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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|---|---|---|---|---|---|---|
ProbabilityTheory.Kernel.IsProper.lintegral_mul
|
Mathlib/Probability/Kernel/Proper.lean
|
lemma IsProper.lintegral_mul (hπ : IsProper π) (h𝓑𝓧 : 𝓑 ≤ 𝓧) (hf : Measurable[𝓧] f)
(hg : Measurable[𝓑] g) (x₀ : X) :
∫⁻ x, g x * f x ∂(π x₀) = g x₀ * ∫⁻ x, f x ∂(π x₀)
|
case refine_1
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
f g : X → ℝ≥0∞
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hg : Measurable g
x₀ : X
c : ℝ≥0∞
A : Set X
hA : MeasurableSet A
⊢ ∫⁻ (x : X), c * (A.indicator 1 x * f x) ∂π x₀ = c * (A.indicator 1 x₀ * ∫⁻ (x : X), f x ∂π x₀)
|
rw [lintegral_const_mul, hπ.lintegral_indicator_mul h𝓑𝓧 hf hA]
|
case refine_1.hf
X : Type u_1
𝓑 𝓧 : MeasurableSpace X
π : Kernel X X
f g : X → ℝ≥0∞
hπ : π.IsProper
h𝓑𝓧 : 𝓑 ≤ 𝓧
hf : Measurable f
hg : Measurable g
x₀ : X
c : ℝ≥0∞
A : Set X
hA : MeasurableSet A
⊢ Measurable fun x => A.indicator 1 x * f x
|
c4118f574b6ba09a
|
LinearMap.BilinForm.nondegenerate_restrict_iff_disjoint_ker
|
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
lemma nondegenerate_restrict_iff_disjoint_ker (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm)
{W : Submodule R M} :
(B.domRestrict₁₂ W W).Nondegenerate ↔ Disjoint W (LinearMap.ker B)
|
R : Type u_1
M : Type u_5
inst✝² : LinearOrderedCommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
W : Submodule R M
hW : Disjoint W (ker B)
hB' : (domRestrict₁₂ B W W).IsRefl
x : M
hx : x ∈ W
h : ∀ (y : ↥W), ((domRestrict₁₂ B W W) ⟨x, hx⟩) y = 0
⊢ ⟨x, hx⟩ = 0
|
simp_rw [Subtype.forall, domRestrict₁₂_apply] at h
|
R : Type u_1
M : Type u_5
inst✝² : LinearOrderedCommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
W : Submodule R M
hW : Disjoint W (ker B)
hB' : (domRestrict₁₂ B W W).IsRefl
x : M
hx : x ∈ W
h : ∀ a ∈ W, (B x) a = 0
⊢ ⟨x, hx⟩ = 0
|
4d1327212668c98c
|
LiouvilleWith.frequently_lt_rpow_neg
|
Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleWith.lean
|
theorem frequently_lt_rpow_neg (h : LiouvilleWith p x) (hlt : q < p) :
∃ᶠ n : ℕ in atTop, ∃ m : ℤ, x ≠ m / n ∧ |x - m / n| < n ^ (-q)
|
case intro.intro
p q x : ℝ
h : LiouvilleWith p x
hlt : q < p
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
this : ∀ᶠ (n : ℕ) in atTop, C < ↑n ^ (p - q)
⊢ ∀ (x_1 : ℕ),
(C < ↑x_1 ^ (p - q) ∧ 1 ≤ x_1 ∧ ∃ m, x ≠ ↑m / ↑x_1 ∧ |x - ↑m / ↑x_1| < C / ↑x_1 ^ p) →
∃ m, x ≠ ↑m / ↑x_1 ∧ |x - ↑m / ↑x_1| < ↑x_1 ^ (-q)
|
rintro n ⟨hnC, hn, m, hne, hlt⟩
|
case intro.intro.intro.intro.intro.intro
p q x : ℝ
h : LiouvilleWith p x
hlt✝ : q < p
C : ℝ
_hC₀ : 0 < C
hC : ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p
this : ∀ᶠ (n : ℕ) in atTop, C < ↑n ^ (p - q)
n : ℕ
hnC : C < ↑n ^ (p - q)
hn : 1 ≤ n
m : ℤ
hne : x ≠ ↑m / ↑n
hlt : |x - ↑m / ↑n| < C / ↑n ^ p
⊢ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < ↑n ^ (-q)
|
ec54ba98d777e83f
|
IsCoprime.of_isCoprime_of_dvd_left
|
Mathlib/RingTheory/Coprime/Basic.lean
|
theorem IsCoprime.of_isCoprime_of_dvd_left (h : IsCoprime y z) (hdvd : x ∣ y) : IsCoprime x z
|
R : Type u
inst✝ : CommSemiring R
x y z : R
h : IsCoprime y z
hdvd : x ∣ y
⊢ IsCoprime x z
|
obtain ⟨d, rfl⟩ := hdvd
|
case intro
R : Type u
inst✝ : CommSemiring R
x z d : R
h : IsCoprime (x * d) z
⊢ IsCoprime x z
|
a10984dfd5fcfbc3
|
Stream'.Seq.of_mem_append
|
Mathlib/Data/Seq/Seq.lean
|
theorem of_mem_append {s₁ s₂ : Seq α} {a : α} (h : a ∈ append s₁ s₂) : a ∈ s₁ ∨ a ∈ s₂
|
α : Type u
s₂ : Seq α
a : α
ss : Seq α
h : a ∈ ss
b : α
s' : Seq α
o : a = b ∨ ∀ {s₁ : Seq α}, a ∈ s₁.append s₂ → s₁.append s₂ = s' → a ∈ s₁ ∨ a ∈ s₂
c : α
t₁ : Seq α
m : a ∈ (cons c t₁).append s₂
e : (cons c t₁).append s₂ = cons b s'
this : ((cons c t₁).append s₂).destruct = (cons b s').destruct
⊢ a = c ∨ a ∈ t₁.append s₂
|
simpa using m
|
no goals
|
616ea8064ee30260
|
RootPairing.isOrthogonal_symm
|
Mathlib/LinearAlgebra/RootSystem/Defs.lean
|
lemma isOrthogonal_symm : IsOrthogonal P i j ↔ IsOrthogonal P j i
|
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
P : RootPairing ι R M N
i j : ι
⊢ P.IsOrthogonal i j ↔ P.IsOrthogonal j i
|
simp only [IsOrthogonal, and_comm]
|
no goals
|
fe6d30cd3cf9a391
|
Int.bmod_add_cancel_right
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
|
theorem bmod_add_cancel_right (i : Int) : bmod (x + i) n = bmod (y + i) n ↔ bmod x n = bmod y n :=
⟨fun H => by
have := add_bmod_eq_add_bmod_right (-i) H
rwa [Int.add_neg_cancel_right, Int.add_neg_cancel_right] at this,
fun H => by rw [← bmod_add_bmod_congr, H, bmod_add_bmod_congr]⟩
|
x : Int
n : Nat
y i : Int
H : x.bmod n = y.bmod n
⊢ (x + i).bmod n = (y + i).bmod n
|
rw [← bmod_add_bmod_congr, H, bmod_add_bmod_congr]
|
no goals
|
b4242a8172e2de9d
|
CategoryTheory.MonoidalOfChosenFiniteProducts.hexagon_forward
|
Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean
|
theorem hexagon_forward (X Y Z : C) :
(BinaryFan.associatorOfLimitCone ℬ X Y Z).hom ≫
(Limits.BinaryFan.braiding (ℬ X (tensorObj ℬ Y Z)).isLimit
(ℬ (tensorObj ℬ Y Z) X).isLimit).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Y Z X).hom =
tensorHom ℬ (Limits.BinaryFan.braiding (ℬ X Y).isLimit (ℬ Y X).isLimit).hom (𝟙 Z) ≫
(BinaryFan.associatorOfLimitCone ℬ Y X Z).hom ≫
tensorHom ℬ (𝟙 Y) (Limits.BinaryFan.braiding (ℬ X Z).isLimit (ℬ Z X).isLimit).hom
|
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X Y Z : C
⊢ ∀ (j : Discrete WalkingPair),
((BinaryFan.associatorOfLimitCone ℬ X Y Z).hom ≫
((ℬ X (tensorObj ℬ Y Z)).isLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ Y Z) X).isLimit.swapBinaryFan).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Y Z X).hom) ≫
(ℬ Y (ℬ Z X).cone.pt).cone.π.app j =
((ℬ (ℬ Y X).cone.pt Z).isLimit.lift
(BinaryFan.mk
(BinaryFan.fst (ℬ (ℬ X Y).cone.pt Z).cone ≫
((ℬ X Y).isLimit.conePointUniqueUpToIso (ℬ Y X).isLimit.swapBinaryFan).hom)
(BinaryFan.snd (ℬ (ℬ X Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
(BinaryFan.associatorOfLimitCone ℬ Y X Z).hom ≫
(ℬ Y (ℬ Z X).cone.pt).isLimit.lift
(BinaryFan.mk (BinaryFan.fst (ℬ Y (ℬ X Z).cone.pt).cone ≫ 𝟙 Y)
(BinaryFan.snd (ℬ Y (ℬ X Z).cone.pt).cone ≫
((ℬ X Z).isLimit.conePointUniqueUpToIso (ℬ Z X).isLimit.swapBinaryFan).hom))) ≫
(ℬ Y (ℬ Z X).cone.pt).cone.π.app j
|
rintro ⟨⟨⟩⟩
|
case mk.left
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X Y Z : C
⊢ ((BinaryFan.associatorOfLimitCone ℬ X Y Z).hom ≫
((ℬ X (tensorObj ℬ Y Z)).isLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ Y Z) X).isLimit.swapBinaryFan).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Y Z X).hom) ≫
(ℬ Y (ℬ Z X).cone.pt).cone.π.app { as := WalkingPair.left } =
((ℬ (ℬ Y X).cone.pt Z).isLimit.lift
(BinaryFan.mk
(BinaryFan.fst (ℬ (ℬ X Y).cone.pt Z).cone ≫
((ℬ X Y).isLimit.conePointUniqueUpToIso (ℬ Y X).isLimit.swapBinaryFan).hom)
(BinaryFan.snd (ℬ (ℬ X Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
(BinaryFan.associatorOfLimitCone ℬ Y X Z).hom ≫
(ℬ Y (ℬ Z X).cone.pt).isLimit.lift
(BinaryFan.mk (BinaryFan.fst (ℬ Y (ℬ X Z).cone.pt).cone ≫ 𝟙 Y)
(BinaryFan.snd (ℬ Y (ℬ X Z).cone.pt).cone ≫
((ℬ X Z).isLimit.conePointUniqueUpToIso (ℬ Z X).isLimit.swapBinaryFan).hom))) ≫
(ℬ Y (ℬ Z X).cone.pt).cone.π.app { as := WalkingPair.left }
case mk.right
C : Type u
inst✝ : Category.{v, u} C
ℬ : (X Y : C) → LimitCone (pair X Y)
X Y Z : C
⊢ ((BinaryFan.associatorOfLimitCone ℬ X Y Z).hom ≫
((ℬ X (tensorObj ℬ Y Z)).isLimit.conePointUniqueUpToIso (ℬ (tensorObj ℬ Y Z) X).isLimit.swapBinaryFan).hom ≫
(BinaryFan.associatorOfLimitCone ℬ Y Z X).hom) ≫
(ℬ Y (ℬ Z X).cone.pt).cone.π.app { as := WalkingPair.right } =
((ℬ (ℬ Y X).cone.pt Z).isLimit.lift
(BinaryFan.mk
(BinaryFan.fst (ℬ (ℬ X Y).cone.pt Z).cone ≫
((ℬ X Y).isLimit.conePointUniqueUpToIso (ℬ Y X).isLimit.swapBinaryFan).hom)
(BinaryFan.snd (ℬ (ℬ X Y).cone.pt Z).cone ≫ 𝟙 Z)) ≫
(BinaryFan.associatorOfLimitCone ℬ Y X Z).hom ≫
(ℬ Y (ℬ Z X).cone.pt).isLimit.lift
(BinaryFan.mk (BinaryFan.fst (ℬ Y (ℬ X Z).cone.pt).cone ≫ 𝟙 Y)
(BinaryFan.snd (ℬ Y (ℬ X Z).cone.pt).cone ≫
((ℬ X Z).isLimit.conePointUniqueUpToIso (ℬ Z X).isLimit.swapBinaryFan).hom))) ≫
(ℬ Y (ℬ Z X).cone.pt).cone.π.app { as := WalkingPair.right }
|
f0efb67d9e1bf6f6
|
DirichletCharacter.Odd.eval_neg
|
Mathlib/NumberTheory/DirichletCharacter/Basic.lean
|
lemma Odd.eval_neg (x : ZMod m) (hψ : ψ.Odd) : ψ (- x) = - ψ x
|
S : Type u_2
inst✝ : CommRing S
m : ℕ
ψ : DirichletCharacter S m
x : ZMod m
hψ : ψ (-1) = -1
⊢ ψ (-1) * ψ x = -ψ x
|
simp [hψ]
|
no goals
|
0aff6b1f1935d13b
|
exists_rat_btwn
|
Mathlib/Algebra/Order/Archimedean/Basic.lean
|
theorem exists_rat_btwn {x y : α} (h : x < y) : ∃ q : ℚ, x < q ∧ (q : α) < y
|
case intro.intro
α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : Archimedean α
x y : α
h : x < y
n : ℕ
nh : (y - x)⁻¹ < ↑n
z : ℤ
zh : ∀ (z_1 : ℤ), z_1 ≤ z ↔ ↑z_1 ≤ x * ↑n
n0' : 0 < ↑n
⊢ x < ↑(↑(z + 1) / ↑n) ∧ ↑(↑(z + 1) / ↑n) < y
|
have n0 := Nat.cast_pos.1 n0'
|
case intro.intro
α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : Archimedean α
x y : α
h : x < y
n : ℕ
nh : (y - x)⁻¹ < ↑n
z : ℤ
zh : ∀ (z_1 : ℤ), z_1 ≤ z ↔ ↑z_1 ≤ x * ↑n
n0' : 0 < ↑n
n0 : 0 < n
⊢ x < ↑(↑(z + 1) / ↑n) ∧ ↑(↑(z + 1) / ↑n) < y
|
876a28cf777622dc
|
List.getElem?_reverse'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem getElem?_reverse' : ∀ {l : List α} (i j), i + j + 1 = length l →
l.reverse[i]? = l[j]?
| [], _, _, _ => rfl
| a::l, i, 0, h => by simp [Nat.succ.injEq] at h; simp [h, getElem?_append_right, Nat.succ.injEq]
| a::l, i, j+1, h => by
have := Nat.succ.inj h; simp at this ⊢
rw [getElem?_append_left, getElem?_reverse' _ _ this]
rw [length_reverse, ← this]; apply Nat.lt_add_of_pos_right (Nat.succ_pos _)
|
α : Type u_1
a : α
l : List α
i j : Nat
h : i + (j + 1) + 1 = (a :: l).length
this : i + (j + 1) = l.length
⊢ (a :: l).reverse[i]? = (a :: l)[j + 1]?
|
simp at this ⊢
|
α : Type u_1
a : α
l : List α
i j : Nat
h : i + (j + 1) + 1 = (a :: l).length
this : i + (j + 1) = l.length
⊢ (l.reverse ++ [a])[i]? = l[j]?
|
c7805acdaa00531e
|
lt_map_inv_iff
|
Mathlib/Order/Hom/Basic.lean
|
theorem lt_map_inv_iff (f : F) {a : α} {b : β} : a < EquivLike.inv f b ↔ f a < b
|
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Preorder α
inst✝² : Preorder β
inst✝¹ : EquivLike F α β
inst✝ : OrderIsoClass F α β
f : F
a : α
b : β
⊢ f a < f (EquivLike.inv f b) ↔ f a < b
|
simp only [EquivLike.apply_inv_apply]
|
no goals
|
32bb5e38adf6ab93
|
BitVec.getElem_zero_ofNat_zero
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem getElem_zero_ofNat_zero (i : Nat) (h : i < w) : (BitVec.ofNat w 0)[i] = false
|
w i : Nat
h : i < w
⊢ (0#w)[i] = false
|
simp
|
no goals
|
0e148b70db0f3910
|
IntermediateField.adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable
|
Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean
|
theorem adjoin_eq_adjoin_pow_expChar_pow_of_isSeparable (S : Set E)
[Algebra.IsSeparable F (adjoin F S)] (q : ℕ) [ExpChar F q] (n : ℕ) :
adjoin F S = adjoin F ((· ^ q ^ n) '' S)
|
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
S : Set E
q : ℕ
inst✝¹ : ExpChar F q
n : ℕ
L : IntermediateField F E := adjoin F S
inst✝ : Algebra.IsSeparable F ↥L
M : IntermediateField F E := adjoin F ((fun x => x ^ q ^ n) '' S)
hi : M ≤ L
this : Algebra ↥M ↥L := (inclusion hi).toAlgebra
⊢ L = M
|
haveI : Algebra.IsSeparable M (extendScalars hi) :=
Algebra.isSeparable_tower_top_of_isSeparable F M L
|
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
S : Set E
q : ℕ
inst✝¹ : ExpChar F q
n : ℕ
L : IntermediateField F E := adjoin F S
inst✝ : Algebra.IsSeparable F ↥L
M : IntermediateField F E := adjoin F ((fun x => x ^ q ^ n) '' S)
hi : M ≤ L
this✝ : Algebra ↥M ↥L := (inclusion hi).toAlgebra
this : Algebra.IsSeparable ↥M ↥(extendScalars hi)
⊢ L = M
|
5d9784243c5f5669
|
CoalgebraCat.MonoidalCategoryAux.comul_tensorObj_tensorObj_left
|
Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
|
theorem comul_tensorObj_tensorObj_left :
Coalgebra.comul (R := R)
(A := ((CoalgebraCat.of R M ⊗ CoalgebraCat.of R N) ⊗ CoalgebraCat.of R P : CoalgebraCat R))
= Coalgebra.comul (A := (M ⊗[R] N) ⊗[R] P)
|
R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom
((comonEquivalence R).symm.inverse.obj (of R M ⊗ of R N) ⊗ (comonEquivalence R).symm.inverse.obj (of R P)).comul =
CoalgebraStruct.comul
|
dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj,
instCoalgebraStruct_comul]
|
R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ ModuleCat.Hom.hom ((of R M ⊗ of R N).toComonObj ⊗ (of R P).toComonObj).comul =
↑(tensorTensorTensorComm R (M ⊗[R] N) (M ⊗[R] N) P P) ∘ₗ
map (↑(tensorTensorTensorComm R M M N N) ∘ₗ map CoalgebraStruct.comul CoalgebraStruct.comul) CoalgebraStruct.comul
|
6198806ba36fecc6
|
Monoid.PushoutI.NormalWord.prod_smul
|
Mathlib/GroupTheory/PushoutI.lean
|
theorem prod_smul (g : PushoutI φ) (w : NormalWord d) :
(g • w).prod = g * w.prod
|
case base
ι : Type u_1
G : ι → Type u_2
H : Type u_3
inst✝³ : (i : ι) → Group (G i)
inst✝² : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (G i)
h : H
w : NormalWord d
⊢ ((base φ) h • w).prod = (base φ) h * w.prod
|
rw [base_smul_eq_smul, prod_base_smul]
|
no goals
|
241b74ad5bbc10a4
|
ContinuousOn.aestronglyMeasurable_of_isSeparable
|
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
theorem ContinuousOn.aestronglyMeasurable_of_isSeparable [TopologicalSpace α]
[PseudoMetrizableSpace α] [OpensMeasurableSpace α] [TopologicalSpace β]
[PseudoMetrizableSpace β] {f : α → β} {s : Set α} {μ : Measure α} (hf : ContinuousOn f s)
(hs : MeasurableSet s) (h's : TopologicalSpace.IsSeparable s) :
AEStronglyMeasurable f (μ.restrict s)
|
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetrizableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
h's : IsSeparable s
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEMeasurable f (μ.restrict s) ∧ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ.restrict s, f x ∈ t
|
refine ⟨hf.aemeasurable hs, f '' s, hf.isSeparable_image h's, ?_⟩
|
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetrizableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
h's : IsSeparable s
this : PseudoMetricSpace α := pseudoMetrizableSpacePseudoMetric α
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∀ᵐ (x : α) ∂μ.restrict s, f x ∈ f '' s
|
1b30cac63b07288d
|
WeierstrassCurve.Jacobian.negAddY_of_Z_eq_zero_right
|
Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
|
lemma negAddY_of_Z_eq_zero_right {P Q : Fin 3 → R} (hQ : W'.Equation Q) (hQz : Q z = 0) :
W'.negAddY P Q = (-(Q x * P z)) ^ 3 * W'.negY P
|
case a.a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hQ : W'.Equation Q
hQz : Q z = 0
⊢ -P y * Q x ^ 3 * P z ^ 3 + 2 * P y * Q y ^ 2 * P z ^ 3 - 3 * P x ^ 2 * Q x * Q y * P z ^ 2 * 0 +
3 * P x * P y * Q x ^ 2 * P z * 0 ^ 2 +
P x ^ 3 * Q y * 0 ^ 3 -
2 * P y ^ 2 * Q y * 0 ^ 3 +
W'.a₁ * P x * Q y ^ 2 * P z ^ 4 +
W'.a₁ * P y * Q x * Q y * P z ^ 3 * 0 -
W'.a₁ * P x * P y * Q y * P z * 0 ^ 3 -
W'.a₁ * P y ^ 2 * Q x * 0 ^ 4 -
2 * W'.a₂ * P x * Q x * Q y * P z ^ 4 * 0 +
2 * W'.a₂ * P x * P y * Q x * P z * 0 ^ 4 +
W'.a₃ * Q y ^ 2 * P z ^ 6 -
W'.a₃ * P y ^ 2 * 0 ^ 6 -
W'.a₄ * Q x * Q y * P z ^ 6 * 0 -
W'.a₄ * P x * Q y * P z ^ 4 * 0 ^ 3 +
W'.a₄ * P y * Q x * P z ^ 3 * 0 ^ 4 +
W'.a₄ * P x * P y * P z * 0 ^ 6 -
2 * W'.a₆ * Q y * P z ^ 6 * 0 ^ 3 +
2 * W'.a₆ * P y * P z ^ 3 * 0 ^ 6 +
(P y - (-P y - W'.a₁ * P x * P z - W'.a₃ * P z ^ 3)) * P z ^ 3 * Q x ^ 3 -
((-(Q x * P z)) ^ 3 * (-P y - W'.a₁ * P x * P z - W'.a₃ * P z ^ 3) +
(P y - (-P y - W'.a₁ * P x * P z - W'.a₃ * P z ^ 3)) * P z ^ 3 * Q y ^ 2) =
0
|
ring1
|
no goals
|
8d8319522de87cc7
|
Polynomial.X_mul
|
Mathlib/Algebra/Polynomial/Basic.lean
|
theorem X_mul : X * p = p * X
|
case ofFinsupp.H
R : Type u
inst✝ : Semiring R
toFinsupp✝ : R[ℕ]
x✝ : ℕ
⊢ (Finsupp.single 1 1 * toFinsupp✝) x✝ = (toFinsupp✝ * Finsupp.single 1 1) x✝
|
simp [AddMonoidAlgebra.mul_apply, AddMonoidAlgebra.sum_single_index, add_comm]
|
no goals
|
2210153ddaf42b3f
|
PSigma.eta
|
Mathlib/.lake/packages/lean4/src/lean/Init/Core.lean
|
theorem PSigma.eta {α : Sort u} {β : α → Sort v} {a₁ a₂ : α} {b₁ : β a₁} {b₂ : β a₂}
(h₁ : a₁ = a₂) (h₂ : Eq.ndrec b₁ h₁ = b₂) : PSigma.mk a₁ b₁ = PSigma.mk a₂ b₂
|
α : Sort u
β : α → Sort v
a₁ : α
b₁ : β a₁
⊢ ⟨a₁, b₁⟩ = ⟨a₁, Eq.ndrec b₁ ⋯⟩
|
exact rfl
|
no goals
|
8bb501dff720abc2
|
Submodule.LinearDisjoint.of_basis_mul'
|
Mathlib/LinearAlgebra/LinearDisjoint.lean
|
theorem of_basis_mul' {κ ι : Type*} (m : Basis κ R M) (n : Basis ι R N)
(H : Function.Injective (Finsupp.linearCombination R fun i : κ × ι ↦ (m i.1 * n i.2 : S))) :
M.LinearDisjoint N
|
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
κ : Type u_1
ι : Type u_2
m : Basis κ R ↥M
n : Basis ι R ↥N
H : Function.Injective ⇑(Finsupp.linearCombination R fun i => ↑(m i.1) * ↑(n i.2))
i0 : (κ × ι →₀ R) ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := (finsuppTensorFinsupp' R κ ι).symm
i1 : ↥M ⊗[R] ↥N ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := TensorProduct.congr m.repr n.repr
i : (κ × ι →₀ R) →ₗ[R] S := M.mulMap N ∘ₗ ↑(i0 ≪≫ₗ i1.symm)
this : i = Finsupp.linearCombination R fun i => ↑(m i.1) * ↑(n i.2)
⊢ M.LinearDisjoint N
|
simp_rw [← this, i, LinearMap.coe_comp, LinearEquiv.coe_coe, EquivLike.injective_comp] at H
|
R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring S
inst✝ : Algebra R S
M N : Submodule R S
κ : Type u_1
ι : Type u_2
m : Basis κ R ↥M
n : Basis ι R ↥N
i0 : (κ × ι →₀ R) ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := (finsuppTensorFinsupp' R κ ι).symm
i1 : ↥M ⊗[R] ↥N ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := TensorProduct.congr m.repr n.repr
i : (κ × ι →₀ R) →ₗ[R] S := M.mulMap N ∘ₗ ↑(i0 ≪≫ₗ i1.symm)
this : i = Finsupp.linearCombination R fun i => ↑(m i.1) * ↑(n i.2)
H : Function.Injective ⇑(M.mulMap N)
⊢ M.LinearDisjoint N
|
2e5b7c565dbbc520
|
GroupExtension.IsConj.trans
|
Mathlib/GroupTheory/GroupExtension/Basic.lean
|
theorem trans {s₁ s₂ s₃ : S.Splitting} (h₁ : S.IsConj s₁ s₂) (h₂ : S.IsConj s₂ s₃) :
S.IsConj s₁ s₃
|
case intro.intro
N : Type u_1
G : Type u_2
inst✝² : Group N
inst✝¹ : Group G
E : Type u_3
inst✝ : Group E
S : GroupExtension N E G
s₁ s₂ s₃ : S.Splitting
n₁ : N
hn₁ : ⇑s₁ = fun g => S.inl n₁ * s₂ g * (S.inl n₁)⁻¹
n₂ : N
hn₂ : ⇑s₂ = fun g => S.inl n₂ * s₃ g * (S.inl n₂)⁻¹
⊢ S.IsConj s₁ s₃
|
exact ⟨n₁ * n₂, by simp only [hn₁, hn₂, map_mul]; group⟩
|
no goals
|
75033f227f23746b
|
Matroid.map_closure_eq
|
Mathlib/Data/Matroid/Closure.lean
|
@[simp] lemma map_closure_eq {β : Type*} (M : Matroid α) (f : α → β) (hf) (X : Set β) :
(M.map f hf).closure X = f '' M.closure (f ⁻¹' X)
|
case mp
α : Type u_2
β : Type u_3
M : Matroid α
f : α → β
hf : InjOn f M.E
X : Set β
I : Set α
hI : M.Indep I
e : β
⊢ ((∃ x ∈ M.E, f x = e) ∧ ∀ (x : Set α), M.Indep x → insert e (f '' I) = f '' x → ∃ x ∈ I, f x = e) →
∃ x, (x ∈ M.E ∧ (M.Indep (insert x I) → x ∈ I)) ∧ f x = e
|
rintro ⟨⟨x, hxE, rfl⟩, h2⟩
|
case mp.intro.intro.intro
α : Type u_2
β : Type u_3
M : Matroid α
f : α → β
hf : InjOn f M.E
X : Set β
I : Set α
hI : M.Indep I
x : α
hxE : x ∈ M.E
h2 : ∀ (x_1 : Set α), M.Indep x_1 → insert (f x) (f '' I) = f '' x_1 → ∃ x_2 ∈ I, f x_2 = f x
⊢ ∃ x_1, (x_1 ∈ M.E ∧ (M.Indep (insert x_1 I) → x_1 ∈ I)) ∧ f x_1 = f x
|
6a1d5e64a78caa46
|
NNReal.IsConjExponent.inv_inv
|
Mathlib/Data/Real/ConjExponents.lean
|
protected lemma inv_inv (ha : a ≠ 0) (hb : b ≠ 0) (hab : a + b = 1) :
a⁻¹.IsConjExponent b⁻¹ :=
⟨(one_lt_inv₀ ha.bot_lt).2 <| by rw [← hab]; exact lt_add_of_pos_right _ hb.bot_lt, by
simpa only [inv_inv] using hab⟩
|
a b : ℝ≥0
ha : a ≠ 0
hb : b ≠ 0
hab : a + b = 1
⊢ a < 1
|
rw [← hab]
|
a b : ℝ≥0
ha : a ≠ 0
hb : b ≠ 0
hab : a + b = 1
⊢ a < a + b
|
f44417bc28a58013
|
SpectrumRestricts.nnreal_iff_spectralRadius_le
|
Mathlib/Analysis/Normed/Algebra/Spectrum.lean
|
lemma nnreal_iff_spectralRadius_le [Algebra ℝ A] {a : A} {t : ℝ≥0} (ht : spectralRadius ℝ a ≤ t) :
SpectrumRestricts a ContinuousMap.realToNNReal ↔
spectralRadius ℝ (algebraMap ℝ A t - a) ≤ t
|
A : Type u_3
inst✝¹ : Ring A
inst✝ : Algebra ℝ A
a : A
t : ℝ≥0
ht : spectralRadius ℝ a ≤ ↑t
this : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t
h : ∀ x ∈ spectrum ℝ a, 0 ≤ x
x : ℝ
hx : x ∈ {↑t} - spectrum ℝ a
⊢ ∃ y ∈ spectrum ℝ a, ↑t - y = x
|
simpa using hx
|
no goals
|
23e88e5a29773e00
|
Submodule.one_le_one_div
|
Mathlib/Algebra/Algebra/Operations.lean
|
theorem one_le_one_div {I : Submodule R A} : 1 ≤ 1 / I ↔ I ≤ 1
|
case mpr
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
I : Submodule R A
hI : I ≤ 1
⊢ 1 ≤ 1 / I
|
rwa [le_div_iff_mul_le, one_mul]
|
no goals
|
fd5f5935d310a5ca
|
WeierstrassCurve.ψ₂_sq
|
Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean
|
lemma ψ₂_sq : W.ψ₂ ^ 2 = C W.Ψ₂Sq + 4 * W.toAffine.polynomial
|
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ W.ψ₂ ^ 2 = C W.Ψ₂Sq + 4 * W.toAffine.polynomial
|
rw [C_Ψ₂Sq, sub_add_cancel]
|
no goals
|
e554a9b2f346690c
|
WittVector.wittSub_zero
|
Mathlib/RingTheory/WittVector/Defs.lean
|
theorem wittSub_zero : wittSub p 0 = X (0, 0) - X (1, 0)
|
p : ℕ
hp : Fact (Nat.Prime p)
⊢ wittSub p 0 = X (0, 0) - X (1, 0)
|
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
|
case a
p : ℕ
hp : Fact (Nat.Prime p)
⊢ (map (Int.castRingHom ℚ)) (wittSub p 0) = (map (Int.castRingHom ℚ)) (X (0, 0) - X (1, 0))
|
5dea8bc048ab8d80
|
IntermediateField.AdjoinSimple.trace_gen_eq_zero
|
Mathlib/RingTheory/Trace/Basic.lean
|
theorem trace_gen_eq_zero {x : L} (hx : ¬IsIntegral K x) :
Algebra.trace K K⟮x⟯ (AdjoinSimple.gen K x) = 0
|
case h.intro.intro.refine_1
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
s : Finset ↥K⟮x⟯
b : Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯
⊢ (Subalgebra.toSubmodule K⟮x⟯.toSubalgebra).FG
|
exact (Submodule.fg_iff_finiteDimensional _).mpr (FiniteDimensional.of_fintype_basis b)
|
no goals
|
e3d62924677daba0
|
Poly.induction
|
Mathlib/NumberTheory/Dioph.lean
|
theorem induction {C : Poly α → Prop} (H1 : ∀ i, C (proj i)) (H2 : ∀ n, C (const n))
(H3 : ∀ f g, C f → C g → C (f - g)) (H4 : ∀ f g, C f → C g → C (f * g)) (f : Poly α) : C f
|
case mk.proj
α : Type u_1
C : Poly α → Prop
H1 : ∀ (i : α), C (proj i)
H2 : ∀ (n : ℤ), C (const n)
H3 : ∀ (f g : Poly α), C f → C g → C (f - g)
H4 : ∀ (f g : Poly α), C f → C g → C (f * g)
f : (α → ℕ) → ℤ
i✝ : α
⊢ C ⟨fun x => ↑(x i✝), ⋯⟩
|
apply H1
|
no goals
|
62d6795f68e83879
|
exists_mem_nhds_zero_mul_subset
|
Mathlib/Topology/Algebra/Monoid.lean
|
theorem exists_mem_nhds_zero_mul_subset
{K U : Set M} (hK : IsCompact K) (hU : U ∈ 𝓝 0) : ∃ V ∈ 𝓝 0, K * V ⊆ U
|
case refine_3.intro.intro.intro.intro
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
K U : Set M
hK : IsCompact K
hU : U ∈ 𝓝 0
s t V : Set M
V_in : V ∈ 𝓝 0
hV' : s * V ⊆ U
W : Set M
W_in : W ∈ 𝓝 0
hW' : t * W ⊆ U
⊢ ∃ V ∈ 𝓝 0, (s ∪ t) * V ⊆ U
|
use V ∩ W, inter_mem V_in W_in
|
case right
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : MulZeroClass M
inst✝ : ContinuousMul M
K U : Set M
hK : IsCompact K
hU : U ∈ 𝓝 0
s t V : Set M
V_in : V ∈ 𝓝 0
hV' : s * V ⊆ U
W : Set M
W_in : W ∈ 𝓝 0
hW' : t * W ⊆ U
⊢ (s ∪ t) * (V ∩ W) ⊆ U
|
75e54ced8d3ab4b1
|
Set.image2_iUnion_left
|
Mathlib/Data/Set/Lattice.lean
|
theorem image2_iUnion_left (s : ι → Set α) (t : Set β) :
image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t
|
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_5
f : α → β → γ
s : ι → Set α
t : Set β
⊢ image2 f (⋃ i, s i) t = ⋃ i, image2 f (s i) t
|
simp only [← image_prod, iUnion_prod_const, image_iUnion]
|
no goals
|
70ab160b17c814e2
|
Lists.Equiv.antisymm_iff
|
Mathlib/SetTheory/Lists.lean
|
theorem Equiv.antisymm_iff {l₁ l₂ : Lists' α true} : of' l₁ ~ of' l₂ ↔ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁
|
α : Type u_1
l₁ l₂ : Lists' α true
h : of' l₁ ~ of' l₂
⊢ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁
|
obtain - | ⟨h₁, h₂⟩ := h
|
case refl
α : Type u_1
l₁ : Lists' α true
⊢ l₁ ⊆ l₁ ∧ l₁ ⊆ l₁
case antisymm
α : Type u_1
l₁ l₂ : Lists' α true
h₁ : l₁.Subset l₂
h₂ : l₂.Subset l₁
⊢ l₁ ⊆ l₂ ∧ l₂ ⊆ l₁
|
b18e28e6f039acda
|
Lean.Order.Array.monotone_allM
|
Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean
|
theorem monotone_allM
{m : Type → Type v} [Monad m] [∀ α, PartialOrder (m α)] [MonoBind m] {α : Type u}
(f : γ → α → m Bool) (xs : Array α) (start stop : Nat) (hmono : monotone f) :
monotone (fun x => xs.allM (f x) start stop)
|
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type u
f : γ → α → m Bool
xs : Array α
start stop : Nat
hmono : monotone f
⊢ monotone fun x => do
let __do_lift ←
Array.anyM
(fun v => do
let __do_lift ← f x v
pure !__do_lift)
xs start stop
pure !__do_lift
|
apply monotone_bind
|
case hmono₁
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type u
f : γ → α → m Bool
xs : Array α
start stop : Nat
hmono : monotone f
⊢ monotone fun x =>
Array.anyM
(fun v => do
let __do_lift ← f x v
pure !__do_lift)
xs start stop
case hmono₂
γ : Type w
inst✝³ : PartialOrder γ
m : Type → Type v
inst✝² : Monad m
inst✝¹ : (α : Type) → PartialOrder (m α)
inst✝ : MonoBind m
α : Type u
f : γ → α → m Bool
xs : Array α
start stop : Nat
hmono : monotone f
⊢ monotone fun x __do_lift => pure !__do_lift
|
cf015d7f59a8c09d
|
MeasureTheory.Measure.prod_swap
|
Mathlib/MeasureTheory/Measure/Prod.lean
|
theorem prod_swap : map Prod.swap (μ.prod ν) = ν.prod μ
|
case h
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
inst✝ : SFinite μ
s : Set (β × α)
hs : MeasurableSet s
⊢ ∑' (i : ℕ × ℕ), (map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) s =
∑' (i : ℕ × ℕ), (map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteSeq ν i.1))) s
|
exact ((Equiv.prodComm ℕ ℕ).tsum_eq _).symm
|
no goals
|
604d2c93da35b541
|
Matrix.toBlock_diagonal_disjoint
|
Mathlib/Data/Matrix/Block.lean
|
theorem toBlock_diagonal_disjoint (d : m → α) {p q : m → Prop} (hpq : Disjoint p q) :
Matrix.toBlock (diagonal d) p q = 0
|
case a.mk.mk
m : Type u_2
α : Type u_12
inst✝¹ : DecidableEq m
inst✝ : Zero α
d : m → α
p q : m → Prop
hpq : Disjoint p q
i : m
hi : p i
j : m
hj : q j
this : i ≠ j
⊢ (diagonal d).toBlock p q ⟨i, hi⟩ ⟨j, hj⟩ = 0 ⟨i, hi⟩ ⟨j, hj⟩
|
simp [diagonal_apply_ne d this]
|
no goals
|
33bcebef30499139
|
Subalgebra.centralizer_coe_image_includeLeft_eq_center_tensorProduct
|
Mathlib/Algebra/Algebra/Subalgebra/Centralizer.lean
|
/--
Let `R` be a commutative ring and `A, B` be `R`-algebras where `B` is free as `R`-module.
For any subset `S ⊆ A`, the centralizer of `S ⊗ 1 ⊆ A ⊗ B` is `C_A(S) ⊗ B` where `C_A(S)` is the
centralizer of `S` in `A`.
-/
lemma centralizer_coe_image_includeLeft_eq_center_tensorProduct
(S : Set A) [Module.Free R B] :
Subalgebra.centralizer R
(Algebra.TensorProduct.includeLeft (S := R) '' S) =
(Algebra.TensorProduct.map (Subalgebra.centralizer R (S : Set A)).val
(AlgHom.id R B)).range
|
case h.mp.intro
R : Type u_1
inst✝⁵ : CommSemiring R
A : Type u_2
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u_3
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Set A
inst✝ : Module.Free R B
ℬ : Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B
b : Module.Free.ChooseBasisIndex R B →₀ A
j : Module.Free.ChooseBasisIndex R B
hj : j ∈ b.support
x : A
hx : x ∈ S
hw : (x ⊗ₜ[R] 1 * b.sum fun i m => m ⊗ₜ[R] ℬ i) = (b.sum fun i m => m ⊗ₜ[R] ℬ i) * x ⊗ₜ[R] 1
⊢ x • b = mapRange (fun x_1 => x_1 * x) ⋯ b
|
simp only [Finsupp.sum, Finset.mul_sum, Algebra.TensorProduct.tmul_mul_tmul, one_mul,
Finset.sum_mul, mul_one] at hw
|
case h.mp.intro
R : Type u_1
inst✝⁵ : CommSemiring R
A : Type u_2
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u_3
inst✝² : Semiring B
inst✝¹ : Algebra R B
S : Set A
inst✝ : Module.Free R B
ℬ : Basis (Module.Free.ChooseBasisIndex R B) R B := Module.Free.chooseBasis R B
b : Module.Free.ChooseBasisIndex R B →₀ A
j : Module.Free.ChooseBasisIndex R B
hj : j ∈ b.support
x : A
hx : x ∈ S
hw : ∑ i ∈ b.support, (x * b i) ⊗ₜ[R] ℬ i = ∑ i ∈ b.support, (b i * x) ⊗ₜ[R] ℬ i
⊢ x • b = mapRange (fun x_1 => x_1 * x) ⋯ b
|
104f26fc78558199
|
powers_eq_top_of_prime_card
|
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
theorem powers_eq_top_of_prime_card {p : ℕ}
[hp : Fact p.Prime] (h : Nat.card G = p) {g : G} (hg : g ≠ 1) : Submonoid.powers g = ⊤
|
case h
G : Type u_2
inst✝ : Group G
p : ℕ
hp : Fact (Nat.Prime p)
h : Nat.card G = p
g : G
hg : g ≠ 1
x : G
⊢ x ∈ Submonoid.powers g ↔ x ∈ ⊤
|
simp [mem_powers_of_prime_card h hg]
|
no goals
|
432db5b83eafddb1
|
Besicovitch.exists_closedBall_covering_tsum_measure_le
|
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ]
[Measure.OuterRegular μ] {ε : ℝ≥0∞} (hε : ε ≠ 0) (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧
(s ⊆ ⋃ x ∈ t, closedBall x (r x)) ∧ (∑' x : t, μ (closedBall x (r x))) ≤ μ s + ε
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0
t0_disj : t0.PairwiseDisjoint fun x => closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty (SatelliteConfig α N τ)
⊢ ∃ t r,
t.Countable ∧
t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧ s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧ ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + ε
|
obtain ⟨v, s'v, v_open, μv⟩ : ∃ v, v ⊇ s' ∧ IsOpen v ∧ μ v ≤ μ s' + ε / 2 / N :=
Set.exists_isOpen_le_add _ _
(by simp only [ne_eq, ENNReal.div_eq_zero_iff, hε, ENNReal.ofNat_ne_top, or_self,
ENNReal.natCast_ne_top, not_false_eq_true])
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterRegular
ε : ℝ≥0∞
hε : ε ≠ 0
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
u : Set α
su : u ⊇ s
u_open : IsOpen u
μu : μ u ≤ μ s + ε / 2
R : α → ℝ
hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u
t0 : Set α
r0 : α → ℝ
t0_count : t0.Countable
t0s : t0 ⊆ s
hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)
μt0 : μ (s \ ⋃ x ∈ t0, closedBall x (r0 x)) = 0
t0_disj : t0.PairwiseDisjoint fun x => closedBall x (r0 x)
s' : Set α := s \ ⋃ x ∈ t0, closedBall x (r0 x)
s's : s' ⊆ s
N : ℕ
τ : ℝ
hτ : 1 < τ
H : IsEmpty (SatelliteConfig α N τ)
v : Set α
s'v : v ⊇ s'
v_open : IsOpen v
μv : μ v ≤ μ s' + ε / 2 / ↑N
⊢ ∃ t r,
t.Countable ∧
t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧ s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧ ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + ε
|
181682c2ed74f98b
|
EReal.tendsto_toReal
|
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
theorem tendsto_toReal {a : EReal} (ha : a ≠ ⊤) (h'a : a ≠ ⊥) :
Tendsto EReal.toReal (𝓝 a) (𝓝 a.toReal)
|
case intro
a : ℝ
ha : ↑a ≠ ⊤
h'a : ↑a ≠ ⊥
⊢ Tendsto toReal (𝓝 ↑a) (𝓝 (↑a).toReal)
|
rw [nhds_coe, tendsto_map'_iff]
|
case intro
a : ℝ
ha : ↑a ≠ ⊤
h'a : ↑a ≠ ⊥
⊢ Tendsto (toReal ∘ Real.toEReal) (𝓝 a) (𝓝 (↑a).toReal)
|
640a6db1a2fb3b69
|
MeasureTheory.maximal_ineq
|
Mathlib/Probability/Martingale/OptionalStopping.lean
|
theorem maximal_ineq [IsFiniteMeasure μ] (hsub : Submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0}
(n : ℕ) : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω} ≤
ENNReal.ofReal (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω},
f n ω ∂μ)
|
case hst
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
⊢ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} ⊓ {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε} ≤ ⊥
|
rintro ω ⟨hω₁, hω₂⟩
|
case hst.intro
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
𝒢 : Filtration ℕ m0
f : ℕ → Ω → ℝ
inst✝ : IsFiniteMeasure μ
hsub : Submartingale f 𝒢 μ
hnonneg : 0 ≤ f
ε : ℝ≥0
n : ℕ
ω : Ω
hω₁ : ω ∈ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω}
hω₂ : ω ∈ {ω | ((range (n + 1)).sup' ⋯ fun k => f k ω) < ↑ε}
⊢ ω ∈ ⊥
|
7389c69afd10c321
|
Polynomial.card_roots_le_derivative
|
Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean
|
theorem card_roots_le_derivative (p : ℝ[X]) :
Multiset.card p.roots ≤ Multiset.card (derivative p).roots + 1 :=
calc
Multiset.card p.roots = ∑ x ∈ p.roots.toFinset, p.roots.count x :=
(Multiset.toFinset_sum_count_eq _).symm
_ = ∑ x ∈ p.roots.toFinset, (p.roots.count x - 1 + 1) :=
(Eq.symm <| Finset.sum_congr rfl fun _ hx => tsub_add_cancel_of_le <|
Nat.succ_le_iff.2 <| Multiset.count_pos.2 <| Multiset.mem_toFinset.1 hx)
_ = (∑ x ∈ p.roots.toFinset, (p.rootMultiplicity x - 1)) + p.roots.toFinset.card
|
p : ℝ[X]
⊢ ∑ x ∈ p.roots.toFinset, rootMultiplicity x (derivative p) +
(((derivative p).roots.toFinset \ p.roots.toFinset).card + 1) ≤
∑ x ∈ p.roots.toFinset, Multiset.count x (derivative p).roots +
(∑ x ∈ (derivative p).roots.toFinset \ p.roots.toFinset, Multiset.count x (derivative p).roots + 1)
|
simp only [← count_roots]
|
p : ℝ[X]
⊢ ∑ x ∈ p.roots.toFinset, Multiset.count x (derivative p).roots +
(((derivative p).roots.toFinset \ p.roots.toFinset).card + 1) ≤
∑ x ∈ p.roots.toFinset, Multiset.count x (derivative p).roots +
(∑ x ∈ (derivative p).roots.toFinset \ p.roots.toFinset, Multiset.count x (derivative p).roots + 1)
|
08e969b6e08acf5a
|
AlgebraicGeometry.IsAffineOpen.fromSpec_image_basicOpen
|
Mathlib/AlgebraicGeometry/AffineScheme.lean
|
theorem fromSpec_image_basicOpen :
hU.fromSpec ''ᵁ (PrimeSpectrum.basicOpen f) = X.basicOpen f
|
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
⊢ hU.fromSpec ''ᵁ hU.fromSpec ⁻¹ᵁ X.basicOpen f = X.basicOpen f
|
ext1
|
case h
X : Scheme
U : X.Opens
hU : IsAffineOpen U
f : ↑Γ(X, U)
⊢ ↑(hU.fromSpec ''ᵁ hU.fromSpec ⁻¹ᵁ X.basicOpen f) = ↑(X.basicOpen f)
|
e9ee2eabe0c4108b
|
Orientation.abs_areaForm_le
|
Mathlib/Analysis/InnerProductSpace/TwoDim.lean
|
theorem abs_areaForm_le (x y : E) : |ω x y| ≤ ‖x‖ * ‖y‖
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E
⊢ |(o.areaForm x) y| ≤ ‖x‖ * ‖y‖
|
simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y]
|
no goals
|
5a9811f410c8d903
|
AddCircle.ae_empty_or_univ_of_forall_vadd_ae_eq_self
|
Mathlib/Dynamics/Ergodic/AddCircle.lean
|
theorem ae_empty_or_univ_of_forall_vadd_ae_eq_self {s : Set <| AddCircle T}
(hs : NullMeasurableSet s volume) {ι : Type*} {l : Filter ι} [l.NeBot] {u : ι → AddCircle T}
(hu₁ : ∀ i, (u i +ᵥ s : Set _) =ᵐ[volume] s) (hu₂ : Tendsto (addOrderOf ∘ u) l atTop) :
s =ᵐ[volume] (∅ : Set <| AddCircle T) ∨ s =ᵐ[volume] univ
|
case inr.h.intro.intro
T : ℝ
hT : Fact (0 < T)
s : Set (AddCircle T)
ι : Type u_1
l : Filter ι
inst✝ : l.NeBot
u : ι → AddCircle T
μ : Measure (AddCircle T) := volume
hs : NullMeasurableSet s μ
hu₁ : ∀ (i : ι), u i +ᵥ s =ᶠ[ae μ] s
n : ι → ℕ := addOrderOf ∘ u
hu₂ : Tendsto n l atTop
hT₀ : 0 < T
hT₁ : ENNReal.ofReal T ≠ 0
h : μ s ≠ 0
d : AddCircle T
I : ι → Set (AddCircle T) := fun j => closedBall d (T / (2 * ↑(n j)))
hd : Tendsto (fun j => μ (s ∩ I j) / μ (I j)) l (𝓝 1)
j : ι
hj : 0 < n j
this : addOrderOf (u j) = n j
huj : IsOfFinAddOrder (u j)
huj' : 1 ≤ ↑(n j)
hI₀ : μ (I j) ≠ 0
hI₁ : μ (I j) ≠ ⊤
hI₂ : μ (I j) * ↑(n j) = ENNReal.ofReal T
⊢ μ (s ∩ I j) / μ (I j) = μ s / ENNReal.ofReal T
|
rw [ENNReal.div_eq_div_iff hT₁ ENNReal.ofReal_ne_top hI₀ hI₁,
volume_of_add_preimage_eq s _ (u j) d huj (hu₁ j) closedBall_ae_eq_ball, nsmul_eq_mul, ←
mul_assoc, this, hI₂]
|
no goals
|
caa1d183a972f27c
|
TopCat.Presheaf.stalkPushforward.id
|
Mathlib/Topology/Sheaves/Stalks.lean
|
theorem id (ℱ : X.Presheaf C) (x : X) :
ℱ.stalkPushforward C (𝟙 X) x = (stalkFunctor C x).map (Pushforward.id ℱ).hom
|
case ih
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : TopCat
ℱ : Presheaf C X
x : ↑X
U✝ : Opens ↑X
hxU✝ : (ConcreteCategory.hom (𝟙 X)) x ∈ U✝
⊢ ℱ.map ((NatTrans.op (OpenNhds.inclusionMapIso (𝟙 X) x).inv).app (op { obj := U✝, property := hxU✝ })) ≫
colimit.ι
((OpenNhds.map (𝟙 X) x).op ⋙
((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ)
(op { obj := U✝, property := hxU✝ }) ≫
colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ)
(OpenNhds.map (𝟙 X) x).op =
colimit.ι ((OpenNhds.inclusion ((ConcreteCategory.hom (𝟙 X)) x)).op ⋙ (pushforward C (𝟙 X)).obj ℱ)
(op { obj := U✝, property := hxU✝ }) ≫
(stalkFunctor C x).map (Pushforward.id ℱ).hom
|
erw [CategoryTheory.Functor.map_id]
|
case ih
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X : TopCat
ℱ : Presheaf C X
x : ↑X
U✝ : Opens ↑X
hxU✝ : (ConcreteCategory.hom (𝟙 X)) x ∈ U✝
⊢ 𝟙
(ℱ.obj
((OpenNhds.inclusion ((ConcreteCategory.hom (𝟙 X)) x) ⋙ Opens.map (𝟙 X)).op.obj
(op { obj := U✝, property := hxU✝ }))) ≫
colimit.ι
((OpenNhds.map (𝟙 X) x).op ⋙
((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ)
(op { obj := U✝, property := hxU✝ }) ≫
colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj ℱ)
(OpenNhds.map (𝟙 X) x).op =
colimit.ι ((OpenNhds.inclusion ((ConcreteCategory.hom (𝟙 X)) x)).op ⋙ (pushforward C (𝟙 X)).obj ℱ)
(op { obj := U✝, property := hxU✝ }) ≫
(stalkFunctor C x).map (Pushforward.id ℱ).hom
|
a8fdc3f0ca56f963
|
List.findIdx?_isSome
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem findIdx?_isSome {xs : List α} {p : α → Bool} :
(xs.findIdx? p).isSome = xs.any p
|
case cons
α : Type u_1
p : α → Bool
x : α
xs : List α
ih : (findIdx? p xs).isSome = xs.any p
⊢ (if p x = true then some 0 else Option.map (fun i => i + 1) (findIdx? p xs)).isSome = (x :: xs).any p
|
split <;> simp_all
|
no goals
|
f5b414e8399ca944
|
MvPolynomial.IsHomogeneous.prod
|
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
theorem prod {ι : Type*} (s : Finset ι) (φ : ι → MvPolynomial σ R) (n : ι → ℕ)
(h : ∀ i ∈ s, IsHomogeneous (φ i) (n i)) : IsHomogeneous (∏ i ∈ s, φ i) (∑ i ∈ s, n i)
|
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
ι : Type u_5
s✝ : Finset ι
φ : ι → MvPolynomial σ R
n : ι → ℕ
i : ι
s : Finset ι
his : i ∉ s
IH : (∀ i ∈ s, (φ i).IsHomogeneous (n i)) → (∏ i ∈ s, φ i).IsHomogeneous (∑ i ∈ s, n i)
h : ∀ i_1 ∈ insert i s, (φ i_1).IsHomogeneous (n i_1)
⊢ ∀ i ∈ s, (φ i).IsHomogeneous (n i)
|
intro j hjs
|
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
ι : Type u_5
s✝ : Finset ι
φ : ι → MvPolynomial σ R
n : ι → ℕ
i : ι
s : Finset ι
his : i ∉ s
IH : (∀ i ∈ s, (φ i).IsHomogeneous (n i)) → (∏ i ∈ s, φ i).IsHomogeneous (∑ i ∈ s, n i)
h : ∀ i_1 ∈ insert i s, (φ i_1).IsHomogeneous (n i_1)
j : ι
hjs : j ∈ s
⊢ (φ j).IsHomogeneous (n j)
|
44b0ae11f9a5b764
|
CStarMatrix.toCLM_injective
|
Mathlib/Analysis/CStarAlgebra/CStarMatrix.lean
|
lemma toCLM_injective : Function.Injective (toCLM (A := A) (m := m) (n := n))
|
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype n
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype m
⊢ ∀ (a : CStarMatrix m n A), toCLM a = 0 → a = 0
|
intro M h
|
m : Type u_1
n : Type u_2
A : Type u_3
inst✝⁴ : Fintype n
inst✝³ : NonUnitalCStarAlgebra A
inst✝² : PartialOrder A
inst✝¹ : StarOrderedRing A
inst✝ : Fintype m
M : CStarMatrix m n A
h : toCLM M = 0
⊢ M = 0
|
bd474ad324000856
|
Sigma.isConnected_iff
|
Mathlib/Topology/Connected/Clopen.lean
|
theorem Sigma.isConnected_iff [∀ i, TopologicalSpace (π i)] {s : Set (Σi, π i)} :
IsConnected s ↔ ∃ i t, IsConnected t ∧ s = Sigma.mk i '' t
|
ι : Type u_1
π : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (π i)
s : Set ((i : ι) × π i)
⊢ IsConnected s ↔ ∃ i t, IsConnected t ∧ s = mk i '' t
|
refine ⟨fun hs => ?_, ?_⟩
|
case refine_1
ι : Type u_1
π : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (π i)
s : Set ((i : ι) × π i)
hs : IsConnected s
⊢ ∃ i t, IsConnected t ∧ s = mk i '' t
case refine_2
ι : Type u_1
π : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (π i)
s : Set ((i : ι) × π i)
⊢ (∃ i t, IsConnected t ∧ s = mk i '' t) → IsConnected s
|
1455e050525a61d2
|
Polynomial.Splits.comp_of_map_degree_le_one
|
Mathlib/Algebra/Polynomial/Splits.lean
|
theorem Splits.comp_of_map_degree_le_one {f : K[X]} {p : K[X]} (hd : (p.map i).degree ≤ 1)
(h : f.Splits i) : (f.comp p).Splits i
|
K : Type v
L : Type w
inst✝¹ : CommRing K
inst✝ : Field L
i : K →+* L
f p : K[X]
hd : (map i p).degree ≤ 1
h : Splits i f
⊢ Splits i (f.comp p)
|
by_cases hzero : map i (f.comp p) = 0
|
case pos
K : Type v
L : Type w
inst✝¹ : CommRing K
inst✝ : Field L
i : K →+* L
f p : K[X]
hd : (map i p).degree ≤ 1
h : Splits i f
hzero : map i (f.comp p) = 0
⊢ Splits i (f.comp p)
case neg
K : Type v
L : Type w
inst✝¹ : CommRing K
inst✝ : Field L
i : K →+* L
f p : K[X]
hd : (map i p).degree ≤ 1
h : Splits i f
hzero : ¬map i (f.comp p) = 0
⊢ Splits i (f.comp p)
|
7862ae74e5525276
|
MeasureTheory.levyProkhorovEDist_le_of_forall
|
Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
|
/-- A simple general sufficient condition for bounding `levyProkhorovEDist` from above. -/
lemma levyProkhorovEDist_le_of_forall (μ ν : Measure Ω) (δ : ℝ≥0∞)
(h : ∀ ε B, δ < ε → ε < ∞ → MeasurableSet B →
μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε) :
levyProkhorovEDist μ ν ≤ δ
|
case pos
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
δ : ℝ≥0∞
h :
∀ (ε : ℝ≥0∞) (B : Set Ω),
δ < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening ε.toReal B) + ε ∧ ν B ≤ μ (thickening ε.toReal B) + ε
δ_top : δ = ⊤
⊢ levyProkhorovEDist μ ν ≤ δ
|
simp only [δ_top, add_top, le_top]
|
no goals
|
4df6916ed43dac71
|
CategoryTheory.Functor.preservesEqualizer_of_preservesKernels
|
Mathlib/CategoryTheory/Preadditive/LeftExact.lean
|
/-- A functor between preadditive categories preserves the equalizer of two
morphisms if it preserves all kernels. -/
lemma preservesEqualizer_of_preservesKernels
[∀ {X Y} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F]
{X Y : C} (f g : X ⟶ Y) : PreservesLimit (parallelPair f g) F
|
case preserves.val
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : Preadditive D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : HasBinaryBiproducts C
inst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
X Y : C
f g : X ⟶ Y
this✝ : PreservesBinaryBiproducts F := preservesBinaryBiproducts_of_preservesBinaryProducts F
this : F.Additive
c : Cone (parallelPair f g)
i : IsLimit c
c' : IsLimit (KernelFork.ofι (Fork.ι c) ⋯) := isLimitKernelForkOfFork (i.ofIsoLimit (Fork.isoForkOfι c))
iFc : IsLimit (KernelFork.ofι (F.map (Fork.ι c)) ⋯) := isLimitForkMapOfIsLimit' F ⋯ c'
⊢ IsLimit (F.mapCone c)
|
apply IsLimit.ofIsoLimit _ ((Cones.functoriality _ F).mapIso (Fork.isoForkOfι c).symm)
|
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : Preadditive D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : HasBinaryBiproducts C
inst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
X Y : C
f g : X ⟶ Y
this✝ : PreservesBinaryBiproducts F := preservesBinaryBiproducts_of_preservesBinaryProducts F
this : F.Additive
c : Cone (parallelPair f g)
i : IsLimit c
c' : IsLimit (KernelFork.ofι (Fork.ι c) ⋯) := isLimitKernelForkOfFork (i.ofIsoLimit (Fork.isoForkOfι c))
iFc : IsLimit (KernelFork.ofι (F.map (Fork.ι c)) ⋯) := isLimitForkMapOfIsLimit' F ⋯ c'
⊢ IsLimit ((Cones.functoriality (parallelPair f g) F).obj (Fork.ofι (Fork.ι c) ⋯))
|
6bda66db73d4b92f
|
Real.le_mk_of_forall_le
|
Mathlib/Data/Real/Basic.lean
|
theorem le_mk_of_forall_le {f : CauSeq ℚ abs} : (∃ i, ∀ j ≥ i, x ≤ f j) → x ≤ mk f
|
case h.h.intro.intro
x✝ : ℝ
f x : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, mk x ≤ ↑(↑f j)
K : ℚ
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ ↑(x - f) j
⊢ False
|
obtain ⟨i, H⟩ := exists_forall_ge_and h (exists_forall_ge_and hK (f.cauchy₃ <| half_pos K0))
|
case h.h.intro.intro.intro
x✝ : ℝ
f x : CauSeq ℚ abs
h : ∃ i, ∀ j ≥ i, mk x ≤ ↑(↑f j)
K : ℚ
K0 : K > 0
hK : ∃ i, ∀ j ≥ i, K ≤ ↑(x - f) j
i : ℕ
H : ∀ j ≥ i, mk x ≤ ↑(↑f j) ∧ K ≤ ↑(x - f) j ∧ ∀ k ≥ j, |↑f k - ↑f j| < K / 2
⊢ False
|
dbfd66513efa46dc
|
ascPochhammer_succ_right
|
Mathlib/RingTheory/Polynomial/Pochhammer.lean
|
theorem ascPochhammer_succ_right (n : ℕ) :
ascPochhammer S (n + 1) = ascPochhammer S n * (X + (n : S[X]))
|
S : Type u
inst✝ : Semiring S
n : ℕ
h : map (algebraMap ℕ S) (ascPochhammer ℕ (n + 1)) = map (algebraMap ℕ S) (ascPochhammer ℕ n * (X + ↑n))
⊢ ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n)
|
simpa only [ascPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_natCast] using h
|
no goals
|
f731d56e5a9bbcf9
|
EuclideanGeometry.angle_eq_angle_of_angle_eq_pi
|
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
theorem angle_eq_angle_of_angle_eq_pi (p₁ : P) {p₂ p₃ p₄ : P} (h : ∠ p₂ p₃ p₄ = π) :
∠ p₁ p₂ p₃ = ∠ p₁ p₂ p₄
|
case h.e'_2.h.e'_5
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
h : InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₄ -ᵥ p₃) = π
left✝ : p₂ -ᵥ p₃ ≠ 0
r : ℝ
hr : r < 0
hpr : p₄ -ᵥ p₃ = r • (p₂ -ᵥ p₃)
⊢ p₄ -ᵥ p₂ = - -(p₄ -ᵥ p₃) + 1 • -(p₂ -ᵥ p₃)
|
simp
|
no goals
|
b4f1b4ad95c44d44
|
MeasureTheory.Measure.sum_extend_zero
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
@[simp] lemma sum_extend_zero {ι ι' : Type*} {f : ι → ι'} (hf : Injective f) (m : ι → Measure α) :
sum (Function.extend f m 0) = sum m
|
case h
α : Type u_1
m0 : MeasurableSpace α
ι : Type u_8
ι' : Type u_9
f : ι → ι'
hf : Injective f
m : ι → Measure α
s : Set α
hs : MeasurableSet s
⊢ (sum (Function.extend f m 0)) s = (sum m) s
|
simp [*, Function.apply_extend (fun μ : Measure α ↦ μ s)]
|
no goals
|
e11ee0592cd1571b
|
LinearMap.toMvPolynomial_eval_eq_apply
|
Mathlib/Algebra/Module/LinearMap/Polynomial.lean
|
lemma toMvPolynomial_eval_eq_apply (f : M₁ →ₗ[R] M₂) (i : ι₂) (c : ι₁ →₀ R) :
eval c (f.toMvPolynomial b₁ b₂ i) = b₂.repr (f (b₁.repr.symm c)) i
|
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
ι₁ : Type u_4
ι₂ : Type u_5
inst✝⁷ : CommRing R
inst✝⁶ : AddCommGroup M₁
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M₁
inst✝³ : Module R M₂
inst✝² : Fintype ι₁
inst✝¹ : Finite ι₂
inst✝ : DecidableEq ι₁
b₁ : Basis ι₁ R M₁
b₂ : Basis ι₂ R M₂
f : M₁ →ₗ[R] M₂
i : ι₂
c : ι₁ →₀ R
⊢ (eval ⇑c) (toMvPolynomial b₁ b₂ f i) = (b₂.repr (f (b₁.repr.symm c))) i
|
rw [toMvPolynomial, Matrix.toMvPolynomial_eval_eq_apply,
← LinearMap.toMatrix_mulVec_repr b₁ b₂, LinearEquiv.apply_symm_apply]
|
no goals
|
0d058754669a5378
|
LucasLehmer.X.ω_mul_ωb
|
Mathlib/NumberTheory/LucasLehmer.lean
|
theorem ω_mul_ωb (q : ℕ+) : (ω : X q) * ωb = 1
|
q : ℕ+
⊢ ω * ωb = 1
|
dsimp [ω, ωb]
|
q : ℕ+
⊢ (2, 1) * (2, -1) = 1
|
f75eba3969b269ac
|
IsClosed.pathComponent
|
Mathlib/Topology/Connected/LocPathConnected.lean
|
theorem IsClosed.pathComponent (x : X) : IsClosed (pathComponent x)
|
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocPathConnectedSpace X
x y : X
hxy : y ∈ (pathComponent x)ᶜ
⊢ (pathComponent x)ᶜ ∈ 𝓝 y
|
rcases (path_connected_basis y).ex_mem with ⟨V, hVy, hVc⟩
|
case intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : LocPathConnectedSpace X
x y : X
hxy : y ∈ (pathComponent x)ᶜ
V : Set X
hVy : V ∈ 𝓝 y
hVc : IsPathConnected V
⊢ (pathComponent x)ᶜ ∈ 𝓝 y
|
1ecf7ca070936f11
|
jacobsonSpace_iff_locallyClosed
|
Mathlib/Topology/JacobsonSpace.lean
|
lemma jacobsonSpace_iff_locallyClosed :
JacobsonSpace X ↔ ∀ Z, Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty
|
case mpr
X : Type u_1
inst✝ : TopologicalSpace X
H : ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty
Z : Set X
hZ : IsClosed Z
H' : ((closure (Z ∩ closedPoints X))ᶜ ∩ Z).Nonempty
this : ¬Z ∩ closedPoints X ⊆ closure (Z ∩ closedPoints X)
⊢ False
|
exact this subset_closure
|
no goals
|
89e2d8a6f5ce61b2
|
SimpleGraph.isNClique_singleton
|
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
@[simp]
lemma isNClique_singleton : G.IsNClique n {a} ↔ n = 1
|
α : Type u_1
G : SimpleGraph α
n : ℕ
a : α
⊢ G.IsNClique n {a} ↔ n = 1
|
simp [isNClique_iff, eq_comm]
|
no goals
|
18f8ce12586f20b9
|
QuasiconvexOn.sup
|
Mathlib/Analysis/Convex/Quasiconvex.lean
|
theorem QuasiconvexOn.sup [SemilatticeSup β] (hf : QuasiconvexOn 𝕜 s f)
(hg : QuasiconvexOn 𝕜 s g) : QuasiconvexOn 𝕜 s (f ⊔ g)
|
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝³ : OrderedSemiring 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
s : Set E
f g : E → β
inst✝ : SemilatticeSup β
hf : QuasiconvexOn 𝕜 s f
hg : QuasiconvexOn 𝕜 s g
⊢ QuasiconvexOn 𝕜 s (f ⊔ g)
|
intro r
|
𝕜 : Type u_1
E : Type u_2
β : Type u_3
inst✝³ : OrderedSemiring 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
s : Set E
f g : E → β
inst✝ : SemilatticeSup β
hf : QuasiconvexOn 𝕜 s f
hg : QuasiconvexOn 𝕜 s g
r : β
⊢ Convex 𝕜 {x | x ∈ s ∧ (f ⊔ g) x ≤ r}
|
156a9e7c30102d0b
|
ContinuousMap.mem_setOfIdeal
|
Mathlib/Topology/ContinuousMap/Ideals.lean
|
theorem mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0
|
X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
I : Ideal C(X, R)
x : X
⊢ (¬∀ f ∈ I, f x = 0) ↔ ∃ f ∈ I, f x ≠ 0
|
push_neg
|
X : Type u_1
R : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : Semiring R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalSemiring R
I : Ideal C(X, R)
x : X
⊢ (∃ f ∈ I, f x ≠ 0) ↔ ∃ f ∈ I, f x ≠ 0
|
b6df98633a7f7742
|
Pell.dvd_of_ysq_dvd
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
theorem dvd_of_ysq_dvd {n t} (h : yn a1 n * yn a1 n ∣ yn a1 t) : yn a1 n ∣ t :=
have nt : n ∣ t := (y_dvd_iff a1 n t).1 <| dvd_of_mul_left_dvd h
n.eq_zero_or_pos.elim (fun n0 => by rwa [n0] at nt ⊢) fun n0l : 0 < n => by
let ⟨k, ke⟩ := nt
have : yn a1 n ∣ k * xn a1 n ^ (k - 1) :=
Nat.dvd_of_mul_dvd_mul_right (strictMono_y a1 n0l) <|
modEq_zero_iff_dvd.1 <| by
have xm := (xy_modEq_yn a1 n k).right; rw [← ke] at xm
exact (xm.of_dvd <| by simp [_root_.pow_succ]).symm.trans h.modEq_zero_nat
rw [ke]
exact dvd_mul_of_dvd_right (((xy_coprime _ _).pow_left _).symm.dvd_of_dvd_mul_right this) _
|
a : ℕ
a1 : 1 < a
n t : ℕ
h : yn a1 n * yn a1 n ∣ yn a1 t
nt : n ∣ t
n0l : 0 < n
k : ℕ
ke : t = n * k
xm : yn a1 t ≡ k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]
⊢ yn a1 n * yn a1 n ∣ yn a1 n ^ 3
|
simp [_root_.pow_succ]
|
no goals
|
0409ee0aa5773315
|
MeasureTheory.StronglyMeasurable.norm_approxBounded_le
|
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
theorem norm_approxBounded_le {β} {f : α → β} [SeminormedAddCommGroup β] [NormedSpace ℝ β]
{m : MeasurableSpace α} {c : ℝ} (hf : StronglyMeasurable[m] f) (hc : 0 ≤ c) (n : ℕ) (x : α) :
‖hf.approxBounded c n x‖ ≤ c
|
case neg.inl
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ ‖1 ⊓ c / ‖(hf.approx n) x‖‖ * ‖(hf.approx n) x‖ ≤ c
|
rw [min_eq_left _]
|
case neg.inl
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ ‖1‖ * ‖(hf.approx n) x‖ ≤ c
α : Type u_1
β : Type u_5
f : α → β
inst✝¹ : SeminormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m : MeasurableSpace α
c : ℝ
hf : StronglyMeasurable f
hc : 0 ≤ c
n : ℕ
x : α
h0 : ¬‖(hf.approx n) x‖ = 0
h : ‖(hf.approx n) x‖ ≤ c
⊢ 1 ≤ c / ‖(hf.approx n) x‖
|
c6dcf159a9df0ec3
|
AccPt.nhds_inter
|
Mathlib/Topology/Perfect.lean
|
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C))
|
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
x : α
U : Set α
h_acc : AccPt x (𝓟 C)
hU : U ∈ 𝓝 x
⊢ AccPt x (𝓟 (U ∩ C))
|
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_principal_iff]
exact mem_nhdsWithin_of_mem_nhds hU
|
α : Type u_1
inst✝ : TopologicalSpace α
C : Set α
x : α
U : Set α
h_acc : AccPt x (𝓟 C)
hU : U ∈ 𝓝 x
this : 𝓝[≠] x ≤ 𝓟 U
⊢ AccPt x (𝓟 (U ∩ C))
|
e1c3c6891dd944be
|
AddCircle.norm_coe_eq_abs_iff
|
Mathlib/Analysis/Normed/Group/AddCircle.lean
|
theorem norm_coe_eq_abs_iff {x : ℝ} (hp : p ≠ 0) : ‖(x : AddCircle p)‖ = |x| ↔ |x| ≤ |p| / 2
|
case inr
p x : ℝ
hp✝ : p ≠ 0
hx : |x| ≤ -p / 2
this : ∀ (p : ℝ), 0 < p → |x| ≤ p / 2 → ‖↑x‖ = |x|
hp : p < 0
⊢ ‖↑x‖ = |x|
|
exact this (-p) (neg_pos.mpr hp) hx
|
no goals
|
1e4f630c7ac9b9d6
|
LinearMap.det_restrictScalars
|
Mathlib/RingTheory/Norm/Transitivity.lean
|
theorem LinearMap.det_restrictScalars [AddCommGroup A] [Module R A] [Module S A]
[IsScalarTower R S A] [Module.Free S A] {f : A →ₗ[S] A} :
(f.restrictScalars R).det = Algebra.norm R f.det
|
case inr
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝ : Nontrivial R
h✝ : Nontrivial A
this✝ : Nontrivial S
ιS : Type u_2
bS : Basis ιS R S
ιA : Type u_3
bA : Basis ιA S A
this : Nonempty ιS
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)
|
have := bA.index_nonempty
|
case inr
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
inst✝⁵ : Module.Free R S
inst✝⁴ : AddCommGroup A
inst✝³ : Module R A
inst✝² : Module S A
inst✝¹ : IsScalarTower R S A
inst✝ : Module.Free S A
f : A →ₗ[S] A
a✝ : Nontrivial R
h✝ : Nontrivial A
this✝¹ : Nontrivial S
ιS : Type u_2
bS : Basis ιS R S
ιA : Type u_3
bA : Basis ιA S A
this✝ : Nonempty ιS
this : Nonempty ιA
⊢ LinearMap.det (↑R f) = (Algebra.norm R) (LinearMap.det f)
|
e61a35e4b25ff2ca
|
contraction_of_isPowMul_of_boundedWrt
|
Mathlib/Analysis/Normed/Ring/IsPowMulFaithful.lean
|
theorem contraction_of_isPowMul_of_boundedWrt {F : Type*} {α : outParam (Type*)} [Ring α]
[FunLike F α ℝ] [RingSeminormClass F α ℝ] {β : Type*} [Ring β] (nα : F) {nβ : β → ℝ}
(hβ : IsPowMul nβ) {f : α →+* β} (hf : f.IsBoundedWrt nα nβ) (x : α) : nβ (f x) ≤ nα x
|
F : Type u_1
α : outParam (Type u_2)
inst✝³ : Ring α
inst✝² : FunLike F α ℝ
inst✝¹ : RingSeminormClass F α ℝ
β : Type u_3
inst✝ : Ring β
nα : F
nβ : β → ℝ
hβ : IsPowMul nβ
f : α →+* β
x : α
C : ℝ
hC0 : 0 < C
hC : ∀ (x : α), nβ (f x) ≤ C * nα x
⊢ Tendsto (fun n => C ^ (1 / ↑n) * nα x) atTop (𝓝 (nα x))
|
nth_rewrite 2 [← one_mul (nα x)]
|
F : Type u_1
α : outParam (Type u_2)
inst✝³ : Ring α
inst✝² : FunLike F α ℝ
inst✝¹ : RingSeminormClass F α ℝ
β : Type u_3
inst✝ : Ring β
nα : F
nβ : β → ℝ
hβ : IsPowMul nβ
f : α →+* β
x : α
C : ℝ
hC0 : 0 < C
hC : ∀ (x : α), nβ (f x) ≤ C * nα x
⊢ Tendsto (fun n => C ^ (1 / ↑n) * nα x) atTop (𝓝 (1 * nα x))
|
585c053a4f683cf1
|
List.erase_orderedInsert_of_not_mem
|
Mathlib/Data/List/Sort.lean
|
theorem erase_orderedInsert_of_not_mem [DecidableEq α]
{x : α} {xs : List α} (hx : x ∉ xs) :
(xs.orderedInsert r x).erase x = xs
|
α : Type u
r : α → α → Prop
inst✝¹ : DecidableRel r
inst✝ : DecidableEq α
x : α
xs : List α
hx : x ∉ xs
⊢ (orderedInsert r x xs).erase x = xs
|
rw [orderedInsert_eq_take_drop, erase_append_right, List.erase_cons_head,
takeWhile_append_dropWhile]
|
case h
α : Type u
r : α → α → Prop
inst✝¹ : DecidableRel r
inst✝ : DecidableEq α
x : α
xs : List α
hx : x ∉ xs
⊢ x ∉ takeWhile (fun b => decide ¬r x b) xs
|
1a0d191d116d3954
|
SimplexCategoryGenRel.isSplitEpi_toSimplexCategory_map_of_P_σ
|
Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/EpiMono.lean
|
lemma isSplitEpi_toSimplexCategory_map_of_P_σ {x y : SimplexCategoryGenRel} {e : x ⟶ y}
(he : P_σ e) : IsSplitEpi <| toSimplexCategory.map e
|
case exists_splitEpi.val.se
x y : SimplexCategoryGenRel
e : x ⟶ y
he : P_σ e
⊢ SplitEpi e
|
exact isSplitEpi_P_σ he |>.exists_splitEpi.some
|
no goals
|
80ea862bb88258c7
|
PerfectClosure.mk_pow
|
Mathlib/FieldTheory/PerfectClosure.lean
|
theorem mk_pow (x : ℕ × K) (n : ℕ) : mk K p x ^ n = mk K p (x.1, x.2 ^ n)
|
case succ
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : ℕ × K
n : ℕ
ih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)
⊢ mk K p x ^ (n + 1) = mk K p (x.1, x.2 ^ (n + 1))
|
rw [pow_succ, pow_succ, ih, mk_mul_mk, mk_eq_iff]
|
case succ
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : ℕ × K
n : ℕ
ih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)
⊢ ∃ z,
(⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + z]
((x.1, x.2 ^ n).1 + x.1,
(⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p))^[(x.1, x.2 ^ n).1] x.2).2 =
(⇑(frobenius K
p))^[((x.1, x.2 ^ n).1 + x.1,
(⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p))^[(x.1, x.2 ^ n).1] x.2).1 +
z]
(x.1, x.2 ^ n * x.2).2
|
7ceaa08111bba2c4
|
Set.exists_mem_image
|
Mathlib/Data/Set/Image.lean
|
theorem exists_mem_image {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x)
|
α : Type u_1
β : Type u_2
f : α → β
s : Set α
p : β → Prop
⊢ (∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x)
|
simp
|
no goals
|
456acbf06282d059
|
Finset.exists_subset_or_subset_of_two_mul_lt_card
|
Mathlib/Data/Finset/Card.lean
|
theorem exists_subset_or_subset_of_two_mul_lt_card [DecidableEq α] {X Y : Finset α} {n : ℕ}
(hXY : 2 * n < #(X ∪ Y)) : ∃ C : Finset α, n < #C ∧ (C ⊆ X ∨ C ⊆ Y)
|
α : Type u_1
inst✝ : DecidableEq α
X Y : Finset α
n : ℕ
hXY : 2 * n < #(X ∪ Y)
h₁ : #(X ∩ (Y \ X)) = 0
h₂ : #(X ∪ Y) = #X + #(Y \ X)
⊢ ∃ C, n < #C ∧ (C ⊆ X ∨ C ⊆ Y)
|
rw [h₂, Nat.two_mul] at hXY
|
α : Type u_1
inst✝ : DecidableEq α
X Y : Finset α
n : ℕ
hXY : n + n < #X + #(Y \ X)
h₁ : #(X ∩ (Y \ X)) = 0
h₂ : #(X ∪ Y) = #X + #(Y \ X)
⊢ ∃ C, n < #C ∧ (C ⊆ X ∨ C ⊆ Y)
|
d2784859d1018aca
|
ZMod.wilsons_lemma
|
Mathlib/NumberTheory/Wilson.lean
|
theorem wilsons_lemma : ((p - 1)! : ZMod p) = -1
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ∏ x : (ZMod p)ˣ, ↑x = -1
|
simp_rw [← Units.coeHom_apply]
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ ∏ x : (ZMod p)ˣ, (Units.coeHom (ZMod p)) x = -1
|
39f6b98ba0c68bfd
|
Dynamics.netMaxcard_zero
|
Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean
|
lemma netMaxcard_zero (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
netMaxcard T F U 0 = 1
|
X : Type u_1
T : X → X
F : Set X
h : F.Nonempty
U : Set (X × X)
s : Finset X
left✝ : ↑s ⊆ F
s_net : (↑s).PairwiseDisjoint fun x => ball x (dynEntourage T U 0)
⊢ ↑s.card ≤ 1
|
simp only [ball, dynEntourage_zero, preimage_univ] at s_net
|
X : Type u_1
T : X → X
F : Set X
h : F.Nonempty
U : Set (X × X)
s : Finset X
left✝ : ↑s ⊆ F
s_net : (↑s).PairwiseDisjoint fun x => univ
⊢ ↑s.card ≤ 1
|
9c03c09b4d97a006
|
ContinuousOn.continuousAt_mulIndicator
|
Mathlib/Topology/Algebra/Indicator.lean
|
theorem ContinuousOn.continuousAt_mulIndicator (hf : ContinuousOn f (interior s)) {x : α}
(hx : x ∉ frontier s) :
ContinuousAt (s.mulIndicator f) x
|
case inr
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
f : α → β
s : Set α
inst✝ : One β
hf : ContinuousOn f (interior s)
x : α
h : x ∈ interior sᶜ
⊢ ContinuousAt (s.mulIndicator f) x
|
exact ContinuousAt.congr continuousAt_const <| Filter.eventuallyEq_iff_exists_mem.mpr
⟨sᶜ, mem_interior_iff_mem_nhds.mp h, Set.eqOn_mulIndicator'.symm⟩
|
no goals
|
6280ff3089489911
|
CategoryTheory.Functor.mem_homologicalKernel_W_iff
|
Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean
|
lemma mem_homologicalKernel_W_iff {X Y : C} (f : X ⟶ Y) :
F.homologicalKernel.W f ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)
|
case intro.intro.intro
C : Type u_1
A : Type u_3
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{u_5, 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
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f g h ∈ distinguishedTriangles
h₁ :
∀ (n : ℤ),
IsZero
(ShortComplex.mk ((F.shift n).map (Triangle.mk f g h).mor₂)
(F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯) ⋯).X₂ ↔
(ShortComplex.mk ((F.shift n).map (Triangle.mk f g h).mor₂) (F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯)
⋯).f =
0 ∧
(ShortComplex.mk ((F.shift n).map (Triangle.mk f g h).mor₂)
(F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯) ⋯).g =
0
⊢ F.homologicalKernel.P (Triangle.mk f g h).obj₃ ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)
|
have h₂ := fun n => F.homologySequence_mono_shift_map_mor₁_iff _ hT n _ rfl
|
case intro.intro.intro
C : Type u_1
A : Type u_3
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{u_5, 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
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f g h ∈ distinguishedTriangles
h₁ :
∀ (n : ℤ),
IsZero
(ShortComplex.mk ((F.shift n).map (Triangle.mk f g h).mor₂)
(F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯) ⋯).X₂ ↔
(ShortComplex.mk ((F.shift n).map (Triangle.mk f g h).mor₂) (F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯)
⋯).f =
0 ∧
(ShortComplex.mk ((F.shift n).map (Triangle.mk f g h).mor₂)
(F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯) ⋯).g =
0
h₂ :
∀ (n : ℤ),
Mono ((F.shift (n + 1)).map (Triangle.mk f g h).mor₁) ↔ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
⊢ F.homologicalKernel.P (Triangle.mk f g h).obj₃ ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)
|
c0e449c7603716db
|
Subgroup.le_normalizer_map
|
Mathlib/Algebra/Group/Subgroup/Basic.lean
|
theorem le_normalizer_map (f : G →* N) : H.normalizer.map f ≤ (H.map f).normalizer := fun _ => by
simp only [and_imp, exists_prop, mem_map, exists_imp, mem_normalizer_iff]
rintro x hx rfl n
constructor
· rintro ⟨y, hy, rfl⟩
use x * y * x⁻¹, (hx y).1 hy
simp
· rintro ⟨y, hyH, hy⟩
use x⁻¹ * y * x
rw [hx]
simp [hy, hyH, mul_assoc]
|
case h
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
N : Type u_5
inst✝ : Group N
f : G →* N
x : G
hx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H
n : N
y : G
hyH : y ∈ H
hy : f y = f x * n * (f x)⁻¹
⊢ x⁻¹ * y * x ∈ H ∧ f (x⁻¹ * y * x) = n
|
rw [hx]
|
case h
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
N : Type u_5
inst✝ : Group N
f : G →* N
x : G
hx : ∀ (h : G), h ∈ H ↔ x * h * x⁻¹ ∈ H
n : N
y : G
hyH : y ∈ H
hy : f y = f x * n * (f x)⁻¹
⊢ x * (x⁻¹ * y * x) * x⁻¹ ∈ H ∧ f (x⁻¹ * y * x) = n
|
d1641f5f705fc8d6
|
Polynomial.Monic.isPrimitive
|
Mathlib/RingTheory/Polynomial/Content.lean
|
theorem Monic.isPrimitive {p : R[X]} (hp : p.Monic) : p.IsPrimitive
|
R : Type u_1
inst✝ : CommSemiring R
p : R[X]
hp : p.Monic
r : R
q : R[X]
h : p = C r * q
⊢ r * q.coeff p.natDegree = 1
|
rwa [← coeff_C_mul, ← h]
|
no goals
|
2e90a779e6a2d05d
|
Complex.IsExpCmpFilter.isLittleO_log_norm_re
|
Mathlib/Analysis/SpecialFunctions/CompareExp.lean
|
theorem isLittleO_log_norm_re (hl : IsExpCmpFilter l) : (fun z => Real.log ‖z‖) =o[l] re :=
calc
(fun z => Real.log ‖z‖) =O[l] fun z => Real.log (√2) + Real.log (max z.re |z.im|) :=
.of_norm_eventuallyLE <|
(hl.tendsto_re.eventually_ge_atTop 1).mono fun z hz => by
have h2 : 0 < √2
|
l : Filter ℂ
hl : IsExpCmpFilter l
z : ℂ
hz : 1 ≤ z.re
h2 : 0 < √2
hz' : 1 ≤ ‖z‖
⊢ (fun x => ‖Real.log ‖x‖‖) z ≤ (fun z => Real.log √2 + Real.log (z.re ⊔ |z.im|)) z
|
have hm₀ : 0 < max z.re |z.im| := lt_max_iff.2 (Or.inl <| one_pos.trans_le hz)
|
l : Filter ℂ
hl : IsExpCmpFilter l
z : ℂ
hz : 1 ≤ z.re
h2 : 0 < √2
hz' : 1 ≤ ‖z‖
hm₀ : 0 < z.re ⊔ |z.im|
⊢ (fun x => ‖Real.log ‖x‖‖) z ≤ (fun z => Real.log √2 + Real.log (z.re ⊔ |z.im|)) z
|
a15b3dd9177e83a7
|
AkraBazziRecurrence.rpow_p_mul_one_add_smoothingFn_ge
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma rpow_p_mul_one_add_smoothingFn_ge :
∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
≤ (r i n) ^ (p a b) * (1 + ε (r i n))
|
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
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
⊢ (fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖
|
refine IsLittleO.eventuallyLE ?_
|
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
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
⊢ (fun x => q ↑(r i x) - q (b i * ↑x)) =o[atTop] fun x => b i ^ p a b * ↑x ^ p a b * (ε (b i * ↑x) - ε ↑x)
|
6b09e1ce84116b26
|
strictConvexOn_rpow
|
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
theorem strictConvexOn_rpow {p : ℝ} (hp : 1 < p) : StrictConvexOn ℝ (Ici 0) fun x : ℝ ↦ x ^ p
|
p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
⊢ (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)
|
have hy' : 0 < y ^ p := rpow_pos_of_pos hy _
|
p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
⊢ (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)
|
003004303f55f1b7
|
CategoryTheory.Limits.Types.colimit_eq
|
Mathlib/CategoryTheory/Limits/Types.lean
|
theorem colimit_eq {j j' : J} {x : F.obj j} {x' : F.obj j'}
(w : colimit.ι F j x = colimit.ι F j' x') :
Relation.EqvGen (Quot.Rel F) ⟨j, x⟩ ⟨j', x'⟩
|
J : Type v
inst✝¹ : Category.{w, v} J
F : J ⥤ Type u
inst✝ : HasColimit F
j j' : J
x : F.obj j
x' : F.obj j'
w : colimit.ι F j x = colimit.ι F j' x'
⊢ Quot.mk (Quot.Rel F) ⟨j, x⟩ = Quot.mk (Quot.Rel F) ⟨j', x'⟩
|
simpa using congr_arg (colimitEquivQuot F) w
|
no goals
|
590ecf05694f85c0
|
Finset.centerMass_insert
|
Mathlib/Analysis/Convex/Combination.lean
|
theorem Finset.centerMass_insert (ha : i ∉ t) (hw : ∑ j ∈ t, w j ≠ 0) :
(insert i t).centerMass w z =
(w i / (w i + ∑ j ∈ t, w j)) • z i +
((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z
|
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
i : ι
t : Finset ι
w : ι → R
z : ι → E
ha : i ∉ t
hw : ∑ j ∈ t, w j ≠ 0
⊢ (w i / (w i + ∑ i ∈ t, w i)) • z i + (w i + ∑ i ∈ t, w i)⁻¹ • ∑ i ∈ t, w i • z i =
(w i / (w i + ∑ i ∈ t, w i)) • z i + ((∑ i ∈ t, w i) / (w i + ∑ i ∈ t, w i) * (∑ i ∈ t, w i)⁻¹) • ∑ i ∈ t, w i • z i
|
congr 2
|
case e_a.e_a
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
i : ι
t : Finset ι
w : ι → R
z : ι → E
ha : i ∉ t
hw : ∑ j ∈ t, w j ≠ 0
⊢ (w i + ∑ i ∈ t, w i)⁻¹ = (∑ i ∈ t, w i) / (w i + ∑ i ∈ t, w i) * (∑ i ∈ t, w i)⁻¹
|
ede647c11966ce4d
|
MeasureTheory.LevyProkhorov.continuous_equiv_probabilityMeasure
|
Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
|
/-- The identity map `LevyProkhorov (ProbabilityMeasure Ω) → ProbabilityMeasure Ω` is continuous. -/
lemma LevyProkhorov.continuous_equiv_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω))
|
Ω : Type u_1
inst✝² : MeasurableSpace Ω
inst✝¹ : PseudoMetricSpace Ω
inst✝ : OpensMeasurableSpace Ω
μs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)
ν : LevyProkhorov (ProbabilityMeasure Ω)
hμs : Tendsto μs atTop (𝓝 ν)
P : ProbabilityMeasure Ω := (equiv (ProbabilityMeasure Ω)) ν
Ps : ℕ → ProbabilityMeasure Ω := fun n => (equiv (ProbabilityMeasure Ω)) (μs n)
f✝ f : Ω →ᵇ ℝ
f_nn : 0 ≤ f
f_zero : ¬‖f‖ = 0
norm_f_pos : 0 < ‖f‖
δ : ℝ
δ_pos : 0 < δ
εs : ℕ → ℝ
left✝ : StrictAnti εs
εs_pos : ∀ (n : ℕ), 0 < εs n
εs_lim : Tendsto εs atTop (𝓝 0)
ε_of_room : Tendsto (fun x => dist (μs x) ν + εs x) atTop (𝓝 (0 + 0))
n : ℕ
⊢ 0 < dist (μs n) ν + εs n
|
linarith [εs_pos n, dist_nonneg (x := μs n) (y := ν)]
|
no goals
|
e3f8df53302ac07c
|
ContinuousMonoidHom.locallyCompactSpace_of_equicontinuousAt
|
Mathlib/Topology/Algebra/Group/CompactOpen.lean
|
theorem locallyCompactSpace_of_equicontinuousAt (U : Set X) (V : Set Y)
(hU : IsCompact U) (hV : V ∈ nhds (1 : Y))
(h : EquicontinuousAt (fun f : {f : X →* Y | Set.MapsTo f U V} ↦ (f : X → Y)) 1) :
LocallyCompactSpace (ContinuousMonoidHom X Y)
|
X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : UniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := {f | Set.MapsTo (⇑f) U W}
S2 : Set (ContinuousMonoidHom X Y) := {f | Set.MapsTo (⇑f) U W}
S3 : Set C(X, Y) := _root_.toContinuousMap '' S2
S4 : Set (X → Y) := DFunLike.coe '' S3
h : Equicontinuous fun x => ⇑↑x
⊢ S4 = DFunLike.coe '' S1
|
ext
|
case h
X : Type u_7
Y : Type u_8
inst✝⁷ : TopologicalSpace X
inst✝⁶ : Group X
inst✝⁵ : IsTopologicalGroup X
inst✝⁴ : UniformSpace Y
inst✝³ : CommGroup Y
inst✝² : UniformGroup Y
inst✝¹ : T0Space Y
inst✝ : CompactSpace Y
U : Set X
V : Set Y
hU : IsCompact U
hV : V ∈ 𝓝 1
W : Set Y
hWo : W ∈ 𝓝 1
hWV : W ⊆ V
hWc : IsCompact W
S1 : Set (X →* Y) := {f | Set.MapsTo (⇑f) U W}
S2 : Set (ContinuousMonoidHom X Y) := {f | Set.MapsTo (⇑f) U W}
S3 : Set C(X, Y) := _root_.toContinuousMap '' S2
S4 : Set (X → Y) := DFunLike.coe '' S3
h : Equicontinuous fun x => ⇑↑x
x✝ : X → Y
⊢ x✝ ∈ S4 ↔ x✝ ∈ DFunLike.coe '' S1
|
20e48f97652adb36
|
CFC.one_rpow
|
Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow/Basic.lean
|
@[simp]
lemma one_rpow {x : ℝ} : (1 : A) ^ x = (1 : A)
|
A : Type u_1
inst✝⁵ : PartialOrder A
inst✝⁴ : Ring A
inst✝³ : StarRing A
inst✝² : TopologicalSpace A
inst✝¹ : Algebra ℝ A
inst✝ : ContinuousFunctionalCalculus ℝ≥0 fun a => 0 ≤ a
x : ℝ
⊢ 1 ^ x = 1
|
simp [rpow_def]
|
no goals
|
06b9de69f2dcad22
|
Finset.Nontrivial.sdiff_singleton_nonempty
|
Mathlib/Data/Finset/SDiff.lean
|
theorem Nontrivial.sdiff_singleton_nonempty {c : α} {s : Finset α} (hS : s.Nontrivial) :
(s \ {c}).Nonempty
|
α : Type u_1
inst✝ : DecidableEq α
c : α
s : Finset α
hS : s.Nontrivial
⊢ s ≠ ∅
|
rintro rfl
|
α : Type u_1
inst✝ : DecidableEq α
c : α
hS : ∅.Nontrivial
⊢ False
|
a339ab238357f678
|
Profinite.NobelingProof.GoodProducts.union
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem GoodProducts.union : range C = ⋃ (e : {o' // o' < o}), (smaller C e.val)
|
case h.refine_2.intro.intro.intro.intro.intro.intro
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
ho : o.IsLimit
hsC : contained C o
o' : Ordinal.{u}
h : o' < o
l : Products I
hl : Products.isGood (π C fun x => ord I x < o') l
⊢ Products.isGood C l
|
rw [contained_eq_proj C o hsC]
|
case h.refine_2.intro.intro.intro.intro.intro.intro
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
inst✝ : WellFoundedLT I
o : Ordinal.{u}
ho : o.IsLimit
hsC : contained C o
o' : Ordinal.{u}
h : o' < o
l : Products I
hl : Products.isGood (π C fun x => ord I x < o') l
⊢ Products.isGood (π C fun x => ord I x < o) l
|
30093b771f1e8664
|
Equiv.Perm.sign_sumCongr
|
Mathlib/GroupTheory/Perm/Sign.lean
|
theorem sign_sumCongr (σa : Perm α) (σb : Perm β) : sign (sumCongr σa σb) = sign σa * sign σb
|
α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
σa : Perm α
σb : Perm β
⊢ sign (σa.sumCongr σb) = sign σa * sign σb
|
suffices sign (sumCongr σa (1 : Perm β)) = sign σa ∧ sign (sumCongr (1 : Perm α) σb) = sign σb
by rw [← this.1, ← this.2, ← sign_mul, sumCongr_mul, one_mul, mul_one]
|
α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
σa : Perm α
σb : Perm β
⊢ sign (σa.sumCongr 1) = sign σa ∧ sign (sumCongr 1 σb) = sign σb
|
c74fcfacb0104818
|
ProbabilityTheory.tsum_prob_mem_Ioi_lt_top
|
Mathlib/Probability/StrongLaw.lean
|
theorem tsum_prob_mem_Ioi_lt_top {X : Ω → ℝ} (hint : Integrable X) (hnonneg : 0 ≤ X) :
(∑' j : ℕ, ℙ {ω | X ω ∈ Set.Ioi (j : ℝ)}) < ∞
|
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K i : ℕ
x✝ : i ∈ range K
⊢ ⋃ N, {ω | X ω ∈ Set.Ioc ↑i ↑N} ⊆ {ω | X ω ∈ Set.Ioi ↑i}
|
simp (config := {contextual := true}) only [Set.mem_Ioc, Set.mem_Ioi,
Set.iUnion_subset_iff, Set.setOf_subset_setOf, imp_true_iff]
|
no goals
|
104c70daf554c26d
|
PartialEquiv.transEquiv_transEquiv
|
Mathlib/Logic/Equiv/PartialEquiv.lean
|
theorem transEquiv_transEquiv (e : PartialEquiv α β) (f' : β ≃ γ) (f'' : γ ≃ δ) :
(e.transEquiv f').transEquiv f'' = e.transEquiv (f'.trans f'')
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
e : PartialEquiv α β
f' : β ≃ γ
f'' : γ ≃ δ
⊢ (e.transEquiv f').transEquiv f'' = e.transEquiv (f'.trans f'')
|
simp only [transEquiv_eq_trans, trans_assoc, Equiv.trans_toPartialEquiv]
|
no goals
|
c7cee9f0e75c6334
|
differentiableAt_apply
|
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
theorem differentiableAt_apply (i : ι) (f : ∀ i, F' i) :
DifferentiableAt (𝕜 := 𝕜) (fun f : ∀ i, F' i => f i) f
|
case a
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
ι : Type u_6
inst✝² : Fintype ι
F' : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
i : ι
f : (i : ι) → F' i
h : DifferentiableAt 𝕜 (fun f i' => f i') f → ∀ (i : ι), DifferentiableAt 𝕜 (fun x => x i) f
⊢ DifferentiableAt 𝕜 (fun f i' => f i') f
|
apply differentiableAt_id
|
no goals
|
7f372669966d314d
|
Std.DHashMap.Internal.Raw₀.Const.get?_of_isEmpty
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean
|
theorem get?_of_isEmpty [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {a : α} :
m.1.isEmpty = true → get? m a = none
|
α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw₀ α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
a : α
⊢ (toListModel m.val.buckets).isEmpty = true → getValue? a (toListModel m.val.buckets) = none
|
empty
|
no goals
|
935d58f9e8fd1d8f
|
Polynomial.IsUnitTrinomial.irreducible_aux1
|
Mathlib/Algebra/Polynomial/UnitTrinomial.lean
|
theorem irreducible_aux1 {k m n : ℕ} (hkm : k < m) (hmn : m < n) (u v w : Units ℤ)
(hp : p = trinomial k m n (u : ℤ) v w) :
C (v : ℤ) * (C (u : ℤ) * X ^ (m + n) + C (w : ℤ) * X ^ (n - m + k + n)) =
⟨Finsupp.filter (· ∈ Set.Ioo (k + n) (n + n)) (p * p.mirror).toFinsupp⟩
|
case h
p : ℤ[X]
k m n : ℕ
hkm : k < m
hmn : m < n
u v w : ℤˣ
hp : p = trinomial k m n ↑u ↑v ↑w
key : n - m + k < n
⊢ k + n ∉ Set.Ioo (k + n) (n + n)
|
exact fun h => h.1.ne rfl
|
no goals
|
fb6ff88b1ff3f5f1
|
Metric.totallyBounded_of_finite_discretization
|
Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
|
theorem totallyBounded_of_finite_discretization {s : Set α}
(H : ∀ ε > (0 : ℝ),
∃ (β : Type u) (_ : Fintype β) (F : s → β), ∀ x y, F x = F y → dist (x : α) y < ε) :
TotallyBounded s
|
case inl
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
H : ∀ ε > 0, ∃ β x F, ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε
hs : s = ∅
⊢ TotallyBounded s
|
rw [hs]
|
case inl
α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
H : ∀ ε > 0, ∃ β x F, ∀ (x y : ↑s), F x = F y → dist ↑x ↑y < ε
hs : s = ∅
⊢ TotallyBounded ∅
|
1adb2e5de1e48f63
|
WeierstrassCurve.b₈_of_isCharTwoJEqZeroNF
|
Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean
|
theorem b₈_of_isCharTwoJEqZeroNF : W.b₈ = -W.a₄ ^ 2
|
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJEqZeroNF
⊢ W.b₈ = -W.a₄ ^ 2
|
rw [b₈, a₁_of_isCharTwoJEqZeroNF, a₂_of_isCharTwoJEqZeroNF]
|
R : Type u_1
inst✝¹ : CommRing R
W : WeierstrassCurve R
inst✝ : W.IsCharTwoJEqZeroNF
⊢ 0 ^ 2 * W.a₆ + 4 * 0 * W.a₆ - 0 * W.a₃ * W.a₄ + 0 * W.a₃ ^ 2 - W.a₄ ^ 2 = -W.a₄ ^ 2
|
70a8d50e66eca8ab
|
AffineSubspace.SSameSide.oangle_sign_eq
|
Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean
|
theorem _root_.AffineSubspace.SSameSide.oangle_sign_eq {s : AffineSubspace ℝ P} {p₁ p₂ p₃ p₄ : P}
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃p₄ : s.SSameSide p₃ p₄) :
(∡ p₁ p₄ p₂).sign = (∡ p₁ p₃ p₂).sign
|
case neg
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)
s : AffineSubspace ℝ P
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃p₄ : s.SSameSide p₃ p₄
h : ¬p₁ = p₂
sp : Set (P × P × P) := (fun p => (p₁, p, p₂)) '' {p | s.SSameSide p₃ p}
hc : IsConnected sp
hf : ContinuousOn (fun p => ∡ p.1 p.2.1 p.2.2) sp
hsp : ∀ p ∈ sp, ∡ p.1 p.2.1 p.2.2 ≠ 0 ∧ ∡ p.1 p.2.1 p.2.2 ≠ ↑π
⊢ (∡ p₁ p₄ p₂).sign = (∡ p₁ p₃ p₂).sign
|
have hp₃ : (p₁, p₃, p₂) ∈ sp :=
Set.mem_image_of_mem _ (sSameSide_self_iff.2 ⟨hp₃p₄.nonempty, hp₃p₄.2.1⟩)
|
case neg
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)
s : AffineSubspace ℝ P
p₁ p₂ p₃ p₄ : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
hp₃p₄ : s.SSameSide p₃ p₄
h : ¬p₁ = p₂
sp : Set (P × P × P) := (fun p => (p₁, p, p₂)) '' {p | s.SSameSide p₃ p}
hc : IsConnected sp
hf : ContinuousOn (fun p => ∡ p.1 p.2.1 p.2.2) sp
hsp : ∀ p ∈ sp, ∡ p.1 p.2.1 p.2.2 ≠ 0 ∧ ∡ p.1 p.2.1 p.2.2 ≠ ↑π
hp₃ : (p₁, p₃, p₂) ∈ sp
⊢ (∡ p₁ p₄ p₂).sign = (∡ p₁ p₃ p₂).sign
|
33ab13a8df4e19ea
|
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