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 |
|---|---|---|---|---|---|---|
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
... | 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... | 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 / ... | 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.appen... | 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)... | 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.associato... | 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... | 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,... | 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
⊢... | 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
... | 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 ⊗... | 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).c... | 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) :
... | 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 : PseudoMetr... | 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 : PseudoMetr... | 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 +
... | 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... | 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... | 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 ← ... | 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 !_... | 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 ... | 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] :... | 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 :... | 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 :... | 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, closedBa... | 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 :... | 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... | 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 <| F... | 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... | 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 ≠ ... | 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).o... | 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✝ }... | 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).IsHomogene... | 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) → ... | 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 ... | 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 ... | 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✝ : P... | 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✝ : PreservesBinaryBipro... | 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 ... | 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
⊢ (e... | 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_mu... | 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 : Semino... | 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 ... | 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... | 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))^[(... | 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.... | 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.... | 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
... | 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)‖)... | 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)) =... | 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,... | 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 => ... | 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... | 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 : IsCompa... | 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
⊢ Differ... | 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 (... | 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 (... | 33ab13a8df4e19ea |
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