name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
ProbabilityTheory.Kernel.integral_withDensity | Mathlib/Probability/Kernel/WithDensity.lean | theorem integral_withDensity {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : β → E} [IsSFiniteKernel κ] {a : α} {g : α → β → ℝ≥0}
(hg : Measurable (Function.uncurry g)) :
∫ b, f b ∂withDensity κ (fun a b => g a b) a = ∫ b, g a b • f b ∂κ a | case hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : β → E
inst✝ : IsSFiniteKernel κ
a : α
g : α → β → ℝ≥0
hg : Measurable (Function.uncurry g)
⊢ Measurable (Function.uncurry fun a b => ↑(g a b)) | fun_prop | no goals | a452ee35b1f07cfc |
Set.star_mem_center | Mathlib/Algebra/Star/Center.lean | theorem Set.star_mem_center (ha : a ∈ Set.center R) : star a ∈ Set.center R where
comm | R : Type u_1
inst✝¹ : Mul R
inst✝ : StarMul R
a : R
ha : a ∈ center R
b c : R
⊢ star (a * star c * star b) = b * star (a * star c) | rw [star_mul, star_star] | no goals | c15bad2514aa2cf1 |
tendsto_tsum_of_dominated_convergence | Mathlib/Analysis/Normed/Group/Tannery.lean | /-- **Tannery's theorem**: topological sums commute with termwise limits, when the norms of the
summands are eventually uniformly bounded by a summable function.
(This is the special case of the Lebesgue dominated convergence theorem for the counting measure
on a discrete set. However, we prove it under somewhat weake... | α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝¹ : Nonempty β
h✝ : 𝓕.NeBot
h_g_le :... | simpa only [sum_add_tsum_compl h_sum, eq_sub_iff_add_eq'] using hS.tsum_eq | no goals | d4198172206cf97e |
WittVector.coeff_p_pow_eq_zero | Mathlib/RingTheory/WittVector/Identities.lean | theorem coeff_p_pow_eq_zero [CharP R p] {i j : ℕ} (hj : j ≠ i) : ((p : 𝕎 R) ^ i).coeff j = 0 | case succ.succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
hi : ∀ {j : ℕ}, j ≠ i → (↑p ^ i).coeff j = 0
n✝ : ℕ
hj : n✝ + 1 ≠ i + 1
⊢ (verschiebung (↑p ^ i)).coeff (n✝ + 1) ^ p = 0 | rw [verschiebung_coeff_succ, hi (ne_of_apply_ne _ hj), zero_pow hp.out.ne_zero] | no goals | 0456d8b61855531e |
PullbackCone.IsLimit.equivPullbackObj_symm_apply_fst | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | @[simp]
lemma equivPullbackObj_symm_apply_fst (x : Types.PullbackObj f g) :
c.fst ((equivPullbackObj hc).symm x) = x.1.1 | X Y S : Type v
f : X ⟶ S
g : Y ⟶ S
c : PullbackCone f g
hc : IsLimit c
x : Types.PullbackObj f g
⊢ c.fst ((equivPullbackObj hc).symm x) = (↑x).1 | obtain ⟨x, rfl⟩ := (equivPullbackObj hc).surjective x | case intro
X Y S : Type v
f : X ⟶ S
g : Y ⟶ S
c : PullbackCone f g
hc : IsLimit c
x : c.pt
⊢ c.fst ((equivPullbackObj hc).symm ((equivPullbackObj hc) x)) = (↑((equivPullbackObj hc) x)).1 | 54db9da2c6aaa66b |
IsPrimitiveRoot.arg | Mathlib/RingTheory/RootsOfUnity/Complex.lean | theorem IsPrimitiveRoot.arg {n : ℕ} {ζ : ℂ} (h : IsPrimitiveRoot ζ n) (hn : n ≠ 0) :
∃ i : ℤ, ζ.arg = i / n * (2 * Real.pi) ∧ IsCoprime i n ∧ i.natAbs < n | case neg.convert_2.refine_2
n : ℕ
hn : n ≠ 0
i : ℕ
h : i < n
hin : i.Coprime n
h₂ : ¬i * 2 ≤ n
⊢ (↑i - ↑n) * (2 * Real.pi / ↑n) ≤ 0 | exact mul_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr <| mod_cast h.le)
(div_nonneg (by simp [Real.pi_pos.le]) <| by simp) | no goals | 27e9c2564262c07b |
prime_factors_unique | Mathlib/RingTheory/UniqueFactorizationDomain/Basic.lean | theorem prime_factors_unique [CancelCommMonoidWithZero α] :
∀ {f g : Multiset α},
(∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g | α : Type u_1
inst✝ : CancelCommMonoidWithZero α
⊢ ∀ {f g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g | intro f | α : Type u_1
inst✝ : CancelCommMonoidWithZero α
f : Multiset α
⊢ ∀ {g : Multiset α}, (∀ x ∈ f, Prime x) → (∀ x ∈ g, Prime x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g | 076a07413cee2004 |
Ordinal.log_opow_mul_add | Mathlib/SetTheory/Ordinal/Exponential.lean | theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v ≠ 0) (hw : w < b ^ u) :
log b (b ^ u * v + w) = u + log b v | case right
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hw : w < b ^ u
⊢ b ^ u * v + b ^ u ≤ b ^ succ (u + log b v) | rw [← mul_succ, ← add_succ, opow_add] | case right
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hw : w < b ^ u
⊢ b ^ u * succ v ≤ b ^ u * b ^ succ (log b v) | fdca12373d77fc9a |
CategoryTheory.PreGaloisCategory.has_decomp_connected_components_aux | Mathlib/CategoryTheory/Galois/Decomposition.lean | private lemma has_decomp_connected_components_aux (F : C ⥤ FintypeCat.{w}) [FiberFunctor F]
(n : ℕ) : ∀ (X : C), n = Nat.card (F.obj X) → ∃ (ι : Type) (f : ι → C)
(g : (i : ι) → (f i) ⟶ X) (_ : IsColimit (Cofan.mk X g)),
(∀ i, IsConnected (f i)) ∧ Finite ι | C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
n : ℕ
⊢ ∀ (X : C), n = Nat.card (F.obj X).carrier → ∃ ι f g x, (∀ (i : ι), IsConnected (f i)) ∧ Finite ι | induction' n using Nat.strongRecOn with n hi | case ind
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
n : ℕ
hi : ∀ m < n, ∀ (X : C), m = Nat.card (F.obj X).carrier → ∃ ι f g x, (∀ (i : ι), IsConnected (f i)) ∧ Finite ι
⊢ ∀ (X : C), n = Nat.card (F.obj X).carrier → ∃ ι f g x, (∀ (i : ι), IsConnected (f i... | 8c6babb797aa0c98 |
Polynomial.mul_scaleRoots_of_noZeroDivisors | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | lemma mul_scaleRoots_of_noZeroDivisors (p q : R[X]) (r : R) [NoZeroDivisors R] :
(p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r | case pos
R : Type u_1
inst✝¹ : CommSemiring R
p q : R[X]
r : R
inst✝ : NoZeroDivisors R
hp : p = 0
⊢ (p * q).scaleRoots r = p.scaleRoots r * q.scaleRoots r | simp [hp] | no goals | 2faef2df62866efd |
CategoryTheory.Functor.isConnected_iff_of_final | Mathlib/CategoryTheory/Limits/IsConnected.lean | theorem isConnected_iff_of_final (F : C ⥤ D) [F.Final] : IsConnected C ↔ IsConnected D | C : Type u
inst✝² : Category.{v, u} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : F.Final
⊢ IsConnected C ↔ IsConnected D | rw [isConnected_iff_colimit_constPUnitFunctor_iso_pUnit.{max v u v₂ u₂} C,
isConnected_iff_colimit_constPUnitFunctor_iso_pUnit.{max v u v₂ u₂} D] | C : Type u
inst✝² : Category.{v, u} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : F.Final
⊢ Nonempty (Limits.colimit (constPUnitFunctor C) ≅ PUnit.{(max v u v₂ u₂) + 1}) ↔
Nonempty (Limits.colimit (constPUnitFunctor D) ≅ PUnit.{(max v u v₂ u₂) + 1}) | 765f67438475d3d8 |
fermatLastTheoremWith'_polynomial | Mathlib/NumberTheory/FLT/Polynomial.lean | theorem fermatLastTheoremWith'_polynomial {n : ℕ} (hn : 3 ≤ n) (chn : (n : k) ≠ 0) :
FermatLastTheoremWith' k[X] n | k : Type u_1
inst✝ : Field k
n : ℕ
hn✝ : 3 ≤ n
chn : ↑n ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
a' b' : k[X]
d : k[X] := gcd a b
heq : d ^ n * (a' ^ n + b' ^ n) = c ^ n
eq_a : a = d * a'
eq_b : b = d * b'
hd : d ≠ 0
hn : 0 < n
hdncn : ∀ (a : k[X]), Multiset.count a (normalizedFactors d) ≤ Multiset.count a (no... | exact hdncn | no goals | 0b2f8f9d4d152b49 |
CategoryTheory.MorphismProperty.IsStableUnderProductsOfShape.mk | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | lemma IsStableUnderProductsOfShape.mk (J : Type*) [W.RespectsIso]
(hW : ∀ (X₁ X₂ : J → C) [HasProduct X₁] [HasProduct X₂]
(f : ∀ j, X₁ j ⟶ X₂ j) (_ : ∀ (j : J), W (f j)),
W (Limits.Pi.map f)) : W.IsStableUnderProductsOfShape J | C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : W.RespectsIso
hW :
∀ (X₁ X₂ : J → C) [inst : HasProduct X₁] [inst_1 : HasProduct X₂] (f : (j : J) → X₁ j ⟶ X₂ j),
(∀ (j : J), W (f j)) → W (Limits.Pi.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cone X₁
c₂ : Cone X₂
hc₁ : IsLimit c₁
hc₂ : IsL... | have : HasLimit X₂ := ⟨c₂, hc₂⟩ | C : Type u
inst✝¹ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝ : W.RespectsIso
hW :
∀ (X₁ X₂ : J → C) [inst : HasProduct X₁] [inst_1 : HasProduct X₂] (f : (j : J) → X₁ j ⟶ X₂ j),
(∀ (j : J), W (f j)) → W (Limits.Pi.map f)
X₁ X₂ : Discrete J ⥤ C
c₁ : Cone X₁
c₂ : Cone X₂
hc₁ : IsLimit c₁
hc₂ : IsL... | e8bcc319c9a1955e |
Stream'.WSeq.exists_of_mem_join | Mathlib/Data/Seq/WSeq.lean | theorem exists_of_mem_join {a : α} : ∀ {S : WSeq (WSeq α)}, a ∈ join S → ∃ s, s ∈ S ∧ a ∈ s | case h2.h3
α : Type u
a : α
ss : WSeq α
h : a ∈ ss
S : WSeq (WSeq α)
s : WSeq α
IH :
∀ (s_1 : WSeq α) (S_1 : WSeq (WSeq α)),
s_1.append S_1.join = s.append S.join → a ∈ s_1.append S_1.join → a ∈ s_1 ∨ ∃ s, s ∈ S_1 ∧ a ∈ s
ej : (s.append S.join).think = (s.append S.join).think
m : a ∈ s.append S.join
⊢ a ∈ s ∨ ∃ s... | apply IH _ _ rfl m | no goals | ec68064248c6eec8 |
integrable_exp_neg_mul_sq | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | theorem integrable_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) :
Integrable fun x : ℝ => exp (-b * x ^ 2) | b : ℝ
hb : 0 < b
⊢ Integrable (fun x => rexp (-b * x ^ 2)) volume | simpa using integrable_rpow_mul_exp_neg_mul_sq hb (by norm_num : (-1 : ℝ) < 0) | no goals | 43cf0bce12faa9c9 |
ProbabilityTheory.integrable_rpow_abs_mul_exp_add_of_integrable_exp_mul | Mathlib/Probability/Moments/IntegrableExpMul.lean | /-- If `exp ((v + t) * X)` and `exp ((v - t) * X)` are integrable
then for nonnegative `p : ℝ` and any `x ∈ [0, |t|)`,
`|X| ^ p * exp (v * X + x * |X|)` is integrable. -/
lemma integrable_rpow_abs_mul_exp_add_of_integrable_exp_mul {x : ℝ}
(h_int_pos : Integrable (fun ω ↦ exp ((v + t) * X ω)) μ)
(h_int_neg : Int... | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω => rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω => rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
⊢ t ≠ 0 | suffices |t| ≠ 0 by simpa | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t v x : ℝ
h_int_pos : Integrable (fun ω => rexp ((v + t) * X ω)) μ
h_int_neg : Integrable (fun ω => rexp ((v - t) * X ω)) μ
h_nonneg : 0 ≤ x
hx : x < |t|
p : ℝ
hp : 0 ≤ p
⊢ |t| ≠ 0 | 6d408f5eb4922de2 |
HomologicalComplex.homotopyCofiber.inlX_desc_f | Mathlib/Algebra/Homology/HomotopyCofiber.lean | @[reassoc (attr := simp)]
lemma inlX_desc_f (i j : ι) (hjk : c.Rel j i) :
inlX φ i j hjk ≫ (desc φ α hα).f j = hα.hom i j | C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
α : G ⟶ K
hα : Homotopy (φ ≫ α) 0
j : ι
hjk : c.Rel j (c.next j)
⊢ inlX φ (c.next j) j hjk ≫ (desc φ α hα).f j = hα.hom (c.... | dsimp [desc] | C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
α : G ⟶ K
hα : Homotopy (φ ≫ α) 0
j : ι
hjk : c.Rel j (c.next j)
⊢ (inlX φ (c.next j) j hjk ≫
if hj : c.Rel j (c.next... | 056c26232b147ff6 |
Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod | Mathlib/LinearAlgebra/FreeModule/Int.lean | /-- Given a submodule `N` in Smith normal form of a free `R`-module, its index as an additive
subgroup is an appropriate power of the cardinality of `R` multiplied by the product of the
indexes of the ideals generated by each basis vector. -/
lemma toAddSubgroup_index_eq_pow_mul_prod [Module R M] {N : Submodule R M}
... | case h.e'_4
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equ... | convert Finset.sum_const_zero with j | case h.e'_2.a
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.e... | 3d7a9cb7455a5c0f |
Nat.card_of_subsingleton | Mathlib/SetTheory/Cardinal/Finite.lean | theorem card_of_subsingleton (a : α) [Subsingleton α] : Nat.card α = 1 | α : Type u_1
a : α
inst✝ : Subsingleton α
this : Fintype α := Fintype.ofSubsingleton a
⊢ Nat.card α = 1 | rw [card_eq_fintype_card, Fintype.card_ofSubsingleton a] | no goals | a1246ff58d2ee7ee |
nodup_permsOfList | Mathlib/Data/Fintype/Perm.lean | theorem nodup_permsOfList : ∀ {l : List α}, l.Nodup → (permsOfList l).Nodup
| [], _ => by simp [permsOfList]
| a :: l, hl => by
have hl' : l.Nodup := hl.of_cons
have hln' : (permsOfList l).Nodup := nodup_permsOfList hl'
have hmeml : ∀ {f : Perm α}, f ∈ permsOfList l → f a = a := fun {f} hf =>
not_... | α : Type u_1
inst✝ : DecidableEq α
a : α
l : List α
hl : (a :: l).Nodup
hl' : l.Nodup
hln' : (permsOfList l).Nodup
hmeml : ∀ {f : Equiv.Perm α}, f ∈ permsOfList l → f a = a
f : Equiv.Perm α
hf₁ : f ∈ permsOfList l
hf₂ : f ∈ flatMap (fun b => List.map (fun f => Equiv.swap a b * f) (permsOfList l)) l
x : α
hx : x ∈ l
hx'... | simp | no goals | 1438bc15e355e5ea |
Module.reflection_mul_reflection_zpow_apply_self | Mathlib/LinearAlgebra/Reflection.lean | /-- A formula for $(r_1 r_2)^m x$, where $m$ is an integer. This is the special case of
`Module.reflection_mul_reflection_zpow_apply` with $z = x$. -/
lemma reflection_mul_reflection_zpow_apply_self (m : ℤ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 | R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
t : optParam R (f y * g x - 2)
ht : autoParam (t = f y * g x - 2) _auto✝
S_eval_t_sub_two :
∀ (k : ℤ),
Polynomial.eval t (S R (k - 2)) = (f y * g x - 2) * Polynomial.eval t (S ... | linear_combination (norm := ring_nf) g x * S_eval_t_sub_two (-m) | no goals | f7647e25720205c5 |
Algebra.TensorProduct.not_isField_of_transcendental | Mathlib/RingTheory/LinearDisjoint.lean | theorem _root_.Algebra.TensorProduct.not_isField_of_transcendental
(A : Type v) [CommRing A] (B : Type w) [CommRing B] [Algebra R A] [Algebra R B]
[Module.Flat R A] [Module.Flat R B] [Algebra.Transcendental R A] [Algebra.Transcendental R B] :
¬IsField (A ⊗[R] B) := fun H ↦ by
letI := H.toField
obtain ⟨a... | R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Module.Flat R A
inst✝² : Module.Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
⊢ False | letI := H.toField | R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Module.Flat R A
inst✝² : Module.Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this : Field (A ⊗[R] B) := H.toField
⊢ Fa... | ae40762afcd6e13c |
summable_indicator_mod_iff_summable | Mathlib/Analysis/SumOverResidueClass.lean | /-- A sequence `f` with values in an additive topological group `R` is summable on the
residue class of `k` mod `m` if and only if `f (m*n + k)` is summable. -/
lemma summable_indicator_mod_iff_summable {R : Type*} [AddCommGroup R] [TopologicalSpace R]
[IsTopologicalAddGroup R] (m : ℕ) [hm : NeZero m] (k : ℕ) (f : ... | R : Type u_1
inst✝² : AddCommGroup R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalAddGroup R
m : ℕ
hm : NeZero m
k : ℕ
f : ℕ → R
g : ℕ → ℕ := fun n => m * n + k
⊢ Summable ({n | ↑n = ↑k ∧ k ≤ n}.indicator f) ↔ Summable fun n => f (m * n + k) | have hg : Function.Injective g := fun m n hmn ↦ by simpa [g, hm.ne] using hmn | R : Type u_1
inst✝² : AddCommGroup R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalAddGroup R
m : ℕ
hm : NeZero m
k : ℕ
f : ℕ → R
g : ℕ → ℕ := fun n => m * n + k
hg : Function.Injective g
⊢ Summable ({n | ↑n = ↑k ∧ k ≤ n}.indicator f) ↔ Summable fun n => f (m * n + k) | ea5dd927c9817359 |
PrimeSpectrum.denseRange_comap_iff_minimalPrimes | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | @[stacks 00FL]
lemma denseRange_comap_iff_minimalPrimes :
DenseRange (comap f) ↔ ∀ I (h : I ∈ minimalPrimes R), ⟨I, h.1.1⟩ ∈ Set.range (comap f) | R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
H : RingHom.ker f ≤ nilradical R
I : Ideal R
hI : Minimal (fun q => q.IsPrime ∧ ⊥ ≤ q) I
⊢ Minimal (fun q => q.IsPrime ∧ RingHom.ker f ≤ q) I | convert hI using 2 with p | case h.e'_3.h.a
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
H : RingHom.ker f ≤ nilradical R
I : Ideal R
hI : Minimal (fun q => q.IsPrime ∧ ⊥ ≤ q) I
p : Ideal R
⊢ p.IsPrime ∧ RingHom.ker f ≤ p ↔ p.IsPrime ∧ ⊥ ≤ p | ee79b196a5fb714f |
Ideal.subset_union_prime' | Mathlib/RingTheory/Ideal/Operations.lean | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
ih :
∀ {s : Finset ι} {a b : ι},
(∀ i ∈ s, (f i).IsPrime) →
s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i
a b i j : ι
hfji : f j ≤ f i
u : Finset ι
hju : j ∉ u
hit : i ∉ insert j u
hn :... | exact ⟨hp.1, hp.2.2⟩ | no goals | fa41770a7d2d9fd9 |
ProbabilityTheory.variance_le_sub_mul_sub | Mathlib/Probability/Variance.lean | /-- **The Bhatia-Davis inequality on variance**
The variance of a random variable `X` satisfying `a ≤ X ≤ b` almost everywhere is at most
`(b - 𝔼 X) * (𝔼 X - a)`. -/
lemma variance_le_sub_mul_sub [IsProbabilityMeasure μ] {a b : ℝ} {X : Ω → ℝ}
(h : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc a b) (hX : AEMeasurable X μ) :
varianc... | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝ : IsProbabilityMeasure μ
a b : ℝ
X : Ω → ℝ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
hX : AEMeasurable X μ
ha : ∀ᵐ (ω : Ω) ∂μ, a ≤ X ω
hb : ∀ᵐ (ω : Ω) ∂μ, X ω ≤ b
hX_int₂ : Integrable (fun ω => -X ω ^ 2) μ
hX_int₁ : Integrable (fun ω => (a + b) * X ω) μ
h0 : 0 ≤ -∫ (x ... | linarith | no goals | d31be09bb6b21fe3 |
CategoryTheory.NonPreadditiveAbelian.σ_comp | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | theorem σ_comp {X Y : C} (f : X ⟶ Y) : σ ≫ f = Limits.prod.map f f ≫ σ | case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
f : X ⟶ Y
g : (CokernelCofork.ofπ σ ⋯).pt ⟶ Y
hg : Cofork.π (CokernelCofork.ofπ σ ⋯) ≫ g = prod.map f f ≫ σ
⊢ σ ≫ f = prod.map f f ≫ σ | suffices hfg : f = g by rw [← hg, Cofork.π_ofπ, hfg] | case mk
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
f : X ⟶ Y
g : (CokernelCofork.ofπ σ ⋯).pt ⟶ Y
hg : Cofork.π (CokernelCofork.ofπ σ ⋯) ≫ g = prod.map f f ≫ σ
⊢ f = g | b7d15c6a05369a0f |
CoxeterSystem.getD_leftInvSeq_mul_wordProd | Mathlib/GroupTheory/Coxeter/Inversion.lean | theorem getD_leftInvSeq_mul_wordProd (ω : List B) (j : ℕ) :
((lis ω).getD j 1) * π ω = π (ω.eraseIdx j) | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
⊢ cs.wordProd (take j ω) * (Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹ *
cs.wordProd (take (j + 1) ω ++ drop (j + 1) ω) =
cs.wordProd (take j ω ++ drop (j + 1) ω) | rw [take_succ] | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
j : ℕ
⊢ cs.wordProd (take j ω) * (Option.map cs.simple ω[j]?).getD 1 * (cs.wordProd (take j ω))⁻¹ *
cs.wordProd (take j ω ++ ω[j]?.toList ++ drop (j + 1) ω) =
cs.wordProd (take j ω ++ drop (j + 1) ω) | 5b8cc6ccfadd733b |
HomologicalComplex.mapBifunctor₂₃.d₁_eq | Mathlib/Algebra/Homology/BifunctorAssociator.lean | lemma d₁_eq {i₁ i₁' : ι₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) :
d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j =
(ComplexShape.ε₁ c₁ c₂₃ c₄ (i₁, ComplexShape.π c₂ c₃ c₂₃ (i₂, i₃))) •
((F.map (K₁.d i₁ i₁'))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫
ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ _ i₂ i₃ j | C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{u_15, u_1} C₁
inst✝²¹ : Category.{u_17, u_2} C₂
inst✝²⁰ : Category.{u_16, u_5} C₃
inst✝¹⁹ : Category.{u_13, u_6} C₄
inst✝¹⁸ : Category.{u_14, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroM... | rfl | no goals | 717697c3dc2065d9 |
Subgroup.exists_pow_mem_of_index_ne_zero | Mathlib/GroupTheory/Index.lean | @[to_additive]
lemma exists_pow_mem_of_index_ne_zero (h : H.index ≠ 0) (a : G) :
∃ n, 0 < n ∧ n ≤ H.index ∧ a ^ n ∈ H | G : Type u_1
inst✝ : Group G
H : Subgroup G
h : H.index ≠ 0
a : G
hc : ∀ (x x_1 : ℕ), ¬(x ≠ x_1 ∧ x ≤ H.index ∧ x_1 ≤ H.index ∧ ↑(a ^ x_1) = ↑(a ^ x))
f : ↑(Set.Icc 0 H.index) → G ⧸ H := fun n => ↑(a ^ ↑n)
hf : Function.Injective f
this : Finite (G ⧸ H)
⊢ False | have hcard := Finite.card_le_of_injective f hf | G : Type u_1
inst✝ : Group G
H : Subgroup G
h : H.index ≠ 0
a : G
hc : ∀ (x x_1 : ℕ), ¬(x ≠ x_1 ∧ x ≤ H.index ∧ x_1 ≤ H.index ∧ ↑(a ^ x_1) = ↑(a ^ x))
f : ↑(Set.Icc 0 H.index) → G ⧸ H := fun n => ↑(a ^ ↑n)
hf : Function.Injective f
this : Finite (G ⧸ H)
hcard : Nat.card ↑(Set.Icc 0 H.index) ≤ Nat.card (G ⧸ H)
⊢ False | e678927c1f4a5913 |
SzemerediRegularity.sum_density_div_card_le_density_add_eps | Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | theorem sum_density_div_card_le_density_add_eps [Nonempty α]
(hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5)
(hε₁ : ε ≤ 1) {hU : U ∈ P.parts} {hV : V ∈ P.parts} {A B : Finset (Finset α)}
(hA : A ⊆ (chunk hP G ε hU).parts) (hB : B ⊆ (chunk hP G ε hV).parts) :
(∑ ab ∈ A.prod... | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
A B : F... | exact mod_cast G.edgeDensity_le_one _ _ | no goals | 15003bb13a21ffca |
Nat.testBit_bit_succ | Mathlib/Data/Nat/Bits.lean | lemma testBit_bit_succ (m b n) : testBit (bit b n) (succ m) = testBit n m | m : ℕ
b : Bool
n : ℕ
⊢ (bit b n).testBit m.succ = n.testBit m | have : bodd (((bit b n) >>> 1) >>> m) = bodd (n >>> m) := by
simp only [shiftRight_eq_div_pow]
simp [← div2_val, div2_bit] | m : ℕ
b : Bool
n : ℕ
this : (bit b n >>> 1 >>> m).bodd = (n >>> m).bodd
⊢ (bit b n).testBit m.succ = n.testBit m | de09e232ea3ace89 |
FractionalIdeal.div_spanSingleton | Mathlib/RingTheory/FractionalIdeal/Operations.lean | theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J | case neg.a
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 0
h_spand : spanSingleton R₁⁰ d ≠ 0
⊢ J / spanSingleton R₁⁰ d ≤ 1 / spanSingleton R₁⁰ d * J | intro x hx | case neg.a
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 0
h_spand : spanSingleton R₁⁰ d ≠ 0
x : K
hx : x ∈ (fun a => ↑a) (J / spanSingleton R₁⁰ d)
⊢ x ∈ (fun a => ↑a) (1 / spanSingleton R... | 3a4f3cf63940a3cf |
FormalMultilinearSeries.radius_right_inv_pos_of_radius_pos_aux1 | Mathlib/Analysis/Analytic/Inverse.lean | theorem radius_right_inv_pos_of_radius_pos_aux1 (n : ℕ) (p : ℕ → ℝ) (hp : ∀ k, 0 ≤ p k) {r a : ℝ}
(hr : 0 ≤ r) (ha : 0 ≤ a) :
∑ k ∈ Ico 2 (n + 1),
a ^ k *
∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
r ^ c.length * ∏ j, p (c.blocksFun j) ≤
∑ j ∈ I... | case a
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
j : ℕ
a✝ : j ∈ Ico 2 (n + 1)
⊢ ∑ s ∈ Fintype.piFinset fun x => Ico 1 n, ∏ j : Fin j, r * (a ^ s j * p (s j)) =
r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j | simp only [← @MultilinearMap.mkPiAlgebra_apply ℝ (Fin j) _ ℝ] | case a
n : ℕ
p : ℕ → ℝ
hp : ∀ (k : ℕ), 0 ≤ p k
r a : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
j : ℕ
a✝ : j ∈ Ico 2 (n + 1)
⊢ (∑ x ∈ Fintype.piFinset fun x => Ico 1 n, (MultilinearMap.mkPiAlgebra ℝ (Fin j) ℝ) fun j => r * (a ^ x j * p (x j))) =
r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j | 5c2dd9dd9e093d01 |
Polynomial.exists_primitive_lcm_of_isPrimitive | Mathlib/RingTheory/Polynomial/Content.lean | theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s | case pos
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
h : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h
rprim : IsUnit (r.coeff 0)
pr : p ∣ r
qr : q ∣ r
con : ∃ n s, s.natDegree = n ∧... | rw [eq_C_of_natDegree_le_zero (le_trans hs sC), ← dvd_content_iff_C_dvd] at rs | case pos
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
h : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h
rprim : IsUnit (r.coeff 0)
pr : p ∣ r
qr : q ∣ r
con : ∃ n s, s.natDegree = n ∧... | 806b283d42e7cf3d |
BoxIntegral.Integrable.tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity | Mathlib/Analysis/BoxIntegral/Basic.lean | theorem tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity (h : Integrable I l f vol) :
Tendsto (fun π : TaggedPrepartition I × TaggedPrepartition I =>
(integralSum f vol π.1, integralSum f vol π.2))
((l.toFilter I ×ˢ l.toFilter I) ⊓ 𝓟 {π | π.1.iUnion = π.2.iUnion}) (𝓤 F) | ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
h : Integrable I l f vol
ε : ℝ
ε0 : 0 < ε / 2
⊢ ∃ ia,
(∀ (c : ℝ≥0), l.RCond (ia c)... | use h.convergenceR (ε / 2), h.convergenceR_cond (ε / 2) | case right
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
h : Integrable I l f vol
ε : ℝ
ε0 : 0 < ε / 2
⊢ ∀
x ∈
{π | ∃ c, l... | 08a8f06d71e014c5 |
Asymptotics.isBigOWith_iff_exists_eq_mul | Mathlib/Analysis/Asymptotics/Lemmas.lean | theorem isBigOWith_iff_exists_eq_mul (hc : 0 ≤ c) :
IsBigOWith c l u v ↔ ∃ φ : α → 𝕜, (∀ᶠ x in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v | case mpr
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
c : ℝ
l : Filter α
u v : α → 𝕜
hc : 0 ≤ c
⊢ (∃ φ, (∀ᶠ (x : α) in l, ‖φ x‖ ≤ c) ∧ u =ᶠ[l] φ * v) → IsBigOWith c l u v | rintro ⟨φ, hφ, h⟩ | case mpr.intro.intro
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
c : ℝ
l : Filter α
u v : α → 𝕜
hc : 0 ≤ c
φ : α → 𝕜
hφ : ∀ᶠ (x : α) in l, ‖φ x‖ ≤ c
h : u =ᶠ[l] φ * v
⊢ IsBigOWith c l u v | c2c8aee6ef1774de |
Polynomial.Gal.mul_splits_in_splittingField_of_mul | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | theorem mul_splits_in_splittingField_of_mul {p₁ q₁ p₂ q₂ : F[X]} (hq₁ : q₁ ≠ 0) (hq₂ : q₂ ≠ 0)
(h₁ : p₁.Splits (algebraMap F q₁.SplittingField))
(h₂ : p₂.Splits (algebraMap F q₂.SplittingField)) :
(p₁ * p₂).Splits (algebraMap F (q₁ * q₂).SplittingField) | case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : Splits (algebraMap F q₁.SplittingField) p₁
h₂ : Splits (algebraMap F q₂.SplittingField) p₂
⊢ Splits ((↑(SplittingField.lift q₁ ⋯)).comp (algebraMap F q₁.SplittingField)) p₁ | exact splits_comp_of_splits _ _ h₁ | no goals | 12f0bbd6ecef4d4b |
LinearMap.commute_pow_left_of_commute | Mathlib/Algebra/Module/LinearMap/End.lean | theorem commute_pow_left_of_commute
[Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂}
{f : M →ₛₗ[σ₁₂] M₂} {g : Module.End R M} {g₂ : Module.End R₂ M₂}
(h : g₂.comp f = f.comp g) (k : ℕ) : (g₂ ^ k).comp f = f.comp (g ^ k) | case zero
R : Type u_1
R₂ : Type u_2
M : Type u_4
M₂ : Type u_6
inst✝⁵ : Semiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : Semiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
σ₁₂ : R →+* R₂
f : M →ₛₗ[σ₁₂] M₂
g : Module.End R M
g₂ : Module.End R₂ M₂
h : comp g₂ f = f.comp g
⊢ comp (g₂ ^ 0) f = f.co... | simp only [pow_zero, one_eq_id, id_comp, comp_id] | no goals | b5769a82a32d21d9 |
AnalyticAt.apply_eq_zero_of_order_toNat_ne_zero | Mathlib/Analysis/Analytic/Order.lean | /-- An analytic function vanishes at a point if its order is nonzero when converted to ℕ. -/
lemma apply_eq_zero_of_order_toNat_ne_zero (hf : AnalyticAt 𝕜 f z₀) :
hf.order.toNat ≠ 0 → f z₀ = 0 | 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
⊢ f z₀ = 0 → ¬hf.order = ⊤ → f z₀ = 0 | tauto | no goals | 9dcb2d5157a801f6 |
Finset.exists_ne_one_of_prod_ne_one | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem exists_ne_one_of_prod_ne_one (h : ∏ x ∈ s, f x ≠ 1) : ∃ a ∈ s, f a ≠ 1 | α : Type u_3
β : Type u_4
s : Finset α
f : α → β
inst✝ : CommMonoid β
h : ∏ x ∈ s, f x ≠ 1
⊢ ∃ a ∈ s, f a ≠ 1 | rw [← prod_filter_ne_one] at h | α : Type u_3
β : Type u_4
s : Finset α
f : α → β
inst✝ : CommMonoid β
h : ∏ x ∈ filter (fun x => f x ≠ 1) s, f x ≠ 1
⊢ ∃ a ∈ s, f a ≠ 1 | 305c9d27d8422aed |
NonUnitalNonAssocRing.ext | Mathlib/Algebra/Ring/Ext.lean | theorem ext ⦃inst₁ inst₂ : NonUnitalNonAssocRing R⦄
(h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_mul : local_hMul[R, inst₁] = local_hMul[R, inst₂]) :
inst₁ = inst₂ | case mk.mk.mk.mk.e_toAddCommGroup
R : Type u
toAddCommGroup✝¹ : AddCommGroup R
mul✝¹ : R → R → R
left_distrib✝¹ : ∀ (a b c : R), a * (b + c) = a * b + a * c
right_distrib✝¹ : ∀ (a b c : R), (a + b) * c = a * c + b * c
zero_mul✝¹ : ∀ (a : R), 0 * a = 0
mul_zero✝¹ : ∀ (a : R), a * 0 = 0
toAddCommGroup✝ : AddCommGroup R
m... | (ext : 1; assumption) | no goals | c47255c0f38a615d |
Subgroup.mem_iSup_of_directed | Mathlib/Algebra/Group/Subgroup/Lattice.lean | theorem mem_iSup_of_directed {ι} [hι : Nonempty ι] {K : ι → Subgroup G} (hK : Directed (· ≤ ·) K)
{x : G} : x ∈ (iSup K : Subgroup G) ↔ ∃ i, x ∈ K i | case refine_2.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
ι : Sort u_2
hι : Nonempty ι
K : ι → Subgroup G
hK : Directed (fun x1 x2 => x1 ≤ x2) K
x✝ : G
hx : x✝ ∈ closure (⋃ i, ↑(K i))
x y : G
hx✝ : x ∈ closure (⋃ i, ↑(K i))
hy✝ : y ∈ closure (⋃ i, ↑(K i))
i : ι
hi : x ∈ K i
j : ι
hj : y ∈ K j
k : ι
hki : K i ≤... | exact ⟨k, mul_mem (hki hi) (hkj hj)⟩ | no goals | 313a31a341bc01d2 |
CategoryTheory.Comonad.ComonadicityInternal.comparisonAdjunction_unit_app | Mathlib/CategoryTheory/Monad/Comonadicity.lean | theorem comparisonAdjunction_unit_app
[∀ A : adj.toComonad.Coalgebra, HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))] (B : C) :
(comparisonAdjunction adj).unit.app B = limit.lift _ (unitFork adj B) | case h
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : ∀ (A : adj.toComonad.Coalgebra), HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))
B : C
⊢ (comparisonAdjunction adj).unit.app B ≫
equalizer.ι (G.map ((comparison adj).obj B).a) (adj.... | change
equalizer.lift ((adj.homEquiv B _) (𝟙 _)) _ ≫ equalizer.ι _ _ =
equalizer.lift _ _ ≫ equalizer.ι _ _ | case h
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : ∀ (A : adj.toComonad.Coalgebra), HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))
B : C
⊢ equalizer.lift ((adj.homEquiv B (F.obj B)) (𝟙 (F.obj B))) ⋯ ≫
equalizer.ι (G.map ((comparis... | 56285f6d00a0cc25 |
CategoryTheory.Grothendieck.eqToHom_eq | Mathlib/CategoryTheory/Grothendieck.lean | lemma eqToHom_eq {X Y : Grothendieck F} (hF : X = Y) :
eqToHom hF = { base := eqToHom (by subst hF; rfl), fiber := eqToHom (by subst hF; simp) } | C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₁
inst✝ : Category.{v₁, u₁} D
F : C ⥤ Cat
X : Grothendieck F
⊢ (F.map (eqToHom ⋯)).obj X.fiber = X.fiber | simp | no goals | 59bdc65a2b8ed05d |
xInTermsOfW_vars_aux | Mathlib/RingTheory/WittVector/WittPolynomial.lean | theorem xInTermsOfW_vars_aux (n : ℕ) :
n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1) | p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1) | induction n using Nat.strongRecOn with | ind n ih => ?_ | case ind
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
ih : ∀ m < n, m ∈ (xInTermsOfW p ℚ m).vars ∧ (xInTermsOfW p ℚ m).vars ⊆ range (m + 1)
⊢ n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1) | 6a6d3bb57935e668 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.mem_of_insertRupUnits | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/Lemmas.lean | theorem mem_of_insertRupUnits {n : Nat} (f : DefaultFormula n) (units : CNF.Clause (PosFin n))
(c : DefaultClause n) :
c ∈ toList (insertRupUnits f units).1 → c ∈ units.map Clause.unit ∨ c ∈ toList f | case isFalse.inr
n : Nat
f : DefaultFormula n
units : CNF.Clause (PosFin n)
c : DefaultClause n
h :
some c ∈ f.clauses.toList ∨
(∃ a,
(a, false) ∈ (List.foldl insertUnit (f.rupUnits, f.assignments, false) units).fst.toList ∧
Std.Tactic.BVDecide.LRAT.Internal.DefaultClause.unit (a, false) = c ∨... | exact Or.inr unit_in_units | no goals | d2213faecfa0fcb1 |
MeasureTheory.norm_integral_sub_setIntegral_le | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem norm_integral_sub_setIntegral_le [IsFiniteMeasure μ] {C : ℝ}
(hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C) {s : Set X} (hs : MeasurableSet s) (hf1 : Integrable f μ) :
‖∫ (x : X), f x ∂μ - ∫ x in s, f x ∂μ‖ ≤ (μ sᶜ).toReal * C | X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : X → E
μ : Measure X
inst✝ : IsFiniteMeasure μ
C : ℝ
hf : ∀ᵐ (x : X) ∂μ, ‖f x‖ ≤ C
s : Set X
hs : MeasurableSet s
hf1 : Integrable f μ
h0 : ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in sᶜ, f x ∂μ
h1 :... | exact le_trans (norm_integral_le_integral_norm f) h1 | no goals | 4ebc5dd21284b746 |
Filter.Tendsto.const_mul_atTop' | Mathlib/Order/Filter/AtTopBot/Archimedean.lean | theorem Tendsto.const_mul_atTop' (hr : 0 < r) (hf : Tendsto f l atTop) :
Tendsto (fun x => r * f x) l atTop | α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝¹ : LinearOrderedSemiring R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
⊢ Tendsto (fun x => r * f x) l atTop | refine tendsto_atTop.2 fun b => ?_ | α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝¹ : LinearOrderedSemiring R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
b : R
⊢ ∀ᶠ (a : α) in l, b ≤ r * f a | ea722137e0e97e5b |
ZMod.neg_eq_self_mod_two | Mathlib/Data/ZMod/Basic.lean | theorem neg_eq_self_mod_two (a : ZMod 2) : -a = a | case «1».h
⊢ ↑(-1) = 1 | rfl | no goals | 98e52898ae8fbe55 |
PadicSeq.norm_mul | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem norm_mul (f g : PadicSeq p) : (f * g).norm = f.norm * g.norm | p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
⊢ (if hf : f * g ≈ 0 then 0 else padicNorm p (↑(f * g) (stationaryPoint hf))) =
(if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) *
if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf)) | have hfg := mul_not_equiv_zero hf hg | p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hf : ¬f ≈ 0
hg : ¬g ≈ 0
hfg : ¬f * g ≈ 0
⊢ (if hf : f * g ≈ 0 then 0 else padicNorm p (↑(f * g) (stationaryPoint hf))) =
(if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf))) *
if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf)) | 5d67169ade4a773a |
μ_limsup_le_one | Mathlib/Analysis/Normed/Ring/SmoothingSeminorm.lean | theorem μ_limsup_le_one {s : ℕ → ℕ} (hs_le : ∀ n : ℕ, s n ≤ n) {x : R} {ψ : ℕ → ℕ}
(hψ_lim : Tendsto ((fun n : ℕ => ↑(s n) / (n : ℝ)) ∘ ψ) atTop (𝓝 0)) :
limsup (fun n : ℕ => μ x ^ ((s (ψ n) : ℝ) * (1 / (ψ n : ℝ)))) atTop ≤ 1 | R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
s : ℕ → ℕ
hs_le : ∀ (n : ℕ), s n ≤ n
x : R
ψ : ℕ → ℕ
hψ_lim : Tendsto ((fun n => ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)
c : ℝ
hc_bd : ∀ (x_1 : ℝ) (x_2 : ℕ), (∀ (b : ℕ), x_2 ≤ b → μ x ^ (↑(s (ψ b)) * (1 / ↑(ψ b))) ≤ x_1) → c ≤ x_1
hμx : ¬μ x < 1
hμ_lim : ∀ (U : Set ℝ), 1 ∈ U → ... | simp only [Set.mem_Ioo, zero_lt_one, lt_add_iff_pos_right, hε, and_self] | no goals | 56e579c9d7dd8cf5 |
BitVec.shiftRight_sub_one_eq_shiftConcat | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem shiftRight_sub_one_eq_shiftConcat (n : BitVec w) (hwn : 0 < wn) :
n >>> (wn - 1) = (n >>> wn).shiftConcat (n.getLsbD (wn - 1)) | case pred.isFalse.e_i
w wn : Nat
n : BitVec w
hwn : 0 < wn
i : Nat
h : i < w
h✝ : ¬i = 0
⊢ wn - 1 + i = wn + (i - 1) | omega | no goals | 50919c461ed9204f |
Nat.finMulAntidiag_one | Mathlib/Algebra/Order/Antidiag/Nat.lean | theorem finMulAntidiag_one {d : ℕ} :
finMulAntidiag d 1 = {fun _ => 1} | case h.mpr
d : ℕ
⊢ ∏ i : Fin d, (fun x => 1) i = 1 ∧ 1 ≠ 0 | simp only [prod_const_one, implies_true, ne_eq, one_ne_zero, not_false_eq_true, and_self] | no goals | 7d8f41c0f8c962c2 |
Real.borel_eq_generateFrom_Iio_rat | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | theorem borel_eq_generateFrom_Iio_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iio (a : ℝ)}) | ⊢ ∀ t ∈ range Iio, MeasurableSet t | rintro _ ⟨a, rfl⟩ | case intro
a : ℝ
⊢ MeasurableSet (Iio a) | dccc6ba8ada3f3ac |
ProbabilityTheory.measure_ge_le_exp_mul_mgf | Mathlib/Probability/Moments/Basic.lean | theorem measure_ge_le_exp_mul_mgf [IsFiniteMeasure μ] (ε : ℝ) (ht : 0 ≤ t)
(h_int : Integrable (fun ω => exp (t * X ω)) μ) :
(μ {ω | ε ≤ X ω}).toReal ≤ exp (-t * ε) * mgf X μ t | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
ε : ℝ
ht : 0 ≤ t
h_int : Integrable (fun ω => rexp (t * X ω)) μ
ht_pos : 0 < t
⊢ (rexp (t * ε))⁻¹ * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ = rexp (-t * ε) * mgf X μ t | rw [neg_mul, exp_neg] | Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ : Measure Ω
t : ℝ
inst✝ : IsFiniteMeasure μ
ε : ℝ
ht : 0 ≤ t
h_int : Integrable (fun ω => rexp (t * X ω)) μ
ht_pos : 0 < t
⊢ (rexp (t * ε))⁻¹ * ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ = (rexp (t * ε))⁻¹ * mgf X μ t | d0b4656bb63daebb |
contDiff_iff_ftaylorSeries | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem contDiff_iff_ftaylorSeries {n : ℕ∞} :
ContDiff 𝕜 n f ↔ HasFTaylorSeriesUpTo n f (ftaylorSeries 𝕜 f) | case mp
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
f : E → F
n : ℕ∞
⊢ ContDiff 𝕜 (↑n) f → HasFTaylorSeriesUpTo (↑n) f (ftaylorSeries 𝕜 f) | rw [← contDiffOn_univ, ← hasFTaylorSeriesUpToOn_univ_iff, ← ftaylorSeriesWithin_univ] | case mp
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
f : E → F
n : ℕ∞
⊢ ContDiffOn 𝕜 (↑n) f univ → HasFTaylorSeriesUpToOn (↑n) f (ftaylorSeriesWithin 𝕜 f univ) univ | 0b68cf261d89baaf |
Localization.mem_range_mapToFractionRing_iff | Mathlib/RingTheory/Localization/AsSubring.lean | theorem mem_range_mapToFractionRing_iff (B : Type*) [CommRing B] [Algebra A B] [IsLocalization S B]
(hS : S ≤ A⁰) (x : K) :
x ∈ (mapToFractionRing K S B hS).range ↔
∃ (a s : A) (hs : s ∈ S), x = IsLocalization.mk' K a ⟨s, hS hs⟩ :=
⟨by
rintro ⟨x, rfl⟩
obtain ⟨a, s, rfl⟩ := IsLocalization.mk'_sur... | case h
A : Type u_1
K : Type u_2
inst✝⁶ : CommRing A
S : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsFractionRing A K
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
hS : S ≤ A⁰
a : A
s : ↥S
⊢ (mapToFractionRing K S B hS).toRingHom (IsLocalization.mk' B a s) = IsLoca... | apply IsLocalization.lift_mk' | no goals | 68d824c2f8020338 |
ProbabilityTheory.IndepFun.integral_mul_of_integrable | Mathlib/Probability/Integration.lean | theorem IndepFun.integral_mul_of_integrable (hXY : IndepFun X Y μ) (hX : Integrable X μ)
(hY : Integrable Y μ) : integral μ (X * Y) = integral μ X * integral μ Y | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : Integrable X μ
hY : Integrable Y μ
pos : ℝ → ℝ := fun x => x ⊔ 0
neg : ℝ → ℝ := fun x => -x ⊔ 0
posm : Measurable pos
negm : Measurable neg
⊢ integral μ (X * Y) = integral μ X * integral μ Y | let Xp := pos ∘ X | Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X Y : Ω → ℝ
hXY : IndepFun X Y μ
hX : Integrable X μ
hY : Integrable Y μ
pos : ℝ → ℝ := fun x => x ⊔ 0
neg : ℝ → ℝ := fun x => -x ⊔ 0
posm : Measurable pos
negm : Measurable neg
Xp : Ω → ℝ := pos ∘ X
⊢ integral μ (X * Y) = integral μ X * integral μ Y | 1836e7e522e593a2 |
exteriorPower.ιMulti_span | Mathlib/LinearAlgebra/ExteriorPower/Basic.lean | /-- The image of `exteriorPower.ιMulti` spans `⋀[R]^n M`. -/
lemma ιMulti_span :
Submodule.span R (Set.range (ιMulti R n)) = (⊤ : Submodule R (⋀[R]^n M)) | case a
R : Type u
inst✝² : CommRing R
n : ℕ
M : Type u_1
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ Submodule.span R (Set.range fun a => (ExteriorAlgebra.ιMulti R n) a) = ⋀[R]^n M | exact ExteriorAlgebra.ιMulti_span_fixedDegree R n | no goals | ad0fa8e0b2d37c81 |
Array.foldr_induction | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem foldr_induction
{as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive as.size init) {f : α → β → β}
(hf : ∀ i : Fin as.size, ∀ b, motive (i.1 + 1) b → motive i.1 (f as[i] b)) :
motive 0 (as.foldr f init) | α : Type u_1
β : Type u_2
as : Array α
motive : Nat → β → Prop
init : β
h0 : motive as.size init
f : α → β → β
hf : ∀ (i : Fin as.size) (b : β), motive (↑i + 1) b → motive (↑i) (f as[i] b)
⊢ motive 0 (if 0 < as.size then foldrM.fold f as 0 as.size ⋯ init else init).run | split | case isTrue
α : Type u_1
β : Type u_2
as : Array α
motive : Nat → β → Prop
init : β
h0 : motive as.size init
f : α → β → β
hf : ∀ (i : Fin as.size) (b : β), motive (↑i + 1) b → motive (↑i) (f as[i] b)
h✝ : 0 < as.size
⊢ motive 0 (foldrM.fold f as 0 as.size ⋯ init).run
case isFalse
α : Type u_1
β : Type u_2
as : Array ... | 36c52e13ffc37b61 |
Subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero | Mathlib/RingTheory/Algebraic/Basic.lean | theorem Subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero {x : A} {p : K[X]}
(aeval_eq : aeval x p = 0) (coeff_zero_ne : p.coeff 0 ≠ 0) : (x⁻¹ : L) ∈ A | K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
A : Subalgebra K L
x : ↥A
p : K[X]
aeval_eq : (aeval x) p = 0
coeff_zero_ne : p.coeff 0 ≠ 0
this : (↑x)⁻¹ = (-p.coeff 0)⁻¹ • ↑((aeval x) p.divX)
⊢ (-p.coeff 0)⁻¹ • ↑((aeval x) p.divX) ∈ A | exact A.smul_mem (aeval x _).2 _ | no goals | 440f3b608db59a1e |
Finset.strictMono_iff_forall_lt_cons | Mathlib/Data/Finset/Interval.lean | /-- A function `f` from `Finset α` is strictly monotone if and only if `f s < f (cons a s ha)` for
all `s` and `a ∉ s`. -/
lemma strictMono_iff_forall_lt_cons : StrictMono f ↔ ∀ s ⦃a⦄ ha, f s < f (cons a s ha) | α : Type u_1
β : Type u_2
inst✝ : Preorder β
f : Finset α → β
⊢ StrictMono f ↔ ∀ (s : Finset α) ⦃a : α⦄ (ha : a ∉ s), f s < f (cons a s ha) | simp [strictMono_iff_forall_covBy, covBy_iff_exists_cons] | no goals | e995ec8ad786815e |
Num.castNum_eq_bitwise | Mathlib/Data/Num/Lemmas.lean | theorem castNum_eq_bitwise {f : Num → Num → Num} {g : Bool → Bool → Bool}
(p : PosNum → PosNum → Num)
(gff : g false false = false) (f00 : f 0 0 = 0)
(f0n : ∀ n, f 0 (pos n) = cond (g false true) (pos n) 0)
(fn0 : ∀ n, f (pos n) 0 = cond (g true false) (pos n) 0)
(fnn : ∀ m n, f (pos m) (pos n) = p ... | case pos.pos.bit1.bit1
f : Num → Num → Num
g : Bool → Bool → Bool
p : PosNum → PosNum → Num
gff : g false false = false
f00 : f 0 0 = 0
f0n : ∀ (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0
fn0 : ∀ (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0
fnn : ∀ (m n : PosNum), f (pos m) (pos n) ... | all_goals
rw [← show ∀ n : PosNum, ↑(p m n) = Nat.bitwise g ↑m ↑n from IH]
rw [← bit_to_nat, pbb] | no goals | d446b0d6506ab74a |
preconnectedSpace_iff_connectedComponent | Mathlib/Topology/Connected/Basic.lean | theorem preconnectedSpace_iff_connectedComponent :
PreconnectedSpace α ↔ ∀ x : α, connectedComponent x = univ | case mpr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
⊢ PreconnectedSpace α | rcases isEmpty_or_nonempty α with hα | hα | case mpr.inl
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
hα : IsEmpty α
⊢ PreconnectedSpace α
case mpr.inr
α : Type u
inst✝ : TopologicalSpace α
h : ∀ (x : α), connectedComponent x = univ
hα : Nonempty α
⊢ PreconnectedSpace α | e6948b2a1277dd55 |
toMul_multiset_sum | Mathlib/Algebra/BigOperators/Group/Finset/Defs.lean | theorem toMul_multiset_sum (s : Multiset (Additive α)) : s.sum.toMul = (s.map toMul).prod | α : Type u_3
inst✝ : CommMonoid α
s : Multiset (Additive α)
⊢ toMul s.sum = (Multiset.map (⇑toMul) s).prod | simp [toMul, ofMul] | α : Type u_3
inst✝ : CommMonoid α
s : Multiset (Additive α)
⊢ s.sum = s.prod | 810c6fd03f48374d |
CategoryTheory.Triangulated.TStructure.le_monotone | Mathlib/CategoryTheory/Triangulated/TStructure/Basic.lean | lemma le_monotone : Monotone t.le | C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
H : ℕ → Prop := fun a => ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
H_zero : H 0
H_one : H 1
H_add : ∀ (a b c : ℕ), a + b = ... | intro a | C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
H : ℕ → Prop := fun a => ∀ (n : ℤ), t.le n ≤ t.le (n + ↑a)
H_zero : H 0
H_one : H 1
H_add : ∀ (a b c : ℕ), a + b = ... | 37a0503483d0a558 |
MDifferentiableWithinAt.clm_apply | Mathlib/Geometry/Manifold/MFDeriv/NormedSpace.lean | theorem MDifferentiableWithinAt.clm_apply {g : M → F₁ →L[𝕜] F₂} {f : M → F₁} {s : Set M} {x : M}
(hg : MDifferentiableWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) g s x)
(hf : MDifferentiableWithinAt I 𝓘(𝕜, F₁) f s x) :
MDifferentiableWithinAt I 𝓘(𝕜, F₂) (fun x => g x (f x)) s x :=
DifferentiableWithinAt.comp_mdi... | 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpac... | apply (Differentiable.differentiableAt _).differentiableWithinAt | 𝕜 : Type u_1
inst✝⁹ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
F₁ : Type u_14
inst✝³ : NormedAddCommGroup F₁
inst✝² : NormedSpac... | a547ab9a62b5927c |
CoxeterSystem.not_isReduced_alternatingWord | Mathlib/GroupTheory/Coxeter/Length.lean | theorem not_isReduced_alternatingWord (i i' : B) {m : ℕ} (hM : M i i' ≠ 0) (hm : m > M i i') :
¬cs.IsReduced (alternatingWord i i' m) | case refl
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
hM : M.M i i' ≠ 0
⊢ ¬cs.IsReduced (alternatingWord i i' (M.M i i').succ) | suffices h : ℓ (π (alternatingWord i i' (M i i' + 1))) < M i i' + 1 by
unfold IsReduced
rw [Nat.succ_eq_add_one, length_alternatingWord]
omega | case refl
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
hM : M.M i i' ≠ 0
⊢ cs.length (cs.wordProd (alternatingWord i i' (M.M i i' + 1))) < M.M i i' + 1 | f64124f674aa1270 |
lipschitzGroup.conjAct_smul_ι_mem_range_ι | Mathlib/LinearAlgebra/CliffordAlgebra/SpinGroup.lean | theorem conjAct_smul_ι_mem_range_ι {x : (CliffordAlgebra Q)ˣ} (hx : x ∈ lipschitzGroup Q)
[Invertible (2 : R)] (m : M) :
ConjAct.toConjAct x • ι Q m ∈ LinearMap.range (ι Q) | R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx : x ∈ lipschitzGroup Q
inst✝ : Invertible 2
m : M
⊢ ConjAct.toConjAct x • (ι Q) m ∈ LinearMap.range (ι Q) | unfold lipschitzGroup at hx | R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x : (CliffordAlgebra Q)ˣ
hx : x ∈ Subgroup.closure (Units.val ⁻¹' Set.range ⇑(ι Q))
inst✝ : Invertible 2
m : M
⊢ ConjAct.toConjAct x • (ι Q) m ∈ LinearMap.range (ι Q) | 5df234339fc72163 |
Compactum.cl_cl | Mathlib/Topology/Category/Compactum.lean | theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A | case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
⊢ X.st... | let C1 := insert AA C0 | case intro.intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : s... | 1c30fb7238f2120d |
IsSemiprimaryRing.induction | Mathlib/RingTheory/HopkinsLevitzki.lean | theorem induction
{P : ∀ (M : Type u) [AddCommGroup M] [Module R₀ M] [Module R M], Prop}
(h0 : ∀ (M) [AddCommGroup M] [Module R₀ M] [Module R M] [IsScalarTower R₀ R M]
[IsSemisimpleModule R M], Module.IsTorsionBySet R M (Ring.jacobson R) → P M)
(h1 : ∀ (M) [AddCommGroup M] [Module R₀ M] [Module R M] [... | case succ.zero
R₀ : Type u_1
R : Type u_2
inst✝⁷ : Ring R₀
inst✝⁶ : Ring R
inst✝⁵ : Module R₀ R
inst✝⁴ : IsSemiprimaryRing R
P : (M : Type u) → [inst : AddCommGroup M] → [inst_1 : Module R₀ M] → [inst : Module R M] → Prop
ss : IsSemisimpleRing (R ⧸ Ring.jacobson R)
Jac : Ideal R := Ring.jacobson R
h0 :
∀ (M : Type u)... | exact this hn | no goals | 18de1097a1dd67d8 |
PadicSeq.norm_nonarchimedean | Mathlib/NumberTheory/Padics/PadicNumbers.lean | theorem norm_nonarchimedean (f g : PadicSeq p) : (f + g).norm ≤ max f.norm g.norm | p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : g ≈ 0
hfg' : f + g ≈ f
⊢ (f + g).norm ≤ f.norm ⊔ g.norm | have hcfg : (f + g).norm = f.norm := norm_equiv hfg' | p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : g ≈ 0
hfg' : f + g ≈ f
hcfg : (f + g).norm = f.norm
⊢ (f + g).norm ≤ f.norm ⊔ g.norm | 4a919ac1cede5d3f |
Ideal.subset_union_prime' | Mathlib/RingTheory/Ideal/Operations.lean | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | case pos.h.inr.inl
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : t.card = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : (f i).IsPrime ∧ ∀ x ∈ t, (f x).IsPrime
Ht : ¬∃ j ∈ t, f j ≤ f i
Ha : f a ≤ f i
h' : ↑I ⊆ ↑(f i) ∪ ↑(f b) ∪ ⋃ ... | exact Or.inl ih | no goals | fa41770a7d2d9fd9 |
Finset.memberSubfamily_union_nonMemberSubfamily | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | theorem memberSubfamily_union_nonMemberSubfamily (a : α) (𝒜 : Finset (Finset α)) :
𝒜.memberSubfamily a ∪ 𝒜.nonMemberSubfamily a = 𝒜.image fun s => s.erase a | case pos
α : Type u_1
inst✝ : DecidableEq α
a : α
𝒜 : Finset (Finset α)
s : Finset α
hs : s ∈ 𝒜
ha : a ∈ s
⊢ insert a (s.erase a) ∈ 𝒜 ∧ a ∉ s.erase a ∨ s.erase a ∈ 𝒜 ∧ a ∉ s.erase a | exact Or.inl ⟨by rwa [insert_erase ha], not_mem_erase _ _⟩ | no goals | e84640d1d865ba15 |
finprod_apply_ne_one | Mathlib/Algebra/BigOperators/Finprod.lean | theorem finprod_apply_ne_one (f : α → M) (a : α) : ∏ᶠ _ : f a ≠ 1, f a = f a | α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
a : α
⊢ ∏ᶠ (_ : f a ≠ 1), f a = f a | rw [← mem_mulSupport, finprod_eq_mulIndicator_apply, mulIndicator_mulSupport] | no goals | 8158e1dabf1a7043 |
IsArtinianRing.isUnit_submonoid_eq_of_isIntegral | Mathlib/RingTheory/Artinian/Algebra.lean | theorem isUnit_submonoid_eq_of_isIntegral [Algebra.IsIntegral R A] : IsUnit.submonoid A = A⁰ | case h
R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsArtinianRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : Algebra.IsIntegral R A
x✝ : A
⊢ x✝ ∈ IsUnit.submonoid A ↔ x✝ ∈ A⁰ | simpa [IsUnit.mem_submonoid_iff] using isUnit_iff_nonZeroDivisor_of_isIntegral' (R := R) | no goals | 4c3b08554612886b |
Multiset.inter_le_right | Mathlib/Data/Multiset/UnionInter.lean | lemma inter_le_right : s ∩ t ≤ t | case neg
α : Type u_1
inst✝ : DecidableEq α
a : α
s : Multiset α
IH : ∀ {t : Multiset α}, s ∩ t ≤ t
t : Multiset α
h : a ∉ t
⊢ (a ::ₘ s) ∩ t ≤ t | simp [h, IH] | no goals | e30103a6354a110c |
Equiv.Perm.cycleOf_mul_of_apply_right_eq_self | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | theorem cycleOf_mul_of_apply_right_eq_self [DecidableRel f.SameCycle]
[DecidableRel (f * g).SameCycle]
(h : Commute f g) (x : α) (hx : g x = x) : (f * g).cycleOf x = f.cycleOf x | case neg
α : Type u_2
f g : Perm α
inst✝¹ : DecidableRel f.SameCycle
inst✝ : DecidableRel (f * g).SameCycle
h : Commute f g
x : α
hx : g x = x
y : α
hxy : f.SameCycle x y
⊢ (f * g).SameCycle x y | obtain ⟨z, rfl⟩ := hxy | case neg.intro
α : Type u_2
f g : Perm α
inst✝¹ : DecidableRel f.SameCycle
inst✝ : DecidableRel (f * g).SameCycle
h : Commute f g
x : α
hx : g x = x
z : ℤ
⊢ (f * g).SameCycle x ((f ^ z) x) | 9f223ade6df5a61f |
Polynomial.coeff_hermite_explicit | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | theorem coeff_hermite_explicit :
∀ n k : ℕ, coeff (hermite (2 * n + k)) k = (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
| 0, _ => by simp
| n + 1, 0 => by
convert coeff_hermite_succ_zero (2 * n + 1) using 1
-- Porting note: ring_nf did not solve the goal on line 165
rw [coeff_hermite_explicit... | n k : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
hermite_explicit_recur :
∀ (n k : ℕ), hermite_explicit (n + 1) (k + 1) = hermite_explicit (n + 1) k - (↑k + 2) * hermite_explicit n (k + 2)
⊢ 2 * (n + 1) + k = 2 * n + (k + 2) | ring | no goals | 76b4bea6433d05b5 |
Polynomial.EisensteinCriterionAux.isUnit_of_natDegree_eq_zero_of_isPrimitive | Mathlib/RingTheory/EisensteinCriterion.lean | theorem isUnit_of_natDegree_eq_zero_of_isPrimitive {p q : R[X]}
-- Porting note: stated using `IsPrimitive` which is defeq to old statement.
(hu : IsPrimitive (p * q)) (hpm : p.natDegree = 0) : IsUnit p | R : Type u_1
inst✝ : CommRing R
p q : R[X]
hu : (p * q).IsPrimitive
hpm : p.natDegree = 0
⊢ C (p.coeff 0) ∣ p * q | rw [← eq_C_of_degree_le_zero (natDegree_eq_zero_iff_degree_le_zero.1 hpm)] | R : Type u_1
inst✝ : CommRing R
p q : R[X]
hu : (p * q).IsPrimitive
hpm : p.natDegree = 0
⊢ p ∣ p * q | 447194fa7b9cf99d |
Fin.map_valEmbedding_Ici | Mathlib/Order/Interval/Finset/Fin.lean | theorem map_valEmbedding_Ici : (Ici a).map Fin.valEmbedding = Icc ↑a (n - 1) | case h.mpr.succ
x n✝ : ℕ
a : Fin (n✝ + 1)
⊢ ↑a ≤ x ∧ x ≤ n✝ + 1 - 1 → ∃ a_2, a ≤ a_2 ∧ valEmbedding a_2 = x | exact fun hx => ⟨⟨x, Nat.lt_succ_iff.2 hx.2⟩, hx.1, rfl⟩ | no goals | 659a5c83d884a7cb |
Finset.iSup_insert_update | Mathlib/Order/CompleteLattice/Finset.lean | theorem iSup_insert_update {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t) :
⨆ i ∈ insert x t, Function.update f x s i = s ⊔ ⨆ i ∈ t, f i | case e_a.e_s.h.e_s.h.h
α : Type u_2
β : Type u_3
inst✝¹ : CompleteLattice β
inst✝ : DecidableEq α
t : Finset α
f : α → β
s : β
i : α
hi : i ∈ t
hx : i ∉ t
⊢ False | exact hx hi | no goals | f80925da6cd69053 |
IsMulFreimanIso.mono | Mathlib/Combinatorics/Additive/FreimanHom.lean | @[to_additive]
lemma IsMulFreimanIso.mono {hmn : m ≤ n} (hf : IsMulFreimanIso n A B f) :
IsMulFreimanIso m A B f where
bijOn := hf.bijOn
map_prod_eq_map_prod s t hsA htA hs ht | case inr.intro.refine_2
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoid α
inst✝ : CancelCommMonoid β
A : Set α
B : Set β
f : α → β
m n : ℕ
hmn : m ≤ n
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = m
ht : t.card = m
a✝ : α
ha✝ : a✝ ∈ A
a : α... | rw [Multiset.mem_add] at ha | case inr.intro.refine_2
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoid α
inst✝ : CancelCommMonoid β
A : Set α
B : Set β
f : α → β
m n : ℕ
hmn : m ≤ n
hf : IsMulFreimanIso n A B f
s t : Multiset α
hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A
htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A
hs : s.card = m
ht : t.card = m
a✝ : α
ha✝ : a✝ ∈ A
a : α... | 4d38848292c1597f |
Nat.div2_bit | Mathlib/Data/Nat/Bits.lean | lemma div2_bit (b n) : div2 (bit b n) = n | b : Bool
n : ℕ
⊢ (bit b n).div2 = n | rw [bit_val, div2_val, Nat.add_comm, add_mul_div_left, div_eq_of_lt, Nat.zero_add]
<;> cases b
<;> decide | no goals | dab866d736182597 |
MeasureTheory.SimpleFunc.eapprox_lt_top | Mathlib/MeasureTheory/Function/SimpleFunc.lean | theorem eapprox_lt_top (f : α → ℝ≥0∞) (n : ℕ) (a : α) : eapprox f n a < ∞ | case pos
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
n : ℕ
a : α
b : ℕ
a✝ : b ∈ Finset.range n
h✝ : MeasurableSet {a | ennrealRatEmbed b ≤ f a}
⊢ {a | ennrealRatEmbed b ≤ f a}.indicator (Function.const α (ennrealRatEmbed b)) a < ⊤ | calc
{ a : α | ennrealRatEmbed b ≤ f a }.indicator (fun _ => ennrealRatEmbed b) a ≤
ennrealRatEmbed b :=
indicator_le_self _ _ a
_ < ⊤ := ENNReal.coe_lt_top | no goals | a31470243a3f9f84 |
List.exists_pw_disjoint_with_card | Mathlib/GroupTheory/Perm/Cycle/PossibleTypes.lean | theorem List.exists_pw_disjoint_with_card {α : Type*} [Fintype α]
{c : List ℕ} (hc : c.sum ≤ Fintype.card α) :
∃ o : List (List α),
o.map length = c ∧ (∀ s ∈ o, s.Nodup) ∧ Pairwise List.Disjoint o | α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
l : List ℕ
hl : l ∈ c.ranges
n : ℕ
hn : n ∈ l
⊢ n < ... | rw [← mem_mem_ranges_iff_lt_sum] | α : Type u_2
inst✝ : Fintype α
c : List ℕ
hc : c.sum ≤ Fintype.card α
klift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn => ⟨n, hn⟩
klift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl => pmap klift l hl
l : List ℕ
hl : l ∈ c.ranges
n : ℕ
hn : n ∈ l
⊢ ∃ s ... | 463233383e1ded3a |
List.unzip_zip_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Zip.lean | theorem unzip_zip_right :
∀ {l₁ : List α} {l₂ : List β}, length l₂ ≤ length l₁ → (unzip (zip l₁ l₂)).2 = l₂
| [], l₂, _ => by simp_all
| l₁, [], _ => by simp
| a :: l₁, b :: l₂, h => by
simp only [zip_cons_cons, unzip_cons, unzip_zip_right (le_of_succ_le_succ h)]
| α : Type u_1
β : Type u_2
l₂ : List β
x✝ : l₂.length ≤ [].length
⊢ ([].zip l₂).unzip.snd = l₂ | simp_all | no goals | f6544f1eafdebf70 |
CategoryTheory.Idempotents.isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent | Mathlib/CategoryTheory/Idempotents/Basic.lean | theorem isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent :
IsIdempotentComplete C ↔ ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p | case create
C : Type u_1
inst✝ : Category.{u_2, u_1} C
a✝ : IsIdempotentComplete C
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : C
i : Y ⟶ X
e : X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p
s : Fork (𝟙 X) p
⊢ (s.ι ≫ e) ≫ (Fork.ofι i ⋯).ι = s.ι ∧
∀
{m :
((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParalle... | constructor | case create.left
C : Type u_1
inst✝ : Category.{u_2, u_1} C
a✝ : IsIdempotentComplete C
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : C
i : Y ⟶ X
e : X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p
s : Fork (𝟙 X) p
⊢ (s.ι ≫ e) ≫ (Fork.ofι i ⋯).ι = s.ι
case create.right
C : Type u_1
inst✝ : Category.{u_2, u_1} C
a✝ : IsIdempotentComplete... | b9521a8a1dfa70fd |
MeasureTheory.hasFiniteIntegral_toReal_iff | Mathlib/MeasureTheory/Function/L1Space/HasFiniteIntegral.lean | lemma hasFiniteIntegral_toReal_iff {f : α → ℝ≥0∞} (hf : ∀ᵐ x ∂μ, f x ≠ ∞) :
HasFiniteIntegral (fun x ↦ (f x).toReal) μ ↔ ∫⁻ x, f x ∂μ ≠ ∞ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤
⊢ HasFiniteIntegral (fun x => (f x).toReal) μ ↔ ∫⁻ (x : α), f x ∂μ ≠ ⊤ | have : ∀ᵐ x ∂μ, .ofReal (f x).toReal = f x := by filter_upwards [hf] with x hx; simp [hx] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤
this : ∀ᵐ (x : α) ∂μ, ENNReal.ofReal (f x).toReal = f x
⊢ HasFiniteIntegral (fun x => (f x).toReal) μ ↔ ∫⁻ (x : α), f x ∂μ ≠ ⊤ | fac8a1e91be27dc4 |
AlgebraicGeometry.Scheme.homOfLE_app | Mathlib/AlgebraicGeometry/Restrict.lean | theorem Scheme.homOfLE_app {U V : X.Opens} (e : U ≤ V) (W : Opens V) :
(X.homOfLE e).app W =
X.presheaf.map (homOfLE <| X.ι_image_homOfLE_le_ι_image e W).op | X : Scheme
U V : X.Opens
e : U ≤ V
W : (↑V).Opens
e₁ : Hom.app (X.homOfLE e ≫ V.ι) (V.ι ''ᵁ W) = Hom.app U.ι (V.ι ''ᵁ W) ≫ (↑U).presheaf.map (eqToHom ⋯).op
this : V.ι ⁻¹ᵁ V.ι ''ᵁ W = W
e₂ :
(↑V).presheaf.map (eqToIso this).hom.op ≫ Hom.app (X.homOfLE e) (V.ι ⁻¹ᵁ V.ι ''ᵁ W) =
Hom.app (X.homOfLE e) W ≫ (↑U).preshea... | rw [e₃, ← Functor.map_comp] | X : Scheme
U V : X.Opens
e : U ≤ V
W : (↑V).Opens
e₁ : Hom.app (X.homOfLE e ≫ V.ι) (V.ι ''ᵁ W) = Hom.app U.ι (V.ι ''ᵁ W) ≫ (↑U).presheaf.map (eqToHom ⋯).op
this : V.ι ⁻¹ᵁ V.ι ''ᵁ W = W
e₂ :
(↑V).presheaf.map (eqToIso this).hom.op ≫ Hom.app (X.homOfLE e) (V.ι ⁻¹ᵁ V.ι ''ᵁ W) =
Hom.app (X.homOfLE e) W ≫ (↑U).preshea... | 389fba341e18e4b3 |
ball_one_eq | Mathlib/Analysis/Normed/Group/Basic.lean | theorem ball_one_eq (r : ℝ) : ball (1 : E) r = { x | ‖x‖ < r } :=
Set.ext fun a => by simp
| E : Type u_5
inst✝ : SeminormedGroup E
r : ℝ
a : E
⊢ a ∈ ball 1 r ↔ a ∈ {x | ‖x‖ < r} | simp | no goals | c36e270c57eaabcd |
CochainComplex.HomComplex.Cochain.leftUnshift_units_smul | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | @[simp]
lemma leftUnshift_units_smul {n' a : ℤ} (γ : Cochain (K⟦a⟧) L n') (n : ℤ)
(hn : n + a = n') (x : Rˣ) :
(x • γ).leftUnshift n hn = x • γ.leftUnshift n hn | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L : CochainComplex C ℤ
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
x : Rˣ
⊢ (x • γ).leftUnshift n hn = x • γ.leftUnshift n hn | apply leftUnshift_smul | no goals | 735cce28459b5483 |
Submodule.coe_dualCoannihilator_span | Mathlib/LinearAlgebra/Dual.lean | @[simp]
lemma coe_dualCoannihilator_span (s : Set (Module.Dual R M)) :
((span R s).dualCoannihilator : Set M) = {x | ∀ f ∈ s, f x = 0} | case h
R : Type u
M : Type v
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
s : Set (Dual R M)
x : M
this : ∀ (φ : Dual R M), x ∈ LinearMap.ker φ ↔ φ ∈ LinearMap.ker ((Dual.eval R M) x)
⊢ (∀ φ ∈ span R s, φ ∈ LinearMap.ker ((Dual.eval R M) x)) ↔ ∀ f ∈ s, f ∈ LinearMap.ker ((Dual.eval R M) x) | exact span_le | no goals | f2343ad38b9ecd7b |
List.Sublist.sym | Mathlib/Data/List/Sym.lean | theorem Sublist.sym (n : ℕ) {xs ys : List α} (h : xs <+ ys) : xs.sym n <+ ys.sym n :=
match n, h with
| 0, _ => by simp [List.sym]
| n + 1, .slnil => by simp only [refl]
| n + 1, .cons a h => by
rw [List.sym, ← nil_append (List.sym (n + 1) xs)]
apply Sublist.append (nil_sublist _)
exact h.sym (n + 1... | α : Type u_1
n✝ : ℕ
xs ys : List α
h✝ : xs <+ ys
n : ℕ
l₁✝ l₂✝ : List α
a : α
h : l₁✝ <+ l₂✝
⊢ map (fun p => a ::ₛ p) (List.sym n (a :: l₁✝)) ++ List.sym (n + 1) l₁✝ <+
map (fun p => a ::ₛ p) (List.sym n (a :: l₂✝)) ++ List.sym (n + 1) l₂✝ | apply Sublist.append | case hl
α : Type u_1
n✝ : ℕ
xs ys : List α
h✝ : xs <+ ys
n : ℕ
l₁✝ l₂✝ : List α
a : α
h : l₁✝ <+ l₂✝
⊢ map (fun p => a ::ₛ p) (List.sym n (a :: l₁✝)) <+ map (fun p => a ::ₛ p) (List.sym n (a :: l₂✝))
case hr
α : Type u_1
n✝ : ℕ
xs ys : List α
h✝ : xs <+ ys
n : ℕ
l₁✝ l₂✝ : List α
a : α
h : l₁✝ <+ l₂✝
⊢ List.sym (n + 1)... | d3891099e3ec45ad |
List.prev_reverse_eq_next | Mathlib/Data/List/Cycle.lean | theorem prev_reverse_eq_next (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) :
prev l.reverse x (mem_reverse.mpr hx) = next l x hx | case intro.intro
α : Type u_1
inst✝ : DecidableEq α
l : List α
h : l.Nodup
k : ℕ
hk : k < l.length
hx : l[k] ∈ l
lpos : 0 < l.length
key : l.length - 1 - k < l.length
⊢ l.reverse.prev l[k] ⋯ = (pmap l.next l ⋯)[k] | simp_rw [getElem_eq_getElem_reverse (l := l), pmap_next_eq_rotate_one _ h] | case intro.intro
α : Type u_1
inst✝ : DecidableEq α
l : List α
h : l.Nodup
k : ℕ
hk : k < l.length
hx : l[k] ∈ l
lpos : 0 < l.length
key : l.length - 1 - k < l.length
⊢ l.reverse.prev l.reverse[l.length - 1 - k] ⋯ = (l.rotate 1)[k] | 223a9f0719f3b161 |
Affine.Simplex.circumsphere_reindex | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem circumsphere_reindex {m n : ℕ} (s : Simplex ℝ P m) (e : Fin (m + 1) ≃ Fin (n + 1)) :
(s.reindex e).circumsphere = s.circumsphere | case refine_1
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
m n : ℕ
s : Simplex ℝ P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ (s.reindex e).circumsphere.center ∈ affineSpan ℝ (Set.range (s.reindex e).points) | exact (s.reindex e).circumsphere_unique_dist_eq.1.1 | no goals | 4450e8b52c9c2999 |
Orientation.oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arctan (‖y‖ / ‖x‖) | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
⊢ o.oangle x (x + y) = ↑(Real.arctan (‖y‖ / ‖x‖)) | have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two] | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle x (x + y)).sign = 1
⊢ o.oangle x (x + y) = ↑(Real.arctan (‖y‖ / ‖x‖)) | 45dcf7e3fc3510ff |
Pi.isAtom_iff_eq_single | Mathlib/Order/Atoms.lean | theorem isAtom_iff_eq_single [DecidableEq ι] [∀ i, PartialOrder (π i)]
[∀ i, OrderBot (π i)] {f : ∀ i, π i} :
IsAtom f ↔ ∃ i a, IsAtom a ∧ f = Function.update ⊥ i a | case mp
ι : Type u_4
π : ι → Type u
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → PartialOrder (π i)
inst✝ : (i : ι) → OrderBot (π i)
f : (i : ι) → π i
⊢ IsAtom f → ∃ i a, IsAtom a ∧ f = Function.update ⊥ i a
case mpr
ι : Type u_4
π : ι → Type u
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → PartialOrder (π i)
inst✝ : (i : ι) ... | case mp =>
intro h
have ⟨i, h, hbot⟩ := isAtom_iff.1 h
refine ⟨_, _, h, funext fun j => if hij : j = i then hij ▸ by simp else ?_⟩
rw [Function.update_of_ne hij, hbot _ hij, bot_apply] | case mpr
ι : Type u_4
π : ι → Type u
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → PartialOrder (π i)
inst✝ : (i : ι) → OrderBot (π i)
f : (i : ι) → π i
⊢ (∃ i a, IsAtom a ∧ f = Function.update ⊥ i a) → IsAtom f | d8e4c77da6e7ba4c |
leOnePart_mul_oneLePart | Mathlib/Algebra/Order/Group/PosPart.lean | @[to_additive] lemma leOnePart_mul_oneLePart (a : α) : a⁻ᵐ * a⁺ᵐ = |a|ₘ | α : Type u_1
inst✝³ : Lattice α
inst✝² : Group α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
a : α
⊢ a⁻ᵐ * a⁺ᵐ = mabs a | rw [oneLePart_def, mul_sup, mul_one, leOnePart_def, sup_mul, one_mul, inv_mul_cancel, sup_assoc,
← @sup_assoc _ _ a, sup_eq_right.2 le_sup_right] | α : Type u_1
inst✝³ : Lattice α
inst✝² : Group α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
a : α
⊢ a ⊔ a⁻¹ ⊔ 1 = mabs a | 17b9e31b8452bb9c |
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