingTheory/WittVector/IsPoly.lean
|
theorem ext [Fact p.Prime] {f g} (hf : IsPoly₂ p f) (hg : IsPoly₂ p g)
(h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ),
ghostComponent n (f x y) = ghostComponent n (g x y)) :
∀ (R) [_Rcr : CommRing R] (x y : 𝕎 R), f x y = g x y
|
case mk'.intro.mk'.intro.h
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n : ℕ
⊢ ∀ (n : ℕ), (bind₁ φ) (wittPolynomial p ℤ n) = (bind₁ ψ) (wittPolynomial p ℤ n)
|
intro k
|
case mk'.intro.mk'.intro.h
p : ℕ
inst✝ : Fact (Nat.Prime p)
f g : ⦃R : Type u⦄ → [inst : CommRing R] → 𝕎 R → 𝕎 R → 𝕎 R
h : ∀ (R : Type u) [_Rcr : CommRing R] (x y : 𝕎 R) (n : ℕ), (ghostComponent n) (f x y) = (ghostComponent n) (g x y)
φ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hf : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (f x y).coeff = fun n => peval (φ n) ![x.coeff, y.coeff]
ψ : ℕ → MvPolynomial (Fin 2 × ℕ) ℤ
hg : ∀ ⦃R : Type u⦄ [inst : CommRing R] (x y : 𝕎 R), (g x y).coeff = fun n => peval (ψ n) ![x.coeff, y.coeff]
R✝ : Type u
_Rcr✝ : CommRing R✝
x✝ y✝ : 𝕎 R✝
n k : ℕ
⊢ (bind₁ φ) (wittPolynomial p ℤ k) = (bind₁ ψ) (wittPolynomial p ℤ k)
|
b9c1b5d0e9359092
|
MeasureTheory.Measure.compProd_congr
|
Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean
|
lemma compProd_congr [IsSFiniteKernel κ] [IsSFiniteKernel η] (h : κ =ᵐ[μ] η) :
μ ⊗ₘ κ = μ ⊗ₘ η
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
h : ⇑κ =ᶠ[ae μ] ⇑η
⊢ μ ⊗ₘ κ = μ ⊗ₘ η
|
rw [compProd, compProd]
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
h : ⇑κ =ᶠ[ae μ] ⇑η
⊢ (Kernel.const Unit μ ⊗ₖ Kernel.prodMkLeft Unit κ) () = (Kernel.const Unit μ ⊗ₖ Kernel.prodMkLeft Unit η) ()
|
e162d78e79d6acec
|
map_nnratCast
|
Mathlib/Data/Rat/Cast/Defs.lean
|
@[simp] lemma map_nnratCast [DivisionSemiring α] [DivisionSemiring β] [RingHomClass F α β] (f : F)
(q : ℚ≥0) : f q = q
|
F : Type u_1
α : Type u_3
β : Type u_4
inst✝³ : FunLike F α β
inst✝² : DivisionSemiring α
inst✝¹ : DivisionSemiring β
inst✝ : RingHomClass F α β
f : F
q : ℚ≥0
⊢ f ↑q = ↑q
|
simp_rw [NNRat.cast_def, map_div₀, map_natCast]
|
no goals
|
a8ce91f6202640c8
|
Ordinal.div_opow_log_pos
|
Mathlib/SetTheory/Ordinal/Exponential.lean
|
theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o)
|
case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ 0 < o / b ^ log b o
|
rw [div_pos (opow_ne_zero _ hb.ne')]
|
case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ b ^ log b o ≤ o
|
2c0e70dc5e0a443e
|
Equiv.Perm.IsCycle.support_pow_eq_iff
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
theorem IsCycle.support_pow_eq_iff (hf : IsCycle f) {n : ℕ} :
support (f ^ n) = support f ↔ ¬orderOf f ∣ n
|
case mpr
α : Type u_2
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
n : ℕ
⊢ ¬f ^ n = 1 → (f ^ n).support = f.support
|
intro H
|
case mpr
α : Type u_2
f : Perm α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
hf : f.IsCycle
n : ℕ
H : ¬f ^ n = 1
⊢ (f ^ n).support = f.support
|
fcbe5eebb49a4140
|
SetTheory.PGame.insertLeft_equiv_of_lf
|
Mathlib/SetTheory/Game/PGame.lean
|
/-- Adding a gift horse left option does not change the value of `x`. A gift horse left option is
a game `x'` with `x' ⧏ x`. It is called "gift horse" because it seems like Left has gotten the
"gift" of a new option, but actually the value of the game did not change. -/
lemma insertLeft_equiv_of_lf {x x' : PGame} (h : x' ⧏ x) : insertLeft x x' ≈ x
|
x x' : PGame
h : x' ⧏ x
⊢ x.insertLeft x' ≤ x ∧ x ≤ x.insertLeft x'
|
constructor
|
case left
x x' : PGame
h : x' ⧏ x
⊢ x.insertLeft x' ≤ x
case right
x x' : PGame
h : x' ⧏ x
⊢ x ≤ x.insertLeft x'
|
a22baa14b9ba85ee
|
tendsto_iff_dist_tendsto_zero
|
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0)
|
α : Type u
β : Type v
inst✝ : PseudoMetricSpace α
f : β → α
x : Filter β
a : α
⊢ Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0)
|
rw [← nhds_comap_dist a, tendsto_comap_iff, Function.comp_def]
|
no goals
|
7dfa88c1fd4326bf
|
Set.subset_pair_iff_eq
|
Mathlib/Data/Set/Insert.lean
|
theorem subset_pair_iff_eq {x y : α} : s ⊆ {x, y} ↔ s = ∅ ∨ s = {x} ∨ s = {y} ∨ s = {x, y}
|
α : Type u
s : Set α
x y : α
h : x ∈ s → {y} = s \ {x} → s = {x, y}
⊢ (s = ∅ ∨ s = {y}) ∨ x ∈ s ∧ ((s = ∅ ∨ s = {x}) ∨ s \ {x} = {y}) → s = ∅ ∨ s = {x} ∨ s = {y} ∨ s = {x, y}
|
tauto
|
no goals
|
73841c1827c5fd17
|
SetTheory.PGame.rightMoves_mul_iff
|
Mathlib/SetTheory/Game/Basic.lean
|
/-- The right options of `x * y` are the left options of `x * (-y)` and of `(-x) * y` of the first
kind, up to equivalence. -/
lemma rightMoves_mul_iff {x y : PGame} (P : Game → Prop) :
(∀ k, P ⟦(x * y).moveRight k⟧) ↔
(∀ i j, P (-⟦mulOption x (-y) i j⟧)) ∧ (∀ i j, P (-⟦mulOption (-x) y i j⟧))
|
case mk.mk.mpr.inr.mk
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h :
(∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),
P (-⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧)) ∧
∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),
P (-⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧)
i : β✝¹
j : α✝
⊢ P ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveRight (Sum.inr (i, j))⟧
|
convert h.2 i j using 1
|
case h.e'_1
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h :
(∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves),
P (-⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧)) ∧
∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves),
P (-⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧)
i : β✝¹
j : α✝
⊢ ⟦(mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveRight (Sum.inr (i, j))⟧ =
-⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧
|
8f7e2212785f2917
|
exists_hasDerivWithinAt_eq_of_gt_of_lt
|
Mathlib/Analysis/Calculus/Darboux.lean
|
theorem exists_hasDerivWithinAt_eq_of_gt_of_lt (hab : a ≤ b)
(hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x) {m : ℝ} (hma : f' a < m)
(hmb : m < f' b) : m ∈ f' '' Ioo a b
|
a b : ℝ
f f' : ℝ → ℝ
hab : a ≤ b
hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x
m : ℝ
hma : f' a < m
hmb : m < f' b
hab' : a < b
g : ℝ → ℝ := fun x => f x - m * x
hg : ∀ x ∈ Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x
c : ℝ
cmem : c ∈ Icc a b
hc : IsMinOn g (Icc a b) c
cmem' : c ∈ Ioo a b
⊢ Icc a b ∈ 𝓝 c
|
rwa [← mem_interior_iff_mem_nhds, interior_Icc]
|
no goals
|
226337358bf5c20c
|
FormalMultilinearSeries.ofScalars_radius_eq_top_of_tendsto
|
Mathlib/Analysis/Analytic/OfScalars.lean
|
theorem ofScalars_radius_eq_top_of_tendsto (hc : ∀ᶠ n in atTop, c n ≠ 0)
(hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)) : (ofScalars E c).radius = ⊤
|
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
⊢ Summable fun n => ‖ofScalars E c n‖ * ↑r' ^ n
|
by_cases hrz : r' = 0
|
case pos
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : r' = 0
⊢ Summable fun n => ‖ofScalars E c n‖ * ↑r' ^ n
case neg
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ Summable fun n => ‖ofScalars E c n‖ * ↑r' ^ n
|
fed995db04eea233
|
CategoryTheory.PreGaloisCategory.card_aut_le_card_fiber_of_connected
|
Mathlib/CategoryTheory/Galois/Basic.lean
|
/-- If `A` is connected, the cardinality of `Aut A` is smaller than the cardinality of the
fiber of `A`. -/
lemma card_aut_le_card_fiber_of_connected (A : C) [IsConnected A] :
Nat.card (Aut A) ≤ Nat.card (F.obj A)
|
C : Type u₁
inst✝³ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝² : PreGaloisCategory C
inst✝¹ : FiberFunctor F
A : C
inst✝ : IsConnected A
⊢ Nat.card (Aut A) ≤ Nat.card (F.obj A).carrier
|
have h : Nonempty (F.obj A) := inferInstance
|
C : Type u₁
inst✝³ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝² : PreGaloisCategory C
inst✝¹ : FiberFunctor F
A : C
inst✝ : IsConnected A
h : Nonempty (F.obj A).carrier
⊢ Nat.card (Aut A) ≤ Nat.card (F.obj A).carrier
|
22d78ca2a5f87417
|
MeasureTheory.MemLp.condExpL2_ae_eq_condExp'
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
|
lemma MemLp.condExpL2_ae_eq_condExp' (hm : m ≤ m₀) (hf1 : Integrable f μ) (hf2 : MemLp f 2 μ)
[SigmaFinite (μ.trim hm)] : condExpL2 E 𝕜 hm hf2.toLp =ᵐ[μ] μ[f | m]
|
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁵ : RCLike 𝕜
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : InnerProductSpace 𝕜 E
hm : m ≤ m₀
hf1 : Integrable f μ
hf2 : MemLp f 2 μ
inst✝ : SigmaFinite (μ.trim hm)
s : Set α
hs : MeasurableSet s
htop : μ s < ⊤
⊢ ∫ (x : α) in s, ↑↑(toLp f hf2) x ∂μ = ∫ (x : α) in s, f x ∂μ
|
refine setIntegral_congr_ae (hm _ hs) ?_
|
α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁵ : RCLike 𝕜
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : InnerProductSpace 𝕜 E
hm : m ≤ m₀
hf1 : Integrable f μ
hf2 : MemLp f 2 μ
inst✝ : SigmaFinite (μ.trim hm)
s : Set α
hs : MeasurableSet s
htop : μ s < ⊤
⊢ ∀ᵐ (x : α) ∂μ, x ∈ s → ↑↑(toLp f hf2) x = f x
|
704d12b97c45b511
|
Topology.isEmbedding_sigmaMap
|
Mathlib/Topology/Constructions.lean
|
lemma Topology.isEmbedding_sigmaMap {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)}
(h : Injective f₁) : IsEmbedding (Sigma.map f₁ f₂) ↔ ∀ i, IsEmbedding (f₂ i)
|
ι : Type u_5
κ : Type u_6
σ : ι → Type u_7
τ : κ → Type u_8
inst✝¹ : (i : ι) → TopologicalSpace (σ i)
inst✝ : (k : κ) → TopologicalSpace (τ k)
f₁ : ι → κ
f₂ : (i : ι) → σ i → τ (f₁ i)
h : Injective f₁
⊢ IsEmbedding (Sigma.map f₁ f₂) ↔ ∀ (i : ι), IsEmbedding (f₂ i)
|
simp only [isEmbedding_iff, Injective.sigma_map, isInducing_sigmaMap h, forall_and,
h.sigma_map_iff]
|
no goals
|
46e9dcc1781a2c29
|
Int.cast_add
|
Mathlib/Data/Int/Cast/Basic.lean
|
theorem cast_add : ∀ m n, ((m + n : ℤ) : R) = m + n
| (m : ℕ), (n : ℕ) => by simp [-Int.natCast_add, ← Int.ofNat_add]
| (m : ℕ), -[n+1] => by erw [cast_subNatNat, cast_natCast, cast_negSucc, sub_eq_add_neg]
| -[m+1], (n : ℕ) => by
erw [cast_subNatNat, cast_natCast, cast_negSucc, sub_eq_iff_eq_add, add_assoc,
eq_neg_add_iff_add_eq, ← Nat.cast_add, ← Nat.cast_add, Nat.add_comm]
| -[m+1], -[n+1] =>
show (-[m + n + 1+1] : R) = _ by
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
|
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ
⊢ ↑-[m + n + 1+1] = ↑-[m+1] + ↑-[n+1]
|
rw [cast_negSucc, cast_negSucc, cast_negSucc, ← neg_add_rev, ← Nat.cast_add,
Nat.add_right_comm m n 1, Nat.add_assoc, Nat.add_comm]
|
no goals
|
1b92f9b12b62a7d0
|
Array.extract_size
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem extract_size (as : Array α) : as.extract 0 as.size = as
|
case h₂.e_i
α : Type u_1
as : Array α
i✝ : Nat
hi₁✝ : i✝ < as.extract.size
hi₂✝ : i✝ < as.size
⊢ 0 + i✝ = i✝
|
rw [Nat.zero_add]
|
no goals
|
e927f7c749316b3a
|
IsPreconnected.iUnion_of_reflTransGen
|
Mathlib/Topology/Connected/Basic.lean
|
theorem IsPreconnected.iUnion_of_reflTransGen {ι : Type*} {s : ι → Set α}
(H : ∀ i, IsPreconnected (s i))
(K : ∀ i j, ReflTransGen (fun i j : ι => (s i ∩ s j).Nonempty) i j) :
IsPreconnected (⋃ n, s n)
|
α : Type u
inst✝ : TopologicalSpace α
ι : Type u_3
s : ι → Set α
H : ∀ (i : ι), IsPreconnected (s i)
K : ∀ (i j : ι), ReflTransGen (fun i j => (s i ∩ s j).Nonempty) i j
i : ι
x✝¹ : i ∈ univ
j : ι
x✝ : j ∈ univ
⊢ ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ univ) i j
|
simpa [mem_univ] using K i j
|
no goals
|
57b13d0a6db6e6e6
|
Projectivization.card
|
Mathlib/LinearAlgebra/Projectivization/Cardinality.lean
|
/-- Fraction free cardinality formula for the points of `ℙ k V` if `k` and `V` are finite
(for silly reasons the formula also holds when `k` and `V` are infinite).
See `Projectivization.card'` and `Projectivization.card''` for other spellings of the formula. -/
lemma card : Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
|
case inr
k : Type u_1
V : Type u_2
inst✝² : DivisionRing k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
a✝ : Nontrivial V
this :
∀ (k : Type u_1) (V : Type u_2) [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V],
Nontrivial V → Finite k → Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
h : ¬Finite k
⊢ Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
|
simp only [not_finite_iff_infinite] at h
|
case inr
k : Type u_1
V : Type u_2
inst✝² : DivisionRing k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
a✝ : Nontrivial V
this :
∀ (k : Type u_1) (V : Type u_2) [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V],
Nontrivial V → Finite k → Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
h : Infinite k
⊢ Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
|
1d5a95497a0b5f13
|
CategoryTheory.extensiveTopology.presheafIsLocallySurjective_iff
|
Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
|
lemma extensiveTopology.presheafIsLocallySurjective_iff [FinitaryPreExtensive C] {F G : Cᵒᵖ ⥤ D}
(f : F ⟶ G) [PreservesFiniteProducts F] [PreservesFiniteProducts G]
[PreservesFiniteProducts (forget D)] : Presheaf.IsLocallySurjective (extensiveTopology C) f ↔
∀ (X : C), Function.Surjective (f.app (op X))
|
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_4, u_1} C
inst✝⁶ : Category.{u_5, u_2} D
FD : D → D → Type u_3
CD : D → Type w
inst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁴ : ConcreteCategory D FD
inst✝³ : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ D
f : F ⟶ G
inst✝² : PreservesFiniteProducts F
inst✝¹ : PreservesFiniteProducts G
inst✝ : PreservesFiniteProducts (forget D)
⊢ Presheaf.IsLocallySurjective (extensiveTopology C) f ↔
∀ (X : C), Function.Surjective ⇑(ConcreteCategory.hom (f.app (op X)))
|
constructor
|
case mp
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_4, u_1} C
inst✝⁶ : Category.{u_5, u_2} D
FD : D → D → Type u_3
CD : D → Type w
inst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁴ : ConcreteCategory D FD
inst✝³ : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ D
f : F ⟶ G
inst✝² : PreservesFiniteProducts F
inst✝¹ : PreservesFiniteProducts G
inst✝ : PreservesFiniteProducts (forget D)
⊢ Presheaf.IsLocallySurjective (extensiveTopology C) f →
∀ (X : C), Function.Surjective ⇑(ConcreteCategory.hom (f.app (op X)))
case mpr
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_4, u_1} C
inst✝⁶ : Category.{u_5, u_2} D
FD : D → D → Type u_3
CD : D → Type w
inst✝⁵ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
inst✝⁴ : ConcreteCategory D FD
inst✝³ : FinitaryPreExtensive C
F G : Cᵒᵖ ⥤ D
f : F ⟶ G
inst✝² : PreservesFiniteProducts F
inst✝¹ : PreservesFiniteProducts G
inst✝ : PreservesFiniteProducts (forget D)
⊢ (∀ (X : C), Function.Surjective ⇑(ConcreteCategory.hom (f.app (op X)))) →
Presheaf.IsLocallySurjective (extensiveTopology C) f
|
49c86ecf3311443d
|
BitVec.forall_zero_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem forall_zero_iff {P : BitVec 0 → Prop} :
(∀ v, P v) ↔ P 0#0
|
P : BitVec 0 → Prop
⊢ (∀ (v : BitVec 0), P v) ↔ P 0#0
|
constructor
|
case mp
P : BitVec 0 → Prop
⊢ (∀ (v : BitVec 0), P v) → P 0#0
case mpr
P : BitVec 0 → Prop
⊢ P 0#0 → ∀ (v : BitVec 0), P v
|
14570500a054ffbe
|
FirstOrder.Language.dlo_age
|
Mathlib/ModelTheory/Order.lean
|
lemma dlo_age [Language.order.Structure M] [Mdlo : M ⊨ Language.order.dlo] [Nonempty M] :
Language.order.age M = {M : CategoryTheory.Bundled.{w'} Language.order.Structure |
Finite M ∧ M ⊨ Language.order.linearOrderTheory}
|
M : Type w'
inst✝¹ : Language.order.Structure M
Mdlo : M ⊨ Language.order.dlo
inst✝ : Nonempty M
⊢ Language.order.age M = {M | Finite ↑M ∧ ↑M ⊨ Language.order.linearOrderTheory}
|
rw [age]
|
M : Type w'
inst✝¹ : Language.order.Structure M
Mdlo : M ⊨ Language.order.dlo
inst✝ : Nonempty M
⊢ {N | FG Language.order ↑N ∧ Nonempty (↑N ↪[Language.order] M)} =
{M | Finite ↑M ∧ ↑M ⊨ Language.order.linearOrderTheory}
|
833eff06b402bf5b
|
CochainComplex.exists_iso_single
|
Mathlib/Algebra/Homology/Embedding/CochainComplex.lean
|
/-- A cochain complex that is both strictly `≤ n` and `≥ n` is isomorphic to
a complex `(single _ _ n).obj M` for some object `M`. -/
lemma exists_iso_single [HasZeroObject C] (n : ℤ) [K.IsStrictlyGE n] [K.IsStrictlyLE n] :
∃ (M : C), Nonempty (K ≅ (single _ _ n).obj M) :=
⟨K.X n, ⟨{
hom := mkHomToSingle (𝟙 _) (fun i (hi : i + 1 = n) ↦
(K.isZero_of_isStrictlyGE n i (by omega)).eq_of_src _ _)
inv := mkHomFromSingle (𝟙 _) (fun i (hi : n + 1 = i) ↦
(K.isZero_of_isStrictlyLE n i (by omega)).eq_of_tgt _ _)
hom_inv_id
|
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : HasZeroMorphisms C
K : CochainComplex C ℤ
inst✝² : HasZeroObject C
n : ℤ
inst✝¹ : K.IsStrictlyGE n
inst✝ : K.IsStrictlyLE n
i : ℤ
hi : n < i
⊢ n < i
|
omega
|
no goals
|
5826bfce8c98b5e6
|
MeasureTheory.hasFDerivAt_convolution_right_with_param
|
Mathlib/Analysis/Convolution.lean
|
theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)) (q₀ : P × G)
(hq₀ : q₀.1 ∈ s) :
HasFDerivAt (fun q : P × G => (f ⋆[L, μ] g q.1) q.2)
((f ⋆[L.precompR (P × G), μ] fun x : G => fderiv 𝕜 (↿g) (q₀.1, x)) q₀.2) q₀
|
case intro.intro.intro.intro
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A : ∀ p ∈ s, Continuous (g p)
A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
ε C : ℝ
εpos : 0 < ε
h₀ε : ball q₀.1 ε ⊆ s
hε : ∀ (p : P) (x : G), ‖p - q₀.1‖ < ε → ‖g' (p, x)‖ ≤ C
I1 : ∀ᶠ (x : P × G) in 𝓝 q₀, AEStronglyMeasurable (fun a => (L (f a)) (g x.1 (x.2 - a))) μ
I2 : Integrable (fun a => (L (f a)) (g q₀.1 (q₀.2 - a))) μ
I3 : AEStronglyMeasurable (fun a => (L (f a)).comp (g' (q₀.1, q₀.2 - a))) μ
K' : Set G := -k + {q₀.2}
K'_def : K' = -k + {q₀.2}
⊢ HasFDerivAt (fun q => (f ⋆[L, μ] g q.1) q.2) ((f ⋆[precompR (P × G) L, μ] fun x => fderiv 𝕜 (↿g) (q₀.1, x)) q₀.2) q₀
|
have hK' : IsCompact K' := hk.neg.add isCompact_singleton
|
case intro.intro.intro.intro
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ univ)
q₀ : P × G
hq₀ : q₀.1 ∈ s
g' : P × G → P × G →L[𝕜] E' := fderiv 𝕜 ↿g
A : ∀ p ∈ s, Continuous (g p)
A' : ∀ (q : P × G), q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q
g'_zero : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g' (p, x) = 0
ε C : ℝ
εpos : 0 < ε
h₀ε : ball q₀.1 ε ⊆ s
hε : ∀ (p : P) (x : G), ‖p - q₀.1‖ < ε → ‖g' (p, x)‖ ≤ C
I1 : ∀ᶠ (x : P × G) in 𝓝 q₀, AEStronglyMeasurable (fun a => (L (f a)) (g x.1 (x.2 - a))) μ
I2 : Integrable (fun a => (L (f a)) (g q₀.1 (q₀.2 - a))) μ
I3 : AEStronglyMeasurable (fun a => (L (f a)).comp (g' (q₀.1, q₀.2 - a))) μ
K' : Set G := -k + {q₀.2}
K'_def : K' = -k + {q₀.2}
hK' : IsCompact K'
⊢ HasFDerivAt (fun q => (f ⋆[L, μ] g q.1) q.2) ((f ⋆[precompR (P × G) L, μ] fun x => fderiv 𝕜 (↿g) (q₀.1, x)) q₀.2) q₀
|
681ec723c7fd0b70
|
Vector.vector₃_induction
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean
|
theorem vector₃_induction (P : Vector (Vector (Vector α n) m) k → Prop)
(of : ∀ (xss : Array (Array (Array α))) (h₁ : xss.size = k) (h₂ : ∀ xs ∈ xss, xs.size = m)
(h₃ : ∀ xs ∈ xss, ∀ x ∈ xs, x.size = n),
P (mk (xss.attach.map (fun ⟨xs, m⟩ =>
mk (xs.attach.map (fun ⟨x, m'⟩ =>
mk x (h₃ xs m x m'))) (by simpa using h₂ xs m))) (by simpa using h₁)))
(ass : Vector (Vector (Vector α n) m) k) : P ass
|
α : Type u_1
n m k : Nat
P : Vector (Vector (Vector α n) m) k → Prop
of :
∀ (xss : Array (Array (Array α))) (h₁ : xss.size = k) (h₂ : ∀ (xs : Array (Array α)), xs ∈ xss → xs.size = m)
(h₃ : ∀ (xs : Array (Array α)), xs ∈ xss → ∀ (x : Array α), x ∈ xs → x.size = n),
P
{
toArray :=
Array.map
(fun x =>
match x with
| ⟨xs, m_1⟩ =>
{
toArray :=
Array.map
(fun x =>
match x with
| ⟨x, m'⟩ => { toArray := x, size_toArray := ⋯ })
xs.attach,
size_toArray := ⋯ })
xss.attach,
size_toArray := ⋯ }
ass : Vector (Vector (Vector α n) m) k
⊢ ∀ (xs : Array (Array α)),
xs ∈ (map (fun as => (map toArray as).toArray) ass).toArray → ∀ (x : Array α), x ∈ xs → x.size = n
|
simp
|
no goals
|
ac402192d6221673
|
CochainComplex.mappingCone.ext_cochain_from_iff
|
Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
|
lemma ext_cochain_from_iff (i j : ℤ) (hij : i + 1 = j)
{K : CochainComplex C ℤ} {γ₁ γ₂ : Cochain (mappingCone φ) K j} :
γ₁ = γ₂ ↔
(inl φ).comp γ₁ (show _ = i by omega) = (inl φ).comp γ₂ (by omega) ∧
(Cochain.ofHom (inr φ)).comp γ₁ (zero_add j) =
(Cochain.ofHom (inr φ)).comp γ₂ (zero_add j)
|
case mpr.intro.h
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
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain (mappingCone φ) K j
p q : ℤ
hpq : p + j = q
h₁ : (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq
h₂ : (inr φ).f p ≫ γ₁.v p q ⋯ = (inr φ).f p ≫ γ₂.v p q ⋯
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
(inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq
|
exact ⟨h₁, h₂⟩
|
no goals
|
dbcc60702c58cd99
|
blimsup_cthickening_ae_le_of_eventually_mul_le_aux
|
Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
|
theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
(hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
{M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
(blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
(blimsup (fun i => cthickening (r₂ i) (s i)) atTop p : Set α)
|
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
⊢ μ W = 0
|
by_contra contra
|
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
⊢ False
|
f93df810108e5d67
|
CategoryTheory.ProjectiveResolution.fromLeftDerivedZero_eq
|
Mathlib/CategoryTheory/Abelian/LeftDerived.lean
|
lemma ProjectiveResolution.fromLeftDerivedZero_eq
{X : C} (P : ProjectiveResolution X) (F : C ⥤ D) [F.Additive] :
F.fromLeftDerivedZero.app X = (P.isoLeftDerivedObj F 0).hom ≫
(ChainComplex.isoHomologyι₀ _).hom ≫
P.fromLeftDerivedZero' F
|
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasProjectiveResolutions C
inst✝¹ : Abelian D
X : C
P : ProjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
⊢ (lift (𝟙 X) P (projectiveResolution X)).f 0 ≫ (projectiveResolution X).π.f 0 = P.π.f 0 ≫ 𝟙 X
|
simp
|
no goals
|
4e0dec9d0923a13a
|
int_prod_range_nonneg
|
Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean
|
theorem int_prod_range_nonneg (m : ℤ) (n : ℕ) (hn : Even n) :
0 ≤ ∏ k ∈ Finset.range n, (m - k)
|
case intro.succ.inr
m : ℤ
n : ℕ
ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)
k : ℕ
hmk : ↑k < m
⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))
|
exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk)
|
no goals
|
845865b3fc92c079
|
Finmap.erase_erase
|
Mathlib/Data/Finmap.lean
|
theorem erase_erase {a a' : α} {s : Finmap β} : erase a (erase a' s) = erase a' (erase a s) :=
induction_on s fun s => ext (by simp only [AList.erase_erase, erase_toFinmap])
|
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a a' : α
s✝ : Finmap β
s : AList β
⊢ (erase a (erase a' ⟦s⟧)).entries = (erase a' (erase a ⟦s⟧)).entries
|
simp only [AList.erase_erase, erase_toFinmap]
|
no goals
|
9578036a56fd462f
|
Matrix.IsHermitian.cfc_eq
|
Mathlib/LinearAlgebra/Matrix/HermitianFunctionalCalculus.lean
|
lemma cfc_eq (f : ℝ → ℝ) : cfc f A = hA.cfc f
|
n : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : Fintype n
inst✝ : DecidableEq n
A : Matrix n n 𝕜
hA : A.IsHermitian
f : ℝ → ℝ
hA' : IsSelfAdjoint A
this : cfcHom hA' = hA.cfcAux
⊢ hA.cfcAux { toFun := (spectrum ℝ A).restrict f, continuous_toFun := ⋯ } = hA.cfc f
|
simp only [cfcAux_apply, ContinuousMap.coe_mk, Function.comp_def, Set.restrict_apply,
IsHermitian.cfc]
|
no goals
|
c4d21dcf8fd1c96c
|
list_mul_sum
|
Mathlib/NumberTheory/Ostrowski.lean
|
private lemma list_mul_sum {R : Type*} [CommSemiring R] {T : Type*} (l : List T) (y : R) (x : R) :
(l.mapIdx fun i _ => x * y ^ i).sum = x * (l.mapIdx fun i _ => y ^ i).sum
|
case e_a
R : Type u_1
inst✝ : CommSemiring R
T : Type u_2
l : List T
y x : R
⊢ List.map (fun x_1 => x • y ^ x_1.2) l.zipIdx = List.map (fun x_1 => x • x_1) (List.map (fun x => y ^ x.2) l.zipIdx)
|
simp
|
no goals
|
7a4121929e1fc1a5
|
analyticAt_inverse
|
Mathlib/Analysis/Analytic/Constructions.lean
|
/-- If `A` is a normed algebra over `𝕜` with summable geometric series, then inversion on `A` is
analytic at any unit. -/
@[fun_prop]
lemma analyticAt_inverse {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{A : Type*} [NormedRing A] [NormedAlgebra 𝕜 A] [HasSummableGeomSeries A] (z : Aˣ) :
AnalyticAt 𝕜 Ring.inverse (z : A)
|
case inr
𝕜 : Type u_9
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_10
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
z : Aˣ
hA : Nontrivial A
f1 : A → A := fun a => a * z.inv
f2 : A → A := fun b => Ring.inverse (1 - b)
f3 : A → A := fun c => 1 - z.inv * c
feq : ∀ᶠ (y : A) in 𝓝 ↑z, (f1 ∘ f2 ∘ f3) y = Ring.inverse y
⊢ AnalyticAt 𝕜 Ring.inverse ↑z
|
apply AnalyticAt.congr _ feq
|
𝕜 : Type u_9
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_10
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
z : Aˣ
hA : Nontrivial A
f1 : A → A := fun a => a * z.inv
f2 : A → A := fun b => Ring.inverse (1 - b)
f3 : A → A := fun c => 1 - z.inv * c
feq : ∀ᶠ (y : A) in 𝓝 ↑z, (f1 ∘ f2 ∘ f3) y = Ring.inverse y
⊢ AnalyticAt 𝕜 (f1 ∘ f2 ∘ f3) ↑z
|
a28e69020b72c97e
|
Std.Sat.AIG.mkGateCached.go_decl_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/CachedLemmas.lean
|
theorem mkGateCached.go_decl_eq (aig : AIG α) (input : GateInput aig) :
∀ (idx : Nat) (h1) (h2), (go aig input).aig.decls[idx]'h1 = aig.decls[idx]'h2
|
case h_2.h_2
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
input : aig.GateInput
res : Entrypoint α
x✝² : Option (CacheHit aig.decls (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv))
heq✝² : aig.cache.get? (Decl.gate input.lhs.ref.gate input.rhs.ref.gate input.lhs.inv input.rhs.inv) = none
x✝¹ x✝ : Decl α
linv✝ rinv✝ : Bool
heq✝¹ : aig.decls[input.lhs.ref.gate] = Decl.const false
heq✝ : input.lhs.inv = false
hres : { decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false = res
idx✝ : Nat
h1✝ : idx✝ < ({ decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false).aig.decls.size
h2✝ : idx✝ < aig.decls.size
⊢ ({ decls := aig.decls, cache := aig.cache, invariant := ⋯ }.mkConstCached false).aig.decls[idx✝] = aig.decls[idx✝]
|
rw [LawfulOperator.decl_eq (f := AIG.mkConstCached)]
|
no goals
|
f550d988487b2b3c
|
IntermediateField.exists_algHom_adjoin_of_splits
|
Mathlib/FieldTheory/Extension.lean
|
theorem exists_algHom_adjoin_of_splits : ∃ φ : adjoin F S →ₐ[F] K, φ.comp (inclusion hL) = f
|
case intro.intro.refine_2.H
F : Type u_1
E : Type u_2
K : Type u_3
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Field K
inst✝¹ : Algebra F E
inst✝ : Algebra F K
S : Set E
hK : ∀ s ∈ S, IsIntegral F s ∧ Splits (algebraMap F K) (minpoly F s)
L : IntermediateField F E
f : ↥L →ₐ[F] K
hL : L ≤ adjoin F S
φ : Lifts F E K
hfφ : { carrier := L, emb := f } ≤ φ
hφ : IsMax φ
x✝ : ↥L
⊢ ((φ.emb.comp (inclusion ⋯)).comp (inclusion hL)) x✝ = f x✝
|
apply hfφ.2
|
no goals
|
8633288b622106c6
|
MvPolynomial.map_mvPolynomial_eq_eval₂
|
Mathlib/RingTheory/Polynomial/Basic.lean
|
theorem map_mvPolynomial_eq_eval₂ {S : Type*} [CommRing S] [Finite σ] (ϕ : MvPolynomial σ R →+* S)
(p : MvPolynomial σ R) :
ϕ p = MvPolynomial.eval₂ (ϕ.comp MvPolynomial.C) (fun s => ϕ (MvPolynomial.X s)) p
|
case intro.e_f.h
R : Type u
σ : Type v
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
inst✝ : Finite σ
ϕ : MvPolynomial σ R →+* S
p : MvPolynomial σ R
val✝ : Fintype σ
x✝ : σ →₀ ℕ
⊢ ϕ ((monomial x✝) (coeff x✝ p)) = (ϕ.comp C) (coeff x✝ p) * ∏ i : σ, ϕ (X i) ^ x✝ i
|
simp only [monomial_eq, ϕ.map_pow, map_prod ϕ, ϕ.comp_apply, ϕ.map_mul, Finsupp.prod_pow]
|
no goals
|
b7d61a3a15b39281
|
Rat.add_nonneg
|
Mathlib/Algebra/Order/Ring/Unbundled/Rat.lean
|
protected lemma add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b :=
numDenCasesOn' a fun n₁ d₁ h₁ ↦ numDenCasesOn' b fun n₂ d₂ h₂ ↦ by
have d₁0 : 0 < (d₁ : ℤ) := mod_cast Nat.pos_of_ne_zero h₁
have d₂0 : 0 < (d₂ : ℤ) := mod_cast Nat.pos_of_ne_zero h₂
simp only [d₁0, d₂0, h₁, h₂, Int.mul_pos, divInt_nonneg_iff_of_pos_right, divInt_add_divInt, Ne,
Nat.cast_eq_zero, not_false_iff]
intro n₁0 n₂0
apply Int.add_nonneg <;> apply Int.mul_nonneg <;> · first | assumption | apply Int.ofNat_zero_le
|
a b : ℚ
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
⊢ 0 ≤ n₁ /. ↑d₁ → 0 ≤ n₂ /. ↑d₂ → 0 ≤ n₁ /. ↑d₁ + n₂ /. ↑d₂
|
have d₁0 : 0 < (d₁ : ℤ) := mod_cast Nat.pos_of_ne_zero h₁
|
a b : ℚ
n₁ : ℤ
d₁ : ℕ
h₁ : d₁ ≠ 0
n₂ : ℤ
d₂ : ℕ
h₂ : d₂ ≠ 0
d₁0 : 0 < ↑d₁
⊢ 0 ≤ n₁ /. ↑d₁ → 0 ≤ n₂ /. ↑d₂ → 0 ≤ n₁ /. ↑d₁ + n₂ /. ↑d₂
|
54edc86bdbe9e3bd
|
MonoidAlgebra.mem_ideal_span_of_image
|
Mathlib/Algebra/MonoidAlgebra/Ideal.lean
|
theorem MonoidAlgebra.mem_ideal_span_of_image [Monoid G] [Semiring k] {s : Set G}
{x : MonoidAlgebra k G} :
x ∈ Ideal.span (MonoidAlgebra.of k G '' s) ↔ ∀ m ∈ x.support, ∃ m' ∈ s, ∃ d, m = d * m'
|
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : MonoidAlgebra k G
RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := {p | ∀ m ∈ p.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
⊢ x ∈ Ideal.span (⇑(of k G) '' s) ↔ x ∈ RHS
|
constructor
|
case mp
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : MonoidAlgebra k G
RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := {p | ∀ m ∈ p.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
⊢ x ∈ Ideal.span (⇑(of k G) '' s) → x ∈ RHS
case mpr
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : MonoidAlgebra k G
RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := {p | ∀ m ∈ p.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
⊢ x ∈ RHS → x ∈ Ideal.span (⇑(of k G) '' s)
|
848b4f1553241180
|
MeasureTheory.OuterMeasure.isometry_comap_mkMetric
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem isometry_comap_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) {f : X → Y} (hf : Isometry f)
(H : Monotone m ∨ Surjective f) : comap f (mkMetric m) = mkMetric m
|
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
⊢ ⨆ i, ⨆ (_ : i > 0), (comap f) (boundedBy (extend fun s x => m (diam s))) =
⨆ r, ⨆ (_ : r > 0), boundedBy (extend fun s x => m (diam s))
|
refine surjective_id.iSup_congr id fun ε => surjective_id.iSup_congr id fun hε => ?_
|
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y
m : ℝ≥0∞ → ℝ≥0∞
f : X → Y
hf : Isometry f
H : Monotone m ∨ Surjective f
ε : ℝ≥0∞
hε : id ε > 0
⊢ (comap f) (boundedBy (extend fun s x => m (diam s))) = boundedBy (extend fun s x => m (diam s))
|
15bf55d09123a3a2
|
WeierstrassCurve.Affine.CoordinateRing.map_mk
|
Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
|
lemma map_mk (x : R[X][Y]) : map W f (mk W x) = mk (W.map f) (x.map <| mapRingHom f)
|
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
W : Affine R
f : R →+* S
x : R[X][Y]
⊢ eval₂ (AdjoinRoot.of (WeierstrassCurve.map W f).toAffine.polynomial)
(AdjoinRoot.root (WeierstrassCurve.map W f).toAffine.polynomial) (Polynomial.map (mapRingHom f) x) =
(mk (WeierstrassCurve.map W f)) (Polynomial.map (mapRingHom f) x)
|
exact AdjoinRoot.aeval_eq <| x.map <| mapRingHom f
|
no goals
|
ac21b53691daba88
|
Module.End.independent_iInf_maxGenEigenspace_of_forall_mapsTo
|
Mathlib/LinearAlgebra/Eigenspace/Pi.lean
|
lemma independent_iInf_maxGenEigenspace_of_forall_mapsTo
(h : ∀ i j φ, MapsTo (f i) ((f j).maxGenEigenspace φ) ((f j).maxGenEigenspace φ)) :
iSupIndep fun χ : ι → R ↦ ⨅ i, (f i).maxGenEigenspace (χ i)
|
ι : Type u_1
R : Type u_2
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
f : ι → End R M
inst✝ : NoZeroSMulDivisors R M
h : ∀ (l : ι) (χ : ι → R), MapsTo (⇑(f l)) (⨅ i, ↑((f i).maxGenEigenspace (χ i))) (⨅ i, ↑((f i).maxGenEigenspace (χ i)))
χ₁ χ₂ : ι → R
s : Finset (ι → R)
_n : χ₂ ∉ s
ih : χ₁ ∉ s → Disjoint (⨅ i, (f i).maxGenEigenspace (χ₁ i)) (s.sup fun χ => ⨅ i, (f i).maxGenEigenspace (χ i))
hχ₁₂ : χ₁ ∉ insert χ₂ s
⊢ ?m.53136
|
rwa [Finset.mem_insert, not_or] at hχ₁₂
|
no goals
|
4dd72b81f13de433
|
DihedralGroup.commProb_reciprocal
|
Mathlib/GroupTheory/CommutingProbability.lean
|
theorem commProb_reciprocal (n : ℕ) :
commProb (Product (reciprocalFactors n)) = 1 / n
|
case neg.inr.hb
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : Odd n
this : n / 4 + 1 < n
key : n % 4 = 1 ∨ n % 4 = 3
hn : Odd (n % 4)
⊢ ¬4 * (n % 4 * n) * (n / 4 + 1) = 0
|
have := hn.pos.ne'
|
case neg.inr.hb
n : ℕ
h0 : ¬n = 0
h1 : ¬n = 1
h2 : Odd n
this✝ : n / 4 + 1 < n
key : n % 4 = 1 ∨ n % 4 = 3
hn : Odd (n % 4)
this : n % 4 ≠ 0
⊢ ¬4 * (n % 4 * n) * (n / 4 + 1) = 0
|
36923ae66db82ec7
|
NonemptyFinLinOrd.mono_iff_injective
|
Mathlib/Order/Category/NonemptyFinLinOrd.lean
|
theorem mono_iff_injective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :
Mono f ↔ Function.Injective f
|
A B : NonemptyFinLinOrd
f : A ⟶ B
a✝ : Mono f
a₁ a₂ : ↑A.toLinOrd
h : (ConcreteCategory.hom f) a₁ = (ConcreteCategory.hom f) a₂
X : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))
x✝² x✝¹ : ULift.{u, 0} (Fin 1)
x✝ : x✝² ≤ x✝¹
⊢ (fun x => a₁) x✝² ≤ (fun x => a₁) x✝¹
|
rfl
|
no goals
|
70ff9efadffc8dad
|
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
|
Mathlib/MeasureTheory/Integral/Layercake.lean
|
theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
(μ : Measure α) [SFinite μ]
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t)
|
case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
(fun x => (Ioc 0 (f x)).indicator (fun t => ENNReal.ofReal (g t)) s) = fun x =>
ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x => 1) s * (Ici s).indicator (fun x => 1) (f x)
h : ¬0 < s
⊢ (Ioi 0).indicator (fun x => 1) s ≠ ⊤
|
simp [h]
|
no goals
|
028c71a79fc18349
|
mdifferentiableWithinAt_iff_target_of_mem_source
|
Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
|
theorem mdifferentiableWithinAt_iff_target_of_mem_source
[IsManifold I' 1 M'] {x : M} {y : M'} (hy : f x ∈ (chartAt H' y).source) :
MDifferentiableWithinAt I I' f s x ↔
ContinuousWithinAt f s x ∧ MDifferentiableWithinAt I 𝓘(𝕜, E') (extChartAt I' y ∘ f) s x
|
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (chartAt H' y).source
hf : ContinuousWithinAt f s x
⊢ True ∧
DifferentiableWithinAtProp I I' ((↑(chartAt H' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
(↑(chartAt H x) x) ↔
ContinuousWithinAt (↑(extChartAt I' y) ∘ f) s x ∧
DifferentiableWithinAtProp I 𝓘(𝕜, E') ((↑(extChartAt I' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
(↑(chartAt H x) x)
|
rw [← extChartAt_source I'] at hy
|
𝕜 : Type u_1
inst✝¹¹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁸ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁷ : TopologicalSpace M
inst✝⁶ : ChartedSpace H M
E' : Type u_5
inst✝⁵ : NormedAddCommGroup E'
inst✝⁴ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝³ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝² : TopologicalSpace M'
inst✝¹ : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
hy : f x ∈ (extChartAt I' y).source
hf : ContinuousWithinAt f s x
⊢ True ∧
DifferentiableWithinAtProp I I' ((↑(chartAt H' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
(↑(chartAt H x) x) ↔
ContinuousWithinAt (↑(extChartAt I' y) ∘ f) s x ∧
DifferentiableWithinAtProp I 𝓘(𝕜, E') ((↑(extChartAt I' y) ∘ f) ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s)
(↑(chartAt H x) x)
|
5253979901641d91
|
Filter.limsup_eq_iInf_iSup_of_nat'
|
Mathlib/Order/LiminfLimsup.lean
|
theorem limsup_eq_iInf_iSup_of_nat' {u : ℕ → α} : limsup u atTop = ⨅ n : ℕ, ⨆ i : ℕ, u (i + n)
|
α : Type u_1
inst✝ : CompleteLattice α
u : ℕ → α
⊢ limsup u atTop = ⨅ n, ⨆ i, u (i + n)
|
simp only [limsup_eq_iInf_iSup_of_nat, iSup_ge_eq_iSup_nat_add]
|
no goals
|
e0daa982b35aac36
|
Substring.ValidFor.prevn
|
Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean
|
theorem prevn : ∀ {s}, ValidFor l (m₁.reverse ++ m₂) r s →
∀ n, s.prevn n ⟨utf8Len m₁⟩ = ⟨utf8Len (m₁.drop n)⟩
| _, _, 0 => by simp [Substring.prevn]
| s, h, n+1 => by
simp only [Substring.prevn]
match m₁ with
| [] => simp
| c::m₁ =>
rw [List.reverse_cons, List.append_assoc] at h
have := h.prev; simp at this; simp [this, h.prevn n]
|
l m₂ r m₁✝ : List Char
s : Substring
n : Nat
c : Char
m₁ : List Char
h : ValidFor l ((c :: m₁).reverse ++ m₂) r s
⊢ s.prevn n (s.prev { byteIdx := utf8Len (c :: m₁) }) = { byteIdx := utf8Len (List.drop (n + 1) (c :: m₁)) }
|
rw [List.reverse_cons, List.append_assoc] at h
|
l m₂ r m₁✝ : List Char
s : Substring
n : Nat
c : Char
m₁ : List Char
h : ValidFor l (m₁.reverse ++ ([c] ++ m₂)) r s
⊢ s.prevn n (s.prev { byteIdx := utf8Len (c :: m₁) }) = { byteIdx := utf8Len (List.drop (n + 1) (c :: m₁)) }
|
335aa626418f50e2
|
Fin.pred_eq_iff_eq_succ
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean
|
theorem pred_eq_iff_eq_succ {n : Nat} {i : Fin (n + 1)} (hi : i ≠ 0) {j : Fin n} :
i.pred hi = j ↔ i = j.succ :=
⟨fun h => by simp only [← h, Fin.succ_pred], fun h => by simp only [h, Fin.pred_succ]⟩
|
n : Nat
i : Fin (n + 1)
hi : i ≠ 0
j : Fin n
h : i.pred hi = j
⊢ i = j.succ
|
simp only [← h, Fin.succ_pred]
|
no goals
|
779f547746d96aea
|
AlgebraicGeometry.reduce_to_affine_global
|
Mathlib/AlgebraicGeometry/Properties.lean
|
theorem reduce_to_affine_global (P : ∀ {X : Scheme} (_ : X.Opens), Prop)
{X : Scheme} (U : X.Opens)
(h₁ : ∀ (X : Scheme) (U : X.Opens),
(∀ x : U, ∃ (V : _) (_ : x.1 ∈ V) (_ : V ⟶ U), P V) → P U)
(h₂ : ∀ (X Y) (f : X ⟶ Y) [IsOpenImmersion f],
∃ (U : X.Opens) (V : Y.Opens), U = ⊤ ∧ V = f.opensRange ∧ (P U → P V))
(h₃ : ∀ R : CommRingCat, P (X := Spec R) ⊤) : P U
|
P : {X : Scheme} → X.Opens → Prop
X : Scheme
U : X.Opens
h₁ : ∀ (X : Scheme) (U : X.Opens), (∀ (x : ↥U), ∃ V, ∃ (_ : ↑x ∈ V), ∃ x, P V) → P U
h₂ : ∀ (X Y : Scheme) (f : X ⟶ Y) [inst : IsOpenImmersion f], ∃ U V, U = ⊤ ∧ V = Scheme.Hom.opensRange f ∧ (P U → P V)
h₃ : ∀ (R : CommRingCat), P ⊤
⊢ P U
|
apply h₁
|
case a
P : {X : Scheme} → X.Opens → Prop
X : Scheme
U : X.Opens
h₁ : ∀ (X : Scheme) (U : X.Opens), (∀ (x : ↥U), ∃ V, ∃ (_ : ↑x ∈ V), ∃ x, P V) → P U
h₂ : ∀ (X Y : Scheme) (f : X ⟶ Y) [inst : IsOpenImmersion f], ∃ U V, U = ⊤ ∧ V = Scheme.Hom.opensRange f ∧ (P U → P V)
h₃ : ∀ (R : CommRingCat), P ⊤
⊢ ∀ (x : ↥U), ∃ V, ∃ (_ : ↑x ∈ V), ∃ x, P V
|
338a13b39aa3aecd
|
Polynomial.Gal.splits_in_splittingField_of_comp
|
Mathlib/FieldTheory/PolynomialGaloisGroup.lean
|
theorem splits_in_splittingField_of_comp (hq : q.natDegree ≠ 0) :
p.Splits (algebraMap F (p.comp q).SplittingField)
|
case pos
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r => Splits (algebraMap F (r.comp q).SplittingField) r
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : ¬p₁.comp q = 0
h₂ : p₂.comp q = 0
⊢ P (p₁ * p₂)
|
rcases comp_eq_zero_iff.mp h₂ with h | h
|
case pos.inl
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r => Splits (algebraMap F (r.comp q).SplittingField) r
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : ¬p₁.comp q = 0
h₂ : p₂.comp q = 0
h : p₂ = 0
⊢ P (p₁ * p₂)
case pos.inr
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r => Splits (algebraMap F (r.comp q).SplittingField) r
key1 : ∀ {r : F[X]}, Irreducible r → P r
p₁ p₂ : F[X]
hp₁ : P p₁
hp₂ : P p₂
h₁ : ¬p₁.comp q = 0
h₂ : p₂.comp q = 0
h : eval (q.coeff 0) p₂ = 0 ∧ q = C (q.coeff 0)
⊢ P (p₁ * p₂)
|
781c423315763045
|
dist_integral_mulExpNegMulSq_comp_le
|
Mathlib/Analysis/SpecialFunctions/MulExpNegMulSqIntegral.lean
|
theorem dist_integral_mulExpNegMulSq_comp_le (f : E →ᵇ ℝ)
{A : Subalgebra ℝ C(E, ℝ)} (hA : A.SeparatesPoints)
(hbound : ∀ g ∈ A, ∃ C, ∀ x y : E, dist (g x) (g y) ≤ C)
(heq : ∀ g ∈ A, ∫ x, (g : E → ℝ) x ∂P = ∫ x, (g : E → ℝ) x ∂P') (hε : 0 < ε) :
|∫ x, mulExpNegMulSq ε (f x) ∂P - ∫ x, mulExpNegMulSq ε (f x) ∂P'| ≤ 6 * sqrt ε
|
ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalgebra ℝ C(E, ℝ)
hA : A.SeparatesPoints
hbound : ∀ g ∈ A, ∃ C, ∀ (x y : E), dist (g x) (g y) ≤ C
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
hε : 0 < ε
⊢ |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E), ε.mulExpNegMulSq (f x) ∂P'| ≤ 6 * √ε
|
by_cases hPP' : P = 0 ∧ P' = 0
|
case pos
ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalgebra ℝ C(E, ℝ)
hA : A.SeparatesPoints
hbound : ∀ g ∈ A, ∃ C, ∀ (x y : E), dist (g x) (g y) ≤ C
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
hε : 0 < ε
hPP' : P = 0 ∧ P' = 0
⊢ |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E), ε.mulExpNegMulSq (f x) ∂P'| ≤ 6 * √ε
case neg
ε : ℝ
E : Type u_2
inst✝⁶ : MeasurableSpace E
inst✝⁵ : PseudoEMetricSpace E
inst✝⁴ : BorelSpace E
inst✝³ : CompleteSpace E
inst✝² : SecondCountableTopology E
P P' : Measure E
inst✝¹ : IsFiniteMeasure P
inst✝ : IsFiniteMeasure P'
f : E →ᵇ ℝ
A : Subalgebra ℝ C(E, ℝ)
hA : A.SeparatesPoints
hbound : ∀ g ∈ A, ∃ C, ∀ (x y : E), dist (g x) (g y) ≤ C
heq : ∀ g ∈ A, ∫ (x : E), g x ∂P = ∫ (x : E), g x ∂P'
hε : 0 < ε
hPP' : ¬(P = 0 ∧ P' = 0)
⊢ |∫ (x : E), ε.mulExpNegMulSq (f x) ∂P - ∫ (x : E), ε.mulExpNegMulSq (f x) ∂P'| ≤ 6 * √ε
|
6c7688b02378f81c
|
finprod_cond_eq_prod_of_cond_iff
|
Mathlib/Algebra/BigOperators/Finprod.lean
|
theorem finprod_cond_eq_prod_of_cond_iff (f : α → M) {p : α → Prop} {t : Finset α}
(h : ∀ {x}, f x ≠ 1 → (p x ↔ x ∈ t)) : (∏ᶠ (i) (_ : p i), f i) = ∏ i ∈ t, f i
|
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
p : α → Prop
t : Finset α
h : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)
s : Set α := {x | p x}
this : mulSupport (s.mulIndicator f) ⊆ ↑t
⊢ ∏ᶠ (i : α) (_ : i ∈ s), f i = ∏ i ∈ t, f i
|
rw [finprod_mem_def, finprod_eq_prod_of_mulSupport_subset _ this]
|
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
p : α → Prop
t : Finset α
h : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)
s : Set α := {x | p x}
this : mulSupport (s.mulIndicator f) ⊆ ↑t
⊢ ∏ i ∈ t, s.mulIndicator f i = ∏ i ∈ t, f i
|
d1687f56373c7d13
|
Ideal.finrank_quotient_map
|
Mathlib/NumberTheory/RamificationInertia/Basic.lean
|
theorem finrank_quotient_map [IsDomain S] [IsDedekindDomain R] [Algebra K L]
[Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L]
[hp : p.IsMaximal] [Module.Finite R S] :
finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L
|
R : Type u
inst✝¹⁴ : CommRing R
S : Type v
inst✝¹³ : CommRing S
p : Ideal R
inst✝¹² : Algebra R S
K : Type u_1
inst✝¹¹ : Field K
inst✝¹⁰ : Algebra R K
L : Type u_2
inst✝⁹ : Field L
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
hRK : IsFractionRing R K
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Algebra K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsScalarTower R S L
hp : p.IsMaximal
inst✝ : Module.Finite R S
⊢ finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L
|
let ι := Module.Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
|
R : Type u
inst✝¹⁴ : CommRing R
S : Type v
inst✝¹³ : CommRing S
p : Ideal R
inst✝¹² : Algebra R S
K : Type u_1
inst✝¹¹ : Field K
inst✝¹⁰ : Algebra R K
L : Type u_2
inst✝⁹ : Field L
inst✝⁸ : Algebra S L
inst✝⁷ : IsFractionRing S L
hRK : IsFractionRing R K
inst✝⁶ : IsDomain S
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Algebra K L
inst✝³ : Algebra R L
inst✝² : IsScalarTower R K L
inst✝¹ : IsScalarTower R S L
hp : p.IsMaximal
inst✝ : Module.Finite R S
ι : Type v := Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
⊢ finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L
|
c261b0ceb1fb92db
|
Equiv.prodCongr_refl_right
|
Mathlib/Logic/Equiv/Basic.lean
|
theorem prodCongr_refl_right (e : β₁ ≃ β₂) :
prodCongr e (Equiv.refl α₁) = prodCongrLeft fun _ => e
|
case H.mk
α₁ : Type u_9
β₁ : Type u_11
β₂ : Type u_12
e : β₁ ≃ β₂
a : β₁
b : α₁
⊢ (e.prodCongr (Equiv.refl α₁)) (a, b) = (prodCongrLeft fun x => e) (a, b)
|
simp
|
no goals
|
c69778327141b887
|
MeasureTheory.exists_le_lowerSemicontinuous_lintegral_ge
|
Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
|
theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf : Measurable f) {ε : ℝ≥0∞}
(εpos : ε ≠ 0) :
∃ g : α → ℝ≥0∞,
(∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε
|
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0∞
hf : Measurable f
ε : ℝ≥0∞
εpos : ε ≠ 0
δ : ℕ → ℝ≥0∞
δpos : ∀ (i : ℕ), 0 < δ i
hδ : ∑' (i : ℕ), δ i < ε
g : ℕ → α → ℝ≥0
f_le_g : ∀ (n : ℕ) (x : α), (SimpleFunc.eapproxDiff f n) x ≤ g n x
gcont : ∀ (n : ℕ), LowerSemicontinuous (g n)
hg : ∀ (n : ℕ), ∫⁻ (x : α), ↑(g n x) ∂μ ≤ ∫⁻ (x : α), ↑((SimpleFunc.eapproxDiff f n) x) ∂μ + δ n
⊢ ∫⁻ (a : α), ∑' (i : ℕ), ↑((SimpleFunc.eapproxDiff f i) a) ∂μ ≤ ∫⁻ (x : α), f x ∂μ
|
simp_rw [SimpleFunc.tsum_eapproxDiff f hf, le_refl]
|
no goals
|
73601cdcac15c7f7
|
CategoryTheory.toNerve₂.mk_naturality_σ00
|
Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean
|
lemma toNerve₂.mk_naturality_σ00 : toNerve₂.mk.naturalityProperty F (σ₂ (n := 0) 0)
|
case h
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
x : X.obj (op { obj := [0], property := ⋯ })
⊢ (X.map (σ₂ 0 ⋯ ⋯).op ≫ (fun n => mk.app F (unop n)) (op { obj := [0 + 1], property := ⋯ })) x =
ComposableArrows.mk₁ (𝟙 (F.obj x))
|
have := ReflPrefunctor.map_id F x
|
case h
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
x : X.obj (op { obj := [0], property := ⋯ })
this : F.map (𝟙rq x) = 𝟙rq (F.obj x)
⊢ (X.map (σ₂ 0 ⋯ ⋯).op ≫ (fun n => mk.app F (unop n)) (op { obj := [0 + 1], property := ⋯ })) x =
ComposableArrows.mk₁ (𝟙 (F.obj x))
|
8a989a466710331b
|
Tendsto.isLindelof_insert_range_of_coLindelof
|
Mathlib/Topology/Compactness/Lindelof.lean
|
theorem Tendsto.isLindelof_insert_range_of_coLindelof {f : X → Y} {y}
(hf : Tendsto f (coLindelof X) (𝓝 y)) (hfc : Continuous f) :
IsLindelof (insert y (range f))
|
case neg.intro.intro.intro.intro.intro.intro.intro.intro
X : Type u
Y : Type v
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
y✝ : Y
hf : Tendsto f (coLindelof X) (𝓝 y✝)
hfc : Continuous f
l : Filter Y
hne : l.NeBot
inst✝ : CountableInterFilter l
hle : l ≤ 𝓟 (insert y✝ (range f))
s : Set Y
hsy : s ∈ 𝓝 y✝
t : Set Y
htl : t ∈ l
hd : Disjoint s t
K : Set X
hKc : IsLindelof K
hKs : Kᶜ ⊆ f ⁻¹' s
this : f '' K ∈ l
y : Y
hy : y ∈ f '' K
hyl : ClusterPt y l
⊢ ∃ x ∈ insert y✝ (range f), ClusterPt x l
|
exact ⟨y, Or.inr <| image_subset_range _ _ hy, hyl⟩
|
no goals
|
272add40fb4ffebd
|
WittVector.isocrystal_classification
|
Mathlib/RingTheory/WittVector/Isocrystal.lean
|
theorem isocrystal_classification (k : Type*) [Field k] [IsAlgClosed k] [CharP k p] (V : Type*)
[AddCommGroup V] [Isocrystal p k V] (h_dim : finrank K(p, k) V = 1) :
∃ m : ℤ, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)
|
case h
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V := LinearEquiv.ofBijective F₀ ⋯
c : StandardOneDimIsocrystal p k m
⊢ (φ(p, k) b * (φ(p, k) c * a)) • x = (b * (↑p ^ m * φ(p, k) c)) • x
|
congr 1
|
case h.e_a
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V := LinearEquiv.ofBijective F₀ ⋯
c : StandardOneDimIsocrystal p k m
⊢ φ(p, k) b * (φ(p, k) c * a) = b * (↑p ^ m * φ(p, k) c)
|
d25976e4ff95234e
|
Trivialization.tendsto_nhds_iff
|
Mathlib/Topology/FiberBundle/Trivialization.lean
|
theorem tendsto_nhds_iff {α : Type*} {l : Filter α} {f : α → Z} {z : Z} (hz : z ∈ e.source) :
Tendsto f l (𝓝 z) ↔
Tendsto (proj ∘ f) l (𝓝 (proj z)) ∧ Tendsto (fun x ↦ (e (f x)).2) l (𝓝 (e z).2)
|
case neg
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
α : Type u_5
l : Filter α
f : α → Z
z : Z
hz : z ∈ proj ⁻¹' e.baseSet
hl : ¬∀ᶠ (x : α) in l, f x ∈ proj ⁻¹' e.baseSet
⊢ Tendsto (proj ∘ f) l (𝓝 (proj z)) → ¬Tendsto (fun x => (↑e (f x)).2) l (𝓝 (↑e z).2)
|
exact fun h _ ↦ hl <| h <| e.open_baseSet.mem_nhds hz
|
no goals
|
fd4173837552b8ba
|
IsProperMap.prodMap
|
Mathlib/Topology/Maps/Proper/Basic.lean
|
/-- A binary product of proper maps is proper. -/
lemma IsProperMap.prodMap {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) :
IsProperMap (Prod.map f g)
|
case right.intro.intro.intro.intro
X : Type u_1
Y : Type u_2
Z : Type u_3
W : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace W
f : X → Y
g : Z → W
hf : Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x
hg : Continuous g ∧ ∀ ⦃𝒰 : Ultrafilter Z⦄ ⦃y : W⦄, Tendsto g (↑𝒰) (𝓝 y) → ∃ x, g x = y ∧ ↑𝒰 ≤ 𝓝 x
𝒰 : Ultrafilter (X × Z)
y : Y
w : W
hyw : Tendsto (fun n => (Prod.map f g n).1) (↑𝒰) (𝓝 y) ∧ Tendsto (fun n => (Prod.map f g n).2) (↑𝒰) (𝓝 w)
x : X
hxy : f x = y
hx : ↑(Ultrafilter.map fst 𝒰) ≤ 𝓝 x
z : Z
hzw : g z = w
hz : ↑(Ultrafilter.map snd 𝒰) ≤ 𝓝 z
⊢ Tendsto fst (↑𝒰) (𝓝 x) ∧ Tendsto snd (↑𝒰) (𝓝 z)
|
exact ⟨hx, hz⟩
|
no goals
|
137a7c14fc720f6e
|
CategoryTheory.Limits.limit.pre_pre
|
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
theorem limit.pre_pre [h : HasLimit (D ⋙ E ⋙ F)] : haveI : HasLimit ((D ⋙ E) ⋙ F) := h
limit.pre F E ≫ limit.pre (E ⋙ F) D = limit.pre F (D ⋙ E)
|
case w
J : Type u₁
inst✝⁵ : Category.{v₁, u₁} J
K : Type u₂
inst✝⁴ : Category.{v₂, u₂} K
C : Type u
inst✝³ : Category.{v, u} C
F : J ⥤ C
inst✝² : HasLimit F
E : K ⥤ J
inst✝¹ : HasLimit (E ⋙ F)
L : Type u₃
inst✝ : Category.{v₃, u₃} L
D : L ⥤ K
h : HasLimit (D ⋙ E ⋙ F)
this : HasLimit ((D ⋙ E) ⋙ F)
j : L
⊢ π F (E.obj (D.obj j)) = π F ((D ⋙ E).obj j)
|
rfl
|
no goals
|
801b9ec35cde80e7
|
MvPFunctor.supp_eq
|
Mathlib/Data/PFunctor/Multivariate/Basic.lean
|
theorem supp_eq {α : TypeVec n} (a : P.A) (f : P.B a ⟹ α) (i) :
@supp.{u} _ P.Obj _ α (⟨a, f⟩ : P α) i = f i '' univ
|
case h.mpr.intro
n : ℕ
P : MvPFunctor.{u} n
α : TypeVec.{u} n
a : P.A
f : P.B a ⟹ α
i : Fin2 n
w✝ : P.B a i
⊢ ∀ ⦃P_1 : (i : Fin2 n) → α i → Prop⦄, (∀ (i : Fin2 n) (x : P.B a i), P_1 i (f i x)) → P_1 i (f i w✝)
|
tauto
|
no goals
|
93ead603ecab0e63
|
ProbabilityTheory.Kernel.sectL_zero
|
Mathlib/Probability/Kernel/Composition/MapComap.lean
|
@[simp] lemma sectL_zero (b : β) : sectL (0 : Kernel (α × β) γ) b = 0
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
b : β
⊢ sectL 0 b = 0
|
simp [sectL]
|
no goals
|
7ceb6da6b75db560
|
MeasureTheory.crossing_eq_crossing_of_upperCrossingTime_lt
|
Mathlib/Probability/Martingale/Upcrossing.lean
|
theorem crossing_eq_crossing_of_upperCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
(h : upperCrossingTime a b f N (n + 1) ω < N) :
upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧
lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
|
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
M : ℕ
hNM : N ≤ M
h : ∃ j ∈ Set.Ico (lowerCrossingTime a b f N n ω) N, f j ω ∈ Set.Ici b
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
⊢ ∃ j ∈ Set.Icc (lowerCrossingTime a b f N n ω) N, f j ω ∈ Set.Ici b
|
obtain ⟨j, hj₁, hj₂⟩ := h
|
case intro.intro
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N n : ℕ
ω : Ω
M : ℕ
hNM : N ≤ M
this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω
j : ℕ
hj₁ : j ∈ Set.Ico (lowerCrossingTime a b f N n ω) N
hj₂ : f j ω ∈ Set.Ici b
⊢ ∃ j ∈ Set.Icc (lowerCrossingTime a b f N n ω) N, f j ω ∈ Set.Ici b
|
4ea6dd2e44976d28
|
ENNReal.toReal_natCast
|
Mathlib/Data/ENNReal/Basic.lean
|
theorem toReal_natCast (n : ℕ) : (n : ℝ≥0∞).toReal = n
|
n : ℕ
⊢ (↑n).toReal = ↑n
|
rw [← ENNReal.ofReal_natCast n, ENNReal.toReal_ofReal (Nat.cast_nonneg _)]
|
no goals
|
8fe83155300b6a55
|
MeasureTheory.Measure.measure_isHaarMeasure_eq_smul_of_isEverywherePos
|
Mathlib/MeasureTheory/Measure/Haar/Unique.lean
|
theorem measure_isHaarMeasure_eq_smul_of_isEverywherePos [LocallyCompactSpace G]
(μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ']
{s : Set G} (hs : MeasurableSet s) (h's : IsEverywherePos μ s) :
μ' s = haarScalarFactor μ' μ • μ s
|
case intro.intro.intro
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed k
k_mem : k ∈ 𝓝 1
⊢ μ' s = ν s
|
have one_k : 1 ∈ k := mem_of_mem_nhds k_mem
|
case intro.intro.intro
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
⊢ μ' s = ν s
|
05951a84047ceabf
|
AlgebraicGeometry.Scheme.IsGermInjective.Spec
|
Mathlib/AlgebraicGeometry/SpreadingOut.lean
|
protected
lemma Scheme.IsGermInjective.Spec
(H : ∀ I : Ideal R, I.IsPrime →
∃ f : R, f ∉ I ∧ ∀ (x y : R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0) :
(Spec R).IsGermInjective
|
case intro.intro.refine_2.intro.intro.intro.mk
R : CommRingCat
H✝ : ∀ (I : Ideal ↑R), I.IsPrime → ∃ f ∉ I, ∀ (x y : ↑R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0
p : ↑↑(Spec R).toPresheafedSpace
f : ↑R
hf : f ∉ p.asIdeal
H : ∀ (x y : ↑R), y * x = 0 → y ∉ p.asIdeal → ∃ n, f ^ n * x = 0
x : ↑R
s : ↥(Submonoid.powers f)
hx : (ConcreteCategory.hom (StructureSheaf.toStalk (↑R) p)) x = 0
y : ↑R
hy : y ∈ p.asIdeal.primeCompl
hy' : ↑⟨y, hy⟩ * x = 0
⊢ IsLocalization.mk' (↑((Spec.structureSheaf ↑R).val.obj (Opposite.op (PrimeSpectrum.basicOpen f)))) x s = 0
|
obtain ⟨n, hn⟩ := H x y hy' hy
|
case intro.intro.refine_2.intro.intro.intro.mk.intro
R : CommRingCat
H✝ : ∀ (I : Ideal ↑R), I.IsPrime → ∃ f ∉ I, ∀ (x y : ↑R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0
p : ↑↑(Spec R).toPresheafedSpace
f : ↑R
hf : f ∉ p.asIdeal
H : ∀ (x y : ↑R), y * x = 0 → y ∉ p.asIdeal → ∃ n, f ^ n * x = 0
x : ↑R
s : ↥(Submonoid.powers f)
hx : (ConcreteCategory.hom (StructureSheaf.toStalk (↑R) p)) x = 0
y : ↑R
hy : y ∈ p.asIdeal.primeCompl
hy' : ↑⟨y, hy⟩ * x = 0
n : ℕ
hn : f ^ n * x = 0
⊢ IsLocalization.mk' (↑((Spec.structureSheaf ↑R).val.obj (Opposite.op (PrimeSpectrum.basicOpen f)))) x s = 0
|
55151e889a7b27d9
|
MeasureTheory.AEEqFun.integrable_iff_mem_L1
|
Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean
|
theorem integrable_iff_mem_L1 {f : α →ₘ[μ] β} : Integrable f ↔ f ∈ (α →₁[μ] β)
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α →ₘ[μ] β
⊢ f.Integrable ↔ f ∈ Lp β 1 μ
|
rw [← integrable_coeFn, ← memLp_one_iff_integrable, Lp.mem_Lp_iff_memLp]
|
no goals
|
7bf211493865f02e
|
Lean.Order.List.monotone_foldrM
|
Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean
|
theorem monotone_foldrM
(f : γ → α → β → m β) (init : β) (xs : List α) (hmono : monotone f) :
monotone (fun x => xs.foldrM (f x) (init := init))
|
case hmono.h.h.h.h
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → β → m β
init : β
xs : List α
hmono : monotone f
s : β
a : α
⊢ monotone f
|
apply hmono
|
no goals
|
a113a300d4f9e2d9
|
Ordinal.ord_cof_eq
|
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
theorem ord_cof_eq (r : α → α → Prop) [IsWellOrder α r] :
∃ S, Unbounded r S ∧ type (Subrel r (· ∈ S)) = (cof (type r)).ord
|
α : Type u
r : α → α → Prop
inst✝ : IsWellOrder α r
S : Set α
hS : Unbounded r S
e : #↑S = (type r).cof
s : ↑S → ↑S → Prop
w✝ : IsWellOrder (↑S) s
e' : (#↑S).ord = type s
T : Set α := {a | ∃ (aS : a ∈ S), ∀ (b : ↑S), s b ⟨a, aS⟩ → r (↑b) a}
a : α
this : {b | ¬r (↑b) a}.Nonempty
b : ↑S := ⋯.min {b | ¬r (↑b) a} this
ba : ¬r (↑b) a
c : ↑S
h : ¬r ↑c ↑b
⊢ ¬s c b
|
exact IsWellFounded.wf.not_lt_min _ this (IsOrderConnected.neg_trans h ba)
|
no goals
|
1935e0c9099cb4dd
|
WeierstrassCurve.Affine.CoordinateRing.XYIdeal_neg_mul
|
Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
|
lemma XYIdeal_neg_mul {x y : F} (h : W.Nonsingular x y) :
XYIdeal W x (C <| W.negY x y) * XYIdeal W x (C y) = XIdeal W x
|
case h.e'_2.h.e'_6
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw :
(Y - C (C y)) * (Y - C (C (W.negY x y))) -
C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) =
W.polynomial * 1
⊢ span
{(mk W) (C (X - C x)), (mk W) (Y - C (C y)), (mk W) (Y - C (C (W.negY x y))),
(AdjoinRoot.mk W.polynomial) (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y)} =
⊤
|
simp_rw [← Set.image_singleton (f := mk W), ← Set.image_insert_eq, ← map_span]
|
case h.e'_2.h.e'_6
F : Type u
inst✝ : Field F
W : Affine F
x y : F
h : W.Nonsingular x y
Y_rw :
(Y - C (C y)) * (Y - C (C (W.negY x y))) -
C (X - C x) * (C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y) =
W.polynomial * 1
⊢ Ideal.map (mk W)
(span
{C (X - C x), Y - C (C y), Y - C (C (W.negY x y)),
C (X ^ 2 + C (x + W.a₂) * X + C (x ^ 2 + W.a₂ * x + W.a₄)) - C (C W.a₁) * Y}) =
⊤
|
a91b7de170ddf2df
|
ENNReal.continuous_pow
|
Mathlib/Topology/Instances/ENNReal/Lemmas.lean
|
theorem continuous_pow (n : ℕ) : Continuous fun a : ℝ≥0∞ => a ^ n
|
case succ
n : ℕ
IH : Continuous fun a => a ^ n
x : ℝ≥0∞
⊢ ContinuousAt (fun a => a ^ n * a) x
|
refine ENNReal.Tendsto.mul (IH.tendsto _) ?_ tendsto_id ?_ <;> by_cases H : x = 0
|
case pos
n : ℕ
IH : Continuous fun a => a ^ n
x : ℝ≥0∞
H : x = 0
⊢ x ^ n ≠ 0 ∨ x ≠ ⊤
case neg
n : ℕ
IH : Continuous fun a => a ^ n
x : ℝ≥0∞
H : ¬x = 0
⊢ x ^ n ≠ 0 ∨ x ≠ ⊤
case pos
n : ℕ
IH : Continuous fun a => a ^ n
x : ℝ≥0∞
H : x = 0
⊢ x ≠ 0 ∨ x ^ n ≠ ⊤
case neg
n : ℕ
IH : Continuous fun a => a ^ n
x : ℝ≥0∞
H : ¬x = 0
⊢ x ≠ 0 ∨ x ^ n ≠ ⊤
|
dd273c5089e505d8
|
TensorProduct.vanishesTrivially_of_sum_tmul_eq_zero
|
Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
|
theorem vanishesTrivially_of_sum_tmul_eq_zero (hm : Submodule.span R (Set.range m) = ⊤)
(hmn : ∑ i, m i ⊗ₜ n i = (0 : M ⊗[R] N)) : VanishesTrivially R m n
|
R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : Fintype ι
m : ι → M
n : ι → N
hm : span R (Set.range m) = ⊤
hmn : ∑ i : ι, m i ⊗ₜ[R] n i = 0
G : (ι →₀ R) →ₗ[R] M := linearCombination R m
hG : G = linearCombination R m
G_basis_eq : ∀ (i : ι), G (Finsupp.single i 1) = m i
⊢ ⊤ ≤ range G
|
rw [← hm]
|
R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : Type u_3
inst✝² : AddCommGroup N
inst✝¹ : Module R N
ι : Type u_4
inst✝ : Fintype ι
m : ι → M
n : ι → N
hm : span R (Set.range m) = ⊤
hmn : ∑ i : ι, m i ⊗ₜ[R] n i = 0
G : (ι →₀ R) →ₗ[R] M := linearCombination R m
hG : G = linearCombination R m
G_basis_eq : ∀ (i : ι), G (Finsupp.single i 1) = m i
⊢ span R (Set.range m) ≤ range G
|
c9130b804ecd308f
|
MeasureTheory.IsProjectiveLimit.isFiniteMeasure
|
Mathlib/MeasureTheory/Constructions/Projective.lean
|
lemma isFiniteMeasure [∀ i, IsFiniteMeasure (P i)] (hμ : IsProjectiveLimit μ P) :
IsFiniteMeasure μ
|
case measure_univ_lt_top
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
μ : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)
hμ : IsProjectiveLimit μ P
⊢ (P ∅) univ < ⊤
|
exact measure_lt_top _ _
|
no goals
|
27fa900db03647ce
|
AEMeasurable.iSup
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
|
theorem AEMeasurable.iSup {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α}
(hf : ∀ i, AEMeasurable (f i) μ) : AEMeasurable (fun b => ⨆ i, f i b) μ
|
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
μ : Measure δ
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), AEMeasurable (f i) μ
b : δ
hb : ∀ (i : ι), f i b = mk (f i) ⋯ b
⊢ ⨆ i, f i b = ⨆ i, mk (f i) ⋯ b
|
simp [hb]
|
no goals
|
bfd3c7492f359749
|
isLocalHomeomorphOn_iff_isOpenEmbedding_restrict
|
Mathlib/Topology/IsLocalHomeomorph.lean
|
theorem isLocalHomeomorphOn_iff_isOpenEmbedding_restrict {f : X → Y} :
IsLocalHomeomorphOn f s ↔ ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsOpenEmbedding (U.restrict f)
|
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
h : ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsOpenEmbedding (U.restrict f)
x : X
hx : x ∈ s
U : Set X
hU : U ∈ 𝓝 x
emb : IsOpenEmbedding (U.restrict f)
⊢ IsOpenEmbedding ((interior U).restrict f)
|
refine emb.comp ⟨.inclusion interior_subset, ?_⟩
|
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set X
f : X → Y
h : ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsOpenEmbedding (U.restrict f)
x : X
hx : x ∈ s
U : Set X
hU : U ∈ 𝓝 x
emb : IsOpenEmbedding (U.restrict f)
⊢ IsOpen (Set.range (Set.inclusion ⋯))
|
87a753b146b13ac6
|
CategoryTheory.SmallObject.SuccStruct.prop.succ_eq
|
Mathlib/CategoryTheory/SmallObject/Iteration/Basic.lean
|
lemma prop.succ_eq {X Y : C} {f : X ⟶ Y} (hf : Φ.prop f) :
Φ.succ X = Y
|
case mk
C : Type u
inst✝ : Category.{v, u} C
Φ : SuccStruct C
X : C
⊢ Φ.succ X = Φ.succ X
|
rfl
|
no goals
|
fd65736fffb252eb
|
FractionalIdeal.le_dual_inv_aux
|
Mathlib/RingTheory/DedekindDomain/Different.lean
|
lemma le_dual_inv_aux (hI : I ≠ 0) (hIJ : I * J ≤ 1) :
J ≤ dual A K I
|
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I J : FractionalIdeal B⁰ L
hI : I ≠ 0
hIJ : I * J ≤ 1
⊢ J ≤ dual A K I
|
rw [dual, dif_neg hI]
|
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst✝⁶ : IsFractionRing A K
inst✝⁵ : FiniteDimensional K L
inst✝⁴ : Algebra.IsSeparable K L
inst✝³ : IsIntegralClosure B A L
inst✝² : IsFractionRing B L
inst✝¹ : IsIntegrallyClosed A
inst✝ : IsDedekindDomain B
I J : FractionalIdeal B⁰ L
hI : I ≠ 0
hIJ : I * J ≤ 1
⊢ J ≤ ⟨(↑I)ᵛ, ⋯⟩
|
af77646439faab8b
|
mul_eq_mul_prime_prod
|
Mathlib/RingTheory/Prime.lean
|
theorem mul_eq_mul_prime_prod {α : Type*} [DecidableEq α] {x y a : R} {s : Finset α} {p : α → R}
(hp : ∀ i ∈ s, Prime (p i)) (hx : x * y = a * ∏ i ∈ s, p i) :
∃ (t u : Finset α) (b c : R),
t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ (x = b * ∏ i ∈ t, p i) ∧ y = c * ∏ i ∈ u, p i
|
R : Type u_1
inst✝¹ : CancelCommMonoidWithZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
i : α
s : Finset α
his : i ∉ s
ih :
∀ {x y a : R},
(∀ i ∈ s, Prime (p i)) →
x * y = a * ∏ i ∈ s, p i →
∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
a : R
hp : ∀ i_1 ∈ insert i s, Prime (p i_1)
hpi : Prime (p i)
t u : Finset α
b : R
htus : t ∪ u = s
htu : Disjoint t u
hit : i ∉ t
hiu : i ∉ u
d : R
hbc : a = b * d
hx : (b * ∏ i ∈ t, p i) * (p i * d * ∏ i ∈ u, p i) = a * p i * ∏ x ∈ s, p x
⊢ t ∪ insert i u = insert i s
|
rw [union_insert, htus]
|
no goals
|
8bb0dd23785fb5d8
|
CliffordAlgebra.toBaseChange_reverse
|
Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean
|
theorem toBaseChange_reverse (Q : QuadraticForm R V) (x : CliffordAlgebra (Q.baseChange A)) :
toBaseChange A Q (reverse x) =
TensorProduct.map LinearMap.id reverse (toBaseChange A Q x)
|
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(opLinearEquiv A).symm ∘ₗ ↑(opLinearEquiv A))
(↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
(↑(toBaseChange A Q) x) =
(TensorProduct.map LinearMap.id (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap)) ((toBaseChange A Q) x)
|
dsimp
|
R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (TensorProduct.AlgebraTensorModule.map (↑(opLinearEquiv A ≪≫ₗ (opLinearEquiv A).symm))
(↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap))
((toBaseChange A Q) x) =
(TensorProduct.map LinearMap.id (↑(opLinearEquiv R).symm ∘ₗ reverseOp.toLinearMap)) ((toBaseChange A Q) x)
|
1b433c5423f6e345
|
List.formPerm_apply_lt_getElem
|
Mathlib/GroupTheory/Perm/List.lean
|
theorem formPerm_apply_lt_getElem (xs : List α) (h : Nodup xs) (n : ℕ) (hn : n + 1 < xs.length) :
formPerm xs xs[n] = xs[n + 1]
|
case succ.cons.cons.a
α : Type u_1
inst✝ : DecidableEq α
n : ℕ
IH✝ : ∀ (xs : List α), xs.Nodup → ∀ (hn : n + 1 < xs.length), xs.formPerm xs[n] = xs[n + 1]
x y : α
l : List α
h : (x :: y :: l).Nodup
hn : n + 1 + 1 < (x :: y :: l).length
IH : (y :: l).formPerm (y :: l)[n] = l[n]
⊢ (y :: l).formPerm (y :: l)[n] ≠ y
|
intro hx
|
case succ.cons.cons.a
α : Type u_1
inst✝ : DecidableEq α
n : ℕ
IH✝ : ∀ (xs : List α), xs.Nodup → ∀ (hn : n + 1 < xs.length), xs.formPerm xs[n] = xs[n + 1]
x y : α
l : List α
h : (x :: y :: l).Nodup
hn : n + 1 + 1 < (x :: y :: l).length
IH : (y :: l).formPerm (y :: l)[n] = l[n]
hx : (y :: l).formPerm (y :: l)[n] = y
⊢ False
|
e8c90107fc7bf71c
|
List.mapFinIdx_eq_append_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean
|
theorem mapFinIdx_eq_append_iff {l : List α} {f : (i : Nat) → α → (h : i < l.length) → β} :
l.mapFinIdx f = l₁ ++ l₂ ↔
∃ (l₁' : List α) (l₂' : List α) (w : l = l₁' ++ l₂'),
l₁'.mapFinIdx (fun i a h => f i a (by simp [w]; omega)) = l₁ ∧
l₂'.mapFinIdx (fun i a h => f (i + l₁'.length) a (by simp [w]; omega)) = l₂
|
α : Type u_1
β : Type u_2
l₁ l₂ : List β
l : List α
f : (i : Nat) → α → i < l.length → β
h✝ : (l₁ ++ l₂).length = l.length
w : ∀ (i : Nat) (h : i < l.length), (l₁ ++ l₂)[i] = f i l[i] h
h : l₁.length + l₂.length = l.length
⊢ l = take l₁.length l ++ drop l₁.length l
|
simp
|
no goals
|
6ee35de95da05087
|
LinearOrderedField.exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self
|
Mathlib/Algebra/Order/CompleteField.lean
|
theorem exists_mem_cutMap_mul_self_of_lt_inducedMap_mul_self (ha : 0 < a) (b : β)
(hba : b < inducedMap α β a * inducedMap α β a) : ∃ c ∈ cutMap β (a * a), b < c
|
case inr
α : Type u_2
β : Type u_3
inst✝² : LinearOrderedField α
inst✝¹ : ConditionallyCompleteLinearOrderedField β
inst✝ : Archimedean α
a : α
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
⊢ ∃ c ∈ cutMap β (a * a), b < c
|
obtain ⟨q, hq, hbq, hqa⟩ := exists_rat_pow_btwn two_ne_zero hba (hb.trans_lt hba)
|
case inr.intro.intro.intro
α : Type u_2
β : Type u_3
inst✝² : LinearOrderedField α
inst✝¹ : ConditionallyCompleteLinearOrderedField β
inst✝ : Archimedean α
a : α
ha : 0 < a
b : β
hba : b < inducedMap α β a * inducedMap α β a
hb : 0 ≤ b
q : ℚ
hq : 0 < q
hbq : b < ↑q ^ 2
hqa : ↑q ^ 2 < inducedMap α β a * inducedMap α β a
⊢ ∃ c ∈ cutMap β (a * a), b < c
|
26fca4418a681185
|
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