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BitVec.msb_eq_getLsbD_last | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem msb_eq_getLsbD_last (x : BitVec w) :
x.msb = x.getLsbD (w - 1) | w : Nat
x : BitVec w
⊢ x.msb = x.getLsbD (w - 1) | simp only [BitVec.msb, getMsbD] | w : Nat
x : BitVec w
⊢ (decide (0 < w) && x.getLsbD (w - 1 - 0)) = x.getLsbD (w - 1) | 4adf3a04a1b670a9 |
MeasureTheory.AEEqFun.coeFn_pair | Mathlib/MeasureTheory/Function/AEEqFun.lean | theorem coeFn_pair (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) : f.pair g =ᵐ[μ] fun x => (f x, g x) | α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : α →ₘ[μ] β
g : α →ₘ[μ] γ
⊢ ↑(mk (fun x => (↑f x, ↑g x)) ⋯) =ᶠ[ae μ] fun x => (↑f x, ↑g x) | apply coeFn_mk | no goals | 05adb7a4a5c38670 |
WeierstrassCurve.natDegree_coeff_Φ_ofNat | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | private lemma natDegree_coeff_Φ_ofNat (n : ℕ) :
(W.Φ n).natDegree ≤ n ^ 2 ∧ (W.Φ n).coeff (n ^ 2) = 1 | case succ.succ
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ},
(W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
(W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' :
(n + 1 + 1) ^ 2 =
WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
⊢ (W.Φ ↑(n + 1 + 1)).natDegree ≤ (n + 1 + 1) ^ 2 ∧ (W.Φ ↑(n + 1 + 1)).coeff ((n + 1 + 1) ^ 2) = 1 | have hc : (1 : ℤ) = 1 * expCoeff (n + 2) ^ 2 * (if Even (n + 1) then 1 else 4) -
expCoeff (n + 3) * expCoeff (n + 1) * (if Even (n + 1) then 4 else 1) := by
push_cast [← @Int.cast_inj ℚ, expCoeff_cast, Nat.even_add_one, ite_not]
split_ifs <;> ring1 | case succ.succ
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
fun {m n} {p q} => natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
fun {m n} {p q} => coeff_mul_of_natDegree_le
h : ∀ {n : ℕ},
(W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
(W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * WeierstrassCurve.expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' :
(n + 1 + 1) ^ 2 =
WeierstrassCurve.expDegree (n + 3) + WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
1 =
(1 * WeierstrassCurve.expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
WeierstrassCurve.expCoeff (n + 3) * WeierstrassCurve.expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ (W.Φ ↑(n + 1 + 1)).natDegree ≤ (n + 1 + 1) ^ 2 ∧ (W.Φ ↑(n + 1 + 1)).coeff ((n + 1 + 1) ^ 2) = 1 | 44ae74066fe30e40 |
PosNum.cast_le | Mathlib/Data/Num/Lemmas.lean | theorem cast_le [LinearOrderedSemiring α] {m n : PosNum} : (m : α) ≤ n ↔ m ≤ n | α : Type u_1
inst✝ : LinearOrderedSemiring α
m n : PosNum
⊢ ¬↑n < ↑m ↔ m ≤ n | exact not_congr cast_lt | no goals | beda75c4ffeeb046 |
MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | theorem LocallyIntegrable.integrable_smul_right_of_hasCompactSupport
[NormedSpace ℝ E] [OpensMeasurableSpace X] [T2Space X] {f : X → ℝ} (hf : LocallyIntegrable f μ)
{g : X → E} (hg : Continuous g) (h'g : HasCompactSupport g) :
Integrable (fun x ↦ f x • g x) μ | X : Type u_1
E : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : NormedAddCommGroup E
μ : Measure X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ (K.indicator fun x => f x • g x) = fun x => f x • g x | apply indicator_eq_self.2 | X : Type u_1
E : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : NormedAddCommGroup E
μ : Measure X
inst✝² : NormedSpace ℝ E
inst✝¹ : OpensMeasurableSpace X
inst✝ : T2Space X
f : X → ℝ
hf : LocallyIntegrable f μ
g : X → E
hg : Continuous g
h'g : HasCompactSupport g
K : Set X := tsupport g
hK : IsCompact K
⊢ (support fun x => f x • g x) ⊆ K | 07b57f28d472aaa4 |
TopCat.Presheaf.pushforwardEq_hom_app | Mathlib/Topology/Sheaves/Presheaf.lean | theorem pushforwardEq_hom_app {X Y : TopCat.{w}} {f g : X ⟶ Y}
(h : f = g) (ℱ : X.Presheaf C) (U) :
(pushforwardEq h ℱ).hom.app U = ℱ.map (eqToHom (by aesop_cat)) | C✝ : Type u
inst✝³ : Category.{v, u} C✝
X✝ : TopCat
C : Type u_1
inst✝² : Category.{?u.46617, u_1} C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝¹ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory C FC
X Y : TopCat
f g : X ⟶ Y
h : f = g
ℱ : Presheaf C X
U : (Opens ↑Y)ᵒᵖ
⊢ (Opens.map f).op.obj U = (Opens.map g).op.obj U | aesop_cat | no goals | 3e968e77c81c4ec6 |
MeasureTheory.Measure.AbsolutelyContinuous.zero | Mathlib/MeasureTheory/Measure/AbsolutelyContinuous.lean | @[simp]
protected lemma zero (μ : Measure α) : 0 ≪ μ := fun _ _ ↦ by simp
| α : Type u_1
mα : MeasurableSpace α
μ : Measure α
x✝¹ : Set α
x✝ : μ x✝¹ = 0
⊢ 0 x✝¹ = 0 | simp | no goals | b65f07860a7b6e9d |
MvQPF.recF_eq_of_wEquiv | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | theorem recF_eq_of_wEquiv (α : TypeVec n) {β : Type u} (u : F (α.append1 β) → β) (x y : q.P.W α) :
WEquiv x y → recF u x = recF u y | case refine_2
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
u : F (α ::: β) → β
x y : (P F).W α
a₀✝ : (P F).A
f'₀✝ : (P F).drop.B a₀✝ ⟹ α
f₀✝ : (P F).last.B a₀✝ → (P F).W α
a₁✝ : (P F).A
f'₁✝ : (P F).drop.B a₁✝ ⟹ α
f₁✝ : (P F).last.B a₁✝ → (P F).W α
h✝ : WEquiv ((P F).wMk a₀✝ f'₀✝ f₀✝) ((P F).wMk a₁✝ f'₁✝ f₁✝)
a₀ : (P F).A
f'₀ : (P F).drop.B a₀ ⟹ α
f₀ : (P F).last.B a₀ → (P F).W α
a₁ : (P F).A
f'₁ : (P F).drop.B a₁ ⟹ α
f₁ : (P F).last.B a₁ → (P F).W α
h : abs ⟨a₀, (P F).appendContents f'₀ f₀⟩ = abs ⟨a₁, (P F).appendContents f'₁ f₁⟩
⊢ recF u ((P F).wMk a₀ f'₀ f₀) = recF u ((P F).wMk a₁ f'₁ f₁) | simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h] | no goals | ad7305cb6e79d41e |
CategoryTheory.SemiadditiveOfBinaryBiproducts.distrib | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k | C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasBinaryBiproducts C
X Y : C
f g h k : X ⟶ Y
diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
hd₁ : biprod.inl ≫ diag = biprod.lift f h
⊢ biprod.inr ≫ diag = biprod.lift g k | ext <;> simp [diag] | no goals | 8859d6f168de6f6c |
Zsqrtd.nonneg_mul | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem nonneg_mul {a b : ℤ√d} (ha : Nonneg a) (hb : Nonneg b) : Nonneg (a * b) :=
match a, b, nonneg_cases ha, nonneg_cases hb, ha, hb with
| _, _, ⟨_, _, Or.inl rfl⟩, ⟨_, _, Or.inl rfl⟩, _, _ => trivial
| _, _, ⟨x, y, Or.inl rfl⟩, ⟨z, w, Or.inr <| Or.inr rfl⟩, _, hb => nonneg_mul_lem hb
| _, _, ⟨x, y, Or.inl rfl⟩, ⟨z, w, Or.inr <| Or.inl rfl⟩, _, hb => nonneg_mul_lem hb
| _, _, ⟨x, y, Or.inr <| Or.inr rfl⟩, ⟨z, w, Or.inl rfl⟩, ha, _ => by
rw [mul_comm]; exact nonneg_mul_lem ha
| _, _, ⟨x, y, Or.inr <| Or.inl rfl⟩, ⟨z, w, Or.inl rfl⟩, ha, _ => by
rw [mul_comm]; exact nonneg_mul_lem ha
| _, _, ⟨x, y, Or.inr <| Or.inr rfl⟩, ⟨z, w, Or.inr <| Or.inr rfl⟩, ha, hb => by
rw [calc
(⟨-x, y⟩ * ⟨-z, w⟩ : ℤ√d) = ⟨_, _⟩ := rfl
_ = ⟨x * z + d * y * w, -(x * w + y * z)⟩ | d : ℕ
a b : ℤ√↑d
ha✝ : a.Nonneg
hb : b.Nonneg
x y z w : ℕ
ha : { re := ↑x, im := -↑y }.Nonneg
x✝ : { re := ↑z, im := ↑w }.Nonneg
⊢ ({ re := ↑x, im := -↑y } * { re := ↑z, im := ↑w }).Nonneg | rw [mul_comm] | d : ℕ
a b : ℤ√↑d
ha✝ : a.Nonneg
hb : b.Nonneg
x y z w : ℕ
ha : { re := ↑x, im := -↑y }.Nonneg
x✝ : { re := ↑z, im := ↑w }.Nonneg
⊢ ({ re := ↑z, im := ↑w } * { re := ↑x, im := -↑y }).Nonneg | 46d597e584ea62c6 |
Matrix.Represents.one | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | theorem Matrix.Represents.one : (1 : Matrix ι ι R).Represents b 1 | ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
⊢ ((LinearMap.llcomp R (ι → R) (ι → R) M) ((Fintype.linearCombination R R) b) ∘ₗ algEquivMatrix'.symm.toLinearMap) 1 =
(PiToModule.fromEnd R b) 1 | rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_one] | ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
b : ι → M
inst✝ : DecidableEq ι
⊢ ((LinearMap.llcomp R (ι → R) (ι → R) M) ((Fintype.linearCombination R R) b)) 1 = (PiToModule.fromEnd R b) 1 | bbbc202286fcc352 |
CategoryTheory.tensorRightHomEquiv_symm_coevaluation_comp_whiskerRight | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | theorem tensorRightHomEquiv_symm_coevaluation_comp_whiskerRight {Y Y' Z : C} [ExactPairing Y Y']
(f : Y ⟶ Z) : (tensorRightHomEquiv _ Y _ _).symm (η_ Y Y' ≫ f ▷ Y') = (λ_ _).hom ≫ f :=
calc
_ = η_ Y Y' ▷ Y ⊗≫ (f ▷ (Y' ⊗ Y) ≫ Z ◁ ε_ Y Y') ⊗≫ 𝟙 _ | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
Y Y' Z : C
inst✝ : ExactPairing Y Y'
f : Y ⟶ Z
⊢ ⊗𝟙.hom ⊗≫ f = (λ_ Y).hom ≫ f | monoidal | no goals | 0f6e5fba767e74b3 |
hasStrictFDerivAt_list_prod_attach' | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | theorem hasStrictFDerivAt_list_prod_attach' {l : List ι} {x : {i // i ∈ l} → 𝔸} :
HasStrictFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.attach.map x).prod)
(∑ i : Fin l.length, ((l.attach.take i).map x).prod •
smulRight (proj l.attach[i.cast List.length_attach.symm])
((l.attach.drop (.succ i)).map x).prod) x | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
ι : Type u_5
𝔸 : Type u_6
inst✝¹ : NormedRing 𝔸
inst✝ : NormedAlgebra 𝕜 𝔸
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ ∀ (i : Fin l.length), i ∈ Finset.univ ↔ (finCongr ⋯) i ∈ Finset.univ | simp | no goals | 57ae8c85eed86688 |
dvd_prime_pow | Mathlib/Algebra/GroupWithZero/Associated.lean | theorem dvd_prime_pow [CancelCommMonoidWithZero M] {p q : M} (hp : Prime p) (n : ℕ) :
q ∣ p ^ n ↔ ∃ i ≤ n, Associated q (p ^ i) | case zero
M : Type u_1
inst✝ : CancelCommMonoidWithZero M
p : M
hp : Prime p
q : M
⊢ q ∣ p ^ 0 ↔ ∃ i, i ≤ 0 ∧ q ~ᵤ p ^ i | simp [← isUnit_iff_dvd_one, associated_one_iff_isUnit] | no goals | 55f0c39be5bdf205 |
LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete_of_ne_zero | Mathlib/GroupTheory/ArchimedeanDensely.lean | lemma LinearOrderedCommGroupWithZero.wellFoundedOn_setOf_ge_gt_iff_nonempty_discrete_of_ne_zero
{G₀ : Type*} [LinearOrderedCommGroupWithZero G₀] [Nontrivial G₀ˣ] {g : G₀} (hg : g ≠ 0) :
Set.WellFoundedOn {x : G₀ | x ≤ g} (· > ·) ↔ Nonempty (G₀ ≃*o ℤₘ₀) | case refine_2
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
hg' : g⁻¹ ≠ 0
h : ({x | x ≤ g} \ {0}).WellFoundedOn fun x1 x2 => x1 > x2
⊢ ∀ a ∈ {x | g⁻¹ ≤ x}, ∀ b ∈ {x | g⁻¹ ≤ x}, a < b → Function.onFun (fun x1 x2 => x1 > x2) (fun x => x⁻¹) a b | simp only [mem_setOf_eq, Function.onFun, gt_iff_lt] | case refine_2
G₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero G₀
inst✝ : Nontrivial G₀ˣ
g : G₀
hg : g ≠ 0
hg' : g⁻¹ ≠ 0
h : ({x | x ≤ g} \ {0}).WellFoundedOn fun x1 x2 => x1 > x2
⊢ ∀ (a : G₀), g⁻¹ ≤ a → ∀ (b : G₀), g⁻¹ ≤ b → a < b → b⁻¹ < a⁻¹ | b1f0a3d6058790d2 |
rat_inv_continuous_lemma | Mathlib/Algebra/Order/CauSeq/Basic.lean | theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
{ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε | α : Type u_1
inst✝² : LinearOrderedField α
β : Type u_3
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
⊢ abv (a⁻¹ - b⁻¹) < ε | have b0 := K0.trans_le hb | α : Type u_1
inst✝² : LinearOrderedField α
β : Type u_3
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
ε K : α
ε0 : 0 < ε
K0 : 0 < K
a b : β
ha : K ≤ abv a
hb : K ≤ abv b
h : abv (a - b) < K * ε * K
a0 : 0 < abv a
b0 : 0 < abv b
⊢ abv (a⁻¹ - b⁻¹) < ε | a1ac660eaca776c2 |
Cubic.disc_eq_prod_three_roots | Mathlib/Algebra/CubicDiscriminant.lean | theorem disc_eq_prod_three_roots (ha : P.a ≠ 0) (h3 : (map φ P).roots = {x, y, z}) :
φ P.disc = (φ P.a * φ P.a * (x - y) * (x - z) * (y - z)) ^ 2 | F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ φ P.disc = (φ P.a * φ P.a * (x - y) * (x - z) * (y - z)) ^ 2 | simp only [disc, RingHom.map_add, RingHom.map_sub, RingHom.map_mul, map_pow, map_ofNat] | F : Type u_3
K : Type u_4
P : Cubic F
inst✝¹ : Field F
inst✝ : Field K
φ : F →+* K
x y z : K
ha : P.a ≠ 0
h3 : (map φ P).roots = {x, y, z}
⊢ φ P.b ^ 2 * φ P.c ^ 2 - 4 * φ P.a * φ P.c ^ 3 - 4 * φ P.b ^ 3 * φ P.d - 27 * φ P.a ^ 2 * φ P.d ^ 2 +
18 * φ P.a * φ P.b * φ P.c * φ P.d =
(φ P.a * φ P.a * (x - y) * (x - z) * (y - z)) ^ 2 | abba4934026db346 |
Polynomial.coeff_eq_zero_of_lt_natTrailingDegree | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | theorem coeff_eq_zero_of_lt_natTrailingDegree {p : R[X]} {n : ℕ} (h : n < p.natTrailingDegree) :
p.coeff n = 0 | R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : n < p.natTrailingDegree
⊢ p.coeff n = 0 | apply coeff_eq_zero_of_lt_trailingDegree | case h
R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
h : n < p.natTrailingDegree
⊢ ↑n < p.trailingDegree | 351ae9308e5c06a2 |
Nat.Simproc.le_add_ge | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Simproc.lean | theorem le_add_ge (a : Nat) {b c : Nat} (h : a ≥ c) : (a ≤ b + c) = (a - c ≤ b) | a b c : Nat
h : a ≥ c
r : (a ≤ b + c) = (a - c ≤ b)
⊢ (a ≤ b + c) = (a - c ≤ b) | exact r | no goals | 3450a843b2f206da |
WeierstrassCurve.Projective.addX_neg | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma addX_neg (P : Fin 3 → R) : W'.addX P (W'.neg P) = 0 | R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ -P x * (-P y - W'.a₁ * P x - W'.a₃ * P z) ^ 2 * P z + P x * P y ^ 2 * P z -
2 * P x * P y * (-P y - W'.a₁ * P x - W'.a₃ * P z) * P z +
2 * P x * P y * (-P y - W'.a₁ * P x - W'.a₃ * P z) * P z -
W'.a₁ * P x ^ 2 * (-P y - W'.a₁ * P x - W'.a₃ * P z) * P z +
W'.a₁ * P x ^ 2 * P y * P z +
W'.a₂ * P x ^ 2 * P x * P z -
W'.a₂ * P x * P x ^ 2 * P z -
W'.a₃ * P x * P y * P z ^ 2 +
W'.a₃ * P x * (-P y - W'.a₁ * P x - W'.a₃ * P z) * P z ^ 2 -
2 * W'.a₃ * P x * (-P y - W'.a₁ * P x - W'.a₃ * P z) * P z * P z +
2 * W'.a₃ * P x * P y * P z * P z +
W'.a₄ * P x ^ 2 * P z ^ 2 -
W'.a₄ * P x ^ 2 * P z ^ 2 +
3 * W'.a₆ * P x * P z * P z ^ 2 -
3 * W'.a₆ * P x * P z ^ 2 * P z =
0 | ring1 | no goals | e87e09a2e4abccff |
ProbabilityTheory.lintegral_exponentialPDF_eq_antiDeriv | Mathlib/Probability/Distributions/Exponential.lean | lemma lintegral_exponentialPDF_eq_antiDeriv {r : ℝ} (hr : 0 < r) (x : ℝ) :
∫⁻ y in Iic x, exponentialPDF r y
= ENNReal.ofReal (if 0 ≤ x then 1 - exp (-(r * x)) else 0) | r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
this : ∫ (a : ℝ) in Ι 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0 ..x, r * rexp (-(r * a))
⊢ ∀ x_1 ∈ Ioo 0 x, HasDerivWithinAt (fun a => -1 * rexp (-(r * a))) (r * rexp (-(r * x_1))) (Ioi x_1) x_1 | simp only [neg_mul, one_mul] | r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
this : ∫ (a : ℝ) in Ι 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0 ..x, r * rexp (-(r * a))
⊢ ∀ x_1 ∈ Ioo 0 x, HasDerivWithinAt (fun a => -rexp (-(r * a))) (r * rexp (-(r * x_1))) (Ioi x_1) x_1 | 8690d2dfbd86b4ca |
ProbabilityTheory.condExp_generateFrom_singleton | Mathlib/Probability/Kernel/Condexp.lean | lemma condExp_generateFrom_singleton (hs : MeasurableSet s) {f : Ω → F} (hf : Integrable f μ) :
μ[f | generateFrom {s}] =ᵐ[μ.restrict s] fun _ ↦ ∫ x, f x ∂μ[|s] | case pos
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : μ s = 0
⊢ μ[f|MeasurableSpace.generateFrom {s}] =ᶠ[ae (μ.restrict s)] fun x => ∫ (x : Ω), f x ∂μ[|s] | rw [Measure.restrict_eq_zero.2 hμs] | case pos
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : μ s = 0
⊢ μ[f|MeasurableSpace.generateFrom {s}] =ᶠ[ae 0] fun x => ∫ (x : Ω), f x ∂μ[|s] | b3ab2e228c69d4b5 |
HasEnoughRootsOfUnity.natCard_rootsOfUnity | Mathlib/RingTheory/RootsOfUnity/EnoughRootsOfUnity.lean | /-- If `M` satisfies `HasEnoughRootsOfUnity`, then the group of `n`th roots of unity
in `M` (is cyclic and) has order `n`. -/
lemma natCard_rootsOfUnity (M : Type*) [CommMonoid M] (n : ℕ) [NeZero n]
[HasEnoughRootsOfUnity M n] :
Nat.card (rootsOfUnity n M) = n | case intro.refine_2
M : Type u_1
inst✝² : CommMonoid M
n : ℕ
inst✝¹ : NeZero n
inst✝ : HasEnoughRootsOfUnity M n
ζ : M
h : IsPrimitiveRoot ζ n
⊢ n ∣ Monoid.exponent ↥(rootsOfUnity n M) | nth_rewrite 1 [h.eq_orderOf] | case intro.refine_2
M : Type u_1
inst✝² : CommMonoid M
n : ℕ
inst✝¹ : NeZero n
inst✝ : HasEnoughRootsOfUnity M n
ζ : M
h : IsPrimitiveRoot ζ n
⊢ orderOf ζ ∣ Monoid.exponent ↥(rootsOfUnity n M) | 0eca4e6e7b4b2527 |
Submodule.fst_map_snd | Mathlib/LinearAlgebra/Prod.lean | theorem fst_map_snd : (Submodule.fst R M M₂).map (LinearMap.snd R M M₂) = ⊥ | R : Type u
M : Type v
M₂ : Type w
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
⊢ map (snd R M M₂) (fst R M M₂) ≤ ⊥ | intro x | R : Type u
M : Type v
M₂ : Type w
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
x : M₂
⊢ x ∈ map (snd R M M₂) (fst R M M₂) → x ∈ ⊥ | d3a1b789547d9be0 |
CategoryTheory.Limits.IndizationClosedUnderFilteredColimitsAux.exists_nonempty_limit_obj_of_isColimit | Mathlib/CategoryTheory/Limits/Indization/FilteredColimits.lean | theorem exists_nonempty_limit_obj_of_isColimit [IsFiltered K] {c : Cocone H} (hc : IsColimit c)
(T : Over (colimit F)) (hT : c.pt ≅ T)
(h : Nonempty <| limit <| 𝒢 ⋙ yoneda.obj T) :
∃ k, Nonempty <| limit <| 𝒢 ⋙ yoneda.obj (H.obj k) | C : Type u
inst✝⁵ : Category.{v, u} C
I : Type v
inst✝⁴ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
J : Type v
inst✝³ : SmallCategory J
inst✝² : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst✝¹ : SmallCategory K
H : K ⥤ Over (colimit F)
inst✝ : IsFiltered K
c : Cocone H
hc : IsColimit c
T : Over (colimit F)
hT : c.pt ≅ T
h : Nonempty (limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj T))
⊢ ∃ k, Nonempty (limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (H.obj k))) | refine exists_nonempty_limit_obj_of_colimit F G H ?_ | C : Type u
inst✝⁵ : Category.{v, u} C
I : Type v
inst✝⁴ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
J : Type v
inst✝³ : SmallCategory J
inst✝² : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst✝¹ : SmallCategory K
H : K ⥤ Over (colimit F)
inst✝ : IsFiltered K
c : Cocone H
hc : IsColimit c
T : Over (colimit F)
hT : c.pt ≅ T
h : Nonempty (limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj T))
⊢ Nonempty (limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (colimit H))) | a0d80d0bd0abda50 |
controlled_closure_of_complete | Mathlib/Analysis/Normed/Group/ControlledClosure.lean | theorem controlled_closure_of_complete {f : NormedAddGroupHom G H} {K : AddSubgroup H} {C ε : ℝ}
(hC : 0 < C) (hε : 0 < ε) (hyp : f.SurjectiveOnWith K C) :
f.SurjectiveOnWith K.topologicalClosure (C + ε) | case pos
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : f.SurjectiveOnWith K C
h : H
h_in : h ∈ K.topologicalClosure
hyp_h : h = 0
⊢ ∃ g, f g = h ∧ ‖g‖ ≤ (C + ε) * ‖h‖ | rw [hyp_h] | case pos
G : Type u_1
inst✝² : NormedAddCommGroup G
inst✝¹ : CompleteSpace G
H : Type u_2
inst✝ : NormedAddCommGroup H
f : NormedAddGroupHom G H
K : AddSubgroup H
C ε : ℝ
hC : 0 < C
hε : 0 < ε
hyp : f.SurjectiveOnWith K C
h : H
h_in : h ∈ K.topologicalClosure
hyp_h : h = 0
⊢ ∃ g, f g = 0 ∧ ‖g‖ ≤ (C + ε) * ‖0‖ | 39e3d4e13f8bf078 |
totallyBounded_iff_filter | Mathlib/Topology/UniformSpace/Cauchy.lean | theorem totallyBounded_iff_filter {s : Set α} :
TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c | case mp
α : Type u
uniformSpace : UniformSpace α
s : Set α
⊢ TotallyBounded s → ∀ (f : Filter α), f.NeBot → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c | exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
(Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩ | no goals | 8d4389700a78bd05 |
List.Sublist.findIdx?_isSome | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem Sublist.findIdx?_isSome {l₁ l₂ : List α} (h : l₁ <+ l₂) :
(l₁.findIdx? p).isSome → (l₂.findIdx? p).isSome | case intro.intro
α : Type u_1
p : α → Bool
l₁ l₂ : List α
h : l₁ <+ l₂
w : α
m : w ∈ l₁
q : p w = true
⊢ ∃ x, x ∈ l₂ ∧ p x = true | exact ⟨w, h.mem m, q⟩ | no goals | 1f65915bbb3a011a |
MeasureTheory.Measure.rnDeriv_pos | Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean | lemma rnDeriv_pos [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν) :
∀ᵐ x ∂μ, 0 < μ.rnDeriv ν x | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
⊢ ∀ᵐ (x : α) ∂ν, μ.rnDeriv ν x ≠ 0 → 0 < μ.rnDeriv ν x | exact ae_of_all _ (fun x hx ↦ lt_of_le_of_ne (zero_le _) hx.symm) | no goals | d070182cd8f5d1b6 |
Std.DHashMap.Internal.Raw₀.getKey?_eq_some_getKey | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem getKey?_eq_some_getKey [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {a : α} {h'} :
m.getKey? a = some (m.getKey a h') | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
a : α
h' : m.contains a = true
⊢ m.getKey? a = some (m.getKey a h') | simp_to_model using List.getKey?_eq_some_getKey | no goals | 03d9f1f761c25bbe |
irreducible_or_factor | Mathlib/Algebra/Prime/Defs.lean | theorem irreducible_or_factor {M} [Monoid M] (x : M) (h : ¬IsUnit x) :
Irreducible x ∨ ∃ a b, ¬IsUnit a ∧ ¬IsUnit b ∧ a * b = x | M : Type u_2
inst✝ : Monoid M
x : M
h✝ : ¬IsUnit x
this : (p : Prop) → Decidable p
H : ∀ (x_1 : M), ¬IsUnit x_1 → ∀ (x_2 : M), ¬IsUnit x_2 → ¬x_1 * x_2 = x
a b : M
h : x = a * b
o : ¬IsUnit a ∧ ¬IsUnit b
⊢ False | exact H _ o.1 _ o.2 h.symm | no goals | f17346327bf6e20c |
BoxIntegral.Box.measurableSet_coe | Mathlib/Analysis/BoxIntegral/Partition/Measure.lean | theorem measurableSet_coe : MeasurableSet (I : Set (ι → ℝ)) | ι : Type u_1
I : Box ι
inst✝ : Countable ι
⊢ MeasurableSet ↑I | rw [coe_eq_pi] | ι : Type u_1
I : Box ι
inst✝ : Countable ι
⊢ MeasurableSet (univ.pi fun i => Ioc (I.lower i) (I.upper i)) | df5e6d98efba6af5 |
Submodule.rank_le | Mathlib/LinearAlgebra/Dimension/Basic.lean | theorem Submodule.rank_le (s : Submodule R M) : Module.rank R s ≤ Module.rank R M | R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Submodule R M
⊢ Module.rank R ↥s ≤ Module.rank R ↥⊤ | exact rank_mono le_top | no goals | 4f7b59dfebaa0f7e |
Cardinal.increasing_cantorFunction | Mathlib/Data/Real/Cardinality.lean | theorem increasing_cantorFunction (h1 : 0 < c) (h2 : c < 1 / 2) {n : ℕ} {f g : ℕ → Bool}
(hn : ∀ k < n, f k = g k) (fn : f n = false) (gn : g n = true) :
cantorFunction c f < cantorFunction c g | case h.e'_4.refine_1.zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn✝ : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n => rec false (fun x x => true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n => rec true (fun x x => false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
this : c / (1 - c) < 1
hn : 0 ≠ 0
⊢ cantorFunctionAux c g_min 0 = 0 | contradiction | no goals | 157da5ded2b82856 |
Localization.awayLift_mk | Mathlib/RingTheory/Localization/Away/Basic.lean | lemma awayLift_mk {A : Type*} [CommRing A] (f : R →+* A) (r : R)
(a : R) (v : A) (hv : f r * v = 1) (j : ℕ) :
Localization.awayLift f r (isUnit_iff_exists_inv.mpr ⟨v, hv⟩)
(Localization.mk a ⟨r ^ j, j, rfl⟩) = f a * v ^ j | R : Type u_1
inst✝¹ : CommSemiring R
A : Type u_4
inst✝ : CommRing A
f : R →+* A
r a : R
v : A
hv : f r * v = 1
j : ℕ
⊢ f a * ↑((IsUnit.liftRight ((↑f).restrict (Submonoid.powers r)) ⋯) ⟨r ^ j, ⋯⟩)⁻¹ = f a * v ^ j | rw [Units.mul_inv_eq_iff_eq_mul] | R : Type u_1
inst✝¹ : CommSemiring R
A : Type u_4
inst✝ : CommRing A
f : R →+* A
r a : R
v : A
hv : f r * v = 1
j : ℕ
⊢ f a = f a * v ^ j * ↑((IsUnit.liftRight ((↑f).restrict (Submonoid.powers r)) ⋯) ⟨r ^ j, ⋯⟩) | 6cf43cbb51580878 |
VitaliFamily.ae_eventually_measure_zero_of_singular | Mathlib/MeasureTheory/Covering/Differentiation.lean | theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) | case h
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
x : α
hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
z : ℝ≥0∞
hz : z > 0
⊢ ∀ᶠ (b : Set α) in v.filterAt x, ρ b / μ b < z | obtain ⟨w, w_pos, w_lt⟩ : ∃ w : ℝ≥0, (0 : ℝ≥0∞) < w ∧ (w : ℝ≥0∞) < z :=
ENNReal.lt_iff_exists_nnreal_btwn.1 hz | case h.intro.intro
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ⟂ₘ μ
A : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑ε * μ a
u : ℕ → ℝ≥0
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
x : α
hx : ∀ (n : ℕ), ∀ᶠ (a : Set α) in v.filterAt x, ρ a < ↑(u n) * μ a
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < ↑w
w_lt : ↑w < z
⊢ ∀ᶠ (b : Set α) in v.filterAt x, ρ b / μ b < z | 27b6e7c43d8d9d57 |
LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | theorem LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul
[Module.Finite R M] (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) :
∃ p : R[X], p.Monic ∧ (∀ k, p.coeff k ∈ I ^ (p.natDegree - k)) ∧ Polynomial.aeval f p = 0 | case inl
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Subsingleton R
⊢ ∃ p, p.Monic ∧ (∀ (k : ℕ), p.coeff k ∈ I ^ (p.natDegree - k)) ∧ (aeval f) p = 0 | exact ⟨0, Polynomial.monic_of_subsingleton _, by simp⟩ | no goals | f240718b63d9a548 |
linearIndependent_fin2 | Mathlib/LinearAlgebra/LinearIndependent/Lemmas.lean | theorem linearIndependent_fin2 {f : Fin 2 → V} :
LinearIndependent K f ↔ f 1 ≠ 0 ∧ ∀ a : K, a • f 1 ≠ f 0 | K : Type u_3
V : Type u
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : Fin 2 → V
⊢ LinearIndependent K f ↔ f 1 ≠ 0 ∧ ∀ (a : K), a • f 1 ≠ f 0 | rw [linearIndependent_fin_succ, linearIndependent_unique_iff, range_unique, mem_span_singleton,
not_exists, show Fin.tail f default = f 1 by rw [← Fin.succ_zero_eq_one]; rfl] | no goals | 361cc2ff5c048802 |
CategoryTheory.Limits.colim.map_mono' | Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Colim.lean | /-- Assume that `colim : (J ⥤ C) ⥤ C` preserves monomorphisms, and
`φ : X₁ ⟶ X₂` is a monomorphism in `J ⥤ C`, then if `f : c₁.pt ⟶ c₂.pt` is a morphism
between the points of colimit cocones for `X₁` and `X₂` in such a way that `f`
idenfities to `colim.map φ`, then `f` is a monomorphism. -/
lemma colim.map_mono' [HasColimitsOfShape J C]
[(colim : (J ⥤ C) ⥤ C).PreservesMonomorphisms]
{X₁ X₂ : J ⥤ C} (φ : X₁ ⟶ X₂) [Mono φ]
{c₁ : Cocone X₁} (hc₁ : IsColimit c₁) {c₂ : Cocone X₂} (hc₂ : IsColimit c₂)
(f : c₁.pt ⟶ c₂.pt) (hf : ∀ j, c₁.ι.app j ≫ f = φ.app j ≫ c₂.ι.app j) : Mono f | C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u'
inst✝³ : Category.{v', u'} J
inst✝² : HasColimitsOfShape J C
inst✝¹ : colim.PreservesMonomorphisms
X₁ X₂ : J ⥤ C
φ : X₁ ⟶ X₂
inst✝ : Mono φ
c₁ : Cocone X₁
hc₁ : IsColimit c₁
c₂ : Cocone X₂
hc₂ : IsColimit c₂
f : c₁.pt ⟶ c₂.pt
hf : ∀ (j : J), c₁.ι.app j ≫ f = φ.app j ≫ c₂.ι.app j
j : J
⊢ c₁.ι.app j ≫ (hc₁.coconePointUniqueUpToIso (colimit.isColimit X₁)).hom ≫ colimMap φ =
c₁.ι.app j ≫ f ≫ (hc₂.coconePointUniqueUpToIso (colimit.isColimit X₂)).hom | rw [IsColimit.comp_coconePointUniqueUpToIso_hom_assoc,
colimit.cocone_ι, ι_colimMap, reassoc_of% (hf j),
IsColimit.comp_coconePointUniqueUpToIso_hom, colimit.cocone_ι] | no goals | a626527c67c68c80 |
Polynomial.Chebyshev.U_neg_sub_one | Mathlib/RingTheory/Polynomial/Chebyshev.lean | theorem U_neg_sub_one (n : ℤ) : U R (-n - 1) = -U R (n - 1) | case add_two
R : Type u_1
inst✝ : CommRing R
n : ℕ
ih1 : U R (-(↑n + 1) - 1) = -U R (↑n + 1 - 1)
ih2 : U R (-↑n - 1) = -U R (↑n - 1)
h₁ : U R (↑n + 1) = 2 * X * U R ↑n - U R (↑n - 1)
⊢ U R (-(↑n + 2) - 1) = -U R (↑n + 2 - 1) | have h₂ := U_sub_two R (-n - 1) | case add_two
R : Type u_1
inst✝ : CommRing R
n : ℕ
ih1 : U R (-(↑n + 1) - 1) = -U R (↑n + 1 - 1)
ih2 : U R (-↑n - 1) = -U R (↑n - 1)
h₁ : U R (↑n + 1) = 2 * X * U R ↑n - U R (↑n - 1)
h₂ : U R (-↑n - 1 - 2) = 2 * X * U R (-↑n - 1 - 1) - U R (-↑n - 1)
⊢ U R (-(↑n + 2) - 1) = -U R (↑n + 2 - 1) | c30da371377b8a1e |
IsCompactlyGenerated.BooleanGenerators.sSup_inter | Mathlib/Order/BooleanGenerators.lean | lemma sSup_inter (hS : BooleanGenerators S) {T₁ T₂ : Set α} (hT₁ : T₁ ⊆ S) (hT₂ : T₂ ⊆ S) :
sSup (T₁ ∩ T₂) = (sSup T₁) ⊓ (sSup T₂) | case a.intro.intro.a.a.left
α : Type u_1
inst✝¹ : CompleteLattice α
S : Set α
inst✝ : IsCompactlyGenerated α
hS : BooleanGenerators S
T₁ T₂ : Set α
hT₁ : T₁ ⊆ S
hT₂ : T₂ ⊆ S
X : Set α
hX : X ⊆ S
hX' : sSup T₁ ⊓ sSup T₂ = sSup X
I : α
hI : I ∈ X
⊢ I ∈ T₁ | apply (hS.mono hT₁).mem_of_isAtom_of_le_sSup_atoms _ _ _ | α : Type u_1
inst✝¹ : CompleteLattice α
S : Set α
inst✝ : IsCompactlyGenerated α
hS : BooleanGenerators S
T₁ T₂ : Set α
hT₁ : T₁ ⊆ S
hT₂ : T₂ ⊆ S
X : Set α
hX : X ⊆ S
hX' : sSup T₁ ⊓ sSup T₂ = sSup X
I : α
hI : I ∈ X
⊢ IsAtom I
α : Type u_1
inst✝¹ : CompleteLattice α
S : Set α
inst✝ : IsCompactlyGenerated α
hS : BooleanGenerators S
T₁ T₂ : Set α
hT₁ : T₁ ⊆ S
hT₂ : T₂ ⊆ S
X : Set α
hX : X ⊆ S
hX' : sSup T₁ ⊓ sSup T₂ = sSup X
I : α
hI : I ∈ X
⊢ I ≤ sSup T₁ | 7804e1cb5854118b |
spectrum.units_conjugate | Mathlib/Algebra/Algebra/Spectrum.lean | /-- Conjugation by a unit preserves the spectrum, inverse on right. -/
@[simp]
lemma spectrum.units_conjugate {a : A} {u : Aˣ} :
spectrum R (u * a * u⁻¹) = spectrum R a | R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Ring A
inst✝ : Algebra R A
a : A
u : Aˣ
this : ∀ (b : A) (v : Aˣ), spectrum R (↑v * b * ↑v⁻¹) ⊆ spectrum R b
⊢ spectrum R a = spectrum R (↑u⁻¹ * (↑u * a * ↑u⁻¹) * ↑u⁻¹⁻¹) | simp [mul_assoc] | no goals | 7561a35e3d28b7df |
geometric_hahn_banach_compact_closed | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | theorem geometric_hahn_banach_compact_closed (hs₁ : Convex ℝ s) (hs₂ : IsCompact s)
(ht₁ : Convex ℝ t) (ht₂ : IsClosed t) (disj : Disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u v : ℝ), (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b | case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s t : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
hs : s.Nonempty
_ht : t.Nonempty
U V : Set E
hU : IsOpen U
hV : IsOpen V
hU₁ : Convex ℝ U
hV₁ : Convex ℝ V
sU : s ⊆ U
tV : t ⊆ V
disj' : Disjoint U V
f : E →L[ℝ] ℝ
u : ℝ
hf₁ : ∀ a ∈ U, f a < u
hf₂ : ∀ b ∈ V, u < f b
x : E
hx₁ : x ∈ s
hx₂ : IsMaxOn (⇑f) s x
⊢ ∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b | have : f x < u := hf₁ x (sU hx₁) | case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
E : Type u_2
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : Module ℝ E
s t : Set E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht₁ : Convex ℝ t
ht₂ : IsClosed t
disj : Disjoint s t
hs : s.Nonempty
_ht : t.Nonempty
U V : Set E
hU : IsOpen U
hV : IsOpen V
hU₁ : Convex ℝ U
hV₁ : Convex ℝ V
sU : s ⊆ U
tV : t ⊆ V
disj' : Disjoint U V
f : E →L[ℝ] ℝ
u : ℝ
hf₁ : ∀ a ∈ U, f a < u
hf₂ : ∀ b ∈ V, u < f b
x : E
hx₁ : x ∈ s
hx₂ : IsMaxOn (⇑f) s x
this : f x < u
⊢ ∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b | 521e7c09704a58a0 |
AddEquiv.isRegular_congr | Mathlib/RingTheory/Regular/RegularSequence.lean | lemma _root_.AddEquiv.isRegular_congr {e : M ≃+ M₂} {as bs}
(h : List.Forall₂ (fun (r : R) (s : S) => ∀ x, e (r • x) = s • e x) as bs) :
IsRegular M as ↔ IsRegular M₂ bs | R : Type u_1
S : Type u_2
M : Type u_3
M₂ : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup M₂
inst✝¹ : Module R M
inst✝ : Module S M₂
e : M ≃+ M₂
as : List R
bs : List S
h : List.Forall₂ (fun r s => ∀ (x : M), e (r • x) = s • e x) as bs
e' : M ⧸ (Ideal.ofList as • ⊤).toAddSubgroup ≃+ M₂ ⧸ (Ideal.ofList bs • ⊤).toAddSubgroup :=
QuotientAddGroup.congr (Ideal.ofList as • ⊤).toAddSubgroup (Ideal.ofList bs • ⊤).toAddSubgroup e ⋯
⊢ IsWeaklyRegular M as ∧ ¬Subsingleton (M ⧸ Ideal.ofList as • ⊤) ↔
IsWeaklyRegular M₂ bs ∧ ¬Subsingleton (M₂ ⧸ Ideal.ofList bs • ⊤) | exact and_congr (e.isWeaklyRegular_congr h) e'.subsingleton_congr.not | no goals | 2e0fe779af8112df |
CategoryTheory.ShortComplex.rightHomologyι_comp_fromOpcycles | Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | @[reassoc (attr := simp)]
lemma rightHomologyι_comp_fromOpcycles :
S.rightHomologyι ≫ S.fromOpcycles = 0 :=
S.rightHomologyι_descOpcycles_π_eq_zero_of_boundary S.g (𝟙 _) (by rw [comp_id])
| C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasRightHomology
⊢ S.g = S.g ≫ 𝟙 S.X₃ | rw [comp_id] | no goals | 3672986cf90eece5 |
Convex.taylor_approx_two_segment | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
(hw : x + v + w ∈ interior s) :
(fun h : ℝ => f (x + h • v + h • w)
- f (x + h • v) - h • f' x w - h ^ 2 • f'' v w - (h ^ 2 / 2) • f'' w w) =o[𝓝[>] 0]
fun h => h ^ 2 | E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 => ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
⊢ ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ | have : Filter.Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] (0 : ℝ)) (𝓝 (0 * (‖v‖ + ‖w‖))) :=
(continuous_id.mul continuous_const).continuousWithinAt | E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 => ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
this : Tendsto (fun h => h * (‖v‖ + ‖w‖)) (𝓝[>] 0) (𝓝 (0 * (‖v‖ + ‖w‖)))
⊢ ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ | b5ca530ac5b123c0 |
SimpleGraph.Walk.IsHamiltonianCycle.length_eq | Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean | /-- The length of a hamiltonian cycle is the number of vertices. -/
lemma IsHamiltonianCycle.length_eq [Fintype α] (hp : p.IsHamiltonianCycle) :
p.length = Fintype.card α | α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
hp : p.IsHamiltonianCycle
⊢ 1 ≤ Fintype.card α | rw [Nat.succ_le, Fintype.card_pos_iff] | α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
hp : p.IsHamiltonianCycle
⊢ Nonempty α | 17b11aabd3f48153 |
IsometryEquiv.midpoint_fixed | Mathlib/Analysis/Normed/Affine/MazurUlam.lean | theorem midpoint_fixed {x y : PE} :
∀ e : PE ≃ᵢ PE, e x = x → e y = y → e (midpoint ℝ x y) = midpoint ℝ x y | case intro
E : Type u_1
PE : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
x y : PE
z : PE := midpoint ℝ x y
s : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}
this : Nonempty ↑s
h_bdd : BddAbove (range fun e => dist (↑e z) z)
R : PE ≃ᵢ PE := (pointReflection ℝ z).toIsometryEquiv
f : PE ≃ᵢ PE → PE ≃ᵢ PE := fun e => ((e.trans R).trans e.symm).trans R
hf_dist : ∀ (e : PE ≃ᵢ PE), dist ((f e) z) z = 2 * dist (e z) z
e : PE ≃ᵢ PE
hx : e x = x
hy : e y = y
⊢ f e ∈ s | constructor <;> simp [f, R, z, hx, hy, e.symm_apply_eq.2 hx.symm, e.symm_apply_eq.2 hy.symm] | no goals | b718555705b87dec |
HomologicalComplex.isSeparating_separatingFamily | Mathlib/CategoryTheory/Generator/HomologicalComplex.lean | lemma isSeparating_separatingFamily :
IsSeparating (Set.range (separatingFamily c X)) | C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type w
inst✝³ : DecidableEq ι
c : ComplexShape ι
inst✝² : c.HasNoLoop
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
α : Type t
X : α → C
hX : IsSeparating (Set.range X)
⊢ IsSeparating (Set.range (separatingFamily c X)) | intro K L f g h | C : Type u
inst✝⁴ : Category.{v, u} C
ι : Type w
inst✝³ : DecidableEq ι
c : ComplexShape ι
inst✝² : c.HasNoLoop
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
α : Type t
X : α → C
hX : IsSeparating (Set.range X)
K L : HomologicalComplex C c
f g : K ⟶ L
h : ∀ G ∈ Set.range (separatingFamily c X), ∀ (h : G ⟶ K), h ≫ f = h ≫ g
⊢ f = g | 5bd607e9a8a5401a |
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