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 |
|---|---|---|---|---|---|---|
Commute.list_prod_right | Mathlib/Algebra/BigOperators/Group/List/Basic.lean | theorem _root_.Commute.list_prod_right (l : List M) (y : M) (h : ∀ x ∈ l, Commute y x) :
Commute y l.prod | case cons
M : Type u_4
inst✝ : Monoid M
y z : M
l : List M
IH : (∀ (x : M), x ∈ l → Commute y x) → Commute y l.prod
h : Commute y z ∧ ∀ (x : M), x ∈ l → Commute y x
⊢ Commute y (z :: l).prod | rw [List.prod_cons] | case cons
M : Type u_4
inst✝ : Monoid M
y z : M
l : List M
IH : (∀ (x : M), x ∈ l → Commute y x) → Commute y l.prod
h : Commute y z ∧ ∀ (x : M), x ∈ l → Commute y x
⊢ Commute y (z * l.prod) | 9202b1a0e267f593 |
ProbabilityTheory.Kernel.compProd_restrict_right | Mathlib/Probability/Kernel/Composition/CompProd.lean | theorem compProd_restrict_right {t : Set γ} (ht : MeasurableSet t) :
κ ⊗ₖ Kernel.restrict η ht = Kernel.restrict (κ ⊗ₖ η) (MeasurableSet.univ.prod ht) | 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
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
t : Set γ
ht : MeasurableSet t
⊢ κ ⊗ₖ η.restrict ht = κ.restrict ?hs ⊗ₖ η.restrict ?ht | congr | case e_κ
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
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
t : Set γ
ht : MeasurableSet t
⊢ κ = κ.restrict ?hs | dbbee6ce13b63200 |
MeromorphicAt.order_smul | Mathlib/Analysis/Meromorphic/Order.lean | theorem order_smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜}
(hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
(hf.smul hg).order = hf.order + hg.order | case coe.coe.intro.intro.intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → 𝕜
g : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : hf.order = ↑m
n : ℤ
h₂g : hg.order = ↑n
F : 𝕜 → 𝕜
h₁F : AnalyticAt 𝕜 F x
h₂F : F... | obtain ⟨G, h₁G, h₂G, h₃G⟩ := (hg.order_eq_int_iff _).1 h₂g | case coe.coe.intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → 𝕜
g : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
hg : MeromorphicAt g x
m : ℤ
h₂f : hf.order = ↑m
n : ℤ
h₂g : hg.order = ↑n
F : 𝕜 → 𝕜
h₁F : Analyti... | 888980cd950630aa |
CategoryTheory.representablyFlat_op_iff | Mathlib/CategoryTheory/Functor/Flat.lean | theorem representablyFlat_op_iff : RepresentablyFlat F.op ↔ RepresentablyCoflat F | case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
x✝¹ : RepresentablyCoflat F
x✝ : Dᵒᵖ
X : D
⊢ IsFiltered (CostructuredArrow F X)ᵒᵖᵒᵖ | apply IsFiltered.of_equivalence (opOpEquivalence _) | no goals | aa1dd31a9d933fd3 |
Set.bounded_le_iff_bounded_lt | Mathlib/Order/Bounded.lean | theorem bounded_le_iff_bounded_lt [Preorder α] [NoMaxOrder α] :
Bounded (· ≤ ·) s ↔ Bounded (· < ·) s | case intro.intro
α : Type u_1
s : Set α
inst✝¹ : Preorder α
inst✝ : NoMaxOrder α
a : α
ha : ∀ b ∈ s, (fun x1 x2 => x1 ≤ x2) b a
b : α
hb : a < b
⊢ Bounded (fun x1 x2 => x1 < x2) s | exact ⟨b, fun c hc => lt_of_le_of_lt (ha c hc) hb⟩ | no goals | e8cc0c8fee08602a |
Algebra.FinitePresentation.mvPolynomial_of_finitePresentation | Mathlib/RingTheory/FinitePresentation.lean | theorem mvPolynomial_of_finitePresentation [FinitePresentation.{w₁, w₂} R A]
(ι : Type v) [Finite ι] :
FinitePresentation.{w₁, max v w₂} R (MvPolynomial ι A) | case intro.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
ι' : Type v
w✝ : Fintype ι'
f : MvPolynomial ι' R →ₐ[R] A
hf_surj : Surjective ⇑f
hf_ker : (RingHom.ker f.toRingHom).FG
g : MvPolynomial (ι ⊕ ι') ... | cases nonempty_fintype (ι ⊕ ι') | case intro.intro.intro.intro.intro
R : Type w₁
A : Type w₂
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Algebra R A
inst✝¹ : FinitePresentation R A
ι : Type v
inst✝ : Finite ι
ι' : Type v
w✝ : Fintype ι'
f : MvPolynomial ι' R →ₐ[R] A
hf_surj : Surjective ⇑f
hf_ker : (RingHom.ker f.toRingHom).FG
g : MvPolynomial (ι ... | 2529be89e207e207 |
FormalMultilinearSeries.ofScalars_radius_ge_inv_of_tendsto | Mathlib/Analysis/Analytic/OfScalars.lean | theorem ofScalars_radius_ge_inv_of_tendsto {r : ℝ≥0} (hr : r ≠ 0)
(hc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 r)) :
(ofScalars E c).radius ≥ ofNNReal r⁻¹ | case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : r' * r < 1
hrz : ¬r' = 0
n : ℕ
hn : n ≠ 0
⊢ ‖‖ofScalars E c n‖ * ↑r' ^ n‖ ≤ ‖‖c n‖ * ↑r' ^ n‖ | simp only [norm_mul, norm_norm, norm_pow, NNReal.norm_eq] | case h
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
r : ℝ≥0
hr : r ≠ 0
hc : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)
r' : ℝ≥0
hr' : r' * r < 1
hrz : ¬r' = 0
n : ℕ
hn : n ≠ 0
⊢ ‖ofScalars E c n‖ * ↑r' ^ n ≤ ‖c n‖ * ↑r' ^ n | bd39968bceb675cb |
NormedAddGroupHom.SurjectiveOnWith.mono | Mathlib/Analysis/Normed/Group/Hom.lean | theorem SurjectiveOnWith.mono {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C C' : ℝ}
(h : f.SurjectiveOnWith K C) (H : C ≤ C') : f.SurjectiveOnWith K C' | case pos
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ‖f g‖ = 0
⊢ ‖g‖ ≤ C' * ‖f g‖ | simpa [Hg] using hg | no goals | 0ec223564f13d644 |
Array.getElem_extract_loop_ge | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem getElem_extract_loop_ge (as bs : Array α) (size start : Nat) (hge : i ≥ bs.size)
(h : i < (extract.loop as size start bs).size)
(h' := getElem_extract_loop_ge_aux as bs size start hge h) :
(extract.loop as size start bs)[i] = as[start + i - bs.size] | case succ
α : Type u_1
i : Nat
as : Array α
size : Nat
ih :
∀ (bs : Array α) (start : Nat) (hge : i ≥ bs.size) (h : i < (extract.loop as size start bs).size)
(h' : optParam (start + i - bs.size < as.size) ⋯), (extract.loop as size start bs)[i] = as[start + i - bs.size]
bs : Array α
start : Nat
hge : i ≥ bs.size
h... | have : start < as.size := by
apply Nat.lt_of_le_of_lt (Nat.le_add_right start (i - bs.size))
rwa [← Nat.add_sub_assoc hge] | case succ
α : Type u_1
i : Nat
as : Array α
size : Nat
ih :
∀ (bs : Array α) (start : Nat) (hge : i ≥ bs.size) (h : i < (extract.loop as size start bs).size)
(h' : optParam (start + i - bs.size < as.size) ⋯), (extract.loop as size start bs)[i] = as[start + i - bs.size]
bs : Array α
start : Nat
hge : i ≥ bs.size
h... | 89ddb2da716cb704 |
root_X_pow_sub_C_ne_zero' | Mathlib/FieldTheory/KummerPolynomial.lean | lemma root_X_pow_sub_C_ne_zero' {n : ℕ} {a : K} (hn : 0 < n) (ha : a ≠ 0) :
(AdjoinRoot.root (X ^ n - C a)) ≠ 0 | K : Type u
inst✝ : Field K
a : K
ha : a ≠ 0
hn : 0 < Nat.succ 0
e : root (X - C a) = 0
⊢ (C a).natDegree < (X - C a).natDegree | simp | no goals | 153688240a8b7f53 |
Valuation.val_le_one_or_val_inv_le_one | Mathlib/RingTheory/Valuation/Basic.lean | theorem val_le_one_or_val_inv_le_one (v : Valuation K Γ₀) (x : K) : v x ≤ 1 ∨ v x⁻¹ ≤ 1 | K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_4
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x : K
⊢ v x ≤ 1 ∨ v x⁻¹ ≤ 1 | by_cases h : x = 0 | case pos
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_4
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x : K
h : x = 0
⊢ v x ≤ 1 ∨ v x⁻¹ ≤ 1
case neg
K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_4
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x : K
h : ¬x = 0
⊢ v x ≤ 1 ∨ v x⁻¹ ≤ 1 | bc0e9caf30feef0a |
Polynomial.isUnit_of_isUnit_leadingCoeff_of_isUnit_map | Mathlib/Algebra/Polynomial/Eval/Degree.lean | lemma isUnit_of_isUnit_leadingCoeff_of_isUnit_map (hf : IsUnit f.leadingCoeff)
(H : IsUnit (map φ f)) : IsUnit f | R : Type u
S : Type v
inst✝² : Semiring R
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
f : R[X]
hf : IsUnit f.leadingCoeff
H : IsUnit (map φ f)
dz : f.degree = 0
⊢ IsUnit f | rw [eq_C_of_degree_eq_zero dz] | R : Type u
S : Type v
inst✝² : Semiring R
inst✝¹ : CommRing S
inst✝ : IsDomain S
φ : R →+* S
f : R[X]
hf : IsUnit f.leadingCoeff
H : IsUnit (map φ f)
dz : f.degree = 0
⊢ IsUnit (C (f.coeff 0)) | 89632f3e986a98c7 |
CategoryTheory.Limits.biproduct.map_eq_map' | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | theorem biproduct.map_eq_map' {f g : J → C} [HasBiproduct f] [HasBiproduct g] (p : ∀ b, f b ⟶ g b) :
biproduct.map p = biproduct.map' p | case neg
J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
f g : J → C
inst✝¹ : HasBiproduct f
inst✝ : HasBiproduct g
p : (b : J) → f b ⟶ g b
j✝¹ j✝ : J
h : ¬j✝ = j✝¹
⊢ 0 ≫ p j✝¹ = p j✝ ≫ 0 | simp | no goals | 058dda90510c70db |
mellin_hasDerivAt_of_isBigO_rpow | Mathlib/Analysis/MellinTransform.lean | theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
{f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a)))
(hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
MellinConvergent (fun t => log t • f t) s ∧
HasDerivAt (mellin ... | case h.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) • f t
F' : ℂ → ℝ →... | rw [sub_re, one_re, sub_le_sub_iff_right] | case h.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) • f t
F' : ℂ → ℝ →... | 3ad5d9a87042b6f9 |
LinearMap.isNilpotent_toMatrix_iff | Mathlib/RingTheory/Nilpotent/Lemmas.lean | @[simp]
lemma isNilpotent_toMatrix_iff (b : Basis ι R M) (f : M →ₗ[R] M) :
IsNilpotent (toMatrix b b f) ↔ IsNilpotent f | R : Type u_1
inst✝⁴ : CommSemiring R
ι : Type u_3
M : Type u_4
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis ι R M
f : M →ₗ[R] M
k : ℕ
⊢ (toMatrix b b) f ^ k = 0 ↔ f ^ k = 0 | rw [toMatrix_pow] | R : Type u_1
inst✝⁴ : CommSemiring R
ι : Type u_3
M : Type u_4
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
b : Basis ι R M
f : M →ₗ[R] M
k : ℕ
⊢ (toMatrix b b) (f ^ k) = 0 ↔ f ^ k = 0 | ad82ecdec514556a |
finrank_orthogonal_span_singleton | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem finrank_orthogonal_span_singleton {n : ℕ} [_i : Fact (finrank 𝕜 E = n + 1)] {v : E}
(hv : v ≠ 0) : finrank 𝕜 (𝕜 ∙ v)ᗮ = n | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
n : ℕ
_i : Fact (finrank 𝕜 E = n + 1)
v : E
hv : v ≠ 0
this : FiniteDimensional 𝕜 E
⊢ finrank 𝕜 ↥(Submodule.span 𝕜 {v})ᗮ = n | exact Submodule.finrank_add_finrank_orthogonal' <| by
simp [finrank_span_singleton hv, _i.elim, add_comm] | no goals | 1b86762dacd0365f |
FirstOrder.Language.DirectLimit.of_f | Mathlib/ModelTheory/DirectLimit.lean | theorem of_f {i j : ι} {hij : i ≤ j} {x : G i} : of L ι G f j (f i j hij x) = of L ι G f i x | L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
i j : ι
hij : i ≤ j
x : G i
⊢ (setoid G f) (Structure.Sigma.mk f j ((f i j ... | refine Setoid.symm ⟨j, hij, refl j, ?_⟩ | L : Language
ι : Type v
inst✝⁴ : Preorder ι
G : ι → Type w
inst✝³ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝² : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝ : Nonempty ι
i j : ι
hij : i ≤ j
x : G i
⊢ (f i j hij) x = (f j j ⋯) ((f i j hij) x) | c47aaa6fe449a12b |
ZNum.gcd_to_nat | Mathlib/Data/Num/Lemmas.lean | theorem gcd_to_nat (a b) : (gcd a b : ℕ) = Int.gcd a b :=
(Num.gcd_to_nat _ _).trans <| by simp only [abs_to_nat]; rfl
| a b : ZNum
⊢ (↑a).natAbs.gcd (↑b).natAbs = (↑a).gcd ↑b | rfl | no goals | eb58f39cfed45117 |
IsPrimitiveRoot.sub_one_norm_isPrimePow | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem sub_one_norm_isPrimePow (hn : IsPrimePow (n : ℕ)) [IsCyclotomicExtension {n} K L]
(hirr : Irreducible (cyclotomic (n : ℕ) K)) (h : n ≠ 2) : norm K (ζ - 1) = (n : ℕ).minFac | n : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
hζ : IsPrimitiveRoot ζ ↑n
hn : IsPrimePow ↑n
inst✝ : IsCyclotomicExtension {n} K L
hirr : Irreducible (cyclotomic (↑n) K)
h : n ≠ 2
this : 2 < n
⊢ (Algebra.norm K) (ζ - 1) = ↑(↑n).minFac | letI hprime : Fact (n : ℕ).minFac.Prime := ⟨minFac_prime (IsPrimePow.ne_one hn)⟩ | n : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
hζ : IsPrimitiveRoot ζ ↑n
hn : IsPrimePow ↑n
inst✝ : IsCyclotomicExtension {n} K L
hirr : Irreducible (cyclotomic (↑n) K)
h : n ≠ 2
this : 2 < n
hprime : Fact (Nat.Prime (↑n).minFac) := { out := minFac_prime (IsPrimePow.ne_one hn)... | c894a93f498c96ce |
intervalIntegral.continuousOn_primitive_Icc | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | theorem continuousOn_primitive_Icc (h_int : IntegrableOn f (Icc a b) μ) :
ContinuousOn (fun x => ∫ t in Icc a x, f t ∂μ) (Icc a b) | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NoAtoms μ
h_int : IntegrableOn f (Icc a b) μ
aux : (fun x => ∫ (t : ℝ) in Icc a x, f t ∂μ) = fun x => ∫ (t : ℝ) in Ioc a x, f t ∂μ
⊢ ContinuousOn (fun x => ∫ (t : ℝ) in Ioc a x, f t ∂μ) (Icc a b) | exact continuousOn_primitive h_int | no goals | f3b3cc2afeb16195 |
SeparationQuotient.normedMk_eq_zero_iff | Mathlib/Analysis/Normed/Group/SeparationQuotient.lean | theorem normedMk_eq_zero_iff : normedMk (M := M) = 0 ↔ ∀ (x : M), ‖x‖ = 0 | case mpr.H
M : Type u_1
inst✝ : SeminormedAddCommGroup M
h : ∀ (x : M), ‖x‖ = 0
x : M
⊢ normedMk x = 0 x | simpa [← norm_eq_zero] using h x | no goals | 75b695b993c50bd5 |
Nat.factors_lemma | Mathlib/Data/Nat/Prime/Defs.lean | theorem factors_lemma {k} : (k + 2) / minFac (k + 2) < k + 2 :=
div_lt_self (Nat.zero_lt_succ _) (minFac_prime (by
apply Nat.ne_of_gt
apply Nat.succ_lt_succ
apply Nat.zero_lt_succ
)).one_lt
| k : ℕ
⊢ k + 2 ≠ 1 | apply Nat.ne_of_gt | case h
k : ℕ
⊢ 1 < k + 2 | 5b01b10ee76daed0 |
Finset.mem_finsupp_iff_of_support_subset | Mathlib/Data/Finset/Finsupp.lean | theorem mem_finsupp_iff_of_support_subset {t : ι →₀ Finset α} (ht : t.support ⊆ s) :
f ∈ s.finsupp t ↔ ∀ i, f i ∈ t i | case refine_2
ι : Type u_1
α : Type u_2
inst✝ : Zero α
s : Finset ι
f : ι →₀ α
t : ι →₀ Finset α
ht : t.support ⊆ s
i : ι
h : f i ∈ t i
hi : i ∈ f.support
H : t i = 0
⊢ f i = 0 | rwa [H, mem_zero] at h | no goals | e4c749f6ee236c88 |
Real.Gamma_two | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | theorem Gamma_two : Gamma 2 = 1 | ⊢ Gamma 2 = 1 | simp [Nat.factorial_one] | no goals | b10282c132926fb3 |
bernsteinPolynomial.sum | Mathlib/RingTheory/Polynomial/Bernstein.lean | theorem sum (n : ℕ) : (∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν) = 1 :=
calc
(∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν) = (X + (1 - X)) ^ n | R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
∑ m ∈ Finset.range (n + 1), X ^ m * (1 - X) ^ (n - m) * ↑(n.choose m) | simp only [bernsteinPolynomial, mul_comm, mul_assoc, mul_left_comm] | no goals | 32f26738733d0f8d |
Array.mem_of_mem_erase | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Erase.lean | theorem mem_of_mem_erase {a b : α} {l : Array α} (h : a ∈ l.erase b) : a ∈ l | α : Type u_1
inst✝ : BEq α
a b : α
l : List α
h : a ∈ { toList := l }.erase b
⊢ a ∈ l.erase ?m.36325 | simpa using h | no goals | e3bee16365b52656 |
Mathlib.Tactic.Module.NF.add_eq_eval₁ | Mathlib/Tactic/Module.lean | theorem add_eq_eval₁ [AddMonoid M] [SMul R M] (a₁ : R × M) {a₂ : R × M} {l₁ l₂ l : NF R M}
(h : l₁.eval + (a₂ ::ᵣ l₂).eval = l.eval) :
(a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval | R : Type u_2
M : Type u_3
inst✝¹ : AddMonoid M
inst✝ : SMul R M
a₁ a₂ : R × M
l₁ l₂ l : NF R M
h : l₁.eval + (a₂ ::ᵣ l₂).eval = l.eval
⊢ (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval | simp only [eval_cons, ← h, add_assoc] | no goals | b91bb36922825f97 |
MonotoneOn.union_right | Mathlib/Order/Monotone/Union.lean | theorem MonotoneOn.union_right {s t : Set α} {c : α} (h₁ : MonotoneOn f s)
(h₂ : MonotoneOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : MonotoneOn f (s ∪ t) | α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : MonotoneOn f s
h₂ : MonotoneOn f t
hs : IsGreatest s c
ht : IsLeast t c
A : ∀ x ∈ s ∪ t, x ≤ c → x ∈ s
x : α
hx✝ : x ∈ s ∪ t
hxc : c ≤ x
hx : x ∈ t
⊢ x ∈ t | exact hx | no goals | 2a16573c6e7c5155 |
UniqueMul.mulHom_map_iff | Mathlib/Algebra/Group/UniqueProds/Basic.lean | theorem mulHom_map_iff (f : G ↪ H) (mul : ∀ x y, f (x * y) = f x * f y) :
UniqueMul (A.map f) (B.map f) (f a0) (f b0) ↔ UniqueMul A B a0 b0 | G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
f : G ↪ H
mul : ∀ (x y : G), f (x * y) = f x * f y
⊢ UniqueMul (map f A) (map f B) (f a0) (f b0) ↔ UniqueMul A B a0 b0 | classical simp_rw [← mulHom_image_iff ⟨f, mul⟩ f.2, Finset.map_eq_image]; rfl | no goals | fe2d4f38e33de6b8 |
Basis.prod_parallelepiped | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | theorem Basis.prod_parallelepiped (v : Basis ι ℝ E) (w : Basis ι' ℝ F) :
(v.prod w).parallelepiped = v.parallelepiped.prod w.parallelepiped | case h.left.left
ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ι'
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
v : Basis ι ℝ E
w : Basis ι' ℝ F
x : E × F
t : ι → ℝ
ht1 : t ∈ Icc 0 1
ht2 : x.1 = ∑ i : ι, t i • v i
... | aesop | no goals | 852108d4f62d51be |
LieHom.range_eq_map | Mathlib/Algebra/Lie/Subalgebra.lean | theorem _root_.LieHom.range_eq_map : f.range = map f ⊤ | case h
R : Type u
L : Type v
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
L₂ : Type w
inst✝¹ : LieRing L₂
inst✝ : LieAlgebra R L₂
f : L →ₗ⁅R⁆ L₂
x✝ : L₂
⊢ x✝ ∈ f.range ↔ x✝ ∈ map f ⊤ | simp | no goals | 4bf19b3192b887ac |
tendsto_tsum_div_pow_atTop_integral | Mathlib/Analysis/BoxIntegral/UnitPartition.lean | theorem _root_.tendsto_tsum_div_pow_atTop_integral (hF : Continuous F) (hs₁ : IsBounded s)
(hs₂ : MeasurableSet s) (hs₃ : volume (frontier s) = 0) :
Tendsto (fun n : ℕ ↦ (∑' x : ↑(s ∩ (n : ℝ)⁻¹ • L), F x) / n ^ card ι)
atTop (nhds (∫ x in s, F x)) | case intro.intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ≤ ↑B
ε : ℝ
hε : ε > 0
h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖s.indicator F x‖ ≤ C
h₂ : ... | rw [show r 0 = fun _ ↦ r 0 0 from funext_iff.mpr (hr₁ 0 rfl)] | case intro.intro.intro.intro.refine_1
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ≤ ↑B
ε : ℝ
hε : ε > 0
h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖s.indicator F x‖ ≤ C
h₂ : ... | e99fafdae745475e |
Real.aux₂ | Mathlib/NumberTheory/DiophantineApproximation/Basic.lean | theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v | ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
hu₀ : 0 ≤ u - ⌊ξ⌋ * v
h : (↑u - ξ * ↑v) * (2 * ↑v - 1) + ↑v * (2 * ↑v - 1) < ↑v * (2 * ↑v - 1) + 1
this : (↑u - ↑⌊ξ⌋ * ↑v) * (2 * ↑v - 1) < (↑u - ξ * ↑v) * (2 * ↑v - 1) + ↑v * (2 * ↑v -... | exact mod_cast this.trans h | no goals | b733bf52b4cb0f24 |
mellin_hasDerivAt_of_isBigO_rpow | Mathlib/Analysis/MellinTransform.lean | theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
{f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a)))
(hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
MellinConvergent (fun t => log t • f t) s ∧
HasDerivAt (mellin ... | case refine_4
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) • f t
F' : ℂ → ... | linarith | no goals | 3ad5d9a87042b6f9 |
Subgroup.map_top_of_surjective | Mathlib/Algebra/Group/Subgroup/Map.lean | theorem map_top_of_surjective (f : G →* N) (h : Function.Surjective f) : Subgroup.map f ⊤ = ⊤ | G : Type u_1
inst✝¹ : Group G
N : Type u_5
inst✝ : Group N
f : G →* N
h : Surjective ⇑f
⊢ map f ⊤ = ⊤ | rw [eq_top_iff] | G : Type u_1
inst✝¹ : Group G
N : Type u_5
inst✝ : Group N
f : G →* N
h : Surjective ⇑f
⊢ ⊤ ≤ map f ⊤ | 7e1142dafafaaaa3 |
PrimeSpectrum.range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | lemma range_comap_algebraMap_localization_compl_eq_range_comap_quotientMk
{R : Type*} [CommRing R] (c : R) :
letI := (mapRingHom (algebraMap R (Away c))).toAlgebra
(range (comap (algebraMap R[X] (Away c)[X])))ᶜ
= range (comap (mapRingHom (Ideal.Quotient.mk (.span {c})))) | R : Type u_1
inst✝ : CommRing R
c : R
this✝ : Algebra R[X] (Away c)[X] := (mapRingHom (algebraMap R (Away c))).toAlgebra
this : IsLocalization (Submonoid.powers (C c)) (Away c)[X]
surj : Function.Surjective ⇑(mapRingHom (Ideal.Quotient.mk (Ideal.span {c})))
⊢ (↑(basicOpen (C c)))ᶜ = zeroLocus ↑(RingHom.ker (mapRingHom ... | simp [Polynomial.ker_mapRingHom, Ideal.map_span] | no goals | 75854298d2f56b5e |
IsDedekindDomain.HeightOneSpectrum.not_mem_adicCompletionIntegers | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | theorem not_mem_adicCompletionIntegers {x : v.adicCompletion K} :
x ∉ v.adicCompletionIntegers K ↔ 1 < Valued.v x | R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : adicCompletion K v
⊢ ¬Valued.v x ≤ 1 ↔ 1 < Valued.v x | exact not_le | no goals | 7079b59e6d5b098b |
Field.primitive_element_inf_aux | Mathlib/FieldTheory/PrimitiveElement.lean | theorem primitive_element_inf_aux [Algebra.IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯ | case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField... | have α_in_Fγ : α ∈ F⟮γ⟯ := by
rw [← add_sub_cancel_right α (c • β)]
exact F⟮γ⟯.sub_mem (mem_adjoin_simple_self F γ) (F⟮γ⟯.toSubalgebra.smul_mem β_in_Fγ c) | case h.a
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField... | c759c025da469f0d |
Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv | Mathlib/Analysis/Complex/Polynomial/Basic.lean | theorem card_complex_roots_eq_card_real_add_card_not_gal_inv (p : ℚ[X]) :
(p.rootSet ℂ).toFinset.card =
(p.rootSet ℝ).toFinset.card +
(galActionHom p ℂ (restrict p ℂ
(AlgEquiv.restrictScalars ℚ Complex.conjAe))).support.card | case neg.refine_2.refine_2.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) ... | have ha : ∀ z : ℂ, z ∈ a ↔ aeval z p = 0 := by
intro z; rw [Set.mem_toFinset, mem_rootSet_of_ne hp] | case neg.refine_2.refine_2.refine_2
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
Finset.image (fun a => ↑a) ((galActionHom p ℂ) ((restrict p ℂ) ... | 8dbcfedb34311abf |
Polynomial.integralNormalization_eval₂_leadingCoeff_mul_of_commute | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | theorem integralNormalization_eval₂_leadingCoeff_mul_of_commute (h : 1 ≤ p.natDegree) (f : R →+* A)
(x : A) (h₁ : Commute (f p.leadingCoeff) x) (h₂ : ∀ {r r'}, Commute (f r) (f r')) :
(integralNormalization p).eval₂ f (f p.leadingCoeff * x) =
f p.leadingCoeff ^ (p.natDegree - 1) * p.eval₂ f x | R : Type u
inst✝¹ : Semiring R
p : R[X]
A : Type u_1
inst✝ : Semiring A
h : 1 ≤ p.natDegree
f : R →+* A
x : A
h₁ : Commute (f p.leadingCoeff) x
h₂ : ∀ {r r' : R}, Commute (f r) (f r')
⊢ eval₂ f (f p.leadingCoeff * x) p.integralNormalization = f p.leadingCoeff ^ (p.natDegree - 1) * eval₂ f x p | rw [eval₂_eq_sum_range, eval₂_eq_sum_range, Finset.mul_sum] | R : Type u
inst✝¹ : Semiring R
p : R[X]
A : Type u_1
inst✝ : Semiring A
h : 1 ≤ p.natDegree
f : R →+* A
x : A
h₁ : Commute (f p.leadingCoeff) x
h₂ : ∀ {r r' : R}, Commute (f r) (f r')
⊢ ∑ i ∈ Finset.range (p.integralNormalization.natDegree + 1),
f (p.integralNormalization.coeff i) * (f p.leadingCoeff * x) ^ i =
... | c0b9bc039024250c |
MeasureTheory.MeasurePreserving.lintegral_map_equiv | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem MeasurePreserving.lintegral_map_equiv [MeasurableSpace β] {ν : Measure β}
(f : β → ℝ≥0∞) (g : α ≃ᵐ β) (hg : MeasurePreserving g μ ν) :
∫⁻ a, f a ∂ν = ∫⁻ a, f (g a) ∂μ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : MeasurableSpace β
ν : Measure β
f : β → ℝ≥0∞
g : α ≃ᵐ β
hg : MeasurePreserving (⇑g) μ ν
⊢ ∫⁻ (a : β), f a ∂ν = ∫⁻ (a : α), f (g a) ∂μ | rw [← MeasureTheory.lintegral_map_equiv f g, hg.map_eq] | no goals | c114a8367d09c6a7 |
MeasureTheory.SimpleFunc.setToSimpleFunc_eq_sum_filter | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToSimpleFunc_eq_sum_filter [DecidablePred fun x ↦ x ≠ (0 : F)]
{m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (f : α →ₛ F) :
setToSimpleFunc T f = ∑ x ∈ f.range with x ≠ 0, T (f ⁻¹' {x}) x | α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : DecidablePred fun x => x ≠ 0
m : MeasurableSpace α
T : Set α → F →L[ℝ] F'
f : α →ₛ F
x : F
x✝ : x ∈ f.range
hx0 : x = 0
⊢ (T (⇑f ⁻¹' {x})) x = 0 | rw [hx0] | α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : DecidablePred fun x => x ≠ 0
m : MeasurableSpace α
T : Set α → F →L[ℝ] F'
f : α →ₛ F
x : F
x✝ : x ∈ f.range
hx0 : x = 0
⊢ (T (⇑f ⁻¹' {0})) 0 = 0 | fb02387342aef168 |
Std.Sat.CNF.sat_def | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/CNF/Basic.lean | theorem sat_def (a : α → Bool) (f : CNF α) : Sat a f ↔ (eval a f = true) | α : Type u_1
a : α → Bool
f : CNF α
⊢ Sat a f ↔ eval a f = true | rfl | no goals | eaa3eb5ddd8e7d51 |
RightDerivMeasurableAux.D_subset_differentiable_set | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem D_subset_differentiable_set {K : Set F} (hK : IsComplete K) :
D f K ⊆ { x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } | F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
x : ℝ
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → F
hn :
∀ (e p q : ℕ),
n e ≤ p →
n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2... | norm_num | no goals | 807a865e58286923 |
HomologicalComplex₂.D₂_totalShift₂XIso_hom | Mathlib/Algebra/Homology/TotalComplexShift.lean | @[reassoc]
lemma D₂_totalShift₂XIso_hom (n₀ n₁ n₀' n₁' : ℤ) (h₀ : n₀ + y = n₀') (h₁ : n₁ + y = n₁') :
((shiftFunctor₂ C y).obj K).D₂ (up ℤ) n₀ n₁ ≫ (K.totalShift₂XIso y n₁ n₁' h₁).hom =
y.negOnePow • ((K.totalShift₂XIso y n₀ n₀' h₀).hom ≫ K.D₂ (up ℤ) n₀' n₁') | case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : ¬(up ℤ).Rel n₀ n₁
⊢ ¬(up ℤ).Rel n₀' n₁' | intro h' | case neg.h₁₂
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : HomologicalComplex₂ C (up ℤ) (up ℤ)
y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + y = n₀'
h₁ : n₁ + y = n₁'
h : ¬(up ℤ).Rel n₀ n₁
h' : (up ℤ).Rel n₀' n₁'
⊢ False | f1ece82a737eae57 |
PowerBasis.adjoin_gen_eq_top | Mathlib/RingTheory/PowerBasis.lean | theorem adjoin_gen_eq_top (B : PowerBasis R S) : adjoin R ({B.gen} : Set S) = ⊤ | case intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
B : PowerBasis R S
i : Fin B.dim
⊢ B.gen ^ ↑i ∈ ↑(Subalgebra.toSubmodule (adjoin R {B.gen})) | exact Subalgebra.pow_mem _ (subset_adjoin (Set.mem_singleton _)) _ | no goals | db45fc28b296c790 |
MeasureTheory.Integrable.ae_eq_of_withDensityᵥ_eq | Mathlib/MeasureTheory/VectorMeasure/WithDensity.lean | theorem Integrable.ae_eq_of_withDensityᵥ_eq [CompleteSpace E] {f g : α → E} (hf : Integrable f μ)
(hg : Integrable g μ) (hfg : μ.withDensityᵥ f = μ.withDensityᵥ g) : f =ᵐ[μ] g | α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f g : α → E
hf : Integrable f μ
hg : Integrable g μ
hfg : μ.withDensityᵥ f = μ.withDensityᵥ g
i : Set α
hi : MeasurableSet i
x✝ : μ i < ⊤
⊢ ∫ (x : α) in i, f x ∂μ = ∫ (x : α) in i... | rw [← withDensityᵥ_apply hf hi, hfg, withDensityᵥ_apply hg hi] | no goals | 97e7521ee6fa6aa9 |
gauge_smul_of_nonneg | Mathlib/Analysis/Convex/Gauge.lean | theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
(ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x | case inr.e_a.h
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
α : Type u_3
inst✝⁴ : LinearOrderedField α
inst✝³ : MulActionWithZero α ℝ
inst✝² : OrderedSMul α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
⊢ r ∈ {r | r ∈ Ioi 0 ∧ r⁻¹ • a • ... | simp_rw [Set.mem_smul_set, Set.mem_sep_iff] | case inr.e_a.h
E : Type u_2
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
α : Type u_3
inst✝⁴ : LinearOrderedField α
inst✝³ : MulActionWithZero α ℝ
inst✝² : OrderedSMul α ℝ
inst✝¹ : MulActionWithZero α E
inst✝ : IsScalarTower α ℝ (Set E)
s : Set E
a : α
ha : 0 ≤ a
x : E
ha' : 0 < a
r : ℝ
⊢ r ∈ Ioi 0 ∧ r⁻¹ • a • x ∈ s ↔ ∃... | 2224562162173f32 |
Real.sin_gt_sub_cube | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | theorem sin_gt_sub_cube {x : ℝ} (h : 0 < x) (h' : x ≤ 1) : x - x ^ 3 / 4 < sin x | x : ℝ
h : 0 < x
h' : x ≤ 1
hx : |x| = x
⊢ |x| ≤ 1 | rwa [hx] | no goals | 02efa6669a36a1b3 |
Set.exists_equiv_range_eq_graphOn_univ | Mathlib/Data/Set/Function.lean | /-- **Line test** for equivalences.
Let `f : α → β × γ` be a homomorphism to a product of monoids. Assume that `f` is surjective on both
factors and that the image of `f` intersects every "vertical line" `{(b, c) | c : γ}` and every
"horizontal line" `{(b, c) | b : β}` at most once. Then the image of `f` is the graph ... | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β × γ
hf₁ : Surjective (Prod.fst ∘ f)
hf₂ : Surjective (Prod.snd ∘ f)
hf : ∀ (g₁ g₂ : α), (f g₁).1 = (f g₂).1 ↔ (f g₁).2 = (f g₂).2
⊢ ∃ e, range f = graphOn (⇑e) univ | obtain ⟨e₁, he₁⟩ := exists_range_eq_graphOn_univ hf₁ fun _ _ ↦ (hf _ _).1 | case intro
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → β × γ
hf₁ : Surjective (Prod.fst ∘ f)
hf₂ : Surjective (Prod.snd ∘ f)
hf : ∀ (g₁ g₂ : α), (f g₁).1 = (f g₂).1 ↔ (f g₁).2 = (f g₂).2
e₁ : β → γ
he₁ : range f = graphOn e₁ univ
⊢ ∃ e, range f = graphOn (⇑e) univ | e623d6d79db3116f |
MeasureTheory.FiniteMeasure.testAgainstNN_zero | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | theorem testAgainstNN_zero (μ : FiniteMeasure Ω) : μ.testAgainstNN 0 = 0 | Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : TopologicalSpace Ω
μ : FiniteMeasure Ω
⊢ μ.testAgainstNN 0 = 0 | simpa only [zero_mul] using μ.testAgainstNN_const 0 | no goals | eb429648d2c23f51 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.rupAdd_sound | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem rupAdd_sound {n : Nat} (f : DefaultFormula n) (c : DefaultClause n) (rupHints : Array Nat)
(f' : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(rupAddSuccess : performRupAdd f c rupHints = (f', true)) :
Liff (PosFin n) f f' | case isFalse.isFalse
n : Nat
f : DefaultFormula n
c : DefaultClause n
rupHints : Array Nat
f' : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
f'_def : f' = f.insert c
h✝¹ : ¬(f.insertRupUnits c.negate).snd = true
h✝ : ¬((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.snd = true
rupAddSuccess :
(i... | split at rupAddSuccess | case isFalse.isFalse.isTrue
n : Nat
f : DefaultFormula n
c : DefaultClause n
rupHints : Array Nat
f' : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
f'_def : f' = f.insert c
h✝² : ¬(f.insertRupUnits c.negate).snd = true
h✝¹ : ¬((f.insertRupUnits c.negate).fst.performRupCheck rupHints).2.2.snd = true
h✝ : ((f.ins... | 54bbe5dd0a90a45f |
Set.PairwiseDisjoint.prod_left | Mathlib/Data/Set/Pairwise/Lattice.lean | theorem PairwiseDisjoint.prod_left {f : ι × ι' → α}
(hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i'))
(ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')) :
(s ×ˢ t : Set (ι × ι')).PairwiseDisjoint f | case mk.mk.inr
α : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝ : CompleteLattice α
s : Set ι
t : Set ι'
f : ι × ι' → α
hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')
ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')
i : ι
i' : ι'
hi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t
j : ι
j' : ι'
hj : (j, j').1 ∈ s ∧ (j, j').2 ∈ t... | refine (hs hi.1 hj.1 hij).mono ?_ ?_ | case mk.mk.inr.refine_1
α : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝ : CompleteLattice α
s : Set ι
t : Set ι'
f : ι × ι' → α
hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')
ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')
i : ι
i' : ι'
hi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t
j : ι
j' : ι'
hj : (j, j').1 ∈ s ∧ (j, ... | 34740219f815d1c5 |
List.sublist_flatten_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem sublist_flatten_iff {L : List (List α)} {l} :
l <+ L.flatten ↔
∃ L' : List (List α), l = L'.flatten ∧ ∀ i (_ : i < L'.length), L'[i] <+ L[i]?.getD [] | case cons.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
l' : List α
L : List (List α)
ih : ∀ {l : List α}, l <+ L.flatten ↔ ∃ L', l = L'.flatten ∧ ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []
l₁ : List α
s : l₁ <+ l'
L' : List (List α)
h : ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []
⊢ ∀ (i ... | intro i lt | case cons.mp.intro.intro.intro.intro.intro.intro
α : Type u_1
l' : List α
L : List (List α)
ih : ∀ {l : List α}, l <+ L.flatten ↔ ∃ L', l = L'.flatten ∧ ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []
l₁ : List α
s : l₁ <+ l'
L' : List (List α)
h : ∀ (i : Nat) (x : i < L'.length), L'[i] <+ L[i]?.getD []
i : Nat... | 2d054bb50b16d9ba |
map_inv₀ | Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ | G₀ : Type u_3
G₀' : Type u_5
F : Type u_6
inst✝³ : GroupWithZero G₀
inst✝² : GroupWithZero G₀'
inst✝¹ : FunLike F G₀ G₀'
inst✝ : MonoidWithZeroHomClass F G₀ G₀'
f : F
a : G₀
⊢ f a⁻¹ = (f a)⁻¹ | by_cases h : a = 0 | case pos
G₀ : Type u_3
G₀' : Type u_5
F : Type u_6
inst✝³ : GroupWithZero G₀
inst✝² : GroupWithZero G₀'
inst✝¹ : FunLike F G₀ G₀'
inst✝ : MonoidWithZeroHomClass F G₀ G₀'
f : F
a : G₀
h : a = 0
⊢ f a⁻¹ = (f a)⁻¹
case neg
G₀ : Type u_3
G₀' : Type u_5
F : Type u_6
inst✝³ : GroupWithZero G₀
inst✝² : GroupWithZero G₀'
inst... | 847261fb18c66710 |
DirichletCharacter.re_log_comb_nonneg | Mathlib/NumberTheory/LSeries/Nonvanishing.lean | private lemma re_log_comb_nonneg {n : ℕ} (hn : 2 ≤ n) {x : ℝ} (hx : 1 < x) (y : ℝ) :
0 ≤ 3 * (-log (1 - (1 : DirichletCharacter ℂ N) n * n ^ (-x : ℂ))).re +
4 * (-log (1 - χ n * n ^ (-(x + I * y)))).re +
(-log (1 - (χ n ^ 2) * n ^ (-(x + 2 * I * y)))).re | case pos
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
ha₀ : 0 ≤ ↑n ^ (-x)
ha₁ : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ 0 ≤
3 * (-log (1 - 1 ↑n * ↑n ^ (-↑x))).re + 4 * (-log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y)))).re +
(-log (1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I... | rw [MulChar.one_apply hn', one_mul] | case pos
N : ℕ
χ : DirichletCharacter ℂ N
n : ℕ
hn : 2 ≤ n
x : ℝ
hx : 1 < x
y : ℝ
hn' : IsUnit ↑n
ha₀ : 0 ≤ ↑n ^ (-x)
ha₁ : ↑n ^ (-x) < 1
hz : ‖χ ↑n * ↑n ^ (-(I * ↑y))‖ = 1
⊢ 0 ≤
3 * (-log (1 - ↑n ^ (-↑x))).re + 4 * (-log (1 - χ ↑n * ↑n ^ (-(↑x + I * ↑y)))).re +
(-log (1 - χ ↑n ^ 2 * ↑n ^ (-(↑x + 2 * I * ↑y))... | 7c14ed735b43068e |
LocalSubring.exists_le_valuationSubring | Mathlib/RingTheory/Valuation/LocalSubring.lean | @[stacks 00IA]
lemma LocalSubring.exists_le_valuationSubring (A : LocalSubring K) :
∃ B : ValuationSubring K, A ≤ B.toLocalSubring | case refine_2
K : Type u_3
inst✝ : Field K
A✝ : LocalSubring K
s : Set (LocalSubring K)
hs : s ⊆ Set.Ici A✝
H : IsChain (fun x1 x2 => x1 ≤ x2) s
y : LocalSubring K
hys : y ∈ s
inst : Nonempty ↑s
hdir : Directed LE.le (toSubring ∘ fun x => ↑x)
A : LocalSubring K
hA : A ∈ s
⊢ A ≤ mk (⨆ i, (↑i).toSubring) | refine ⟨le_iSup (fun i : s ↦ i.1.toSubring) ⟨A, hA⟩, ⟨?_⟩⟩ | case refine_2
K : Type u_3
inst✝ : Field K
A✝ : LocalSubring K
s : Set (LocalSubring K)
hs : s ⊆ Set.Ici A✝
H : IsChain (fun x1 x2 => x1 ≤ x2) s
y : LocalSubring K
hys : y ∈ s
inst : Nonempty ↑s
hdir : Directed LE.le (toSubring ∘ fun x => ↑x)
A : LocalSubring K
hA : A ∈ s
⊢ ∀ (a : ↥A.toSubring), IsUnit ((Subring.inclus... | 926576b935296dd2 |
Equiv.Perm.isCycle_swap_mul_aux₁ | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem isCycle_swap_mul_aux₁ {α : Type*} [DecidableEq α] :
∀ (n : ℕ) {b x : α} {f : Perm α} (_ : (swap x (f x) * f) b ≠ b) (_ : (f ^ n) (f x) = b),
∃ i : ℤ, ((swap x (f x) * f) ^ i) (f x) = b | α : Type u_4
inst✝ : DecidableEq α
n : ℕ
hn : ∀ {b x : α} {f : Perm α}, (swap x (f x) * f) b ≠ b → (f ^ n) (f x) = b → ∃ i, ((swap x (f x) * f) ^ i) (f x) = b
b x : α
f : Perm α
hb : (swap x (f x) * f) b ≠ b
h : (f ^ (n + 1)) (f x) = b
hfbx : ¬f x = b
this : f b ≠ b ∧ b ≠ x
⊢ ¬f b = b | exact this.1 | no goals | 788ae8872959a724 |
Fin.addRothNumber_le_rothNumberNat | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | lemma Fin.addRothNumber_le_rothNumberNat (k n : ℕ) (hkn : k ≤ n) :
addRothNumber (Iio k : Finset (Fin n.succ)) ≤ rothNumberNat k | case refine_2
k n : ℕ
hkn : k ≤ n
⊢ Set.SurjOn Nat.cast ↑(range k) ↑(Iio ↑k) | simp only [Set.SurjOn, coe_Iio, Set.subset_def, Set.mem_Iio, Set.mem_image, lt_iff_val_lt_val,
val_cast_of_lt, Nat.lt_succ_iff.2 hkn, coe_range] | case refine_2
k n : ℕ
hkn : k ≤ n
⊢ ∀ (x : Fin n.succ), ↑x < k → ∃ x_1 < k, ↑x_1 = x | 7e145f3c7da155cb |
FormalMultilinearSeries.comp_coeff_zero | Mathlib/Analysis/Analytic/Composition.lean | theorem comp_coeff_zero (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
(v : Fin 0 → E) (v' : Fin 0 → F) : (q.comp p) 0 v = q 0 v' | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologica... | symm | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁵ : CommRing 𝕜
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : AddCommGroup F
inst✝¹² : AddCommGroup G
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : Module 𝕜 F
inst✝⁹ : Module 𝕜 G
inst✝⁸ : TopologicalSpace E
inst✝⁷ : TopologicalSpace F
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologica... | 447f3c3d95e7a73f |
PowerSeries.rescale_mk | Mathlib/RingTheory/PowerSeries/Basic.lean | theorem rescale_mk (f : ℕ → R) (a : R) : rescale a (mk f) = mk fun n : ℕ => a ^ n * f n | R : Type u_1
inst✝ : CommSemiring R
f : ℕ → R
a : R
⊢ (rescale a) (mk f) = mk fun n => a ^ n * f n | ext | case h
R : Type u_1
inst✝ : CommSemiring R
f : ℕ → R
a : R
n✝ : ℕ
⊢ (coeff R n✝) ((rescale a) (mk f)) = (coeff R n✝) (mk fun n => a ^ n * f n) | 301d4f27ad0331ce |
Matrix.charpoly.optionEquivLeft_symm_univ_isHomogeneous | Mathlib/LinearAlgebra/Matrix/Charpoly/Univ.lean | lemma optionEquivLeft_symm_univ_isHomogeneous :
((optionEquivLeft R (n × n)).symm (univ R n)).IsHomogeneous (Fintype.card n) | case h
R : Type u_1
n : Type u_3
inst✝² : CommRing R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
aux : Fintype.card n = 0 + ∑ i : n, 1
⊢ ∀ i ∈ Finset.univ,
(↑↑(Equiv.Perm.sign i) *
∏ x : n,
((Polynomial.aevalTower (rename some) (X none)) (if i x = x then Polynomial.X else 0) -
X (so... | rintro i - | case h
R : Type u_1
n : Type u_3
inst✝² : CommRing R
inst✝¹ : Fintype n
inst✝ : DecidableEq n
aux : Fintype.card n = 0 + ∑ i : n, 1
i : Equiv.Perm n
⊢ (↑↑(Equiv.Perm.sign i) *
∏ x : n,
((Polynomial.aevalTower (rename some) (X none)) (if i x = x then Polynomial.X else 0) -
X (some (i x, x))... | 642ea7d2efa7e9c8 |
fixed_of_fixed2 | Mathlib/RingTheory/Invariant.lean | theorem fixed_of_fixed2 (f : L ≃ₐ[K] L) (x : L)
(hx : ∀ g : MulAction.stabilizer G Q, IsFractionRing.stabilizerHom G P Q K L g x = x) :
f x = x | case intro.intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹² : Q.IsPrime
inst✝¹¹ : Q.LiesOver P
K : ... | replace hy : algebraMap (B ⧸ Q) L y ≠ 0 :=
mt (algebraMap_eq_zero_iff (B ⧸ Q) L).mp (nonZeroDivisors.ne_zero hy) | case intro.intro.intro.intro.intro.intro.intro
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Group G
inst✝¹⁵ : Finite G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : SMulCommClass G A B
P : Ideal A
Q : Ideal B
inst✝¹² : Q.IsPrime
inst✝¹¹ : Q.LiesOver P
K : ... | b1e4ff70683aeec1 |
Array.countP_false | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Count.lean | theorem countP_false : (countP fun (_ : α) => false) = Function.const _ 0 | case h
α : Type u_1
l : Array α
⊢ countP (fun x => false) l = Function.const (Array α) 0 l | simp | no goals | 8bea6f16e284c95c |
bernstein.probability | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | theorem probability (n : ℕ) (x : I) : (∑ k : Fin (n + 1), bernstein n k x) = 1 | n : ℕ
x : ↑I
this : (Polynomial.aeval ↑x) (∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial ℝ n ν) = (Polynomial.aeval ↑x) 1
⊢ ∑ k : Fin (n + 1), (bernstein n ↑k) x = 1 | simp? [map_sum, Finset.sum_range] at this says
simp only [Finset.sum_range, map_sum, Polynomial.coe_aeval_eq_eval, Polynomial.eval_one] at this | n : ℕ
x : ↑I
this : ∑ x_1 : Fin (n + 1), Polynomial.eval (↑x) (bernsteinPolynomial ℝ n ↑x_1) = 1
⊢ ∑ k : Fin (n + 1), (bernstein n ↑k) x = 1 | 3fa61dc5eb7bad44 |
SimpleGraph.farFromTriangleFree_of_disjoint_triangles_aux | Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean | private lemma farFromTriangleFree_of_disjoint_triangles_aux {tris : Finset (Finset α)}
(htris : tris ⊆ G.cliqueFinset 3)
(pd : (tris : Set (Finset α)).Pairwise fun x y ↦ (x ∩ y : Set α).Subsingleton) (hHG : H ≤ G)
(hH : H.CliqueFree 3) : #tris ≤ #G.edgeFinset - #H.edgeFinset | case intro.mk.intro.intro.mk.intro.intro.inl
α : Type u_1
G H : SimpleGraph α
inst✝³ : Fintype α
inst✝² : DecidableRel G.Adj
inst✝¹ : DecidableRel H.Adj
inst✝ : DecidableEq α
tris : Finset (Finset α)
htris : tris ⊆ G.cliqueFinset 3
pd : (↑tris).Pairwise fun x y => (↑x ∩ ↑y).Subsingleton
hHG : H ≤ G
hH : H.CliqueFree 3
... | exact hfne _ (i ⟨hfx ht₁, t.1.symm ▸ hfx ht₂⟩ ⟨hfy ht₁, t.2.symm ▸ hfy ht₂⟩) | no goals | 4bd61c8a2996b1e3 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_fold_fn_preserves_induction_motive | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem reduce_fold_fn_preserves_induction_motive {c_arr : Array (Literal (PosFin n))}
{assignment : Array Assignment}
(idx : Fin c_arr.size) (res : ReduceResult (PosFin n))
(ih : ReducePostconditionInductionMotive c_arr assignment idx.1 res) :
ReducePostconditionInductionMotive c_arr assignment (idx.1 + ... | case right.h_2.h_4.inl
n : Nat
c_arr : Array (Literal (PosFin n))
assignment : Array Assignment
idx : Fin c_arr.size
i : PosFin n
b : Bool
p : PosFin n → Bool
hp : p ⊨ assignment
j : Fin c_arr.size
j_lt_idx_add_one : ↑j < ↑idx + 1
p_entails_c_arr_j : p ⊨ c_arr[↑j]
acc✝ : ReduceResult (PosFin n)
ih : ReducePostcondition... | rcases ih.1 rfl p with ih1 | ih1 | case right.h_2.h_4.inl.inl
n : Nat
c_arr : Array (Literal (PosFin n))
assignment : Array Assignment
idx : Fin c_arr.size
i : PosFin n
b : Bool
p : PosFin n → Bool
hp : p ⊨ assignment
j : Fin c_arr.size
j_lt_idx_add_one : ↑j < ↑idx + 1
p_entails_c_arr_j : p ⊨ c_arr[↑j]
acc✝ : ReduceResult (PosFin n)
ih : ReducePostcondi... | 681a60500ce41fce |
Matroid.Indep.closure_sInter_eq_biInter_closure_of_forall_subset | Mathlib/Data/Matroid/Closure.lean | /-- For a nonempty collection of subsets of a given independent set,
the closure of the intersection is the intersection of the closure. -/
lemma Indep.closure_sInter_eq_biInter_closure_of_forall_subset {Js : Set (Set α)} (hI : M.Indep I)
(hne : Js.Nonempty) (hIs : ∀ J ∈ Js, J ⊆ I) : M.closure (⋂₀ Js) = (⋂ J ∈ Js, ... | α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
e : α
he : e ∈ ⋂ J ∈ Js, M.closure J
he' : e ∉ M.closure (⋂₀ Js)
⊢ False | rw [mem_iInter₂] at he | α : Type u_2
M : Matroid α
I : Set α
Js : Set (Set α)
hI : M.Indep I
hne : Js.Nonempty
hIs : ∀ J ∈ Js, J ⊆ I
hiX : ⋂₀ Js ⊆ I
hiI : M.Indep (⋂₀ Js)
e : α
he : ∀ i ∈ Js, e ∈ M.closure i
he' : e ∉ M.closure (⋂₀ Js)
⊢ False | ce51522815543212 |
CategoryTheory.Localization.exists_leftFraction₂ | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean | lemma exists_leftFraction₂ {X Y : C} (f f' : L.obj X ⟶ L.obj Y) :
∃ (φ : W.LeftFraction₂ X Y), f = φ.fst.map L (inverts L W) ∧
f' = φ.snd.map L (inverts L W) | C : Type u_1
D : Type u_2
inst✝³ : Category.{u_4, u_1} C
inst✝² : Category.{u_3, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map L ⋯
α : W.LeftFra... | infer_instance | no goals | 97a00df262ce14bf |
Array.getElem_ofFn_go | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem getElem_ofFn_go (f : Fin n → α) (i) {acc k}
(hki : k < n) (hin : i ≤ n) (hi : i = acc.size)
(hacc : ∀ j, ∀ hj : j < acc.size, acc[j] = f ⟨j, Nat.lt_of_lt_of_le hj (hi ▸ hin)⟩) :
haveI : acc.size + (n - acc.size) = n := Nat.add_sub_cancel' (hi ▸ hin)
(ofFn.go f i acc)[k]'(by simp [*]) = f ⟨k, hki... | n : Nat
α : Type u_1
f : Fin n → α
i : Nat
acc : Array α
k : Nat
hki : k < n
hin✝ : i ≤ n
hi : i = acc.size
hacc : ∀ (j : Nat) (hj : j < acc.size), acc[j] = f ⟨j, ⋯⟩
hin : i < n
⊢ (if h : i < n then ofFn.go f (i + 1) (acc.push (f ⟨i, h⟩)) else acc)[k] = f ⟨k, hki⟩ | have : 1 + (n - (i + 1)) = n - i :=
Nat.sub_sub .. ▸ Nat.add_sub_cancel' (Nat.le_sub_of_add_le (Nat.add_comm .. ▸ hin)) | n : Nat
α : Type u_1
f : Fin n → α
i : Nat
acc : Array α
k : Nat
hki : k < n
hin✝ : i ≤ n
hi : i = acc.size
hacc : ∀ (j : Nat) (hj : j < acc.size), acc[j] = f ⟨j, ⋯⟩
hin : i < n
this : 1 + (n - (i + 1)) = n - i
⊢ (if h : i < n then ofFn.go f (i + 1) (acc.push (f ⟨i, h⟩)) else acc)[k] = f ⟨k, hki⟩ | f1c59ef786bbde82 |
MeromorphicAt.inv | Mathlib/Analysis/Meromorphic/Basic.lean | @[fun_prop]
lemma inv {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) : MeromorphicAt f⁻¹ x | case h
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f : 𝕜 → 𝕜
x : 𝕜
m : ℕ
hf : AnalyticAt 𝕜 (fun z => (z - x) ^ m • f z) x
h_eq : ¬(fun z => (z - x) ^ m • f z) =ᶠ[𝓝 x] 0
n : ℕ
g : 𝕜 → 𝕜
hg_an : AnalyticAt 𝕜 g x
hg_ne : g x ≠ 0
hg_eq : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ m • f z = (z - x) ^ n • g z
this : Analyti... | rcases eq_or_ne z x with rfl | hz_ne | case h.inl
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
f : 𝕜 → 𝕜
m n : ℕ
g : 𝕜 → 𝕜
z : 𝕜
hg_ne' : g z ≠ 0
hf : AnalyticAt 𝕜 (fun z_1 => (z_1 - z) ^ m • f z_1) z
h_eq : ¬(fun z_1 => (z_1 - z) ^ m • f z_1) =ᶠ[𝓝 z] 0
hg_an : AnalyticAt 𝕜 g z
hg_ne : g z ≠ 0
hg_eq : ∀ᶠ (z_1 : 𝕜) in 𝓝 z, (z_1 - z) ^ m • f z_1... | 1228140cb221b1ad |
hasFPowerSeriesWithinAt_univ | Mathlib/Analysis/Analytic/Basic.lean | @[simp] lemma hasFPowerSeriesWithinAt_univ :
HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
⊢ HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x | simp only [HasFPowerSeriesWithinAt, hasFPowerSeriesWithinOnBall_univ, HasFPowerSeriesAt] | no goals | 3b4260906663fc14 |
WeierstrassCurve.toCharTwoJNeZeroNF_spec | Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean | theorem toCharTwoJNeZeroNF_spec (ha₁ : W.a₁ ≠ 0) :
(W.variableChange (W.toCharTwoJNeZeroNF ha₁)).IsCharTwoJNeZeroNF | case a₁
F : Type u_2
inst✝¹ : Field F
inst✝ : CharP F 2
W : WeierstrassCurve F
ha₁ : W.a₁ ≠ 0
⊢ (W.variableChange (W.toCharTwoJNeZeroNF ha₁)).a₁ = 1 | simp [toCharTwoJNeZeroNF, ha₁] | no goals | 4fddc7c74e73ca2b |
Asymptotics.superpolynomialDecay_iff_isLittleO | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | theorem superpolynomialDecay_iff_isLittleO (hk : Tendsto k l atTop) :
SuperpolynomialDecay l k f ↔ ∀ z : ℤ, f =o[l] fun a : α => k a ^ z | α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝¹ : NormedLinearOrderedField β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
this : (fun x => 1) =o[l] k
⊢ f =o[l] fun a => k a ^ z | have : f =o[l] fun x : α => k x * k x ^ (z - 1) := by
simpa using this.mul_isBigO ((superpolynomialDecay_iff_isBigO f hk).1 h <| z - 1) | α : Type u_1
β : Type u_2
l : Filter α
k f : α → β
inst✝¹ : NormedLinearOrderedField β
inst✝ : OrderTopology β
hk : Tendsto k l atTop
h : SuperpolynomialDecay l k f
z : ℤ
hk0 : ∀ᶠ (x : α) in l, k x ≠ 0
this✝ : (fun x => 1) =o[l] k
this : f =o[l] fun x => k x * k x ^ (z - 1)
⊢ f =o[l] fun a => k a ^ z | a094d01544660196 |
Equiv.Perm.eq_on_support_mem_disjoint | Mathlib/GroupTheory/Perm/Support.lean | theorem eq_on_support_mem_disjoint {l : List (Perm α)} (h : f ∈ l) (hl : l.Pairwise Disjoint) :
∀ x ∈ f.support, f x = l.prod x | case cons.inr
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f hd : Perm α
tl : List (Perm α)
IH : f ∈ tl → List.Pairwise Disjoint tl → ∀ x ∈ f.support, f x = tl.prod x
hl : (∀ a' ∈ tl, hd.Disjoint a') ∧ List.Pairwise Disjoint tl
x : α
hx : x ∈ f.support
h : f ∈ tl
⊢ f x ∈ f.support | simpa using hx | no goals | 1eb7dcf2895333eb |
Multiset.covBy_cons | Mathlib/Data/Finset/Grade.lean | @[simp] lemma covBy_cons (s : Multiset α) (a : α) : s ⋖ a ::ₘ s :=
⟨lt_cons_self _ _, fun t hst hts ↦ (covBy_succ _).2 (card_lt_card hst) <| by
simpa using card_lt_card hts⟩
| α : Type u_1
s : Multiset α
a : α
t : Multiset α
hst : s < t
hts : t < a ::ₘ s
⊢ t.card < succ s.card | simpa using card_lt_card hts | no goals | 57bde0a1685b86fd |
isOpen_setOf_nat_le_rank | Mathlib/Analysis/Normed/Module/FiniteDimension.lean | theorem isOpen_setOf_nat_le_rank (n : ℕ) :
IsOpen { f : E →L[𝕜] F | ↑n ≤ (f : E →ₗ[𝕜] F).rank } | 𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type w
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace 𝕜
n : ℕ
t : Finset E
x✝ : t ∈ fun i => i.card = n
⊢ IsOpen {x | LinearIndependent 𝕜 fun (x_1 : ↑↑t) => ↑x ↑x_1} | have : Continuous fun f : E →L[𝕜] F => fun x : (t : Set E) => f x :=
continuous_pi fun x => (ContinuousLinearMap.apply 𝕜 F (x : E)).continuous | 𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type w
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace 𝕜
n : ℕ
t : Finset E
x✝ : t ∈ fun i => i.card = n
this : Continuous fun f x => f ↑x
⊢ IsOpen {x | LinearIndependent... | 17ea30aca388f58f |
Set.inclusion_inclusion | Mathlib/Data/Set/Basic.lean | theorem inclusion_inclusion (hst : s ⊆ t) (htu : t ⊆ u) (x : s) :
inclusion htu (inclusion hst x) = inclusion (hst.trans htu) x | case mk
α : Type u_1
s t u : Set α
hst : s ⊆ t
htu : t ⊆ u
val✝ : α
property✝ : val✝ ∈ s
⊢ inclusion htu (inclusion hst ⟨val✝, property✝⟩) = inclusion ⋯ ⟨val✝, property✝⟩ | rfl | no goals | 8813dc4000a63973 |
PNat.XgcdType.step_wf | Mathlib/Data/PNat/Xgcd.lean | theorem step_wf (hr : u.r ≠ 0) : SizeOf.sizeOf u.step < SizeOf.sizeOf u | u : XgcdType
hr : u.r ≠ 0
⊢ sizeOf u.step < sizeOf u | change u.r - 1 < u.bp | u : XgcdType
hr : u.r ≠ 0
⊢ u.r - 1 < u.bp | 78c5ae280a13ad01 |
ENNReal.tsum_geometric | Mathlib/Analysis/SpecificLimits/Basic.lean | theorem ENNReal.tsum_geometric (r : ℝ≥0∞) : ∑' n : ℕ, r ^ n = (1 - r)⁻¹ | r : ℝ≥0∞
hr : 1 ≤ r
a : ℝ≥0∞
ha : a < ⊤
n : ℕ
hn : a < ↑n
⊢ ↑n = ∑ i ∈ Finset.range n, 1 | rw [sum_const, nsmul_one, card_range] | no goals | acfc256855063efc |
Real.hasDerivAt_Gamma_one_half | Mathlib/NumberTheory/Harmonic/GammaDeriv.lean | lemma hasDerivAt_Gamma_one_half : HasDerivAt Gamma (-√π * (γ + 2 * log 2)) (1 / 2) | case e_a.e_a.e_a.h.refine_1
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s => Gamma (2 * s)) s
⊢ HasDerivAt (fun s => 1 - s * 2) (-2) (1 / 2) | apply HasDerivAt.const_sub | case e_a.e_a.e_a.h.refine_1.hf
h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s
h_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s => Gamma (2 * s)) s
⊢ HasDerivAt (fun x => x * 2) 2 (1 / 2) | 1ee63d35de254cb4 |
Class.classToCong_empty | Mathlib/SetTheory/ZFC/Basic.lean | theorem classToCong_empty : classToCong ∅ = ∅ | case h
x✝ : Class.{u_1}
⊢ x✝ ∈ ∅.classToCong ↔ x✝ ∈ ∅ | simp [classToCong] | no goals | dfb3844d2d352b4f |
RingHom.surjective_localRingHom_iff | Mathlib/RingTheory/SurjectiveOnStalks.lean | /--
`R_p →+* S_q` is surjective if and only if
every `x : S` is of the form `f x / f r` for some `f r ∉ q`.
This is useful when proving `SurjectiveOnStalks`.
-/
lemma surjective_localRingHom_iff (P : Ideal S) [P.IsPrime] :
Function.Surjective (Localization.localRingHom _ P f rfl) ↔
∀ s : S, ∃ x r : R, ∃ c ∉ P... | case mp.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f : R →+* S
P : Ideal S
inst✝ : P.IsPrime
H : Function.Surjective ⇑(Localization.localRingHom (Ideal.comap f P) P f ⋯)
y : S
a : R
t : ↥(Ideal.comap f P).primeCompl
ha : ∃ c, ↑c * f a = ↑c * (↑⟨f ↑t, ⋯⟩ * y)
⊢ ∃ x r, ∃ c ∉ P, f ... | obtain ⟨c, hc⟩ := ha | case mp.intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
S : Type u_2
inst✝¹ : CommRing S
f : R →+* S
P : Ideal S
inst✝ : P.IsPrime
H : Function.Surjective ⇑(Localization.localRingHom (Ideal.comap f P) P f ⋯)
y : S
a : R
t : ↥(Ideal.comap f P).primeCompl
c : ↥P.primeCompl
hc : ↑c * f a = ↑c * (↑⟨f ↑t, ⋯⟩ * y)
⊢... | b80db46892fd2c98 |
Batteries.UnionFind.equiv_union | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean | theorem equiv_union {self : UnionFind} {x y : Fin self.size} :
Equiv (union self x y) a b ↔
Equiv self a b ∨ Equiv self a x ∧ Equiv self y b ∨ Equiv self a y ∧ Equiv self x b | a b : Nat
self : UnionFind
x y : Fin self.size
⊢ ((self.find x).fst.find ⟨↑y, ⋯⟩).fst.parent ↑⟨self.rootD ↑x, ⋯⟩ = ↑⟨self.rootD ↑x, ⋯⟩ | simp [← rootD_eq_self, rootD_rootD] | no goals | 1090fa5f00490799 |
WeierstrassCurve.exists_variableChange_of_char_ne_two_or_three | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_ne_two_or_three
{p : ℕ} [CharP F p] (hchar2 : p ≠ 2) (hchar3 : p ≠ 3) (heq : E.j = E'.j) :
∃ C : VariableChange F, E.variableChange C = E' | case h.a₂
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 :=... | simp | no goals | 5599f8482dc6841e |
Cardinal.mk_Ioi_real | Mathlib/Data/Real/Cardinality.lean | theorem mk_Ioi_real (a : ℝ) : #(Ioi a) = 𝔠 | a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
⊢ #↑(Iio a) + #↑{a} + #↑(Ioi a) < 𝔠 | have h2 : (fun x => a + a - x) '' Ioi a = Iio a := by
convert @image_const_sub_Ioi ℝ _ _ _
simp | a : ℝ
h : #↑(Ioi a) < 𝔠
hu : Iio a ∪ {a} ∪ Ioi a = Set.univ
h2 : (fun x => a + a - x) '' Ioi a = Iio a
⊢ #↑(Iio a) + #↑{a} + #↑(Ioi a) < 𝔠 | 564ae2d1d94f837f |
Ideal.Filtration.submodule_eq_span_le_iff_stable_ge | Mathlib/RingTheory/Filtration.lean | theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
F.submodule = Submodule.span _ (⋃ i ≤ n₀, single R i '' (F.N i : Set M)) ↔
∀ n ≥ n₀, I • F.N n = F.N (n + 1) | case mpr.succ
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n ... | by_cases hj' : j.succ ≤ n₀ | case pos
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
n₀ : ℕ
F' : Submodule (↥(reesAlgebra I)) (PolynomialModule R M) :=
Submodule.span (↥(reesAlgebra I)) (⋃ i, ⋃ (_ : i ≤ n₀), ⇑(single R i) '' ↑(F.N i))
hF : ∀ n ≥ n₀, I • F.N n = F.N (n + 1)
... | fba02d553ffa8877 |
Ideal.range_cotangentToQuotientSquare | Mathlib/RingTheory/Ideal/Cotangent.lean | theorem range_cotangentToQuotientSquare :
LinearMap.range I.cotangentToQuotientSquare = I.cotangentIdeal.restrictScalars R | R : Type u
inst✝ : CommRing R
I : Ideal R
⊢ LinearMap.range (I.cotangentToQuotientSquare ∘ₗ I.toCotangent) = Submodule.restrictScalars R I.cotangentIdeal | rw [to_quotient_square_comp_toCotangent, LinearMap.range_comp, I.range_subtype] | R : Type u
inst✝ : CommRing R
I : Ideal R
⊢ Submodule.map (Submodule.mkQ (I ^ 2)) I = Submodule.restrictScalars R I.cotangentIdeal | c42965bde81eee75 |
not_wcovBy_iff | Mathlib/Order/Cover.lean | theorem not_wcovBy_iff (h : a ≤ b) : ¬a ⩿ b ↔ ∃ c, a < c ∧ c < b | α : Type u_1
inst✝ : Preorder α
a b : α
h : a ≤ b
⊢ ¬a ⩿ b ↔ ∃ c, a < c ∧ c < b | simp_rw [WCovBy, h, true_and, not_forall, exists_prop, not_not] | no goals | 1cc6bea37c0e72c4 |
IsDenseInducing.closure_image_mem_nhds | Mathlib/Topology/DenseEmbedding.lean | theorem closure_image_mem_nhds {s : Set α} {a : α} (di : IsDenseInducing i) (hs : s ∈ 𝓝 a) :
closure (i '' s) ∈ 𝓝 (i a) | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
s : Set α
a : α
di : IsDenseInducing i
hs : ∃ i_1, (i a ∈ i_1 ∧ IsOpen i_1) ∧ i ⁻¹' i_1 ⊆ s
⊢ closure (i '' s) ∈ 𝓝 (i a) | rcases hs with ⟨U, ⟨haU, hUo⟩, sub : i ⁻¹' U ⊆ s⟩ | case intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
i : α → β
s : Set α
a : α
di : IsDenseInducing i
U : Set β
sub : i ⁻¹' U ⊆ s
haU : i a ∈ U
hUo : IsOpen U
⊢ closure (i '' s) ∈ 𝓝 (i a) | 766d1483a6910d14 |
MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
(f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ | case h
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
A : (fun x => (f x)⁻¹ * i x) ≤ g
x : α
h'x : f x < ⊤
⊢ i x ≤ (f * fun a => (f a)⁻¹ * i a) x | rcases eq_or_ne (f x) 0 with (hx | hx) | case h.inl
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
A : (fun x => (f x)⁻¹ * i x) ≤ g
x : α
h'x : f x < ⊤
hx : f x = 0
⊢ i x ≤ (f * fun a => (f a)⁻¹ * i a) x
case h.inr
α : Type u_1
m0 : Measurabl... | fdfc11b58110e190 |
LieModule.coe_lowerCentralSeries_ideal_le | Mathlib/Algebra/Lie/Nilpotent.lean | theorem LieModule.coe_lowerCentralSeries_ideal_le {I : LieIdeal R L} (k : ℕ) :
LieSubmodule.toSubmodule (lowerCentralSeries R I I k) ≤ lowerCentralSeries R L I k | case succ
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
k : ℕ
ih : ↑(lowerCentralSeries R (↥I) (↥I) k) ≤ ↑(lowerCentralSeries R L (↥I) k)
⊢ Submodule.span R {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ≤ Submodule.span R {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} | apply Submodule.span_mono | case succ.h
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
k : ℕ
ih : ↑(lowerCentralSeries R (↥I) (↥I) k) ≤ ↑(lowerCentralSeries R L (↥I) k)
⊢ {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} ⊆ {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x} | e5e433f33f1be4b1 |
ContDiff.prod_map | Mathlib/Analysis/Calculus/ContDiff/Operations.lean | theorem ContDiff.prod_map {f : E → F} {g : E' → F'} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
ContDiff 𝕜 n (Prod.map f g) | 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
n : WithTop ℕ∞
E' : Type u_3
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
F' : Type u_4
inst✝¹ : NormedAddCommGroup F'
in... | rw [contDiff_iff_contDiffAt] at * | 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
n : WithTop ℕ∞
E' : Type u_3
inst✝³ : NormedAddCommGroup E'
inst✝² : NormedSpace 𝕜 E'
F' : Type u_4
inst✝¹ : NormedAddCommGroup F'
in... | 29d90c98c8839a6c |
Polynomial.isJacobsonRing_polynomial_of_isJacobsonRing | Mathlib/RingTheory/Jacobson/Ring.lean | theorem isJacobsonRing_polynomial_of_isJacobsonRing (hR : IsJacobsonRing R) :
IsJacobsonRing R[X] | R : Type u_1
inst✝ : CommRing R
hR : IsJacobsonRing R
I : Ideal R[X]
hI : I.IsPrime
R' : Subring (R[X] ⧸ I) := ((Ideal.Quotient.mk I).comp C).range
i : R →+* ↥R' := ((Ideal.Quotient.mk I).comp C).rangeRestrict
hi : Function.Surjective ⇑i
f : R[X]
hf : f ∈ RingHom.ker (mapRingHom i)
n : ℕ
⊢ f.coeff n ∈ comap C I | replace hf := congrArg (fun g : Polynomial ((Ideal.Quotient.mk I).comp C).range => g.coeff n) hf | R : Type u_1
inst✝ : CommRing R
hR : IsJacobsonRing R
I : Ideal R[X]
hI : I.IsPrime
R' : Subring (R[X] ⧸ I) := ((Ideal.Quotient.mk I).comp C).range
i : R →+* ↥R' := ((Ideal.Quotient.mk I).comp C).rangeRestrict
hi : Function.Surjective ⇑i
f : R[X]
n : ℕ
hf : (fun g => g.coeff n) ((mapRingHom i) f) = (fun g => g.coeff n)... | 60e32d0d54e0756e |
Zsqrtd.norm_eq_zero_iff | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem norm_eq_zero_iff {d : ℤ} (hd : d < 0) (z : ℤ√d) : z.norm = 0 ↔ z = 0 | case mp
d : ℤ
hd : d < 0
z : ℤ√d
h : z.re * z.re + -(d * (z.im * z.im)) = 0
left : 0 ≤ z.re * z.re
right : 0 ≤ -(d * (z.im * z.im))
⊢ z = 0 | obtain ⟨ha, hb⟩ := (add_eq_zero_iff_of_nonneg left right).mp h | case mp.intro
d : ℤ
hd : d < 0
z : ℤ√d
h : z.re * z.re + -(d * (z.im * z.im)) = 0
left : 0 ≤ z.re * z.re
right : 0 ≤ -(d * (z.im * z.im))
ha : z.re * z.re = 0
hb : -(d * (z.im * z.im)) = 0
⊢ z = 0 | 0bccb488fc86b113 |
Matrix.det_ne_zero_of_sum_col_pos | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | /-- A matrix whose nondiagonal entries are negative with the sum of the entries of each
column positive has nonzero determinant. -/
lemma det_ne_zero_of_sum_col_pos [DecidableEq n] {S : Type*} [LinearOrderedCommRing S]
{A : Matrix n n S} (h1 : Pairwise fun i j => A i j < 0) (h2 : ∀ j, 0 < ∑ i, A i j) :
A.det ≠ ... | case inr.intro.intro.inr.refine_3.intro
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
S : Type u_2
inst✝ : LinearOrderedCommRing S
A : Matrix n n S
h1 : Pairwise fun i j => A i j < 0
h✝ : Nonempty n
h2 : A.det = 0
v : n → S
h_vnz : v ≠ 0
h_vA : v ᵥ* A = 0
this :
∀ {n : Type u_1} [inst : Fintype n] [inst_1 : ... | rw [Pi.smul_apply, neg_smul, one_smul, Left.neg_pos_iff] | case inr.intro.intro.inr.refine_3.intro
n : Type u_1
inst✝² : Fintype n
inst✝¹ : DecidableEq n
S : Type u_2
inst✝ : LinearOrderedCommRing S
A : Matrix n n S
h1 : Pairwise fun i j => A i j < 0
h✝ : Nonempty n
h2 : A.det = 0
v : n → S
h_vnz : v ≠ 0
h_vA : v ᵥ* A = 0
this :
∀ {n : Type u_1} [inst : Fintype n] [inst_1 : ... | 3754ab49091301b2 |
MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | theorem tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ) {l : Filter ι}
(hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)) : TendstoInMeasure μ f l g | α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
f : ι → α → E
g : α → E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
hg : AEStronglyMeasurable g μ
l : Filter ι
hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)
th... | rw [this] | α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
f : ι → α → E
g : α → E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
hg : AEStronglyMeasurable g μ
l : Filter ι
hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) l (𝓝 0)
th... | 413d92a74acc8796 |
CategoryTheory.Localization.Monoidal.triangle_aux₂ | Mathlib/CategoryTheory/Localization/Monoidal.lean | lemma triangle_aux₂ {X Y : LocalizedMonoidal L W ε} {X' Y' : C}
(e₁ : (L').obj X' ≅ X) (e₂ : (L').obj Y' ≅ Y) :
e₁.hom ⊗ (ε.hom ⊗ e₂.hom) ≫ (λ_ Y).hom =
(L').obj X' ◁ ((ε' L W ε).hom ▷ (L').obj Y' ≫
𝟙_ _ ◁ e₂.hom ≫ (λ_ Y).hom) ≫ e₁.hom ▷ Y | case e_g.e_a.e_f
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{u_4, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝² : MonoidalCategory C
inst✝¹ : W.IsMonoidal
inst✝ : L.IsLocalization W
unit : D
ε : L.obj (𝟙_ C) ≅ unit
X Y : LocalizedMonoidal L W ε
X' Y' : C
e₁ : L'.obj X' ≅ X
e₂ : L'.obj ... | exact (comp_id _).symm | no goals | cd7cc339a01b1060 |
IndepMatroid.ofFinset_indep | Mathlib/Data/Matroid/IndepAxioms.lean | theorem ofFinset_indep [DecidableEq α] (E : Set α) Indep indep_empty indep_subset indep_aug
subset_ground {I : Finset α} : (IndepMatroid.ofFinset
E Indep indep_empty indep_subset indep_aug subset_ground).Indep I ↔ Indep I | α : Type u_1
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I : Finset ... | simp only [IndepMatroid.ofFinset, ofFinitaryCardAugment_indep, Finset.coe_subset] | α : Type u_1
inst✝ : DecidableEq α
E : Set α
Indep : Finset α → Prop
indep_empty : Indep ∅
indep_subset : ∀ ⦃I J : Finset α⦄, Indep J → I ⊆ J → Indep I
indep_aug : ∀ ⦃I J : Finset α⦄, Indep I → Indep J → I.card < J.card → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)
subset_ground : ∀ ⦃I : Finset α⦄, Indep I → ↑I ⊆ E
I : Finset ... | b7e097637ffa8b4c |
List.getElem_insertIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/InsertIdx.lean | theorem getElem_insertIdx {l : List α} {x : α} {n k : Nat} (h : k < (insertIdx n x l).length) :
(insertIdx n x l)[k] =
if h₁ : k < n then
l[k]'(by simp [length_insertIdx] at h; split at h <;> omega)
else
if h₂ : k = n then
x
else
l[k-1]'(by simp [length_insert... | case isFalse.isTrue
α : Type u
l : List α
x : α
k : Nat
h : k < (insertIdx k x l).length
h₁ : ¬k < k
⊢ (insertIdx k x l)[k] = x | rw [getElem_insertIdx_self h] | no goals | 2303cbe5977d4288 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.