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
|
|---|---|---|---|---|---|---|
RingHom.finitePresentation_isStableUnderBaseChange
|
Mathlib/RingTheory/RingHom/FinitePresentation.lean
|
theorem finitePresentation_isStableUnderBaseChange :
IsStableUnderBaseChange @FinitePresentation
|
case h.e'_5.h.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FinitePresentation R T
this : Algebra.FinitePresentation S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
r✝ : S
x✝ : S ⊗[R] T
⊢ (let_fun I := Algebra.TensorProduct.includeLeftRingHom.toAlgebra;
r✝ • x✝) =
r✝ • x✝
|
simp_rw [Algebra.smul_def]
|
case h.e'_5.h.h
R S T : Type u_1
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.FinitePresentation R T
this : Algebra.FinitePresentation S (S ⊗[R] T)
e_4✝ : CommSemiring.toSemiring = Algebra.TensorProduct.instSemiring
r✝ : S
x✝ : S ⊗[R] T
⊢ (algebraMap S (S ⊗[R] T)) r✝ * x✝ = (algebraMap S (S ⊗[R] T)) r✝ * x✝
|
ecddf4004478cb5e
|
Complex.norm_exp_sub_one_sub_id_le
|
Mathlib/Data/Complex/Exponential.lean
|
theorem norm_exp_sub_one_sub_id_le {x : ℂ} (hx : ‖x‖ ≤ 1) : ‖exp x - 1 - x‖ ≤ ‖x‖ ^ 2 :=
calc
‖exp x - 1 - x‖ = ‖exp x - ∑ m ∈ range 2, x ^ m / m.factorial‖
|
case h
x : ℂ
hx : ‖x‖ ≤ 1
⊢ ↑(Nat.succ 2) * (↑(Nat.factorial 2) * ↑2)⁻¹ ≤ 1
|
norm_num [Nat.factorial]
|
no goals
|
4cb40c26287901ee
|
List.prod_eq_zero_iff
|
Mathlib/Algebra/BigOperators/Ring/List.lean
|
/-- Product of elements of a list `l` equals zero if and only if `0 ∈ l`. See also
`List.prod_eq_zero` for an implication that needs weaker typeclass assumptions. -/
@[simp] lemma prod_eq_zero_iff : ∀ {l : List M₀}, l.prod = 0 ↔ (0 : M₀) ∈ l
| [] => by simp
| a :: l => by rw [prod_cons, mul_eq_zero, prod_eq_zero_iff, mem_cons, eq_comm]
|
M₀ : Type u_4
inst✝² : MonoidWithZero M₀
inst✝¹ : Nontrivial M₀
inst✝ : NoZeroDivisors M₀
a : M₀
l : List M₀
⊢ (a :: l).prod = 0 ↔ 0 ∈ a :: l
|
rw [prod_cons, mul_eq_zero, prod_eq_zero_iff, mem_cons, eq_comm]
|
no goals
|
899eba2e1ff6cc62
|
Polynomial.mul_scaleRoots
|
Mathlib/RingTheory/Polynomial/ScaleRoots.lean
|
/-- Multiplication and `scaleRoots` commute up to a power of `r`. The factor disappears if we
assume that the product of the leading coeffs does not vanish. See `Polynomial.mul_scaleRoots'`. -/
lemma mul_scaleRoots (p q : R[X]) (r : R) :
r ^ (natDegree p + natDegree q - natDegree (p * q)) • (p * q).scaleRoots r =
p.scaleRoots r * q.scaleRoots r
|
case inr
R : Type u_1
inst✝ : CommSemiring R
p q : R[X]
r : R
n a b : ℕ
e : a + b = n
ha : a ≤ p.natDegree
⊢ p.coeff a * q.coeff b * r ^ (p.natDegree + q.natDegree - n) =
p.coeff a * r ^ (p.natDegree - a) * (q.coeff b * r ^ (q.natDegree - b))
|
cases lt_or_le (natDegree q) b with
| inl h => simp only [coeff_eq_zero_of_natDegree_lt h, zero_mul, mul_zero]
| inr hb =>
simp only [← e, mul_assoc, mul_comm (r ^ (_ - a)), ← pow_add]
rw [add_comm (_ - _), tsub_add_tsub_comm ha hb]
|
no goals
|
ac865a3551bb16a7
|
IntermediateField.sSup_toSubfield
|
Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.lean
|
theorem sSup_toSubfield (S : Set (IntermediateField F E)) (hS : S.Nonempty) :
(sSup S).toSubfield = sSup (toSubfield '' S)
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set (IntermediateField F E)
hS : S.Nonempty
⊢ toSubfield '' S = Subfield.closure '' (SetLike.coe '' S)
|
rw [Set.image_image]
|
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set (IntermediateField F E)
hS : S.Nonempty
⊢ toSubfield '' S = (fun x => Subfield.closure ↑x) '' S
|
28cd4459b9a2dade
|
Set.limsup_eq_tendsto_sum_indicator_atTop
|
Mathlib/Algebra/Order/Archimedean/IndicatorCard.lean
|
lemma limsup_eq_tendsto_sum_indicator_atTop {α R : Type*} [OrderedAddCommMonoid R]
[AddLeftStrictMono R] [Archimedean R] {r : R} (h : 0 < r) (s : ℕ → Set α) :
atTop.limsup s = { ω | atTop.Tendsto
(fun n ↦ ∑ k ∈ Finset.range n, (s k).indicator (fun _ ↦ r) ω) atTop }
|
α : Type u_1
R : Type u_2
inst✝² : OrderedAddCommMonoid R
inst✝¹ : AddLeftStrictMono R
inst✝ : Archimedean R
r : R
h : 0 < r
s : ℕ → Set α
⊢ limsup s atTop = {ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s k).indicator (fun x => r) ω) atTop atTop}
|
nth_rw 1 [← Nat.cofinite_eq_atTop, cofinite.limsup_set_eq]
|
α : Type u_1
R : Type u_2
inst✝² : OrderedAddCommMonoid R
inst✝¹ : AddLeftStrictMono R
inst✝ : Archimedean R
r : R
h : 0 < r
s : ℕ → Set α
⊢ {x | {n | x ∈ s n}.Infinite} =
{ω | Tendsto (fun n => ∑ k ∈ Finset.range n, (s k).indicator (fun x => r) ω) atTop atTop}
|
c2ebfc8a516e427b
|
AlgebraicGeometry.finite_appTop_of_universallyClosed
|
Mathlib/AlgebraicGeometry/Morphisms/Proper.lean
|
theorem finite_appTop_of_universallyClosed (f : X ⟶ Spec (.of K))
[IsIntegral X] [UniversallyClosed f] [LocallyOfFiniteType f] :
f.appTop.hom.Finite
|
case intro.intro.intro.intro.intro
X : Scheme
K : Type u
inst✝³ : Field K
f : X ⟶ Spec (CommRingCat.of K)
inst✝² : IsIntegral X
inst✝¹ : UniversallyClosed f
inst✝ : LocallyOfFiniteType f
x : ↑↑X.toPresheafedSpace
U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace
hU : U ∈ X.affineOpens
hxU : x ∈ ↑U
this✝¹ : Field ↑Γ(Spec (CommRingCat.of K), ⊤) := ⋯.toField
this✝ : Field ↑Γ(X, ⊤) := ⋯.toField
this : Nonempty ↑↑(↑U).toPresheafedSpace
⊢ (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).Finite
|
apply RingHom.finite_of_algHom_finiteType_of_isJacobsonRing (A := Γ(X, U))
(g := (X.presheaf.map (homOfLE le_top).op).hom)
|
case intro.intro.intro.intro.intro.hfg
X : Scheme
K : Type u
inst✝³ : Field K
f : X ⟶ Spec (CommRingCat.of K)
inst✝² : IsIntegral X
inst✝¹ : UniversallyClosed f
inst✝ : LocallyOfFiniteType f
x : ↑↑X.toPresheafedSpace
U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace
hU : U ∈ X.affineOpens
hxU : x ∈ ↑U
this✝¹ : Field ↑Γ(Spec (CommRingCat.of K), ⊤) := ⋯.toField
this✝ : Field ↑Γ(X, ⊤) := ⋯.toField
this : Nonempty ↑↑(↑U).toPresheafedSpace
⊢ ((CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)).comp (CommRingCat.Hom.hom (Scheme.Hom.appTop f))).FiniteType
|
29395717bc249029
|
CategoryTheory.Limits.limit.lift_pre
|
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
theorem limit.lift_pre (c : Cone F) :
limit.lift F c ≫ limit.pre F E = limit.lift (E ⋙ F) (c.whisker E)
|
J : Type u₁
inst✝⁴ : Category.{v₁, u₁} J
K : Type u₂
inst✝³ : Category.{v₂, u₂} K
C : Type u
inst✝² : Category.{v, u} C
F : J ⥤ C
inst✝¹ : HasLimit F
E : K ⥤ J
inst✝ : HasLimit (E ⋙ F)
c : Cone F
⊢ lift F c ≫ pre F E = lift (E ⋙ F) (Cone.whisker E c)
|
ext
|
case w
J : Type u₁
inst✝⁴ : Category.{v₁, u₁} J
K : Type u₂
inst✝³ : Category.{v₂, u₂} K
C : Type u
inst✝² : Category.{v, u} C
F : J ⥤ C
inst✝¹ : HasLimit F
E : K ⥤ J
inst✝ : HasLimit (E ⋙ F)
c : Cone F
j✝ : K
⊢ (lift F c ≫ pre F E) ≫ π (E ⋙ F) j✝ = lift (E ⋙ F) (Cone.whisker E c) ≫ π (E ⋙ F) j✝
|
fc8ff14e6ea27b23
|
CategoryTheory.Sieve.functorPushforward_extend_eq
|
Mathlib/CategoryTheory/Sites/Sieves.lean
|
theorem functorPushforward_extend_eq {R : Presieve X} :
(generate R).arrows.functorPushforward F = R.functorPushforward F
|
case h.h.mp.intro.intro.intro.intro.intro.intro.intro.intro
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
X : C
R : Presieve X
Y : D
X' : C
f' : Y ⟶ F.obj X'
X'' : C
g' : X' ⟶ X''
f'' : X'' ⟶ X
h₁ : R f''
⊢ f' ≫ F.map (g' ≫ f'') ∈ Presieve.functorPushforward F R
|
exact ⟨X'', f'', f' ≫ F.map g', h₁, by simp⟩
|
no goals
|
b61d2485d4b68090
|
Polynomial.div_eq_zero_iff
|
Mathlib/Algebra/Polynomial/FieldDivision.lean
|
theorem div_eq_zero_iff (hq0 : q ≠ 0) : p / q = 0 ↔ degree p < degree q :=
⟨fun h => by
have := EuclideanDomain.div_add_mod p q
rwa [h, mul_zero, zero_add, mod_eq_self_iff hq0] at this,
fun h => by
have hlt : degree p < degree (q * C (leadingCoeff q)⁻¹)
|
R : Type u
inst✝ : Field R
p q : R[X]
hq0 : q ≠ 0
h : p.degree < q.degree
⊢ p / q = 0
|
have hlt : degree p < degree (q * C (leadingCoeff q)⁻¹) := by
rwa [degree_mul_leadingCoeff_inv q hq0]
|
R : Type u
inst✝ : Field R
p q : R[X]
hq0 : q ≠ 0
h : p.degree < q.degree
hlt : p.degree < (q * C q.leadingCoeff⁻¹).degree
⊢ p / q = 0
|
5b5a4fcac6cf08e7
|
Equiv.biSup_comp
|
Mathlib/Order/CompleteLattice.lean
|
lemma Equiv.biSup_comp {ι ι' : Type*} {g : ι' → α} (e : ι ≃ ι') (s : Set ι') :
⨆ i ∈ e.symm '' s, g (e i) = ⨆ i ∈ s, g i
|
α : Type u_1
inst✝ : CompleteLattice α
ι : Type u_8
ι' : Type u_9
g : ι' → α
e : ι ≃ ι'
s : Set ι'
⊢ ⨆ i ∈ ⇑e.symm '' s, g (e i) = ⨆ i ∈ s, g i
|
simpa only [iSup_subtype'] using (image e.symm s).symm.iSup_comp (g := g ∘ (↑))
|
no goals
|
506addd85f641603
|
SimpleGraph.IsSRGWith.top
|
Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean
|
theorem IsSRGWith.top :
(⊤ : SimpleGraph V).IsSRGWith (Fintype.card V) (Fintype.card V - 1) (Fintype.card V - 2) μ where
card := rfl
regular := IsRegularOfDegree.top
of_adj := fun v w h => by
rw [card_commonNeighbors_top]
exact h
of_not_adj := fun v w h h' => False.elim (h' ((top_adj v w).2 h))
|
V : Type u
inst✝¹ : Fintype V
μ : ℕ
inst✝ : DecidableEq V
v w : V
h : ⊤.Adj v w
⊢ Fintype.card ↑(⊤.commonNeighbors v w) = Fintype.card V - 2
|
rw [card_commonNeighbors_top]
|
V : Type u
inst✝¹ : Fintype V
μ : ℕ
inst✝ : DecidableEq V
v w : V
h : ⊤.Adj v w
⊢ v ≠ w
|
ecd0ebdbd9afb1e4
|
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0)
|
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
x : ℝ
hxlb : n₀ ⊔ 2 ≤ x
hxub : x < 2 * (n₀ ⊔ 2)
⊢ f x = f (n₀ ⊔ 2)
|
have h₁ := calc n₀ ≤ 1 * max n₀ 2 := by simp
_ ≤ 2 * max n₀ 2 := by gcongr; norm_num
|
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
x : ℝ
hxlb : n₀ ⊔ 2 ≤ x
hxub : x < 2 * (n₀ ⊔ 2)
h₁ : n₀ ≤ 2 * (n₀ ⊔ 2)
⊢ f x = f (n₀ ⊔ 2)
|
69674f7264744f71
|
IsCyclotomicExtension.neZero
|
Mathlib/NumberTheory/Cyclotomic/Basic.lean
|
theorem neZero [h : IsCyclotomicExtension {n} A B] [IsDomain B] : NeZero ((n : ℕ) : B)
|
n : ℕ+
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
h : IsCyclotomicExtension {n} A B
inst✝ : IsDomain B
⊢ NeZero ↑↑n
|
obtain ⟨⟨r, hr⟩, -⟩ := (iff_singleton n A B).1 h
|
case intro.intro
n : ℕ+
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
h : IsCyclotomicExtension {n} A B
inst✝ : IsDomain B
r : B
hr : IsPrimitiveRoot r ↑n
⊢ NeZero ↑↑n
|
a3f693af1abdecc7
|
Computation.eq_of_bisim
|
Mathlib/Data/Seq/Computation.lean
|
theorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂
|
case right
α : Type u
R : Computation α → Computation α → Prop
bisim : IsBisimulation R
s₁ s₂ : Computation α
r✝ : R s₁ s₂
t₁ t₂ : Stream' (Option α)
e : ∃ s s', ↑s = t₁ ∧ ↑s' = t₂ ∧ R s s'
s s' : Computation α
r' a' : α
r : R (pure a') (pure a')
h : r' = a'
⊢ R (pure a') (pure a')
|
assumption
|
no goals
|
9050670d391b06a4
|
Set.inv_smul_set_distrib₀
|
Mathlib/Data/Set/Pointwise/SMul.lean
|
@[simp] lemma inv_smul_set_distrib₀ (a : α) (s : Set α) : (a • s)⁻¹ = op a⁻¹ • s⁻¹
|
α : Type u_2
inst✝ : GroupWithZero α
a : α
s : Set α
⊢ (a • s)⁻¹ = op a⁻¹ • s⁻¹
|
obtain rfl | ha := eq_or_ne a 0
|
case inl
α : Type u_2
inst✝ : GroupWithZero α
s : Set α
⊢ (0 • s)⁻¹ = op 0⁻¹ • s⁻¹
case inr
α : Type u_2
inst✝ : GroupWithZero α
a : α
s : Set α
ha : a ≠ 0
⊢ (a • s)⁻¹ = op a⁻¹ • s⁻¹
|
1ad0de5f9629fc44
|
MeasureTheory.FinMeasAdditive.map_iUnion_fin_meas_set_eq_sum
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem map_iUnion_fin_meas_set_eq_sum (T : Set α → β) (T_empty : T ∅ = 0)
(h_add : FinMeasAdditive μ T) {ι} (S : ι → Set α) (sι : Finset ι)
(hS_meas : ∀ i, MeasurableSet (S i)) (hSp : ∀ i ∈ sι, μ (S i) ≠ ∞)
(h_disj : ∀ᵉ (i ∈ sι) (j ∈ sι), i ≠ j → Disjoint (S i) (S j)) :
T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i)
|
case refine_2.e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ⊤) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ⊤
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ ⋃ i ∈ insert a s, S i = S a ∪ ⋃ i ∈ s, S i
|
convert Finset.iSup_insert a s S
|
no goals
|
6e1e5c5f22e5eba3
|
Orthonormal.inner_right_sum
|
Mathlib/Analysis/InnerProductSpace/Orthonormal.lean
|
theorem Orthonormal.inner_right_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι → 𝕜) {s : Finset ι}
{i : ι} (hi : i ∈ s) : ⟪v i, ∑ i ∈ s, l i • v i⟫ = l i
|
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Orthonormal 𝕜 v
l : ι → 𝕜
s : Finset ι
i : ι
hi : i ∈ s
⊢ inner (v i) (∑ i ∈ s, l i • v i) = l i
|
classical
simp [inner_sum, inner_smul_right, orthonormal_iff_ite.mp hv, hi]
|
no goals
|
0b90144be34025e9
|
Nat.prod_range_succ_factorial
|
Mathlib/Data/Nat/Factorial/SuperFactorial.lean
|
theorem prod_range_succ_factorial : ∀ n : ℕ, ∏ x ∈ range (n + 1), x ! = sf n
| 0 => rfl
| n + 1 => by
rw [prod_range_succ, prod_range_succ_factorial n, mul_comm, superFactorial]
|
n : ℕ
⊢ ∏ x ∈ range (n + 1 + 1), x ! = sf n + 1
|
rw [prod_range_succ, prod_range_succ_factorial n, mul_comm, superFactorial]
|
no goals
|
e40f8d38db0fc4df
|
HasDerivAt.pow
|
Mathlib/Analysis/Calculus/Deriv/Pow.lean
|
theorem HasDerivAt.pow (hc : HasDerivAt c c' x) :
HasDerivAt (fun y => c y ^ n) ((n : 𝕜) * c x ^ (n - 1) * c') x
|
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
c : 𝕜 → 𝕜
c' : 𝕜
n : ℕ
hc : HasDerivAt c c' x
⊢ HasDerivAt (fun y => c y ^ n) (↑n * c x ^ (n - 1) * c') x
|
rw [← hasDerivWithinAt_univ] at *
|
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
x : 𝕜
c : 𝕜 → 𝕜
c' : 𝕜
n : ℕ
hc : HasDerivWithinAt c c' Set.univ x
⊢ HasDerivWithinAt (fun y => c y ^ n) (↑n * c x ^ (n - 1) * c') Set.univ x
|
04e02f29d0f4b15a
|
EReal.preimage_coe_Ioi
|
Mathlib/Data/Real/EReal.lean
|
@[simp]
lemma preimage_coe_Ioi (x : ℝ) : Real.toEReal ⁻¹' Ioi x = Ioi x
|
x : ℝ
⊢ WithTop.some ⁻¹' (WithBot.some ⁻¹' Ioi ↑↑x) = Ioi x
|
simp only [WithBot.preimage_coe_Ioi, WithTop.preimage_coe_Ioi]
|
no goals
|
7d2244a627048bcf
|
OrderedFinpartition.applyOrderedFinpartition_update_left
|
Mathlib/Analysis/Calculus/ContDiff/FaaDiBruno.lean
|
theorem applyOrderedFinpartition_update_left (p : ∀ (i : Fin c.length), E[×c.partSize i]→L[𝕜] F)
(m : Fin c.length) (v : Fin n → E) (q : E[×c.partSize m]→L[𝕜] F) :
c.applyOrderedFinpartition (update p m q) v
= update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m))
|
case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → ContinuousMultilinearMap 𝕜 (fun i => E) F
m : Fin c.length
v : Fin n → E
q : ContinuousMultilinearMap 𝕜 (fun i => E) F
d : Fin c.length
h : d = m
⊢ c.applyOrderedFinpartition (update p m q) v d = update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m)) d
|
rw [h]
|
case pos
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
c : OrderedFinpartition n
p : (i : Fin c.length) → ContinuousMultilinearMap 𝕜 (fun i => E) F
m : Fin c.length
v : Fin n → E
q : ContinuousMultilinearMap 𝕜 (fun i => E) F
d : Fin c.length
h : d = m
⊢ c.applyOrderedFinpartition (update p m q) v m = update (c.applyOrderedFinpartition p v) m (q (v ∘ c.emb m)) m
|
c6f9cb743bcacd68
|
LieSubalgebra.normalizer_eq_self_of_engel_le
|
Mathlib/Algebra/Lie/EngelSubalgebra.lean
|
/-- A Lie-subalgebra of an Artinian Lie algebra is self-normalizing
if it contains an Engel subalgebra.
See `LieSubalgebra.normalizer_engel` for a proof that Engel subalgebras are self-normalizing,
avoiding the Artinian condition. -/
lemma normalizer_eq_self_of_engel_le [IsArtinian R L]
(H : LieSubalgebra R L) (x : L) (h : engel R x ≤ H) :
normalizer H = H
|
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsArtinian R L
H : LieSubalgebra R L
x : L
h : engel R x ≤ H
N : LieSubalgebra R L := H.normalizer
⊢ (engel R x).toSubmodule ⊔ H.toSubmodule = H.toSubmodule
|
rwa [sup_eq_right]
|
no goals
|
253d8e16fa48a5a4
|
contDiffWithinAt_succ_iff_hasFDerivWithinAt
|
Mathlib/Analysis/Calculus/ContDiff/Defs.lean
|
theorem contDiffWithinAt_succ_iff_hasFDerivWithinAt (hn : n ≠ ∞) :
ContDiffWithinAt 𝕜 (n + 1) f s x ↔ ∃ u ∈ 𝓝[insert x s] x, (n = ω → AnalyticOn 𝕜 f u) ∧
∃ f' : E → E →L[𝕜] F,
(∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffWithinAt 𝕜 n f' u x
|
case a
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : WithTop ℕ∞
hn : n ≠ ∞
h'n : n + 1 ≠ ∞
u : Set E
hu : u ∈ 𝓝[insert x s] x
hf : n = ω → AnalyticOn 𝕜 f u
f' : E → E →L[𝕜] F
f'_eq_deriv : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x
Hf' : ContDiffWithinAt 𝕜 n f' u x
v : Set E
hv : v ∈ 𝓝[insert x u] x
p' : E → FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)
Hp' : HasFTaylorSeriesUpToOn n f' p' v
p'_an : n = ω → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x => p' x i) v
⊢ v ∈ 𝓝[u] x
|
exact nhdsWithin_mono _ (subset_insert x u) hv
|
no goals
|
954f60033717bd9b
|
Monoid.CoprodI.lift_word_prod_nontrivial_of_not_empty
|
Mathlib/GroupTheory/CoprodI.lean
|
theorem lift_word_prod_nontrivial_of_not_empty {i j} (w : NeWord H i j) :
lift f w.prod ≠ 1
|
case neg
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
k : ι
hcard : 3 ≤ #(H k)
hl : ¬j = k
hh : i ≠ k
⊢ (lift f) w.prod ≠ 1
|
change j ≠ k at hl
|
case neg
ι : Type u_1
G : Type u_4
inst✝³ : Group G
H : ι → Type u_5
inst✝² : (i : ι) → Group (H i)
f : (i : ι) → H i →* G
α : Type u_6
inst✝¹ : MulAction G α
X : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hpp : Pairwise fun i j => ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i
inst✝ : Nontrivial ι
i j : ι
w : NeWord H i j
k : ι
hcard : 3 ≤ #(H k)
hh : i ≠ k
hl : j ≠ k
⊢ (lift f) w.prod ≠ 1
|
3e8798a67334cdb3
|
List.chain'_attachWith
|
Mathlib/Data/List/Chain.lean
|
theorem chain'_attachWith {l : List α} {p : α → Prop} (h : ∀ x ∈ l, p x)
{r : {a // p a} → {a // p a} → Prop} :
(l.attachWith p h).Chain' r ↔ l.Chain' fun a b ↦ ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩
|
case cons.mp
α : Type u
p : α → Prop
r : { a // p a } → { a // p a } → Prop
a : α
l : List α
IH : ∀ (h : ∀ (x : α), x ∈ l → p x), Chain' r (l.attachWith p h) ↔ Chain' (fun a b => ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩) l
h : ∀ (x : α), x ∈ a :: l → p x
a✝ : Chain' (fun a b => ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩) l
hc : ∀ (y : { a // p a }), (y ∈ l.head?.pbind fun a_1 h_1 => some ⟨a_1, ⋯⟩) → r ⟨a, ⋯⟩ y
b : α
hb : l.head? = some b
⊢ ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩
|
simp_rw [hb, Option.pbind_some] at hc
|
case cons.mp
α : Type u
p : α → Prop
r : { a // p a } → { a // p a } → Prop
a : α
l : List α
IH : ∀ (h : ∀ (x : α), x ∈ l → p x), Chain' r (l.attachWith p h) ↔ Chain' (fun a b => ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩) l
h : ∀ (x : α), x ∈ a :: l → p x
a✝ : Chain' (fun a b => ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩) l
b : α
hb : l.head? = some b
hc : ∀ (y : { a // p a }), y ∈ some ⟨b, ⋯⟩ → r ⟨a, ⋯⟩ y
⊢ ∃ ha hb, r ⟨a, ha⟩ ⟨b, hb⟩
|
cc5a32ae10ad5ad1
|
ZMod.val_sub
|
Mathlib/Data/ZMod/Basic.lean
|
theorem val_sub {n : ℕ} [NeZero n] {a b : ZMod n} (h : b.val ≤ a.val) :
(a - b).val = a.val - b.val
|
case pos
n : ℕ
inst✝ : NeZero n
a b : ZMod n
h : b.val ≤ a.val
hb : b = 0
⊢ (a - b).val = a.val - b.val
|
cases hb
|
case pos.refl
n : ℕ
inst✝ : NeZero n
a : ZMod n
h : val 0 ≤ a.val
⊢ (a - 0).val = a.val - val 0
|
9aaf26d2276bd4fb
|
Lean.Order.List.monotone_foldrM
|
Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean
|
theorem monotone_foldrM
(f : γ → α → β → m β) (init : β) (xs : List α) (hmono : monotone f) :
monotone (fun x => xs.foldrM (f x) (init := init))
|
case hmono.h
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → β → m β
init : β
xs : List α
hmono : monotone f
⊢ ∀ (y : β), monotone fun x a => f x a y
|
intro s
|
case hmono.h
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
α β : Type u
γ : Type w
inst✝ : PartialOrder γ
f : γ → α → β → m β
init : β
xs : List α
hmono : monotone f
s : β
⊢ monotone fun x a => f x a s
|
a113a300d4f9e2d9
|
Combinatorics.Line.exists_mono_in_high_dimension'
|
Mathlib/Combinatorics/HalesJewett.lean
|
theorem exists_mono_in_high_dimension' :
∀ (α : Type u) [Finite α] (κ : Type max v u) [Finite κ],
∃ (ι : Type) (_ : Fintype ι), ∀ C : (ι → α) → κ, ∃ l : Line α ι, l.IsMono C :=
-- The proof proceeds by induction on `α`.
Finite.induction_empty_option
(-- We have to show that the theorem is invariant under `α ≃ α'` for the induction to work.
fun {α α'} e =>
forall_imp fun κ =>
forall_imp fun _ =>
Exists.imp fun ι =>
Exists.imp fun _ h C =>
let ⟨l, c, lc⟩ := h fun v => C (e ∘ v)
⟨l.map e, c, e.forall_congr_right.mp fun x => by rw [← lc x, Line.map_apply]⟩)
(by
-- This deals with the degenerate case where `α` is empty.
intro κ _
by_cases h : Nonempty κ
· refine ⟨Unit, inferInstance, fun C => ⟨default, Classical.arbitrary _, PEmpty.rec⟩⟩
· exact ⟨Empty, inferInstance, fun C => (h ⟨C (Empty.rec)⟩).elim⟩)
(by
-- Now we have to show that the theorem holds for `Option α` if it holds for `α`.
intro α _ ihα κ _
cases nonempty_fintype κ
-- Later we'll need `α` to be nonempty. So we first deal with the trivial case where `α` is
-- empty.
-- Then `Option α` has only one element, so any line is monochromatic.
by_cases h : Nonempty α
case neg =>
refine ⟨Unit, inferInstance, fun C => ⟨diagonal _ Unit, C fun _ => none, ?_⟩⟩
rintro (_ | ⟨a⟩)
· rfl
· exact (h ⟨a⟩).elim
-- The key idea is to show that for every `r`, in high dimension we can either find
-- `r` color focused lines or a monochromatic line.
suffices key :
∀ r : ℕ,
∃ (ι : Type) (_ : Fintype ι),
∀ C : (ι → Option α) → κ,
(∃ s : ColorFocused C, Multiset.card s.lines = r) ∨ ∃ l, IsMono C l by
-- Given the key claim, we simply take `r = |κ| + 1`. We cannot have this many distinct colors
-- so we must be in the second case, where there is a monochromatic line.
obtain ⟨ι, _inst, hι⟩ := key (Fintype.card κ + 1)
refine ⟨ι, _inst, fun C => (hι C).resolve_left ?_⟩
rintro ⟨s, sr⟩
apply Nat.not_succ_le_self (Fintype.card κ)
rw [← Nat.add_one, ← sr, ← Multiset.card_map, ← Finset.card_mk]
exact Finset.card_le_univ ⟨_, s.distinct_colors⟩
-- We now prove the key claim, by induction on `r`.
intro r
induction r with
-- The base case `r = 0` is trivial as the empty collection is color-focused.
| zero => exact ⟨Empty, inferInstance, fun C => Or.inl ⟨default, Multiset.card_zero⟩⟩
| succ r ihr =>
-- Supposing the key claim holds for `r`, we need to show it for `r+1`. First pick a high
-- enough dimension `ι` for `r`.
obtain ⟨ι, _inst, hι⟩ := ihr
-- Then since the theorem holds for `α` with any number of colors, pick a dimension `ι'` such
-- that `ι' → α` always has a monochromatic line whenever it is `(ι → Option α) → κ`-colored.
specialize ihα ((ι → Option α) → κ)
obtain ⟨ι', _inst, hι'⟩ := ihα
-- We claim that `ι ⊕ ι'` works for `Option α` and `κ`-coloring.
refine ⟨ι ⊕ ι', inferInstance, ?_⟩
intro C
-- A `κ`-coloring of `ι ⊕ ι' → Option α` induces an `(ι → Option α) → κ`-coloring of `ι' → α`.
specialize hι' fun v' v => C (Sum.elim v (some ∘ v'))
-- By choice of `ι'` this coloring has a monochromatic line `l'` with color class `C'`, where
-- `C'` is a `κ`-coloring of `ι → α`.
obtain ⟨l', C', hl'⟩ := hι'
-- If `C'` has a monochromatic line, then so does `C`. We use this in two places below.
have mono_of_mono : (∃ l, IsMono C' l) → ∃ l, IsMono C l
|
case neg
κ : Type (max v u)
inst✝ : Finite κ
h : ¬Nonempty κ
⊢ ∃ ι x, ∀ (C : (ι → PEmpty.{u + 1}) → κ), ∃ l, IsMono C l
|
exact ⟨Empty, inferInstance, fun C => (h ⟨C (Empty.rec)⟩).elim⟩
|
no goals
|
458a3139c96a1a38
|
LieIdeal.comap_bracket_incl_of_le
|
Mathlib/Algebra/Lie/IdealOperations.lean
|
theorem comap_bracket_incl_of_le {I₁ I₂ : LieIdeal R L} (h₁ : I₁ ≤ I) (h₂ : I₂ ≤ I) :
⁅comap I.incl I₁, comap I.incl I₂⁆ = comap I.incl ⁅I₁, I₂⁆
|
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I I₁ I₂ : LieIdeal R L
h₁ : I ⊓ I₁ = I₁
h₂ : I ⊓ I₂ = I₂
⊢ comap I.incl ⁅I ⊓ I₁, I ⊓ I₂⁆ = comap I.incl ⁅I₁, I₂⁆
|
rw [h₁, h₂]
|
no goals
|
e4e33afabd2782f8
|
Set.inj_on_iUnion_of_directed
|
Mathlib/Data/Set/Lattice.lean
|
theorem inj_on_iUnion_of_directed {s : ι → Set α} (hs : Directed (· ⊆ ·) s) {f : α → β}
(hf : ∀ i, InjOn f (s i)) : InjOn f (⋃ i, s i)
|
case intro
α : Type u_1
β : Type u_2
ι : Sort u_5
s : ι → Set α
hs : Directed (fun x1 x2 => x1 ⊆ x2) s
f : α → β
hf : ∀ (i : ι), InjOn f (s i)
x : α
hx✝ : x ∈ ⋃ i, s i
y : α
hy : y ∈ ⋃ i, s i
hxy : f x = f y
i : ι
hx : x ∈ s i
⊢ x = y
|
rcases mem_iUnion.1 hy with ⟨j, hy⟩
|
case intro.intro
α : Type u_1
β : Type u_2
ι : Sort u_5
s : ι → Set α
hs : Directed (fun x1 x2 => x1 ⊆ x2) s
f : α → β
hf : ∀ (i : ι), InjOn f (s i)
x : α
hx✝ : x ∈ ⋃ i, s i
y : α
hy✝ : y ∈ ⋃ i, s i
hxy : f x = f y
i : ι
hx : x ∈ s i
j : ι
hy : y ∈ s j
⊢ x = y
|
35c4c97714d42f6b
|
ENat.toNat_le_of_le_coe
|
Mathlib/Data/ENat/Basic.lean
|
lemma toNat_le_of_le_coe {m : ℕ∞} {n : ℕ} (h : m ≤ n) : toNat m ≤ n
|
m : ℕ∞
n : ℕ
h : m ≤ ↑n
⊢ m.toNat ≤ n
|
lift m to ℕ using ne_top_of_le_ne_top (coe_ne_top n) h
|
case intro
n m : ℕ
h : ↑m ≤ ↑n
⊢ (↑m).toNat ≤ n
|
90dc0428cc06666a
|
Besicovitch.exist_disjoint_covering_families
|
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
theorem exist_disjoint_covering_families {N : ℕ} {τ : ℝ} (hτ : 1 < τ)
(hN : IsEmpty (SatelliteConfig α N τ)) (q : BallPackage β α) :
∃ s : Fin N → Set β,
(∀ i : Fin N, (s i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) ∧
range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, ball (q.c j) (q.r j)
|
case inr
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
⊢ ∃ s,
(∀ (i : Fin N), (s i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) ∧
range q.c ⊆ ⋃ i, ⋃ j ∈ s i, ball (q.c j) (q.r j)
|
let s := fun i : Fin N =>
⋃ (k : Ordinal.{u}) (_ : k < p.lastStep) (_ : p.color k = i), ({p.index k} : Set β)
|
case inr
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i => ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
⊢ ∃ s,
(∀ (i : Fin N), (s i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) ∧
range q.c ⊆ ⋃ i, ⋃ j ∈ s i, ball (q.c j) (q.r j)
|
416e48a035cdf3d5
|
CategoryTheory.Limits.biproduct.map_matrix
|
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
theorem biproduct.map_matrix {f : J → C} {g : J → C} {h : K → C} (m : ∀ k, f k ⟶ g k)
(n : ∀ j k, g j ⟶ h k) :
biproduct.map m ≫ biproduct.matrix n = biproduct.matrix fun j k => m j ≫ n j k
|
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
J K : Type
inst✝² : Finite J
inst✝¹ : HasFiniteBiproducts C
inst✝ : Finite K
f g : J → C
h : K → C
m : (k : J) → f k ⟶ g k
n : (j : J) → (k : K) → g j ⟶ h k
⊢ map m ≫ matrix n = matrix fun j k => m j ≫ n j k
|
ext
|
case w.w
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
J K : Type
inst✝² : Finite J
inst✝¹ : HasFiniteBiproducts C
inst✝ : Finite K
f g : J → C
h : K → C
m : (k : J) → f k ⟶ g k
n : (j : J) → (k : K) → g j ⟶ h k
j✝¹ : K
j✝ : J
⊢ ι f j✝ ≫ (map m ≫ matrix n) ≫ π h j✝¹ = ι f j✝ ≫ (matrix fun j k => m j ≫ n j k) ≫ π h j✝¹
|
ebc2335d1e01283f
|
RingHom.finiteType_ofLocalizationSpan
|
Mathlib/RingTheory/RingHom/FiniteType.lean
|
theorem finiteType_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.FiniteType
|
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), (Localization.awayMap f ↑r).FiniteType
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) :=
fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ :
∀ (r : { x // x ∈ s }),
IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
⊢ f.FiniteType
|
constructor
|
case out
R S : Type u_1
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Finset R
hs : Ideal.span ↑s = ⊤
H : ∀ (r : { x // x ∈ s }), (Localization.awayMap f ↑r).FiniteType
this✝² : Algebra R S := f.toAlgebra
this✝¹ : (r : { x // x ∈ s }) → Algebra (Localization.Away ↑r) (Localization.Away (f ↑r)) :=
fun r => (Localization.awayMap f ↑r).toAlgebra
this✝ :
∀ (r : { x // x ∈ s }),
IsLocalization (Submonoid.map (algebraMap R S) (Submonoid.powers ↑r)) (Localization.Away (f ↑r))
this : ∀ (r : { x // x ∈ s }), IsScalarTower R (Localization.Away ↑r) (Localization.Away (f ↑r))
⊢ ⊤.FG
|
99d09e8c0e75eabf
|
Algebra.TensorProduct.includeRight_map_center_le
|
Mathlib/Algebra/Central/TensorProduct.lean
|
lemma Algebra.TensorProduct.includeRight_map_center_le :
(Subalgebra.center K C).map includeRight ≤ Subalgebra.center K (B ⊗[K] C) := fun x hx ↦ by
simp only [Subalgebra.mem_map, Subalgebra.mem_center_iff] at hx ⊢
obtain ⟨c, hc0, rfl⟩ := hx
intro bc
induction bc using TensorProduct.induction_on with
| zero => simp
| tmul b c' => simp [hc0]
| add _ _ _ _ => simp_all [add_mul, mul_add]
|
case intro.intro.tmul
K : Type u_1
B : Type u_2
C : Type u_3
inst✝⁴ : CommSemiring K
inst✝³ : Semiring B
inst✝² : Semiring C
inst✝¹ : Algebra K B
inst✝ : Algebra K C
c : C
hc0 : ∀ (b : C), b * c = c * b
b : B
c' : C
⊢ b ⊗ₜ[K] c' * includeRight c = includeRight c * b ⊗ₜ[K] c'
|
simp [hc0]
|
no goals
|
48e55bd585cb4dd0
|
Set.SurjOn.prodMap
|
Mathlib/Data/Set/Function.lean
|
lemma SurjOn.prodMap (h₁ : SurjOn f₁ s₁ t₁) (h₂ : SurjOn f₂ s₂ t₂) :
SurjOn (fun x ↦ (f₁ x.1, f₂ x.2)) (s₁ ×ˢ s₂) (t₁ ×ˢ t₂)
|
case intro.intro
α₁ : Type u_7
α₂ : Type u_8
β₁ : Type u_9
β₂ : Type u_10
s₁ : Set α₁
s₂ : Set α₂
t₁ : Set β₁
t₂ : Set β₂
f₁ : α₁ → β₁
f₂ : α₂ → β₂
h₁ : SurjOn f₁ s₁ t₁
h₂ : SurjOn f₂ s₂ t₂
x : β₁ × β₂
hx : x ∈ t₁ ×ˢ t₂
a₁ : α₁
ha₁ : a₁ ∈ s₁
hx₁ : f₁ a₁ = x.1
⊢ x ∈ (fun x => (f₁ x.1, f₂ x.2)) '' s₁ ×ˢ s₂
|
obtain ⟨a₂, ha₂, hx₂⟩ := h₂ hx.2
|
case intro.intro.intro.intro
α₁ : Type u_7
α₂ : Type u_8
β₁ : Type u_9
β₂ : Type u_10
s₁ : Set α₁
s₂ : Set α₂
t₁ : Set β₁
t₂ : Set β₂
f₁ : α₁ → β₁
f₂ : α₂ → β₂
h₁ : SurjOn f₁ s₁ t₁
h₂ : SurjOn f₂ s₂ t₂
x : β₁ × β₂
hx : x ∈ t₁ ×ˢ t₂
a₁ : α₁
ha₁ : a₁ ∈ s₁
hx₁ : f₁ a₁ = x.1
a₂ : α₂
ha₂ : a₂ ∈ s₂
hx₂ : f₂ a₂ = x.2
⊢ x ∈ (fun x => (f₁ x.1, f₂ x.2)) '' s₁ ×ˢ s₂
|
d3ef2f2b010a9b6e
|
Fermat42.not_minimal
|
Mathlib/NumberTheory/FLT/Four.lean
|
theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False
|
case intro.intro.intro.intro.intro.intro.intro
a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
ht : PythagoreanTriple (a ^ 2) (b ^ 2) c
h2 : (a ^ 2).gcd (b ^ 2) = 1
ha22 : a ^ 2 % 2 = 1
m n : ℤ
ht1 : a ^ 2 = m ^ 2 - n ^ 2
ht2 : b ^ 2 = 2 * m * n
ht3 : c = m ^ 2 + n ^ 2
ht4 : m.gcd n = 1
ht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
ht6 : 0 ≤ m
htt : PythagoreanTriple a n m
h3 : a.gcd n = 1
hb20 : b ^ 2 ≠ 0
h4 : 0 < m
⊢ False
|
obtain ⟨r, s, _, htt2, htt3, htt4, htt5, htt6⟩ := htt.coprime_classification' h3 ha2 h4
|
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
a b c : ℤ
h : Minimal a b c
ha2 : a % 2 = 1
hc : 0 < c
ht : PythagoreanTriple (a ^ 2) (b ^ 2) c
h2 : (a ^ 2).gcd (b ^ 2) = 1
ha22 : a ^ 2 % 2 = 1
m n : ℤ
ht1 : a ^ 2 = m ^ 2 - n ^ 2
ht2 : b ^ 2 = 2 * m * n
ht3 : c = m ^ 2 + n ^ 2
ht4 : m.gcd n = 1
ht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
ht6 : 0 ≤ m
htt : PythagoreanTriple a n m
h3 : a.gcd n = 1
hb20 : b ^ 2 ≠ 0
h4 : 0 < m
r s : ℤ
left✝ : a = r ^ 2 - s ^ 2
htt2 : n = 2 * r * s
htt3 : m = r ^ 2 + s ^ 2
htt4 : r.gcd s = 1
htt5 : r % 2 = 0 ∧ s % 2 = 1 ∨ r % 2 = 1 ∧ s % 2 = 0
htt6 : 0 ≤ r
⊢ False
|
58647b498e4824ae
|
Padic.exi_rat_seq_conv_cauchy
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
theorem exi_rat_seq_conv_cauchy : IsCauSeq (padicNorm p) (limSeq f) := fun ε hε ↦ by
have hε3 : 0 < ε / 3 := div_pos hε (by norm_num)
let ⟨N, hN⟩ := exi_rat_seq_conv f hε3
let ⟨N2, hN2⟩ := f.cauchy₂ hε3
exists max N N2
intro j hj
suffices
padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : ℚ_[p]) < ε by
ring_nf at this ⊢
rw [← padicNormE.eq_padic_norm']
exact mod_cast this
apply lt_of_le_of_lt
· apply padicNormE.add_le
· rw [← add_thirds ε]
apply _root_.add_lt_add
· suffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : ℚ_[p]) < ε / 3 + ε / 3 by
simpa only [sub_add_sub_cancel]
apply lt_of_le_of_lt
· apply padicNormE.add_le
· apply _root_.add_lt_add
· rw [padicNormE.map_sub]
apply mod_cast hN j
exact le_of_max_le_left hj
· exact hN2 _ (le_of_max_le_right hj) _ (le_max_right _ _)
· apply mod_cast hN (max N N2)
apply le_max_left
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : CauSeq ℚ_[p] ⇑padicNormE
ε : ℚ
hε : ε > 0
hε3 : 0 < ε / 3
N : ℕ
hN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3
N2 : ℕ
hN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3
j : ℕ
hj : j ≥ N ⊔ N2
this : padicNormE (↑(limSeq f j) - ↑(limSeq f (N ⊔ N2))) < ε
⊢ padicNormE ↑(limSeq f j - limSeq f (N ⊔ N2)) < ε
|
exact mod_cast this
|
no goals
|
5532e11826254be6
|
Basis.repr_linearCombination
|
Mathlib/LinearAlgebra/Basis/Defs.lean
|
theorem repr_linearCombination (v) : b.repr (Finsupp.linearCombination _ b v) = v
|
ι : Type u_1
R : Type u_3
M : Type u_6
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis ι R M
v : ι →₀ R
⊢ b.repr (↑b.repr.symm v) = v
|
exact b.repr.apply_symm_apply v
|
no goals
|
43fa574ab40e2a3e
|
MonoidHom.comp_inv
|
Mathlib/Algebra/Group/Hom/Basic.lean
|
theorem comp_inv (φ : G →* H) (ψ : M →* G) : φ.comp ψ⁻¹ = (φ.comp ψ)⁻¹
|
case h
M : Type u_2
G : Type u_5
H : Type u_6
inst✝² : MulOneClass M
inst✝¹ : CommGroup G
inst✝ : CommGroup H
φ : G →* H
ψ : M →* G
x✝ : M
⊢ (φ.comp ψ⁻¹) x✝ = (φ.comp ψ)⁻¹ x✝
|
simp only [Function.comp_apply, inv_apply, map_inv, coe_comp]
|
no goals
|
8f141575d36f49f7
|
MvPolynomial.schwartz_zippel_totalDegree
|
Mathlib/Algebra/MvPolynomial/SchwartzZippel.lean
|
/-- The **Schwartz-Zippel lemma**
For a nonzero multivariable polynomial `p` over an integral domain, the probability that `p`
evaluates to zero at points drawn at random from some finite subset `S` of the integral domain is
bounded by the degree of `p` over `#S`. This version presents this lemma in terms of `Finset`. -/
lemma schwartz_zippel_totalDegree {n} {p : MvPolynomial (Fin n) R} (hp : p ≠ 0) (S : Finset R) :
#{f ∈ piFinset fun _ ↦ S | eval f p = 0} / (#S ^ n : ℚ≥0) ≤ p.totalDegree / #S :=
calc
_ = #{f ∈ piFinset fun _ ↦ S | eval f p = 0} / (∏ i : Fin n, #S : ℚ≥0)
|
case inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ≠ 0
S : Finset R
hs : S.Nonempty
⊢ (p.support.sup fun s => ∑ i : Fin n, ↑(s i) / ↑(#S)) = ↑p.totalDegree / ↑(#S)
|
simp_rw [totalDegree, Nat.cast_finsetSup]
|
case inr
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : DecidableEq R
n : ℕ
p : MvPolynomial (Fin n) R
hp : p ≠ 0
S : Finset R
hs : S.Nonempty
⊢ (p.support.sup fun s => ∑ i : Fin n, ↑(s i) / ↑(#S)) = (p.support.sup fun i => ↑(i.sum fun x e => e)) / ↑(#S)
|
1b17c44f783f01c3
|
Subgroup.IsComplement.equiv_fst_eq_iff_leftCosetEquivalence
|
Mathlib/GroupTheory/Complement.lean
|
theorem equiv_fst_eq_iff_leftCosetEquivalence {g₁ g₂ : G} :
(hSK.equiv g₁).fst = (hSK.equiv g₂).fst ↔ LeftCosetEquivalence K g₁ g₂
|
case mpr
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ g₁⁻¹ * g₂ ∈ K → (hSK.equiv g₁).1 = (hSK.equiv g₂).1
|
intro h
|
case mpr
G : Type u_1
inst✝ : Group G
K : Subgroup G
S : Set G
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1
|
5a99414234979aa3
|
Array.filterMap_eq_append_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem filterMap_eq_append_iff {f : α → Option β} :
filterMap f l = L₁ ++ L₂ ↔ ∃ l₁ l₂, l = l₁ ++ l₂ ∧ filterMap f l₁ = L₁ ∧ filterMap f l₂ = L₂
|
case mk.mk.mk.mpr.intro.mk.intro.mk.intro.intro
α : Type u_1
β : Type u_2
f : α → Option β
L₁ L₂ : List β
l₁ l₂ : List α
h₁ : filterMap f { toList := l₁ } = { toList := L₁ }
h₂ : filterMap f { toList := l₂ } = { toList := L₂ }
⊢ ∃ l₁_1 l₂_1,
{ toList := l₁ }.toList ++ { toList := l₂ }.toList = l₁_1 ++ l₂_1 ∧
List.filterMap f l₁_1 = L₁ ∧ List.filterMap f l₂_1 = L₂
|
exact ⟨l₁, l₂, by simp_all⟩
|
no goals
|
83277e61f8ec9954
|
Real.exists_rat_eq_convergent
|
Mathlib/NumberTheory/DiophantineApproximation/Basic.lean
|
theorem exists_rat_eq_convergent {q : ℚ} (h : |ξ - q| < 1 / (2 * (q.den : ℝ) ^ 2)) :
∃ n, q = ξ.convergent n
|
case refine_1
ξ : ℝ
q : ℚ
h : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)
⊢ IsCoprime q.num ↑q.den
|
exact isCoprime_iff_nat_coprime.mpr (natAbs_ofNat q.den ▸ q.reduced)
|
no goals
|
3b91ecc0a433d3e5
|
ONote.fundamentalSequence_has_prop
|
Mathlib/SetTheory/Ordinal/Notation.lean
|
theorem fundamentalSequence_has_prop (o) : FundamentalSequenceProp o (fundamentalSequence o)
|
case oadd
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b.FundamentalSequenceProp b.fundamentalSequence
⊢ (a.oadd m b).FundamentalSequenceProp
(match b.fundamentalSequence with
| Sum.inr f => Sum.inr fun i => a.oadd m (f i)
| Sum.inl (some b') => Sum.inl (some (a.oadd m b'))
| Sum.inl none =>
match a.fundamentalSequence, m.natPred with
| Sum.inl none, 0 => Sum.inl (some zero)
| Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
| Sum.inl (some a'), 0 => Sum.inr fun i => a'.oadd i.succPNat zero
| Sum.inl (some a'), m.succ => Sum.inr fun i => a.oadd m.succPNat (a'.oadd i.succPNat zero)
| Sum.inr f, 0 => Sum.inr fun i => (f i).oadd 1 zero
| Sum.inr f, m.succ => Sum.inr fun i => a.oadd m.succPNat ((f i).oadd 1 zero))
|
rcases e : b.fundamentalSequence with (⟨_ | b'⟩ | f) <;>
simp only [FundamentalSequenceProp] <;>
rw [e, FundamentalSequenceProp] at ihb
|
case oadd.inl.none
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
ihb : b = 0
e : b.fundamentalSequence = Sum.inl none
⊢ match
match a.fundamentalSequence, m.natPred with
| Sum.inl none, 0 => Sum.inl (some zero)
| Sum.inl none, m.succ => Sum.inl (some (zero.oadd m.succPNat zero))
| Sum.inl (some a'), 0 => Sum.inr fun i => a'.oadd i.succPNat zero
| Sum.inl (some a'), m.succ => Sum.inr fun i => a.oadd m.succPNat (a'.oadd i.succPNat zero)
| Sum.inr f, 0 => Sum.inr fun i => (f i).oadd 1 zero
| Sum.inr f, m.succ => Sum.inr fun i => a.oadd m.succPNat ((f i).oadd 1 zero) with
| Sum.inl none => a.oadd m b = 0
| Sum.inl (some a_1) => (a.oadd m b).repr = succ a_1.repr ∧ ((a.oadd m b).NF → a_1.NF)
| Sum.inr f =>
(a.oadd m b).repr.IsLimit ∧
(∀ (i : ℕ), f i < f (i + 1) ∧ f i < a.oadd m b ∧ ((a.oadd m b).NF → (f i).NF)) ∧
∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (f i).repr
case oadd.inl.some
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
b' : ONote
ihb : b.repr = succ b'.repr ∧ (b.NF → b'.NF)
e : b.fundamentalSequence = Sum.inl (some b')
⊢ (a.oadd m b).repr = succ (a.oadd m b').repr ∧ ((a.oadd m b).NF → (a.oadd m b').NF)
case oadd.inr
a : ONote
m : ℕ+
b : ONote
iha : a.FundamentalSequenceProp a.fundamentalSequence
f : ℕ → ONote
ihb : b.repr.IsLimit ∧ (∀ (i : ℕ), f i < f (i + 1) ∧ f i < b ∧ (b.NF → (f i).NF)) ∧ ∀ a < b.repr, ∃ i, a < (f i).repr
e : b.fundamentalSequence = Sum.inr f
⊢ (a.oadd m b).repr.IsLimit ∧
(∀ (i : ℕ),
a.oadd m (f i) < a.oadd m (f (i + 1)) ∧ a.oadd m (f i) < a.oadd m b ∧ ((a.oadd m b).NF → (a.oadd m (f i)).NF)) ∧
∀ a_1 < (a.oadd m b).repr, ∃ i, a_1 < (a.oadd m (f i)).repr
|
9de22d9b6d39b1e0
|
cpow_mul_div_cpow_eq_div_div_cpow
|
Mathlib/NumberTheory/LSeries/Injectivity.lean
|
private
lemma cpow_mul_div_cpow_eq_div_div_cpow (m n : ℕ) (z : ℂ) (x : ℝ) :
(n + 1) ^ (x : ℂ) * (z / m ^ (x : ℂ)) = z / (m / (n + 1)) ^ (x : ℂ)
|
m n : ℕ
z : ℂ
x : ℝ
⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
|
have Hn : (0 : ℝ) ≤ (n + 1 : ℝ)⁻¹ := by positivity
|
m n : ℕ
z : ℂ
x : ℝ
Hn : 0 ≤ (↑n + 1)⁻¹
⊢ (↑n + 1) ^ ↑x * (z / ↑m ^ ↑x) = z / (↑m / (↑n + 1)) ^ ↑x
|
f0b6aba071251211
|
MeasureTheory.Measure.haar.le_index_mul
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
theorem le_index_mul (K₀ : PositiveCompacts G) (K : Compacts G) {V : Set G}
(hV : (interior V).Nonempty) :
index (K : Set G) V ≤ index (K : Set G) K₀ * index (K₀ : Set G) V
|
case intro.intro.intro.intro.hm.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
⊢ ∀ (a : G), g₁ * a ∈ ↑K₀ → a ∈ ⋃ g ∈ t * s, (fun h => g * h) ⁻¹' V
|
intro g₂ hg₂
|
case intro.intro.intro.intro.hm.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : (interior V).Nonempty
s : Finset G
h1s : ↑K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' ↑K₀
h2s : s.card = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' V
h2t : t.card = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
⊢ g₂ ∈ ⋃ g ∈ t * s, (fun h => g * h) ⁻¹' V
|
35e42d3f2b29e653
|
UniformConvexOn.add
|
Mathlib/Analysis/Convex/Strong.lean
|
lemma UniformConvexOn.add (hf : UniformConvexOn s φ f) (hg : UniformConvexOn s ψ g) :
UniformConvexOn s (φ + ψ) (f + g)
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
φ ψ : ℝ → ℝ
s : Set E
f g : E → ℝ
hf : UniformConvexOn s φ f
hg : UniformConvexOn s ψ g
⊢ UniformConvexOn s (φ + ψ) (f + g)
|
refine ⟨hf.1, fun x hx y hy a b ha hb hab ↦ ?_⟩
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
φ ψ : ℝ → ℝ
s : Set E
f g : E → ℝ
hf : UniformConvexOn s φ f
hg : UniformConvexOn s ψ g
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y - a * b * (φ + ψ) ‖x - y‖
|
bf2ce77815b18108
|
Set.star_mem_star
|
Mathlib/Algebra/Star/Pointwise.lean
|
theorem star_mem_star [InvolutiveStar α] : a⋆ ∈ s⋆ ↔ a ∈ s
|
α : Type u_1
s : Set α
a : α
inst✝ : InvolutiveStar α
⊢ a⋆ ∈ s⋆ ↔ a ∈ s
|
simp only [mem_star, star_star]
|
no goals
|
83f08f3026bc803a
|
separatedNhds_iff_disjoint
|
Mathlib/Topology/Separation/SeparatedNhds.lean
|
theorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t)
|
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
⊢ SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t)
|
simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ←
exists_and_left, and_assoc, and_comm, and_left_comm]
|
no goals
|
1c7a9758531fe014
|
Real.qaryEntropy_strictMonoOn
|
Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean
|
/-- Qary entropy is strictly increasing in the interval [0, 1 - q⁻¹]. -/
lemma qaryEntropy_strictMonoOn (qLe2 : 2 ≤ q) :
StrictMonoOn (qaryEntropy q) (Icc 0 (1 - 1/q))
|
case x
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc 0 (1 - 1 / ↑q)
p2 : ℝ
hp2 : p2 ∈ Icc 0 (1 - 1 / ↑q)
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
zero_le_qinv : 0 < (↑q)⁻¹
this : 0 < 1 - p
hp : 0 < p ∧ p < 1 - (↑q)⁻¹
⊢ (↑q - 1) * (1 - p) ∈ Ioi 0
|
simp_all only [mem_Ioi, mul_pos_iff_of_pos_left, show 0 < (q : ℝ) - 1 by linarith]
|
no goals
|
9408e1386437c471
|
HomologicalComplex.isSeparator_coproduct_separatingFamily
|
Mathlib/CategoryTheory/Generator/HomologicalComplex.lean
|
lemma isSeparator_coproduct_separatingFamily {X : C} (hX : IsSeparator X) :
IsSeparator (∐ (fun i ↦ separatingFamily c (fun (_ : Unit) ↦ X) ⟨⟨⟩, i⟩))
|
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type w
inst✝⁴ : DecidableEq ι
c : ComplexShape ι
inst✝³ : c.HasNoLoop
inst✝² : HasCoproductsOfShape ι C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
hX : IsSeparator X
φ : ι → HomologicalComplex C c := fun i => separatingFamily c (fun x => X) (PUnit.unit, i)
⊢ IsSeparator (∐ fun i => separatingFamily c (fun x => X) (PUnit.unit, i))
|
refine isSeparator_of_isColimit_cofan
(isSeparating_separatingFamily c (X := fun (_ : Unit) ↦ X) (by simpa using hX))
(c := Cofan.mk (∐ φ) (fun ⟨_, i⟩ ↦ Sigma.ι φ i)) ?_
|
C : Type u
inst✝⁵ : Category.{v, u} C
ι : Type w
inst✝⁴ : DecidableEq ι
c : ComplexShape ι
inst✝³ : c.HasNoLoop
inst✝² : HasCoproductsOfShape ι C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
hX : IsSeparator X
φ : ι → HomologicalComplex C c := fun i => separatingFamily c (fun x => X) (PUnit.unit, i)
⊢ IsColimit
(Cofan.mk (∐ φ) fun x =>
match x with
| (fst, i) => Sigma.ι φ i)
|
e4d66debf27e70ae
|
CategoryTheory.GrothendieckTopology.WEqualsLocallyBijective.transport
|
Mathlib/CategoryTheory/Sites/Equivalence.lean
|
lemma WEqualsLocallyBijective.transport (hG : CoverPreserving K J G) :
J.WEqualsLocallyBijective A where
iff f
|
C : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C
J : GrothendieckTopology C
D : Type u₂
inst✝⁹ : Category.{v₂, u₂} D
K : GrothendieckTopology D
G : D ⥤ C
A : Type u₃
inst✝⁸ : Category.{v₃, u₃} A
inst✝⁷ : G.IsCoverDense J
inst✝⁶ : G.Full
inst✝⁵ : G.IsContinuous K J
inst✝⁴ : (G.sheafPushforwardContinuous A K J).EssSurj
inst✝³ : G.IsCocontinuous K J
FA : A → A → Type u_1
CA : A → Type u_2
inst✝² : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝¹ : ConcreteCategory A FA
inst✝ : K.WEqualsLocallyBijective A
hG : CoverPreserving K J G
X✝ Y✝ : Cᵒᵖ ⥤ A
f : X✝ ⟶ Y✝
⊢ J.W f ↔ Presheaf.IsLocallyInjective J f ∧ Presheaf.IsLocallySurjective J f
|
rw [← W_whiskerLeft_iff J K G f, ← Presheaf.isLocallyInjective_whisker_iff K J G f hG,
← Presheaf.isLocallySurjective_whisker_iff K J G f hG, W_iff_isLocallyBijective]
|
no goals
|
a54f26529a4779b4
|
continuousOn_extendFrom
|
Mathlib/Topology/ExtendFrom.lean
|
theorem continuousOn_extendFrom [RegularSpace Y] {f : X → Y} {A B : Set X} (hB : B ⊆ closure A)
(hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)) : ContinuousOn (extendFrom A f) B
|
case intro.intro.intro.intro
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : RegularSpace Y
f : X → Y
A B : Set X
hB : B ⊆ closure A
hf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)
φ : X → Y := extendFrom A f
x : X
x_in : x ∈ B
V' : Set Y
V'_in : V' ∈ 𝓝 (φ x)
V'_closed : IsClosed V'
V : Set X
V_in : V ∈ 𝓝 x
V_op : IsOpen V
hV : V ∩ A ⊆ f ⁻¹' V'
y : X
hyV : y ∈ V
hyB : y ∈ B
this✝ : (𝓝[A] y).NeBot
limy : Tendsto f (𝓝[A] y) (𝓝 (φ y))
hVy : V ∈ 𝓝 y
this : V ∩ A ∈ 𝓝[A] y
⊢ φ y ∈ V'
|
exact V'_closed.mem_of_tendsto limy (mem_of_superset this hV)
|
no goals
|
a5bc0b035e892139
|
Module.End.independent_genEigenspace
|
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
theorem independent_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) (k : ℕ∞) :
iSupIndep (f.genEigenspace · k)
|
case refine_2
R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
k : ℕ∞
μ₁ μ₂ : R
s : Finset R
a✝ : μ₂ ∉ s
hμ₁₂✝ : μ₁ ∉ insert μ₂ s
hμ₁₂ : μ₁ ≠ μ₂
hμ₁ : μ₁ ∉ s
ih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ => (f.genEigenspace μ) k)
y z : M
hz : z ∈ s.sup fun μ => (f.genEigenspace μ) k
hx : y + z ∈ (f.genEigenspace μ₁) k
g : End R M := f - μ₂ • 1
hy : ∃ l, ∃ (_ : ↑l ≤ k), y ∈ LinearMap.ker ((f - μ₂ • 1) ^ l)
l : ℕ
hlk : ↑l ≤ k
hl : ((f - μ₂ • 1) ^ l) y = 0
⊢ (g ^ l) z ∈ s.sup fun μ => (f.genEigenspace μ) k
|
suffices (s.sup fun μ ↦ f.genEigenspace μ k).map (g ^ l) ≤
s.sup fun μ ↦ f.genEigenspace μ k by exact this (Submodule.mem_map_of_mem hz)
|
case refine_2
R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
k : ℕ∞
μ₁ μ₂ : R
s : Finset R
a✝ : μ₂ ∉ s
hμ₁₂✝ : μ₁ ∉ insert μ₂ s
hμ₁₂ : μ₁ ≠ μ₂
hμ₁ : μ₁ ∉ s
ih : Disjoint ((f.genEigenspace μ₁) k) (s.sup fun μ => (f.genEigenspace μ) k)
y z : M
hz : z ∈ s.sup fun μ => (f.genEigenspace μ) k
hx : y + z ∈ (f.genEigenspace μ₁) k
g : End R M := f - μ₂ • 1
hy : ∃ l, ∃ (_ : ↑l ≤ k), y ∈ LinearMap.ker ((f - μ₂ • 1) ^ l)
l : ℕ
hlk : ↑l ≤ k
hl : ((f - μ₂ • 1) ^ l) y = 0
⊢ Submodule.map (g ^ l) (s.sup fun μ => (f.genEigenspace μ) k) ≤ s.sup fun μ => (f.genEigenspace μ) k
|
bcd042a300b83f15
|
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0)
|
case base
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
⊢ ∀ x ∈ Set.Ico (n₀ ⊔ 2) (2 * (n₀ ⊔ 2)), f x = f (n₀ ⊔ 2)
case step
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
⊢ ∀ n ≥ 1,
(∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)) →
∀ z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
|
case base =>
intro x ⟨hxlb, hxub⟩
have h₁ := calc n₀ ≤ 1 * max n₀ 2 := by simp
_ ≤ 2 * max n₀ 2 := by gcongr; norm_num
have h₂ := hn₀ (2 * max n₀ 2) h₁ (max n₀ 2) ⟨by simp [hxlb], by linarith⟩
rw [h₂]
exact hn₀ (2 * max n₀ 2) h₁ x ⟨by simp [hxlb], le_of_lt hxub⟩
|
case step
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
heq : c₁ = c₂
n₀ : ℝ
hn₀ : ∀ b ≥ n₀, ∀ u ∈ Set.Icc (1 / 2 * b) b, f u = c₂ * f b
⊢ ∀ n ≥ 1,
(∀ z ∈ Set.Ico (n₀ ⊔ 2) (2 ^ n * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)) →
∀ z ∈ Set.Ico (2 ^ n * (n₀ ⊔ 2)) (2 ^ (n + 1) * (n₀ ⊔ 2)), f z = f (n₀ ⊔ 2)
|
69674f7264744f71
|
EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
|
Mathlib/Topology/UniformSpace/Ascoli.lean
|
theorem EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformOnFun.ofFun 𝔖 ∘ F) ℱ (𝓝 <| UniformOnFun.ofFun 𝔖 f) ↔
Tendsto ((⋃₀ 𝔖).restrict ∘ F) ℱ (𝓝 <| (⋃₀ 𝔖).restrict f)
|
case a
ι : Type u_1
X : Type u_2
α : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : UniformSpace α
F : ι → X → α
𝔖 : Set (Set X)
𝔖_compact : ∀ K ∈ 𝔖, IsCompact K
F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K
ℱ : Filter ι
f : X → α
K : Set X
hK : K ∈ 𝔖
this : CompactSpace ↑K
⊢ Tendsto ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ∘ ⇑(UniformOnFun.ofFun 𝔖) ∘ F) ℱ
(𝓝 ((⇑UniformFun.ofFun ∘ K.restrict ∘ ⇑(UniformOnFun.toFun 𝔖)) ((UniformOnFun.ofFun 𝔖) f))) ↔
Tendsto (⇑UniformFun.ofFun ∘ K.restrict ∘ F) ℱ (𝓝 (UniformFun.ofFun (K.restrict f)))
|
rfl
|
no goals
|
df8bc1e377ab037c
|
Order.krullDim_nonpos_iff_forall_isMax
|
Mathlib/Order/KrullDimension.lean
|
lemma krullDim_nonpos_iff_forall_isMax : krullDim α ≤ 0 ↔ ∀ x : α, IsMax x
|
case mk.succ
α : Type u_1
inst✝ : Preorder α
H : ∀ (x b : α), ¬x < b
n : ℕ
l : Fin (n + 1 + 1) → α
h : ∀ (i : Fin (n + 1)), l i.castSucc < l i.succ
⊢ ↑{ length := n + 1, toFun := l, step := h }.length ≤ 0
|
cases H (l 0) (l 1) (h 0)
|
no goals
|
31ff17d08be8193f
|
Nat.dvd_of_forall_prime_mul_dvd
|
Mathlib/Data/Nat/Prime/Basic.lean
|
theorem dvd_of_forall_prime_mul_dvd {a b : ℕ}
(hdvd : ∀ p : ℕ, p.Prime → p ∣ a → p * a ∣ b) : a ∣ b
|
case inr.intro
a b : ℕ
hdvd : ∀ (p : ℕ), Prime p → p ∣ a → p * a ∣ b
ha : a ≠ 1
p : ℕ
hp : Prime p ∧ p ∣ a
⊢ a ∣ b
|
exact _root_.trans (dvd_mul_left a p) (hdvd p hp.1 hp.2)
|
no goals
|
293564ce2c599e74
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.denote_blastDivSubtractShift_r
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean
|
theorem denote_blastDivSubtractShift_r (aig : AIG α) (assign : α → Bool) (lhs rhs : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d : AIG.RefVec aig w) (wn wr : Nat)
(q r : AIG.RefVec aig w) (qbv rbv : BitVec w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, d.get idx hidx, assign⟧ = rhs.getLsbD idx)
(hr : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, r.get idx hidx, assign⟧ = rbv.getLsbD idx)
(hfalse : ⟦aig, falseRef, assign⟧ = false)
:
∀ (idx : Nat) (hidx : idx < w),
⟦
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).aig,
(blastDivSubtractShift aig falseRef trueRef n d wn wr q r).r.get idx hidx,
assign
⟧
=
(BitVec.divSubtractShift { n := lhs, d := rhs } { wn := wn, wr := wr, q := qbv, r := rbv }).r.getLsbD idx
|
case hleft
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
idx✝ : Nat
hidx✝ : idx✝ < w
idx : Nat
hidx : idx < w
⊢ ⟦assign,
{
aig :=
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig,
ref :=
{
gate :=
({
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }.lhs.get
idx hidx).gate,
hgate := ?hleft } }⟧ =
(rbv.shiftConcat (lhs.getLsbD (wn - 1))).getLsbD idx
case hleft
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
idx✝ : Nat
hidx✝ : idx✝ < w
idx : Nat
hidx : idx < w
⊢ ({ lhs := (((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast ⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }.lhs.get
idx hidx).gate <
(blastShiftConcat
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig
{ lhs := (q.cast ⋯).cast ⋯, bit := (trueRef.cast ⋯).cast ⋯ }).aig.decls.size
|
rw [AIG.LawfulVecOperator.denote_mem_prefix (f := blastShiftConcat)]
|
case hleft
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
idx✝ : Nat
hidx✝ : idx✝ < w
idx : Nat
hidx : idx < w
⊢ ⟦assign,
{
aig :=
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig,
ref :=
{
gate :=
({
lhs :=
(((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast
⋯).cast
⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }.lhs.get
idx hidx).gate,
hgate := ?hleft } }⟧ =
(rbv.shiftConcat (lhs.getLsbD (wn - 1))).getLsbD idx
case hleft
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := d.get idx hidx }⟧ = rhs.getLsbD idx
hr : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := r.get idx hidx }⟧ = rbv.getLsbD idx
hfalse : ⟦assign, { aig := aig, ref := falseRef }⟧ = false
idx✝ : Nat
hidx✝ : idx✝ < w
idx : Nat
hidx : idx < w
⊢ ({ lhs := (((blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).vec.cast ⋯).cast ⋯).cast ⋯,
rhs := (((d.cast ⋯).cast ⋯).cast ⋯).cast ⋯ }.lhs.get
idx hidx).gate <
(blastShiftConcat (blastShiftConcat aig { lhs := r, bit := n.getD (wn - 1) falseRef }).aig
{ lhs := q.cast ⋯, bit := falseRef.cast ⋯ }).aig.decls.size
|
249a96ab3a68ac7e
|
hasSum_one_div_pow_mul_fourier_mul_bernoulliFun
|
Mathlib/NumberTheory/ZetaValues.lean
|
theorem hasSum_one_div_pow_mul_fourier_mul_bernoulliFun {k : ℕ} (hk : 2 ≤ k) {x : ℝ}
(hx : x ∈ Icc (0 : ℝ) 1) :
HasSum (fun n : ℤ => 1 / (n : ℂ) ^ k * fourier n (x : 𝕌))
(-(2 * π * I) ^ k / k ! * bernoulliFun k x)
|
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
|
let B : C(𝕌, ℂ) :=
ContinuousMap.mk ((↑) ∘ periodizedBernoulli k)
(continuous_ofReal.comp (periodizedBernoulli.continuous (by omega)))
|
k : ℕ
hk : 2 ≤ k
x : ℝ
hx : x ∈ Icc 0 1
y : ℝ
hy : y ∈ Ico 0 1
B : C(𝕌, ℂ) := { toFun := ofReal ∘ periodizedBernoulli k, continuous_toFun := ⋯ }
⊢ HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑y) (-(2 * ↑π * I) ^ k / ↑k ! * ↑(bernoulliFun k y))
|
2ac81cd59536af31
|
Lagrange.natDegree_basis
|
Mathlib/LinearAlgebra/Lagrange.lean
|
theorem natDegree_basis (hvs : Set.InjOn v s) (hi : i ∈ s) :
(Lagrange.basis s v i).natDegree = #s - 1
|
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v : ι → F
i : ι
hvs : Set.InjOn v ↑s
hi : i ∈ s
⊢ ∀ j ∈ s.erase i, basisDivisor (v i) (v j) ≠ 0
|
simp_rw [Ne, mem_erase, basisDivisor_eq_zero_iff]
|
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v : ι → F
i : ι
hvs : Set.InjOn v ↑s
hi : i ∈ s
⊢ ∀ (j : ι), j ≠ i ∧ j ∈ s → ¬v i = v j
|
228af99d92b7c491
|
IsAlgebraic.restrictScalars
|
Mathlib/RingTheory/Algebraic/Integral.lean
|
theorem restrictScalars [Algebra.IsAlgebraic R S]
{a : A} (h : IsAlgebraic S a) : IsAlgebraic R a
|
case intro.intro
R : Type u_1
S : Type u_2
A : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Ring A
inst✝⁵ : Algebra R S
inst✝⁴ : Algebra R A
inst✝³ : Algebra S A
inst✝² : IsScalarTower R S A
inst✝¹ : NoZeroDivisors S
inst✝ : Algebra.IsAlgebraic R S
a : A
h : IsAlgebraic S a
p : S[X]
hp : p ≠ 0
eval0 : (aeval a) p = 0
hRS : Function.Injective ⇑(algebraMap R S)
this : NoZeroDivisors R
r : R
hr : r ≠ 0
int : ∀ z ∈ Finset.image p.coeff p.support, IsIntegral R (r • z)
n : ℕ
hn : n ∈ (r • p).support
hs : (r • p).coeff n ∈ ↑(r • p).coeffs
⊢ (r • p).coeff n ∈ ↑(integralClosure R S).toSubring
|
exact int _ (Finset.mem_image_of_mem _ <| support_smul _ _ hn)
|
no goals
|
ea0da1ad0b2e05b0
|
Set.PairwiseDisjoint.exists_mem_filter_basis
|
Mathlib/Order/Filter/Bases.lean
|
theorem _root_.Set.PairwiseDisjoint.exists_mem_filter_basis {I : Type*} {l : I → Filter α}
{ι : I → Sort*} {p : ∀ i, ι i → Prop} {s : ∀ i, ι i → Set α} {S : Set I}
(hd : S.PairwiseDisjoint l) (hS : S.Finite) (h : ∀ i, (l i).HasBasis (p i) (s i)) :
∃ ind : ∀ i, ι i, (∀ i, p i (ind i)) ∧ S.PairwiseDisjoint fun i => s i (ind i)
|
case intro.intro
α : Type u_1
I : Type u_6
l : I → Filter α
ι : I → Sort u_7
p : (i : I) → ι i → Prop
s : (i : I) → ι i → Set α
S : Set I
hd✝ : S.PairwiseDisjoint l
hS : S.Finite
h : ∀ (i : I), (l i).HasBasis (p i) (s i)
t : I → Set α
htl : ∀ (i : I), t i ∈ l i
hd : S.PairwiseDisjoint t
⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ S.PairwiseDisjoint fun i => s i (ind i)
|
choose ind hp ht using fun i => (h i).mem_iff.1 (htl i)
|
case intro.intro
α : Type u_1
I : Type u_6
l : I → Filter α
ι : I → Sort u_7
p : (i : I) → ι i → Prop
s : (i : I) → ι i → Set α
S : Set I
hd✝ : S.PairwiseDisjoint l
hS : S.Finite
h : ∀ (i : I), (l i).HasBasis (p i) (s i)
t : I → Set α
htl : ∀ (i : I), t i ∈ l i
hd : S.PairwiseDisjoint t
ind : (i : I) → ι i
hp : ∀ (i : I), p i (ind i)
ht : ∀ (i : I), s i (ind i) ⊆ t i
⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ S.PairwiseDisjoint fun i => s i (ind i)
|
532ac9cccdd2fcfe
|
GenContFract.IntFractPair.exists_succ_get?_stream_of_gcf_of_get?_eq_some
|
Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
|
theorem IntFractPair.exists_succ_get?_stream_of_gcf_of_get?_eq_some {gp_n : Pair K}
(s_nth_eq : (of v).s.get? n = some gp_n) :
∃ ifp : IntFractPair K, IntFractPair.stream v (n + 1) = some ifp ∧ (ifp.b : K) = gp_n.b
|
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
gp_n : Pair K
s_nth_eq :
(match (IntFractPair.of v, Stream'.Seq.tail ⟨IntFractPair.stream v, ⋯⟩) with
| (h, s) => { h := ↑h.b, s := Stream'.Seq.map (fun p => { a := 1, b := ↑p.b }) s }).s.get?
n =
some gp_n
⊢ ∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ { a := 1, b := ↑ifp.b } = gp_n
|
simpa [Stream'.Seq.get?_tail, Stream'.Seq.map_get?] using s_nth_eq
|
no goals
|
0ff50d081728fd39
|
contDiff_norm_rpow
|
Mathlib/Analysis/InnerProductSpace/NormPow.lean
|
theorem contDiff_norm_rpow {p : ℝ} (hp : 1 < p) : ContDiff ℝ 1 (fun x : E ↦ ‖x‖ ^ p)
|
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
p : ℝ
hp : 1 < p
x : E
hx : x = 0
⊢ Filter.Tendsto (fun x => ‖fderiv ℝ (fun x => ‖x‖ ^ p) x‖) (𝓝 0) (𝓝 0)
|
refine tendsto_of_tendsto_of_tendsto_of_le_of_le (tendsto_const_nhds) ?_
(fun _ ↦ norm_nonneg _) (fun _ ↦ norm_fderiv_norm_id_rpow _ hp |>.le)
|
case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
p : ℝ
hp : 1 < p
x : E
hx : x = 0
⊢ Filter.Tendsto (fun x => p * ‖x‖ ^ (p - 1)) (𝓝 0) (𝓝 0)
|
fdf902c8d0df4633
|
Equidecomp.IsDecompOn.comp'
|
Mathlib/Algebra/Group/Action/Equidecomp.lean
|
theorem IsDecompOn.comp' {g f : X → X} {B A : Set X} {T S : Finset G}
(hg : IsDecompOn g B T) (hf : IsDecompOn f A S) :
IsDecompOn (g ∘ f) (A ∩ f ⁻¹' B) (T * S)
|
case right
X : Type u_1
G : Type u_2
inst✝¹ : Monoid G
inst✝ : MulAction G X
g f : X → X
B A : Set X
T S : Finset G
hg : IsDecompOn g B T
hf : IsDecompOn f A S
a✝ : X
aA : a✝ ∈ A
aB : a✝ ∈ f ⁻¹' B
γ : G
γ_mem : γ ∈ S
hγ : f a✝ = γ • a✝
δ : G
δ_mem : δ ∈ T
hδ : g (f a✝) = δ • f a✝
⊢ (g ∘ f) a✝ = (δ * γ) • a✝
|
rwa [mul_smul, ← hγ]
|
no goals
|
af10be9c9f1e3929
|
Nat.psp_from_prime_psp
|
Mathlib/NumberTheory/FermatPsp.lean
|
theorem psp_from_prime_psp {b : ℕ} (b_ge_two : 2 ≤ b) {p : ℕ} (p_prime : p.Prime)
(p_gt_two : 2 < p) (not_dvd : ¬p ∣ b * (b ^ 2 - 1)) : FermatPsp (psp_from_prime b p) b
|
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
⊢ (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
|
apply_fun fun x => x * (b ^ 2 - 1) at AB_id
|
b : ℕ
b_ge_two : 2 ≤ b
p : ℕ
p_prime : Prime p
p_gt_two : 2 < p
not_dvd : ¬p ∣ b * (b ^ 2 - 1)
A : ℕ := (b ^ p - 1) / (b - 1)
B : ℕ := (b ^ p + 1) / (b + 1)
hi_A : 1 < A
hi_B : 1 < B
hi_AB : 1 < A * B
hi_b : 0 < b
hi_p : 1 ≤ p
hi_bsquared : 0 < b ^ 2 - 1
hi_bpowtwop : 1 ≤ b ^ (2 * p)
hi_bpowpsubone : 1 ≤ b ^ (p - 1)
p_odd : Odd p
AB_not_prime : ¬Prime (A * B)
hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1
AB_id : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1)
⊢ (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)
|
6f5ab4acf8539f9d
|
Real.hasStrictFDerivAt_rpow_of_neg
|
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
theorem hasStrictFDerivAt_rpow_of_neg (p : ℝ × ℝ) (hp : p.1 < 0) :
HasStrictFDerivAt (fun x : ℝ × ℝ => x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
(p.1 ^ p.2 * log p.1 - exp (log p.1 * p.2) * sin (p.2 * π) * π) •
ContinuousLinearMap.snd ℝ ℝ ℝ) p
|
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2) * cos (x.2 * π)
⊢ HasStrictFDerivAt (fun x => x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
(p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ)
p
|
refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm
|
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2) * cos (x.2 * π)
⊢ HasStrictFDerivAt (fun x => rexp (log x.1 * x.2) * cos (x.2 * π))
((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
(p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ)
p
|
617bbb9c0f819a1d
|
CochainComplex.HomComplex.Cochain.δ_shift
|
Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
|
@[simp]
lemma δ_shift (a m : ℤ) :
δ n m (γ.shift a) = a.negOnePow • (δ n m γ).shift a
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a m : ℤ
hnm : n + 1 = m
p q : ℤ
hpq : p + m = q
⊢ p + a + m = q + a
|
omega
|
no goals
|
d67c3e8b16ad671d
|
Set.ncard_eq_of_bijective
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_eq_of_bijective {n : ℕ} (f : ∀ i, i < n → α)
(hf : ∀ a ∈ s, ∃ i, ∃ h : i < n, f i h = a) (hf' : ∀ (i) (h : i < n), f i h ∈ s)
(f_inj : ∀ (i j) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j) : s.ncard = n
|
α : Type u_1
s : Set α
n : ℕ
f : (i : ℕ) → i < n → α
hf : ∀ a ∈ s, ∃ i, ∃ (h : i < n), f i h = a
hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s
f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j
f' : Fin n → α := fun i => f ↑i ⋯
himage : s = f' '' univ
⊢ (f' '' univ).ncard = univ.ncard
|
exact ncard_image_of_injOn <| fun i _hi j _hj h ↦ Fin.ext <| f_inj i.val j.val i.is_lt j.is_lt h
|
no goals
|
e5caade2630c707b
|
Complex.norm_max_aux₁
|
Mathlib/Analysis/Complex/AbsMax.lean
|
theorem norm_max_aux₁ [CompleteSpace F] {f : ℂ → F} {z w : ℂ}
(hd : DiffContOnCl ℂ f (ball z (dist w z)))
(hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖
|
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
this : ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * r * (‖f z‖ / r)
⊢ ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖
|
rwa [mul_assoc, mul_div_cancel₀ _ hr.ne'] at this
|
no goals
|
5ca8e31b32469418
|
Module.projective_lifting_property
|
Mathlib/Algebra/Module/Projective.lean
|
theorem projective_lifting_property [h : Projective R P] (f : M →ₗ[R] N) (g : P →ₗ[R] N)
(hf : Function.Surjective f) : ∃ h : P →ₗ[R] M, f ∘ₗ h = g
|
R : Type u_1
inst✝⁶ : Semiring R
P : Type u_2
inst✝⁵ : AddCommMonoid P
inst✝⁴ : Module R P
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_4
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
h : Projective R P
f : M →ₗ[R] N
g : P →ₗ[R] N
hf : Function.Surjective ⇑f
φ : (P →₀ R) →ₗ[R] M := linearCombination R fun p => Function.surjInv hf (g p)
⊢ ∃ h, f ∘ₗ h = g
|
obtain ⟨s, hs⟩ := h.out
|
case intro
R : Type u_1
inst✝⁶ : Semiring R
P : Type u_2
inst✝⁵ : AddCommMonoid P
inst✝⁴ : Module R P
M : Type u_3
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_4
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
h : Projective R P
f : M →ₗ[R] N
g : P →ₗ[R] N
hf : Function.Surjective ⇑f
φ : (P →₀ R) →ₗ[R] M := linearCombination R fun p => Function.surjInv hf (g p)
s : P →ₗ[R] P →₀ R
hs : Function.LeftInverse ⇑(linearCombination R id) ⇑s
⊢ ∃ h, f ∘ₗ h = g
|
9671cfe0ecf83b71
|
Polynomial.Gal.prime_degree_dvd_card
|
Mathlib/FieldTheory/PolynomialGaloisGroup.lean
|
theorem prime_degree_dvd_card [CharZero F] (p_irr : Irreducible p) (p_deg : p.natDegree.Prime) :
p.natDegree ∣ Fintype.card p.Gal
|
case a
F : Type u_1
inst✝¹ : Field F
p : F[X]
inst✝ : CharZero F
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
hp : p.degree ≠ 0
α : p.SplittingField := rootOfSplits (algebraMap F p.SplittingField) ⋯ hp
hα : IsIntegral F α
this : minpoly F α ∣ p
key : p ∣ minpoly F α
⊢ p.natDegree ≤ (minpoly F α).natDegree
|
exact natDegree_le_of_dvd key (minpoly.ne_zero hα)
|
no goals
|
9be8caa417d64950
|
Asymptotics.isBigO_atTop_natCast_rpow_of_tendsto_div_rpow
|
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
|
theorem isBigO_atTop_natCast_rpow_of_tendsto_div_rpow {𝕜 : Type*} [RCLike 𝕜] {g : ℕ → 𝕜}
{a : 𝕜} {r : ℝ} (hlim : Tendsto (fun n ↦ g n / (n ^ r : ℝ)) atTop (𝓝 a)) :
g =O[atTop] fun n ↦ (n : ℝ) ^ r
|
case refine_1
𝕜 : Type u_2
inst✝ : RCLike 𝕜
g : ℕ → 𝕜
a : 𝕜
r : ℝ
hlim : Tendsto (fun n => g n / ↑(↑n ^ r)) atTop (𝓝 a)
⊢ ∀ᶠ (x : ℕ) in atTop, ↑x ^ r = 0 → ‖g x‖ = 0
|
filter_upwards [eventually_ne_atTop 0] with _ h
|
case h
𝕜 : Type u_2
inst✝ : RCLike 𝕜
g : ℕ → 𝕜
a : 𝕜
r : ℝ
hlim : Tendsto (fun n => g n / ↑(↑n ^ r)) atTop (𝓝 a)
a✝ : ℕ
h : a✝ ≠ 0
⊢ ↑a✝ ^ r = 0 → ‖g a✝‖ = 0
|
326db2a9e8788b5a
|
IsUnifLocDoublingMeasure.closedBall_mem_vitaliFamily_of_dist_le_mul
|
Mathlib/MeasureTheory/Covering/DensityTheorem.lean
|
theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r)
(rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x
|
case pos.h.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
⊢ μ (closedBall x (3 * (R / 4))) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)
|
have : closedBall x (3 * (R / 4)) ⊆ closedBall y r := by
apply closedBall_subset_closedBall'
have A : y ∈ closedBall y r := mem_closedBall_self rpos.le
have B := mem_closedBall'.1 (H A)
linarith
|
case pos.h.inr
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : MeasurableSpace α
μ : Measure α
inst✝³ : IsUnifLocDoublingMeasure μ
inst✝² : SecondCountableTopology α
inst✝¹ : BorelSpace α
inst✝ : IsLocallyFiniteMeasure μ
K : ℝ
x y : α
r : ℝ
h : dist x y ≤ K * r
rpos : 0 < r
R : ℝ := scalingScaleOf μ ((4 * K + 3) ⊔ 3)
H : closedBall y r ⊆ closedBall x (R / 4)
hr : R < r
this : closedBall x (3 * (R / 4)) ⊆ closedBall y r
⊢ μ (closedBall x (3 * (R / 4))) ≤ ↑(scalingConstantOf μ ((4 * K + 3) ⊔ 3)) * μ (closedBall y r)
|
60c120fcf237fc0d
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.clear_insert_inductive_case
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
|
theorem clear_insert_inductive_case {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(units : Array (Literal (PosFin n))) (units_nodup : ∀ i : Fin units.size, ∀ j : Fin units.size, i ≠ j → units[i] ≠ units[j])
(idx : Fin units.size) (assignments : Array Assignment)
(ih : ClearInsertInductionMotive f f_assignments_size units idx.1 assignments) :
ClearInsertInductionMotive f f_assignments_size units (idx.1 + 1) (clearUnit assignments units[idx])
|
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ↑j2 ≥ ↑idx + 1 ∧
units[j2] = (⟨↑i, ⋯⟩, false) ∧
(clearUnit assignments units[idx])[↑i] = addAssignment false f.assignments[↑i] ∧
¬hasAssignment false f.assignments[↑i] = true ∧
∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
|
constructor
|
case left
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ ↑j2 ≥ ↑idx + 1
case right
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
j1 : Fin units.size
j1_ge_idx : ↑j1 ≥ ↑idx
j2 : Fin units.size
j2_ge_idx : ↑j2 ≥ ↑idx
i_gt_zero : ↑i > 0
ih1 : units[↑j1] = (⟨↑i, ⋯⟩, true)
ih2 : units[↑j2] = (⟨↑i, ⋯⟩, false)
ih3 : assignments[↑i] = both
ih4 : f.assignments[↑i] = unassigned
ih5 : ∀ (k : Fin units.size), ↑k ≥ ↑idx → ¬k = j1 → ¬k = j2 → ¬units[↑k].fst.val = ↑i
idx_eq_j1 : idx = j1
idx_ne_j2 : idx ≠ j2
⊢ units[j2] = (⟨↑i, ⋯⟩, false) ∧
(clearUnit assignments units[idx])[↑i] = addAssignment false f.assignments[↑i] ∧
¬hasAssignment false f.assignments[↑i] = true ∧
∀ (k : Fin units.size), ↑k ≥ ↑idx + 1 → k ≠ j2 → units[k].fst.val ≠ ↑i
|
c90be518eb885cb1
|
lcm_dvd_iff
|
Mathlib/Algebra/GCDMonoid/Basic.lean
|
theorem lcm_dvd_iff [GCDMonoid α] {a b c : α} : lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c
|
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
⊢ lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c
|
by_cases h : a = 0 ∨ b = 0
|
case pos
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
h : a = 0 ∨ b = 0
⊢ lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c
case neg
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
h : ¬(a = 0 ∨ b = 0)
⊢ lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c
|
472d003f064a3536
|
ih_0
|
Mathlib/NumberTheory/Padics/Hensel.lean
|
theorem ih_0 : ih 0 a :=
⟨rfl, by simp [T_def, mul_div_cancel₀ _ (ne_of_gt (deriv_sq_norm_pos hnorm))]⟩
|
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
⊢ ‖Polynomial.eval a F‖ ≤ ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ 0
|
simp [T_def, mul_div_cancel₀ _ (ne_of_gt (deriv_sq_norm_pos hnorm))]
|
no goals
|
2d9ea65d66b5ac8b
|
Matrix.submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det
|
Mathlib/LinearAlgebra/Matrix/Determinant/Misc.lean
|
theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det {n : ℕ}
(M : Matrix (Fin (n + 1)) (Fin n) R) (hv : ∑ j, M j = 0) (j₁ j₂ : Fin (n + 1)) :
(M.submatrix (Fin.succAbove j₁) id).det =
Int.negOnePow (j₁ - j₂) • (M.submatrix (Fin.succAbove j₂) id).det
|
R : Type u_1
inst✝ : CommRing R
n : ℕ
M : Matrix (Fin (n + 1)) (Fin n) R
hv : ∑ j : Fin (n + 1), M j = 0
j₁ j₂ : Fin (n + 1)
⊢ (M.submatrix j₁.succAbove id).det = (↑↑j₁ - ↑↑j₂).negOnePow • (M.submatrix j₂.succAbove id).det
|
suffices ∀ j, (M.submatrix (Fin.succAbove j) id).det =
Int.negOnePow j • (M.submatrix (Fin.succAbove 0) id).det by
rw [this j₁, this j₂, smul_smul, ← Int.negOnePow_add, sub_add_cancel]
|
R : Type u_1
inst✝ : CommRing R
n : ℕ
M : Matrix (Fin (n + 1)) (Fin n) R
hv : ∑ j : Fin (n + 1), M j = 0
j₁ j₂ : Fin (n + 1)
⊢ ∀ (j : Fin (n + 1)), (M.submatrix j.succAbove id).det = (↑↑j).negOnePow • (M.submatrix (Fin.succAbove 0) id).det
|
f724e374e85a14e0
|
ApproximatesLinearOn.exists_homeomorph_extension
|
Mathlib/Analysis/Calculus/InverseFunctionTheorem/FiniteDimensional.lean
|
theorem exists_homeomorph_extension {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {s : Set E}
{f : E → F} {f' : E ≃L[ℝ] F} {c : ℝ≥0} (hf : ApproximatesLinearOn f (f' : E →L[ℝ] F) s c)
(hc : Subsingleton E ∨ lipschitzExtensionConstant F * c < ‖(f'.symm : F →L[ℝ] E)‖₊⁻¹) :
∃ g : E ≃ₜ F, EqOn f g s
|
case intro.intro
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : FiniteDimensional ℝ F
s : Set E
f : E → F
f' : E ≃L[ℝ] F
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ lipschitzExtensionConstant F * c < ‖↑f'.symm‖₊⁻¹
u : E → F
hu : LipschitzWith (lipschitzExtensionConstant F * c) u
uf : EqOn (f - ⇑f') u s
g : E → F := fun x => f' x + u x
fg : EqOn f g s
hg : ApproximatesLinearOn g (↑f') univ (lipschitzExtensionConstant F * c)
this : FiniteDimensional ℝ E
⊢ ∃ g, EqOn f (⇑g) s
|
exact ⟨hg.toHomeomorph g hc, fg⟩
|
no goals
|
ddf03427b5c8213e
|
Module.Flat.of_linearEquiv
|
Mathlib/RingTheory/Flat/Basic.lean
|
/-- A `R`-module linearly equivalent to a flat `R`-module is flat. -/
lemma of_linearEquiv [Flat R M] (e : N ≃ₗ[R] M) : Flat R N :=
of_retract e.toLinearMap e.symm (by simp)
|
R : Type u
M : Type v
N : Type u_1
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
inst✝ : Flat R M
e : N ≃ₗ[R] M
⊢ ↑e.symm ∘ₗ ↑e = LinearMap.id
|
simp
|
no goals
|
a01b56410a95cb3c
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastMul.go_le_size
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Operations/Mul.lean
|
theorem go_le_size {w : Nat} (aig : AIG BVBit) (curr : Nat) (acc : AIG.RefVec aig w)
(lhs rhs : AIG.RefVec aig w) :
aig.decls.size ≤ (go aig lhs rhs curr acc).aig.decls.size
|
w : Nat
aig : AIG BVBit
curr : Nat
acc lhs rhs : aig.RefVec w
⊢ aig.decls.size ≤ (go aig lhs rhs curr acc).aig.decls.size
|
unfold go
|
w : Nat
aig : AIG BVBit
curr : Nat
acc lhs rhs : aig.RefVec w
⊢ aig.decls.size ≤
(if h : curr < w then
if aig.isConstant (rhs.get curr h) false = true then go aig lhs rhs (curr + 1) acc
else
let res := blastShiftLeftConst aig { vec := lhs, distance := curr };
let aig_1 := res.aig;
let shifted := res.vec;
let_fun this := ⋯;
let lhs_1 := lhs.cast this;
let rhs_1 := rhs.cast this;
let acc_1 := acc.cast this;
let res := blastAdd aig_1 { lhs := acc_1, rhs := shifted };
let aig_2 := res.aig;
let added := res.vec;
let_fun this_1 := ⋯;
let lhs_2 := lhs_1.cast this_1;
let rhs_2 := rhs_1.cast this_1;
let acc_2 := acc_1.cast this_1;
let res := AIG.RefVec.ite aig_2 { discr := rhs_2.get curr h, lhs := added, rhs := acc_2 };
let aig_3 := res.aig;
let acc_3 := res.vec;
let_fun this := ⋯;
let lhs := lhs_2.cast this;
let rhs := rhs_2.cast this;
go aig_3 lhs rhs (curr + 1) acc_3
else { aig := aig, vec := acc }).aig.decls.size
|
599196acca342e56
|
Real.cos_sq_le_one
|
Mathlib/Data/Complex/Trigonometric.lean
|
theorem cos_sq_le_one : cos x ^ 2 ≤ 1
|
x : ℝ
⊢ cos x ^ 2 ≤ sin x ^ 2 + cos x ^ 2
|
exact le_add_of_nonneg_left (sq_nonneg _)
|
no goals
|
c469e1701b709dfa
|
LinearMap.polyCharpoly_coeff_nilRankAux_ne_zero
|
Mathlib/Algebra/Module/LinearMap/Polynomial.lean
|
lemma polyCharpoly_coeff_nilRankAux_ne_zero [Nontrivial R] :
(polyCharpoly φ b).coeff (nilRankAux φ b) ≠ 0
|
R : Type u_1
L : Type u_2
M : Type u_3
ι : Type u_5
inst✝⁹ : CommRing R
inst✝⁸ : AddCommGroup L
inst✝⁷ : Module R L
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
φ : L →ₗ[R] End R M
inst✝⁴ : Fintype ι
inst✝³ : DecidableEq ι
inst✝² : Free R M
inst✝¹ : Module.Finite R M
b : Basis ι R L
inst✝ : Nontrivial R
⊢ φ.polyCharpoly b ≠ 0
|
apply polyCharpoly_ne_zero
|
no goals
|
4e51294121129066
|
essSup_map_measure_of_measurable
|
Mathlib/MeasureTheory/Function/EssSup.lean
|
theorem essSup_map_measure_of_measurable (hg : Measurable g) (hf : AEMeasurable f μ) :
essSup g (Measure.map f μ) = essSup (g ∘ f) μ
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : CompleteLattice β
γ : Type u_3
mγ : MeasurableSpace γ
f : α → γ
g : γ → β
inst✝⁴ : MeasurableSpace β
inst✝³ : TopologicalSpace β
inst✝² : SecondCountableTopology β
inst✝¹ : OrderClosedTopology β
inst✝ : OpensMeasurableSpace β
hg : Measurable g
hf : AEMeasurable f μ
c : β
h_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c
⊢ ∀ᵐ (x : α) ∂μ, g (f x) ≤ c
|
exact h_le
|
no goals
|
f24de263ac49b4ec
|
Turing.PartrecToTM2.succ_ok
|
Mathlib/Computability/TMToPartrec.lean
|
theorem succ_ok {q s n} {c d : List Γ'} :
Reaches₁ (TM2.step tr) ⟨some (Λ'.succ q), s, K'.elim (trList [n]) [] c d⟩
⟨some q, none, K'.elim (trList [n.succ]) [] c d⟩
|
case pos
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
a : PosNum
⊢ Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) [] c d }
{ l := some q, var := none, stk := elim (trPosNum a.succ ++ [Γ'.cons]) [] c d }
|
suffices ∀ l₁, ∃ l₁' l₂' s',
List.reverseAux l₁ (trPosNum a.succ) = List.reverseAux l₁' l₂' ∧
Reaches₁ (TM2.step tr) ⟨some q.succ, s, K'.elim (trPosNum a ++ [Γ'.cons]) l₁ c d⟩
⟨some (unrev q), s', K'.elim (l₂' ++ [Γ'.cons]) l₁' c d⟩ by
obtain ⟨l₁', l₂', s', e, h⟩ := this []
simp? [List.reverseAux] at e says simp only [List.reverseAux, List.reverseAux_eq] at e
refine h.trans ?_
convert unrev_ok using 2
simp [e, List.reverseAux_eq]
|
case pos
q : Λ'
s : Option Γ'
n : ℕ
c d : List Γ'
a : PosNum
⊢ ∀ (l₁ : List Γ'),
∃ l₁' l₂' s',
l₁.reverseAux (trPosNum a.succ) = l₁'.reverseAux l₂' ∧
Reaches₁ (TM2.step tr) { l := some q.succ, var := s, stk := elim (trPosNum a ++ [Γ'.cons]) l₁ c d }
{ l := some (unrev q), var := s', stk := elim (l₂' ++ [Γ'.cons]) l₁' c d }
|
d0cdec3eaf1b4e03
|
OrderIso.preimage_Ioi
|
Mathlib/Order/Interval/Set/OrderIso.lean
|
theorem preimage_Ioi (e : α ≃o β) (b : β) : e ⁻¹' Ioi b = Ioi (e.symm b)
|
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
b : β
⊢ ⇑e ⁻¹' Ioi b = Ioi (e.symm b)
|
ext x
|
case h
α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
b : β
x : α
⊢ x ∈ ⇑e ⁻¹' Ioi b ↔ x ∈ Ioi (e.symm b)
|
ace664e977e8de93
|
ConvexOn.smul''
|
Mathlib/Analysis/Convex/Mul.lean
|
lemma ConvexOn.smul'' [OrderedSMul 𝕜 E] (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
(hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x⦄, x ∈ s → g x ≤ 0) (hfg : AntivaryOn f g s) :
ConcaveOn 𝕜 s (f • g)
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : LinearOrderedCommRing 𝕜
inst✝⁹ : LinearOrderedCommRing E
inst✝⁸ : LinearOrderedAddCommGroup F
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : Module E F
inst✝⁴ : IsScalarTower 𝕜 E F
inst✝³ : SMulCommClass 𝕜 E F
inst✝² : OrderedSMul 𝕜 F
inst✝¹ : OrderedSMul E F
s : Set 𝕜
f : 𝕜 → E
g : 𝕜 → F
inst✝ : OrderedSMul 𝕜 E
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → f x ≤ 0
hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → g x ≤ 0
hfg : AntivaryOn f g s
⊢ ConcaveOn 𝕜 s (-f • -g)
|
exact hf.neg.smul' hg.neg (fun x hx ↦ neg_nonneg.2 <| hf₀ hx) (fun x hx ↦ neg_nonneg.2 <| hg₀ hx)
hfg.neg
|
no goals
|
2e4e949e2864e0e5
|
SimpleGraph.triangle_counting
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean
|
/-- The **Triangle Counting Lemma**. If `G` is a graph and `s`, `t`, `u` are disjoint sets of
vertices such that each pair is `ε`-uniform and `2 * ε`-dense, then `G` contains at least
`(1 - 2 * ε) * ε ^ 3 * |s| * |t| * |u|` triangles. -/
lemma triangle_counting
(dst : 2 * ε ≤ G.edgeDensity s t) (ust : G.IsUniform ε s t) (hst : Disjoint s t)
(dsu : 2 * ε ≤ G.edgeDensity s u) (usu : G.IsUniform ε s u) (hsu : Disjoint s u)
(dtu : 2 * ε ≤ G.edgeDensity t u) (utu : G.IsUniform ε t u) (htu : Disjoint t u) :
(1 - 2 * ε) * ε ^ 3 * #s * #t * #u ≤ #(G.cliqueFinset 3)
|
case refine_1.mk.mk
α : Type u_1
G : SimpleGraph α
inst✝² : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝¹ : DecidableEq α
inst✝ : Fintype α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
ust : G.IsUniform ε s t
hst : Disjoint s t
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
usu : G.IsUniform ε s u
hsu : Disjoint s u
dtu : 2 * ε ≤ ↑(G.edgeDensity t u)
utu : G.IsUniform ε t u
htu : Disjoint t u
x y z : α
⊢ x ∈ s → y ∈ t → z ∈ u → G.Adj x y → G.Adj x z → G.Adj y z → G.Adj x y ∧ G.Adj x z ∧ G.Adj y z
|
exact fun _ _ _ hxy hxz hyz ↦ ⟨hxy, hxz, hyz⟩
|
no goals
|
cb6c419615fd1361
|
CategoryTheory.Abelian.Pseudoelement.pseudo_pullback
|
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
theorem pseudo_pullback {P Q R : C} {f : P ⟶ R} {g : Q ⟶ R} {p : P} {q : Q} :
f p = g q →
∃ s, pullback.fst f g s = p ∧ pullback.snd f g s = q :=
Quotient.inductionOn₂ p q fun x y h => by
obtain ⟨Z, a, b, ea, eb, comm⟩ := Quotient.exact h
obtain ⟨l, hl₁, hl₂⟩ := @pullback.lift' _ _ _ _ _ _ f g _ (a ≫ x.hom) (b ≫ y.hom) (by
simp only [Category.assoc]
exact comm)
exact ⟨l, ⟨Quotient.sound ⟨Z, 𝟙 Z, a, inferInstance, ea, by rwa [Category.id_comp]⟩,
Quotient.sound ⟨Z, 𝟙 Z, b, inferInstance, eb, by rwa [Category.id_comp]⟩⟩⟩
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasPullbacks C
P Q R : C
f : P ⟶ R
g : Q ⟶ R
p : Pseudoelement P
q : Pseudoelement Q
x : Over P
y : Over Q
h : pseudoApply f ⟦x⟧ = pseudoApply g ⟦y⟧
Z : C
a : Z ⟶ ((fun g => app f g) x).left
b : Z ⟶ ((fun g_1 => app g g_1) y).left
ea : Epi a
eb : Epi b
comm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom
⊢ a ≫ x.hom ≫ f = b ≫ y.hom ≫ g
|
exact comm
|
no goals
|
92b3b5dba8e659f2
|
Set.ncard_diff_singleton_le
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_diff_singleton_le (s : Set α) (a : α) : (s \ {a}).ncard ≤ s.ncard
|
α : Type u_1
s : Set α
a : α
hs : s.Infinite
⊢ {a}.Finite
|
simp
|
no goals
|
0875fa66bd9bb07f
|
Basis.map_addHaar
|
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
theorem Basis.map_addHaar {ι E F : Type*} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F]
[NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E]
[BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F]
(b : Basis ι ℝ E) (f : E ≃L[ℝ] F) :
map f b.addHaar = (b.map f.toLinearEquiv).addHaar
|
ι : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Fintype ι
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace E
inst✝⁴ : MeasurableSpace F
inst✝³ : BorelSpace E
inst✝² : BorelSpace F
inst✝¹ : SecondCountableTopology F
inst✝ : SigmaCompactSpace F
b : Basis ι ℝ E
f : E ≃L[ℝ] F
this : (Measure.map (⇑f) b.addHaar).IsAddHaarMeasure
⊢ Measure.map (⇑f) b.addHaar = (b.map f.toLinearEquiv).addHaar
|
rw [eq_comm, Basis.addHaar_eq_iff, Measure.map_apply f.continuous.measurable
(PositiveCompacts.isCompact _).measurableSet, Basis.coe_parallelepiped, Basis.coe_map]
|
ι : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁰ : Fintype ι
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : MeasurableSpace E
inst✝⁴ : MeasurableSpace F
inst✝³ : BorelSpace E
inst✝² : BorelSpace F
inst✝¹ : SecondCountableTopology F
inst✝ : SigmaCompactSpace F
b : Basis ι ℝ E
f : E ≃L[ℝ] F
this : (Measure.map (⇑f) b.addHaar).IsAddHaarMeasure
⊢ b.addHaar (⇑f ⁻¹' _root_.parallelepiped (⇑f.toLinearEquiv ∘ ⇑b)) = 1
|
abb088abaf1d66a7
|
ProbabilityTheory.Kernel.compProdFun_iUnion
|
Mathlib/Probability/Kernel/Composition/CompProd.lean
|
theorem compProdFun_iUnion (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel η] (a : α)
(f : ℕ → Set (β × γ)) (hf_meas : ∀ i, MeasurableSet (f i))
(hf_disj : Pairwise (Disjoint on f)) :
compProdFun κ η a (⋃ i, f i) = ∑' i, compProdFun κ η a (f i)
|
case h.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : ℕ → Set (β × γ)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_disj : Pairwise (Disjoint on f)
h_Union : (fun b => (η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => (η (a, b)) (⋃ i, {c | (b, c) ∈ f i})
b : β
⊢ Pairwise (Disjoint on fun i => {c | (b, c) ∈ f i})
|
intro i j hij s hsi hsj c hcs
|
case h.hn
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : ℕ → Set (β × γ)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf_disj : Pairwise (Disjoint on f)
h_Union : (fun b => (η (a, b)) {c | (b, c) ∈ ⋃ i, f i}) = fun b => (η (a, b)) (⋃ i, {c | (b, c) ∈ f i})
b : β
i j : ℕ
hij : i ≠ j
s : Set γ
hsi : s ≤ (fun i => {c | (b, c) ∈ f i}) i
hsj : s ≤ (fun i => {c | (b, c) ∈ f i}) j
c : γ
hcs : c ∈ s
⊢ c ∈ ⊥
|
a1a5d4db1e3c8048
|
equalizerCondition_yonedaPresheaf
|
Mathlib/Condensed/TopComparison.lean
|
theorem equalizerCondition_yonedaPresheaf
[∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) G]
(hq : ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], IsQuotientMap (G.map π)) :
EqualizerCondition (yonedaPresheaf G X)
|
C : Type u
inst✝² : Category.{v, u} C
G : C ⥤ TopCat
X : Type w'
inst✝¹ : TopologicalSpace X
inst✝ : ∀ (Z B : C) (π : Z ⟶ B) [inst : EffectiveEpi π], PreservesLimit (cospan π π) G
hq : ∀ (Z B : C) (π : Z ⟶ B) [inst : EffectiveEpi π], IsQuotientMap ⇑(ConcreteCategory.hom (G.map π))
⊢ EqualizerCondition (yonedaPresheaf G X)
|
apply EqualizerCondition.mk
|
case hP
C : Type u
inst✝² : Category.{v, u} C
G : C ⥤ TopCat
X : Type w'
inst✝¹ : TopologicalSpace X
inst✝ : ∀ (Z B : C) (π : Z ⟶ B) [inst : EffectiveEpi π], PreservesLimit (cospan π π) G
hq : ∀ (Z B : C) (π : Z ⟶ B) [inst : EffectiveEpi π], IsQuotientMap ⇑(ConcreteCategory.hom (G.map π))
⊢ ∀ (X_1 B : C) (π : X_1 ⟶ B) [inst : EffectiveEpi π] [inst : HasPullback π π],
Function.Bijective (MapToEqualizer (yonedaPresheaf G X) π (pullback.fst π π) (pullback.snd π π) ⋯)
|
1d0a8262c2d7e5b6
|
Set.image2_insert_right
|
Mathlib/Data/Set/NAry.lean
|
theorem image2_insert_right : image2 f s (insert b t) = (fun a => f a b) '' s ∪ image2 f s t
|
α : Type u_1
β : Type u_3
γ : Type u_5
f : α → β → γ
s : Set α
t : Set β
b : β
⊢ image2 f s (insert b t) = (fun a => f a b) '' s ∪ image2 f s t
|
rw [insert_eq, image2_union_right, image2_singleton_right]
|
no goals
|
f9ac43bb0f122233
|
CategoryTheory.StrongMono.iff_of_arrow_iso
|
Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean
|
theorem StrongMono.iff_of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) : StrongMono f ↔ StrongMono g
|
C : Type u
inst✝ : Category.{v, u} C
A B A' B' : C
f : A ⟶ B
g : A' ⟶ B'
e : Arrow.mk f ≅ Arrow.mk g
⊢ StrongMono f ↔ StrongMono g
|
constructor <;> intro
|
case mp
C : Type u
inst✝ : Category.{v, u} C
A B A' B' : C
f : A ⟶ B
g : A' ⟶ B'
e : Arrow.mk f ≅ Arrow.mk g
a✝ : StrongMono f
⊢ StrongMono g
case mpr
C : Type u
inst✝ : Category.{v, u} C
A B A' B' : C
f : A ⟶ B
g : A' ⟶ B'
e : Arrow.mk f ≅ Arrow.mk g
a✝ : StrongMono g
⊢ StrongMono f
|
988baae87a725eb2
|
tendsto_comp_of_locally_uniform_limit
|
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
theorem tendsto_comp_of_locally_uniform_limit (h : ContinuousAt f x) (hg : Tendsto g p (𝓝 x))
(hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ n in p, ∀ y ∈ t, (f y, F n y) ∈ u) :
Tendsto (fun n => F n (g n)) p (𝓝 (f x))
|
α : Type u
β : Type v
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
x : α
p : Filter ι
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝 x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))
|
rw [← nhdsWithin_univ] at hunif hg
|
α : Type u
β : Type v
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
x : α
p : Filter ι
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝[univ] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))
|
c49b6ba895f95c85
|
Orientation.exists_linearIsometryEquiv_eq_of_det_pos
|
Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
|
theorem exists_linearIsometryEquiv_eq_of_det_pos {f : V ≃ₗᵢ[ℝ] V}
(hd : 0 < LinearMap.det (f.toLinearEquiv : V →ₗ[ℝ] V)) :
∃ θ : Real.Angle, f = o.rotation θ
|
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
f : V ≃ₗᵢ[ℝ] V
hd : 0 < LinearMap.det ↑f.toLinearEquiv
this : Nontrivial V
⊢ ∃ θ, f = o.rotation θ
|
obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
|
case intro
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
f : V ≃ₗᵢ[ℝ] V
hd : 0 < LinearMap.det ↑f.toLinearEquiv
this : Nontrivial V
x : V
hx : x ≠ 0
⊢ ∃ θ, f = o.rotation θ
|
6545af5aebabad28
|
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