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 |
|---|---|---|---|---|---|---|
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
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 : ∀ᵐ (x : ι... | refine ⟨⌈(r 0 0 : ℝ)⁻¹⌉₊, fun n hn ↦ lt_of_le_of_lt ?_ (half_lt_self_iff.mpr hε)⟩ | case intro.intro.intro.intro
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 : ∀ᵐ (x : ι... | e99fafdae745475e |
List.partition_eq_filter_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem partition_eq_filter_filter (p : α → Bool) (l : List α) :
partition p l = (filter p l, filter (not ∘ p) l) | α : Type u_1
p : α → Bool
l : List α
⊢ partition p l = (filter p l, filter (not ∘ p) l) | simp [partition, aux] | no goals | 1b206b7b210bcfe8 |
Num.ofNat'_eq | Mathlib/Data/Num/Lemmas.lean | theorem ofNat'_eq : ∀ n, Num.ofNat' n = n :=
Nat.binaryRec (by simp) fun b n IH => by tauto
| ⊢ ofNat' 0 = ↑0 | simp | no goals | a8dda501c3f53f4e |
Nat.primeFactors_pow_succ | Mathlib/Data/Nat/PrimeFin.lean | lemma primeFactors_pow_succ (n k : ℕ) : (n ^ (k + 1)).primeFactors = n.primeFactors | n k : ℕ
⊢ (n ^ (k + 1)).primeFactors = n.primeFactors | rcases eq_or_ne n 0 with (rfl | hn) | case inl
k : ℕ
⊢ (0 ^ (k + 1)).primeFactors = primeFactors 0
case inr
n k : ℕ
hn : n ≠ 0
⊢ (n ^ (k + 1)).primeFactors = n.primeFactors | 1853618c5cac81f8 |
WeierstrassCurve.ΨSq_four | Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | @[simp]
lemma ΨSq_four : W.ΨSq 4 = W.preΨ₄ ^ 2 * W.Ψ₂Sq | R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ Even 4 | decide | no goals | 7ca1b640087f381b |
Finset.Colex.trans_aux | Mathlib/Combinatorics/Colex.lean | private lemma trans_aux (hst : toColex s ≤ toColex t) (htu : toColex t ≤ toColex u)
(has : a ∈ s) (hat : a ∉ t) : ∃ b, b ∈ u ∧ b ∉ s ∧ a ≤ b | α : Type u_1
inst✝ : PartialOrder α
s t u : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
htu : { ofColex := t } ≤ { ofColex := u }
has : a ∈ s
hat : a ∉ t
s' : Finset α := filter (fun b => b ∉ t ∧ a ≤ b) s
b : α
hb : b ∈ s ∧ b ∉ t ∧ a ≤ b
hbmax : ∀ x ∈ s, x ∉ t → a ≤ x → ¬b < x
⊢ ∃ b ∈ u, b ∉ s ∧ a ≤ b | have ⟨c, hct, hcs, hbc⟩ := hst hb.1 hb.2.1 | α : Type u_1
inst✝ : PartialOrder α
s t u : Finset α
a : α
hst : { ofColex := s } ≤ { ofColex := t }
htu : { ofColex := t } ≤ { ofColex := u }
has : a ∈ s
hat : a ∉ t
s' : Finset α := filter (fun b => b ∉ t ∧ a ≤ b) s
b : α
hb : b ∈ s ∧ b ∉ t ∧ a ≤ b
hbmax : ∀ x ∈ s, x ∉ t → a ≤ x → ¬b < x
c : α
hct : c ∈ { ofColex := ... | d44eef7b38eb8d38 |
Rat.uniformSpace_eq | Mathlib/Topology/UniformSpace/CompareReals.lean | theorem Rat.uniformSpace_eq :
(AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace | ⊢ AbsoluteValue.abs.uniformSpace = PseudoMetricSpace.toUniformSpace | ext s | case h.h
s : Set (ℚ × ℚ)
⊢ s ∈ uniformity ℚ ↔ s ∈ uniformity ℚ | 860aee39be103d1d |
MeasureTheory.analyticSet_empty | Mathlib/MeasureTheory/Constructions/Polish/Basic.lean | theorem analyticSet_empty : AnalyticSet (∅ : Set α) | α : Type u_1
inst✝ : TopologicalSpace α
⊢ AnalyticSet ∅ | rw [AnalyticSet] | α : Type u_1
inst✝ : TopologicalSpace α
⊢ ∅ = ∅ ∨ ∃ f, Continuous f ∧ range f = ∅ | 96d6590387ca3fc0 |
TopologicalSpace.IsOpenCover.isOpenMap_iff_restrictPreimage | Mathlib/Topology/LocalAtTarget.lean | theorem isOpenMap_iff_restrictPreimage :
IsOpenMap f ↔ ∀ i, IsOpenMap ((U i).1.restrictPreimage f) | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_3
U : ι → Opens β
hU : IsOpenCover U
⊢ IsOpenMap f ↔ ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f) | refine ⟨fun h i ↦ h.restrictPreimage _, fun H s hs ↦ ?_⟩ | α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
ι : Type u_3
U : ι → Opens β
hU : IsOpenCover U
H : ∀ (i : ι), IsOpenMap ((U i).carrier.restrictPreimage f)
s : Set α
hs : IsOpen s
⊢ IsOpen (f '' s) | 344d9e2126901f20 |
uniformCauchySeqOn_ball_of_fderiv | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | theorem uniformCauchySeqOn_ball_of_fderiv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
(hf : ∀ n : ι, ∀ y : E, y ∈ Metric.ball x r → HasFDerivAt (f n) (f' n y) y)
(hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) | case inr
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ... | constructor | case inr.left
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n :... | 982ef3c3832b9fcd |
MultilinearMap.norm_image_sub_le_of_bound | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | theorem norm_image_sub_le_of_bound (f : MultilinearMap 𝕜 E G)
{C : ℝ} (hC : 0 ≤ C) (H : ∀ m, ‖f m‖ ≤ C * ∏ i, ‖m i‖) (m₁ m₂ : ∀ i, E i) :
‖f m₁ - f m₂‖ ≤ C * Fintype.card ι * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖ | case h.h
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i :... | apply A | no goals | f3950b21519cddc8 |
rexp_neg_quadratic_isLittleO_rpow_atTop | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | lemma rexp_neg_quadratic_isLittleO_rpow_atTop {a : ℝ} (ha : a < 0) (b s : ℝ) :
(fun x ↦ rexp (a * x ^ 2 + b * x)) =o[atTop] (· ^ s) | a : ℝ
ha : a < 0
b s : ℝ
⊢ Tendsto (fun x => -x - (a * x ^ 2 + b * x)) atTop atTop | have : (fun x ↦ -x - (a * x ^ 2 + b * x)) = fun x ↦ x * (-a * x - (b + 1)) := by
ext1 x; ring_nf | a : ℝ
ha : a < 0
b s : ℝ
this : (fun x => -x - (a * x ^ 2 + b * x)) = fun x => x * (-a * x - (b + 1))
⊢ Tendsto (fun x => -x - (a * x ^ 2 + b * x)) atTop atTop | 729446a2306c563c |
ZetaAsymptotics.term_one | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | lemma term_one {n : ℕ} (hn : 0 < n) :
term n 1 = (log (n + 1) - log n) - 1 / (n + 1) | case e_a.e_a
n : ℕ
hn : 0 < n
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ 1 / x ^ 2 = x ^ (-2) | rw [rpow_neg, one_div, ← Nat.cast_two (R := ℝ), rpow_natCast] | case e_a.e_a.hx
n : ℕ
hn : 0 < n
hv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x
x : ℝ
hx : x ∈ uIcc (↑n) (↑n + 1)
⊢ 0 ≤ x | b352dfbb4af79ad9 |
BitVec.mul_eq_and | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem mul_eq_and {a b : BitVec 1} : a * b = a &&& b | a b : BitVec 1
ha : a = 0 ∨ a = 1
hb : b = 0 ∨ b = 1
⊢ a * b = a &&& b | rcases ha with h | h <;> (rcases hb with h' | h' <;> (simp [h, h'])) | no goals | 65f931d1a2379105 |
Set.chainHeight_image | Mathlib/Order/Height.lean | theorem chainHeight_image (f : α → β) (hf : ∀ {x y}, x < y ↔ f x < f y) (s : Set α) :
(f '' s).chainHeight = s.chainHeight | case a.cons
α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
f : α → β
hf : ∀ {x y : α}, x < y ↔ f x < f y
s : Set α
x : β
xs : List β
hx : xs ∈ (f '' s).subchain → ∃ l' ∈ s.subchain, map f l' = xs
h : x ∈ f '' s ∧ xs ∈ (f '' s).subchain ∧ ∀ b ∈ xs.head?, x < b
⊢ ∃ l' ∈ s.subchain, map f l' = x :: xs | obtain ⟨⟨x, hx', rfl⟩, h₁, h₂⟩ := h | case a.cons.intro.intro.intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
f : α → β
hf : ∀ {x y : α}, x < y ↔ f x < f y
s : Set α
xs : List β
hx : xs ∈ (f '' s).subchain → ∃ l' ∈ s.subchain, map f l' = xs
x : α
hx' : x ∈ s
h₁ : xs ∈ (f '' s).subchain
h₂ : ∀ b ∈ xs.head?, f x < b
⊢ ∃ l' ∈ s.subchain, map ... | 0c10e60540ca25b5 |
Real.ceil_logb_natCast | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | theorem ceil_logb_natCast {b : ℕ} {r : ℝ} (hr : 0 ≤ r) :
⌈logb b r⌉ = Int.clog b r | case pos.a
b : ℕ
r : ℝ
hr✝ : 0 ≤ r
hr : 0 < r
hb : 1 < b
hb1' : 1 < ↑b
⊢ ↑b ^ logb (↑b) r ≤ ↑b ^ ↑⌈logb (↑b) r⌉ | exact rpow_le_rpow_of_exponent_le hb1'.le (Int.le_ceil _) | no goals | f353f45d6f6f2c29 |
LieAlgebra.lie_mem_genWeightSpace_of_mem_genWeightSpace | Mathlib/Algebra/Lie/Weights/Cartan.lean | theorem lie_mem_genWeightSpace_of_mem_genWeightSpace {χ₁ χ₂ : H → R} {x : L} {m : M}
(hx : x ∈ rootSpace H χ₁) (hm : m ∈ genWeightSpace M χ₂) :
⁅x, m⁆ ∈ genWeightSpace M (χ₁ + χ₂) | R : Type u_1
L : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
H : LieSubalgebra R L
inst✝⁴ : LieRing.IsNilpotent ↥H
M : Type u_3
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
χ₁ χ₂ : ↥H → R
x : L
m : M
y : ↥H
hx : x ∈ genWeightSpaceOf L (χ₁ y) ... | exact lie_mem_maxGenEigenspace_toEnd hx hm | no goals | 4c37ae78eab20823 |
CoalgebraCat.MonoidalCategoryAux.counit_tensorObj_tensorObj_right | Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean | theorem counit_tensorObj_tensorObj_right :
Coalgebra.counit (R := R)
(A := (CoalgebraCat.of R M ⊗ (CoalgebraCat.of R N ⊗ CoalgebraCat.of R P) : CoalgebraCat R))
= Coalgebra.counit (A := M ⊗[R] (N ⊗[R] P)) | R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
⊢ CoalgebraStruct.counit = CoalgebraStruct.counit | ext | case h
R : Type u
inst✝⁹ : CommRing R
M N P : Type u
inst✝⁸ : AddCommGroup M
inst✝⁷ : AddCommGroup N
inst✝⁶ : AddCommGroup P
inst✝⁵ : Module R M
inst✝⁴ : Module R N
inst✝³ : Module R P
inst✝² : Coalgebra R M
inst✝¹ : Coalgebra R N
inst✝ : Coalgebra R P
x✝ : ↑(of R M ⊗ of R N ⊗ of R P).toModuleCat
⊢ CoalgebraStruct.coun... | 0c920a2c3480b08c |
Polynomial.derivative_mul | Mathlib/Algebra/Polynomial/Derivative.lean | theorem derivative_mul {f g : R[X]} : derivative (f * g) = derivative f * g + f * derivative g | case h_monomial.h_monomial.succ.succ
R : Type u
inst✝ : Semiring R
a b : R
m n : ℕ
⊢ (monomial (m + 1 + n)) (a * (b * ↑(m + 1))) + (monomial (m + 1 + n)) (a * (b * ↑(n + 1))) =
(monomial (m + (n + 1))) (a * (b * ↑(m + 1))) + (monomial (m + 1 + n)) (a * (b * ↑(n + 1))) | rw [add_assoc, add_comm n 1] | no goals | a66065c70ef80a14 |
controlled_prod_of_mem_closure | Mathlib/Analysis/Normed/Group/Continuity.lean | theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
(b_pos : ∀ n, 0 < b n) :
∃ v : ℕ → E,
Tendsto (fun n => ∏ i ∈ range (n + 1), v i) atTop (𝓝 a) ∧
(∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n | case intro.intro.intro
E : Type u_5
inst✝ : SeminormedCommGroup E
a : E
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
⊢ ∃ v,
Tendsto (fun n => ∏ i ∈ ra... | have lim_z : Tendsto z atTop (𝓝 a) := lim_u.comp (tendsto_add_atTop_nat n₀) | case intro.intro.intro
E : Type u_5
inst✝ : SeminormedCommGroup E
a : E
s : Subgroup E
hg : a ∈ closure ↑s
b : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < b n
u : ℕ → E
u_in : ∀ (n : ℕ), u n ∈ s
lim_u : Tendsto u atTop (𝓝 a)
n₀ : ℕ
hn₀ : ∀ n ≥ n₀, ‖u n / a‖ < b 0
z : ℕ → E := fun n => u (n + n₀)
lim_z : Tendsto z atTop (𝓝 a)
⊢ ∃ v,... | 647a783958e68b9a |
List.eraseIdx_modify_of_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Modify.lean | theorem eraseIdx_modify_of_eq (f : α → α) (n) (l : List α) :
(modify f n l).eraseIdx n = l.eraseIdx n | case h
α : Type u_1
f : α → α
n : Nat
l : List α
⊢ ∀ (i : Nat) (h₁ : i < ((modify f n l).eraseIdx n).length) (h₂ : i < (l.eraseIdx n).length),
((modify f n l).eraseIdx n)[i] = (l.eraseIdx n)[i] | intro m h₁ h₂ | case h
α : Type u_1
f : α → α
n : Nat
l : List α
m : Nat
h₁ : m < ((modify f n l).eraseIdx n).length
h₂ : m < (l.eraseIdx n).length
⊢ ((modify f n l).eraseIdx n)[m] = (l.eraseIdx n)[m] | 0e1ed36136351a9f |
MeasureTheory.ae_nonneg_of_forall_setIntegral_nonneg | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
⊢ 0 ≤ᶠ[ae μ] f | simp_rw [EventuallyLE, Pi.zero_apply] | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : Integrable f μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → 0 ≤ ∫ (x : α) in s, f x ∂μ
⊢ ∀ᵐ (x : α) ∂μ, 0 ≤ f x | d52a385303ec1354 |
ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace | Mathlib/Probability/Integration.lean | theorem lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace
{Mf Mg mΩ : MeasurableSpace Ω} {μ : Measure Ω} (hMf : Mf ≤ mΩ) (hMg : Mg ≤ mΩ)
(h_ind : Indep Mf Mg μ) (h_meas_f : Measurable[Mf] f) (h_meas_g : Measurable[Mg] g) :
∫⁻ ω, f ω * g ω ∂μ = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ | case h_add
Ω : Type u_1
f : Ω → ℝ≥0∞
Mf Mg mΩ : MeasurableSpace Ω
μ : Measure Ω
hMf : Mf ≤ mΩ
hMg : Mg ≤ mΩ
h_ind : Indep Mf Mg μ
h_meas_f : Measurable f
h_measM_f : Measurable f
f' g : Ω → ℝ≥0∞
a✝¹ : Disjoint (Function.support f') (Function.support g)
h_measMg_f' : Measurable f'
a✝ : Measurable g
h_ind_f' : ∫⁻ (ω : Ω)... | rw [lintegral_add_left h_measM_f', lintegral_add_left (h_measM_f.mul h_measM_f'), left_distrib,
h_ind_f', h_ind_g'] | no goals | bad3bf0b684e4cf2 |
PNat.factorMultiset_le_iff' | Mathlib/Data/PNat/Factors.lean | theorem factorMultiset_le_iff' {m : ℕ+} {v : PrimeMultiset} :
factorMultiset m ≤ v ↔ m ∣ v.prod | m : ℕ+
v : PrimeMultiset
h : m.factorMultiset ≤ v.prod.factorMultiset ↔ m ∣ v.prod := factorMultiset_le_iff
⊢ m.factorMultiset ≤ v ↔ m ∣ v.prod | rw [v.factorMultiset_prod] at h | m : ℕ+
v : PrimeMultiset
h : m.factorMultiset ≤ v ↔ m ∣ v.prod
⊢ m.factorMultiset ≤ v ↔ m ∣ v.prod | 38271c385288e7e9 |
Polynomial.Gal.splits_in_splittingField_of_comp | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | theorem splits_in_splittingField_of_comp (hq : q.natDegree ≠ 0) :
p.Splits (algebraMap F (p.comp q).SplittingField) | case neg.intro
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r => Splits (algebraMap F (r.comp q).SplittingField) r
r : F[X]
hr : Irreducible r
hr' : ¬r.natDegree = 0
x : (r.comp q).SplittingField
hx : eval₂ (algebraMap F (r.comp q).SplittingField) x (r.comp q) = 0
⊢ P r | rw [← aeval_def, aeval_comp] at hx | case neg.intro
F : Type u_1
inst✝ : Field F
p q : F[X]
hq : q.natDegree ≠ 0
P : F[X] → Prop := fun r => Splits (algebraMap F (r.comp q).SplittingField) r
r : F[X]
hr : Irreducible r
hr' : ¬r.natDegree = 0
x : (r.comp q).SplittingField
hx : (aeval ((aeval x) q)) r = 0
⊢ P r | 781c423315763045 |
Ordering.isGT_swap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Ord.lean | theorem isGT_swap {o : Ordering} : o.swap.isGT = o.isLT | o : Ordering
⊢ o.swap.isGT = o.isLT | cases o <;> simp | no goals | 46ace378b3355311 |
List.chain'_of_mem_splitByLoop | Mathlib/Data/List/SplitBy.lean | theorem chain'_of_mem_splitByLoop {r : α → α → Bool} {l : List α} {a : α} {g : List α}
(hga : ∀ b ∈ g.head?, r b a) (hg : g.Chain' fun y x ↦ r x y)
(h : m ∈ splitBy.loop r l a g []) : m.Chain' fun x y ↦ r x y | case cons.h_2.inl
α : Type u_1
r : α → α → Bool
b : α
l : List α
a : α
g : List α
hga : ∀ (b : α), b ∈ g.head? → r b a = true
hg : Chain' (fun y x => r x y = true) g
x✝ : Bool
heq✝ : r a b = false
IH :
∀ {a_1 : α} {g_1 : List α},
(∀ (b : α), b ∈ g_1.head? → r b a_1 = true) →
Chain' (fun y x => r x y = true)... | exact chain'_cons'.2 ⟨hga, hg⟩ | no goals | f1b83979d934bdc8 |
iteratedFDeriv_comp | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem iteratedFDeriv_comp (hg : ContDiffAt 𝕜 n g (f x)) (hf : ContDiffAt 𝕜 n f x)
{i : ℕ} (hi : i ≤ n) :
iteratedFDeriv 𝕜 i (g ∘ f) x =
(ftaylorSeries 𝕜 g (f x)).taylorComp (ftaylorSeries 𝕜 f x) i | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
g : F → G
x : E
n : WithTop ℕ∞
hg : ContDiffAt 𝕜 n g (f ... | simp only [← iteratedFDerivWithin_univ, ← ftaylorSeriesWithin_univ] | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → F
g : F → G
x : E
n : WithTop ℕ∞
hg : ContDiffAt 𝕜 n g (f ... | ec1777b9bb089ded |
not_summable_of_antitone_of_neg | Mathlib/Analysis/SumOverResidueClass.lean | /-- If `f : ℕ → ℝ` is decreasing and has a negative term, then `f` is not summable. -/
lemma not_summable_of_antitone_of_neg {f : ℕ → ℝ} (hf : Antitone f) {n : ℕ} (hn : f n < 0) :
¬ Summable f | f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
⊢ ¬Summable f | intro hs | f : ℕ → ℝ
hf : Antitone f
n : ℕ
hn : f n < 0
hs : Summable f
⊢ False | bd42ef63684ae4ee |
Real.Icc_mem_vitaliFamily_at_right | Mathlib/MeasureTheory/Covering/OneDim.lean | theorem Icc_mem_vitaliFamily_at_right {x y : ℝ} (hxy : x < y) :
Icc x y ∈ (vitaliFamily (volume : Measure ℝ) 1).setsAt x | x y : ℝ
hxy : x < y
⊢ dist x ((x + y) / 2) ≤ 1 * ((y - x) / 2) | rw [dist_comm, Real.dist_eq, abs_of_nonneg] <;> linarith | no goals | f44bb90059fd7aa2 |
Polynomial.monomial_pow | Mathlib/Algebra/Polynomial/Basic.lean | theorem monomial_pow (n : ℕ) (r : R) (k : ℕ) : monomial n r ^ k = monomial (n * k) (r ^ k) | case zero
R : Type u
inst✝ : Semiring R
n : ℕ
r : R
⊢ (monomial n) r ^ 0 = (monomial (n * 0)) (r ^ 0) | simp [pow_zero, monomial_zero_one] | no goals | bbe2b6146920ba11 |
LawfulFunctor.map_inj_right_of_nonempty | Mathlib/.lake/packages/batteries/Batteries/Control/Monad.lean | theorem _root_.LawfulFunctor.map_inj_right_of_nonempty [Functor f] [LawfulFunctor f] [Nonempty α]
{g : α → β} (h : ∀ {x y : α}, g x = g y → x = y) {x y : f α} :
g <$> x = g <$> y ↔ x = y | case mpr
f : Type u_1 → Type u_2
α β : Type u_1
inst✝² : Functor f
inst✝¹ : LawfulFunctor f
inst✝ : Nonempty α
g : α → β
h : ∀ {x y : α}, g x = g y → x = y
x y : f α
⊢ x = y → g <$> x = g <$> y | intro h' | case mpr
f : Type u_1 → Type u_2
α β : Type u_1
inst✝² : Functor f
inst✝¹ : LawfulFunctor f
inst✝ : Nonempty α
g : α → β
h : ∀ {x y : α}, g x = g y → x = y
x y : f α
h' : x = y
⊢ g <$> x = g <$> y | 9ff9346398dc10b4 |
List.idxOf_mem_indexesOf | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem idxOf_mem_indexesOf [BEq α] [LawfulBEq α] {xs : List α} (m : x ∈ xs) :
xs.idxOf x ∈ xs.indexesOf x | case cons
α : Type u_1
x : α
inst✝¹ : BEq α
inst✝ : LawfulBEq α
h : α
t : List α
ih : x ∈ t → idxOf x t ∈ indexesOf x t
m : x ∈ h :: t
⊢ idxOf x (h :: t) ∈ indexesOf x (h :: t) | simp [idxOf_cons, indexesOf_cons, cond_eq_if] | case cons
α : Type u_1
x : α
inst✝¹ : BEq α
inst✝ : LawfulBEq α
h : α
t : List α
ih : x ∈ t → idxOf x t ∈ indexesOf x t
m : x ∈ h :: t
⊢ (if (h == x) = true then 0 else idxOf x t + 1) ∈
if (h == x) = true then 0 :: map (fun x => x + 1) (indexesOf x t) else map (fun x => x + 1) (indexesOf x t) | e8cb18c97ac0bdad |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_postcondition | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem reduce_postcondition {n : Nat} (c : DefaultClause n) (assignment : Array Assignment) :
(reduce c assignment = reducedToEmpty → Incompatible (PosFin n) c assignment) ∧
(∀ l : Literal (PosFin n), reduce c assignment = reducedToUnit l → ∀ (p : (PosFin n) → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l) | case intro.right.intro.inr.intro
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmp... | rw [c_clause_rw] at pc1 | case intro.right.intro.inr.intro
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reducedToEmp... | bc01533c0ab5cdb8 |
CategoryTheory.Functor.IsCocartesian.map_self | Mathlib/CategoryTheory/FiberedCategory/Cocartesian.lean | @[simp]
lemma map_self : IsCocartesian.map p f φ φ = 𝟙 b | case map
𝒮 : Type u₁
𝒳 : Type u₂
inst✝² : Category.{v₁, u₁} 𝒮
inst✝¹ : Category.{v₂, u₂} 𝒳
p : 𝒳 ⥤ 𝒮
a✝ b✝ : 𝒳
φ : a✝ ⟶ b✝
R S : 𝒮
a b : 𝒳
inst✝ : p.IsCocartesian (p.map φ) φ
⊢ 𝟙 b✝ = IsCocartesian.map p (p.map φ) φ φ | apply map_uniq | case map.hψ
𝒮 : Type u₁
𝒳 : Type u₂
inst✝² : Category.{v₁, u₁} 𝒮
inst✝¹ : Category.{v₂, u₂} 𝒳
p : 𝒳 ⥤ 𝒮
a✝ b✝ : 𝒳
φ : a✝ ⟶ b✝
R S : 𝒮
a b : 𝒳
inst✝ : p.IsCocartesian (p.map φ) φ
⊢ φ ≫ 𝟙 b✝ = φ | 833ab7024cb503ea |
SimpleGraph.chromaticNumber_le_of_forall_imp | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem chromaticNumber_le_of_forall_imp {V' : Type*} {G' : SimpleGraph V'}
(h : ∀ n, G'.Colorable n → G.Colorable n) :
G.chromaticNumber ≤ G'.chromaticNumber | V : Type u
G : SimpleGraph V
V' : Type u_3
G' : SimpleGraph V'
h : ∀ (n : ℕ), G'.Colorable n → G.Colorable n
m : ℕ
hc : G'.Colorable m
this : G.Colorable m
⊢ ⨅ n, ⨅ (_ : G.Colorable n), ↑n ≤ ↑m | rw [← chromaticNumber_le_iff_colorable] at this | V : Type u
G : SimpleGraph V
V' : Type u_3
G' : SimpleGraph V'
h : ∀ (n : ℕ), G'.Colorable n → G.Colorable n
m : ℕ
hc : G'.Colorable m
this : G.chromaticNumber ≤ ↑m
⊢ ⨅ n, ⨅ (_ : G.Colorable n), ↑n ≤ ↑m | 5f4f3f6887a3f266 |
CategoryTheory.Abelian.Ext.add_hom | Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean | @[simp]
lemma add_hom (α β : Ext X Y n) : (α + β).hom = α.hom + β.hom | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
⊢ α' + β' = homEquiv.... | apply biprod_ext | case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
⊢ (mk₀ biprod... | df159b910b840aef |
List.Vector.get_ofFn | Mathlib/Data/Vector/Basic.lean | theorem get_ofFn {n} (f : Fin n → α) (i) : get (ofFn f) i = f i | α : Type u_1
n : ℕ
f : Fin n → α
i : Fin n
⊢ (ofFn f).toList.get (Fin.cast ⋯ i) = (List.ofFn f).get ⟨↑i, ⋯⟩ | congr <;> simp [Fin.heq_ext_iff] | no goals | 0313928d8f9dd4c8 |
MeasureTheory.tendsto_of_lintegral_tendsto_of_antitone | Mathlib/MeasureTheory/Integral/Lebesgue.lean | /-- If an antitone sequence of functions has a lower bound and the sequence of integrals of these
functions tends to the integral of the lower bound, then the sequence of functions converges
almost everywhere to the lower bound. -/
lemma tendsto_of_lintegral_tendsto_of_antitone {α : Type*} {mα : MeasurableSpace α}
... | α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α),... | suffices F' =ᵐ[μ] F by
filter_upwards [this, hF'_tendsto] with a h_eq h_tendsto using h_eq ▸ h_tendsto | α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α),... | 4ccce525cdd179a4 |
Set.chainHeight_insert_of_forall_gt | Mathlib/Order/Height.lean | theorem chainHeight_insert_of_forall_gt (a : α) (hx : ∀ b ∈ s, a < b) :
(insert a s).chainHeight = s.chainHeight + 1 | case a.refine_1
α : Type u_1
s : Set α
inst✝ : Preorder α
a : α
hx : ∀ b ∈ s, a < b
l : List α
hl : l ∈ s.subchain
⊢ Chain' (fun x1 x2 => x1 < x2) (a :: l) | rw [chain'_cons'] | case a.refine_1
α : Type u_1
s : Set α
inst✝ : Preorder α
a : α
hx : ∀ b ∈ s, a < b
l : List α
hl : l ∈ s.subchain
⊢ (∀ y ∈ l.head?, a < y) ∧ Chain' (fun x1 x2 => x1 < x2) l | abd1373dde588775 |
ModularGroup.det_coe | Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | theorem det_coe {g : SL(2, ℤ)} : det (Units.val <| Subtype.val <| coe g) = 1 | g : SL(2, ℤ)
⊢ (↑↑↑g).det = 1 | simp only [SpecialLinearGroup.coe_GLPos_coe_GL_coe_matrix, SpecialLinearGroup.det_coe, coe] | no goals | c80b6f191a45697d |
WeierstrassCurve.exists_variableChange_of_char_two_of_j_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_two_of_j_ne_zero
[E.IsCharTwoJNeZeroNF] [E'.IsCharTwoJNeZeroNF] (heq : E.a₆ = E'.a₆) :
∃ C : VariableChange F, E.variableChange C = E' | F : Type u_1
inst✝⁴ : Field F
inst✝³ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝² : CharP F 2
inst✝¹ : E.IsCharTwoJNeZeroNF
inst✝ : E'.IsCharTwoJNeZeroNF
heq : E.a₆ = E'.a₆
⊢ 2 ∣ 2 | norm_num | no goals | 515774c34fadda64 |
HilbertBasis.hasSum_repr_symm | Mathlib/Analysis/InnerProductSpace/l2Space.lean | theorem hasSum_repr_symm (b : HilbertBasis ι 𝕜 E) (f : ℓ²(ι, 𝕜)) :
HasSum (fun i => f i • b i) (b.repr.symm f) | case h.a
ι : Type u_1
𝕜 : Type u_2
inst✝² : RCLike 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
b : HilbertBasis ι 𝕜 E
f : ↥(lp (fun i => 𝕜) 2)
i : ι
this✝ : NormedSpace 𝕜 ↥(lp (fun _i => 𝕜) 2) := inferInstance
this : lp.single 2 i (↑f i) = ↑f i • lp.single 2 i 1
⊢ lp.single 2 i (↑f... | exact (b.repr.apply_symm_apply (lp.single 2 i (f i))).symm | no goals | 611317bc4a18ea4f |
Nat.testBit_mod_two_pow | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean | theorem testBit_mod_two_pow (x j i : Nat) :
testBit (x % 2^j) i = (decide (i < j) && testBit x i) | case ind
x : Nat
hyp : ∀ (m : Nat), m < x → ∀ (j i : Nat), (m % 2 ^ j).testBit i = (decide (i < j) && m.testBit i)
j i : Nat
⊢ (x % 2 ^ j).testBit i = (decide (i < j) && x.testBit i) | rw [mod_eq] | case ind
x : Nat
hyp : ∀ (m : Nat), m < x → ∀ (j i : Nat), (m % 2 ^ j).testBit i = (decide (i < j) && m.testBit i)
j i : Nat
⊢ (if 0 < 2 ^ j ∧ 2 ^ j ≤ x then (x - 2 ^ j) % 2 ^ j else x).testBit i = (decide (i < j) && x.testBit i) | f8ba27e4751df3a5 |
AlgebraicGeometry.Scheme.IdealSheafData.zeroLocus_inter_subset_support | Mathlib/AlgebraicGeometry/IdealSheaf.lean | lemma zeroLocus_inter_subset_support (I : IdealSheafData X) (U : X.affineOpens) :
X.zeroLocus (U := U.1) (I.ideal U) ∩ U ⊆ I.support | case intro.intro.intro.intro.intro
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑↑X.toPresheafedSpace
hxV : x ∈ ↑↑V
hxU : x ∈ ↑↑U
hx : ∀ f ∈ I.ideal U, x ∉ X.basicOpen f
s : ↑Γ(X, ↑V)
hfU : s ∈ I.ideal V
hxs : x ∈ X.basicOpen s
f : ↑Γ(X, ↑U)
g : ↑Γ(X, ↑V)
hfg : X.basicOpen f = X.basicOpen g
hxf : x ∈ X.basi... | have inst := U.2.isLocalization_basicOpen f | case intro.intro.intro.intro.intro
X : Scheme
I : X.IdealSheafData
U V : ↑X.affineOpens
x : ↑↑X.toPresheafedSpace
hxV : x ∈ ↑↑V
hxU : x ∈ ↑↑U
hx : ∀ f ∈ I.ideal U, x ∉ X.basicOpen f
s : ↑Γ(X, ↑V)
hfU : s ∈ I.ideal V
hxs : x ∈ X.basicOpen s
f : ↑Γ(X, ↑U)
g : ↑Γ(X, ↑V)
hfg : X.basicOpen f = X.basicOpen g
hxf : x ∈ X.basi... | ac346fd7f1a4fa9f |
NormedField.continuousAt_zpow | Mathlib/Analysis/Normed/Field/Lemmas.lean | @[simp]
protected lemma continuousAt_zpow : ContinuousAt (fun x ↦ x ^ n) x ↔ x ≠ 0 ∨ 0 ≤ n | 𝕜 : Type u_4
inst✝ : NontriviallyNormedField 𝕜
n : ℤ
x : 𝕜
⊢ ContinuousAt (fun x => x ^ n) x ↔ x ≠ 0 ∨ 0 ≤ n | refine ⟨?_, continuousAt_zpow₀ _ _⟩ | 𝕜 : Type u_4
inst✝ : NontriviallyNormedField 𝕜
n : ℤ
x : 𝕜
⊢ ContinuousAt (fun x => x ^ n) x → x ≠ 0 ∨ 0 ≤ n | 25b0e3752eb9042c |
LieAlgebra.IsKilling.isSemisimple_ad_of_mem_isCartanSubalgebra | Mathlib/Algebra/Lie/Weights/Killing.lean | lemma isSemisimple_ad_of_mem_isCartanSubalgebra {x : L} (hx : x ∈ H) :
(ad K L x).IsSemisimple | case hN
K : Type u_2
L : Type u_3
inst✝⁷ : LieRing L
inst✝⁶ : Field K
inst✝⁵ : LieAlgebra K L
inst✝⁴ : FiniteDimensional K L
H : LieSubalgebra K L
inst✝³ : H.IsCartanSubalgebra
inst✝² : IsKilling K L
inst✝¹ : IsTriangularizable K (↥H) L
inst✝ : PerfectField K
x : L
hx : x ∈ H
N S : End K L
hN : _root_.IsNilpotent ((ad ... | induction hz using LieSubmodule.iSup_induction' with
| hN β z hz => exact h_der y z α β hy hz
| h0 => simp
| hadd _ _ _ _ h h' => simp only [lie_add, map_add, h, h']; abel | no goals | b080016ae19cad5d |
MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem haveLebesgueDecomposition_of_finiteMeasure [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
HaveLebesgueDecomposition μ ν where
lebesgue_decomposition | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g ... | refine isFiniteMeasure_withDensity ?_ | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g ... | 0e340e69a0abebdf |
bernsteinApproximation_uniform | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | theorem bernsteinApproximation_uniform (f : C(I, ℝ)) :
Tendsto (fun n : ℕ => bernsteinApproximation n f) atTop (𝓝 f) | case h.calc_2
f : C(↑I, ℝ)
ε : ℝ
h : 0 < ε
δ : ℝ := bernsteinApproximation.δ f ε h
nhds_zero : Tendsto (fun n => 2 * ‖f‖ * δ ^ (-2) / ↑n) atTop (𝓝 0)
n : ℕ
nh : 2 * ‖f‖ * δ ^ (-2) / ↑n < ε / 2
npos' : 0 < n
npos : 0 < ↑n
x : ↑I
S : Finset (Fin (n + 1)) := bernsteinApproximation.S f ε h n x
⊢ ∑ k ∈ Sᶜ, |f k/ₙ - f x| * ... | calc
∑ k ∈ Sᶜ, |f k/ₙ - f x| * bernstein n k x ≤ ∑ k ∈ Sᶜ, 2 * ‖f‖ * bernstein n k x := by
gcongr
apply f.dist_le_two_norm
_ = 2 * ‖f‖ * ∑ k ∈ Sᶜ, bernstein n k x := by rw [Finset.mul_sum]
_ ≤ 2 * ‖f‖ * ∑ k ∈ Sᶜ, δ ^ (-2 : ℤ) * ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by
gcongr with _ m
conv_lhs =... | no goals | 404c705b844529e4 |
LinearMap.re_inner_adjoint_mul_self_nonneg | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | theorem re_inner_adjoint_mul_self_nonneg (T : E →ₗ[𝕜] E) (x : E) :
0 ≤ re ⟪x, (LinearMap.adjoint T * T) x⟫ | 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
x : E
⊢ 0 ≤ re ⟪x, (adjoint T * T) x⟫_𝕜 | simp only [mul_apply, adjoint_inner_right, inner_self_eq_norm_sq_to_K] | 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
x : E
⊢ 0 ≤ re (↑‖T x‖ ^ 2) | 59c20261fe17c950 |
ProbabilityTheory.Kernel.indepSets_piiUnionInter_of_disjoint | Mathlib/Probability/Independence/Kernel.lean | theorem indepSets_piiUnionInter_of_disjoint {s : ι → Set (Set Ω)}
{S T : Set ι} (h_indep : iIndepSets s κ μ) (hST : Disjoint S T) :
IndepSets (piiUnionInter s S) (piiUnionInter s T) κ μ | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset... | intro i hi_mem_union | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
s : ι → Set (Set Ω)
S T : Set ι
h_indep : iIndepSets s κ μ
hST : Disjoint S T
t1 t2 : Set Ω
p1 : Finset ι
hp1 : ↑p1 ⊆ S
f1 : ι → Set Ω
ht1_m : ∀ x ∈ p1, f1 x ∈ s x
ht1_eq : t1 = ⋂ x ∈ p1, f1 x
p2 : Finset... | 139cefbe3a1213c8 |
LocalizedModule.subsingleton_iff_support_subset | Mathlib/RingTheory/Support.lean | lemma LocalizedModule.subsingleton_iff_support_subset {f : R} :
Subsingleton (LocalizedModule (.powers f) M) ↔
Module.support R M ⊆ PrimeSpectrum.zeroLocus {f} | case mpr
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : R
H : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}
m : M
⊢ ∃ r ∈ Submonoid.powers f, r • m = 0 | by_cases h : (Submodule.span R {m}).annihilator = ⊤ | case pos
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : R
H : Module.support R M ⊆ PrimeSpectrum.zeroLocus {f}
m : M
h : (Submodule.span R {m}).annihilator = ⊤
⊢ ∃ r ∈ Submonoid.powers f, r • m = 0
case neg
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGrou... | 75188a38276e1985 |
CategoryTheory.compatiblePreservingOfFlat | Mathlib/CategoryTheory/Sites/CoverPreserving.lean | theorem compatiblePreservingOfFlat {C : Type u₁} [Category.{v₁} C] {D : Type u₁} [Category.{v₁} D]
(K : GrothendieckTopology D) (G : C ⥤ D) [RepresentablyFlat G] : CompatiblePreserving K G | case compatible
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : Sheaf K (Type u_1)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
hx : x.Compatible
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ : ... | apply congr_arg | case compatible.h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₁
inst✝¹ : Category.{v₁, u₁} D
K : GrothendieckTopology D
G : C ⥤ D
inst✝ : RepresentablyFlat G
ℱ : Sheaf K (Type u_1)
Z : C
T : Presieve Z
x : FamilyOfElements (G.op ⋙ ℱ.val) T
hx : x.Compatible
Y₁ Y₂ : C
X : D
f₁ : X ⟶ G.obj Y₁
f₂ : X ⟶ G.obj Y₂
g₁ ... | 8ad53d89c26ce87e |
ModuleCat.FreeMonoidal.εIso_inv_freeMk | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | @[simp]
lemma εIso_inv_freeMk (x : PUnit) : (εIso R).inv (freeMk x) = 1 | R : Type u
inst✝ : CommRing R
x : PUnit.{u + 1}
⊢ (Finsupp.single x 1) PUnit.unit = 1 | rw [Finsupp.single_eq_same] | no goals | 930ba52caeff00d7 |
LieAlgebra.derivedSeriesOfIdeal_add_le_add | Mathlib/Algebra/Lie/Solvable.lean | theorem derivedSeriesOfIdeal_add_le_add (J : LieIdeal R L) (k l : ℕ) :
D (k + l) (I + J) ≤ D k I + D l J | R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I J : LieIdeal R L
k l : ℕ
D₁ : LieIdeal R L →o LieIdeal R L := { toFun := fun I => ⁅I, I⁆, monotone' := ⋯ }
⊢ D (k + l) (I + J) ≤ D k I + D l J | have h₁ : ∀ I J : LieIdeal R L, D₁ (I ⊔ J) ≤ D₁ I ⊔ J := by
simp [D₁, LieSubmodule.lie_le_right, LieSubmodule.lie_le_left, le_sup_of_le_right] | R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I J : LieIdeal R L
k l : ℕ
D₁ : LieIdeal R L →o LieIdeal R L := { toFun := fun I => ⁅I, I⁆, monotone' := ⋯ }
h₁ : ∀ (I J : LieIdeal R L), D₁ (I ⊔ J) ≤ D₁ I ⊔ J
⊢ D (k + l) (I + J) ≤ D k I + D l J | c99f5499945f2a44 |
Submodule.apply_mem_map₂ | Mathlib/Algebra/Module/Submodule/Bilinear.lean | theorem apply_mem_map₂ (f : M →ₗ[R] N →ₗ[R] P) {m : M} {n : N} {p : Submodule R M}
{q : Submodule R N} (hm : m ∈ p) (hn : n ∈ q) : f m n ∈ map₂ f p q :=
(le_iSup _ ⟨m, hm⟩ : _ ≤ map₂ f p q) ⟨n, hn, by rfl⟩
| R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N →ₗ[R] P
m : M
n : N
p : Submodule R M
q : Submodule R N
hm : m ∈ p
hn : n ∈ q
⊢ (f ↑⟨m, hm⟩) n =... | rfl | no goals | 6f9e3c11fa4bc728 |
MeasureTheory.tendsto_measure_symmDiff_preimage_nhds_zero | Mathlib/MeasureTheory/Measure/ContinuousPreimage.lean | theorem tendsto_measure_symmDiff_preimage_nhds_zero
{l : Filter α} {f : α → C(X, Y)} {g : C(X, Y)} {s : Set Y} (hfg : Tendsto f l (𝓝 g))
(hf : ∀ᶠ a in l, MeasurePreserving (f a) μ ν) (hg : MeasurePreserving g μ ν)
(hs : NullMeasurableSet s ν) (hνs : ν s ≠ ∞) :
Tendsto (fun a ↦ μ ((f a ⁻¹' s) ∆ (g ⁻¹' s... | α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMe... | refine (measure_symmDiff_le _ (g ⁻¹' U) _).trans ?_ | α : Type u_1
X : Type u_2
Y : Type u_3
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
inst✝⁶ : R1Space X
inst✝⁵ : TopologicalSpace Y
inst✝⁴ : MeasurableSpace Y
inst✝³ : BorelSpace Y
inst✝² : R1Space Y
μ : Measure X
ν : Measure Y
inst✝¹ : μ.InnerRegularCompactLTTop
inst✝ : IsLocallyFiniteMe... | 050b449bb51827ad |
exists_mul_right_lt₀ | Mathlib/Algebra/Order/Field/Basic.lean | private lemma exists_mul_right_lt₀ {a b c : α} (hc : a * b < c) : ∃ b' > b, a * b' < c | α : Type u_2
inst✝ : LinearOrderedField α
a b c : α
hc : a * b < c
⊢ ∃ b' > b, a * b' < c | simp_rw [mul_comm a] at hc ⊢ | α : Type u_2
inst✝ : LinearOrderedField α
a b c : α
hc : b * a < c
⊢ ∃ b' > b, b' * a < c | 65fa0f9f99da1238 |
CategoryTheory.LaxMonoidalFunctor.hom_ext | Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean | @[ext]
lemma hom_ext {F G : LaxMonoidalFunctor C D} {α β : F ⟶ G} (h : α.hom = β.hom) : α = β | case mk.mk
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
inst✝ : MonoidalCategory D
F G : LaxMonoidalFunctor C D
hom✝ : F.toFunctor ⟶ G.toFunctor
isMonoidal✝¹ : NatTrans.IsMonoidal hom✝
isMonoidal✝ : NatTrans.IsMonoidal { hom := hom✝, isMonoidal := isMonoi... | rfl | no goals | 8848a8c64a10a477 |
MeasureTheory.integral_eq_zero_iff_of_nonneg_ae | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_eq_zero_iff_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : Integrable f μ) :
∫ x, f x ∂μ = 0 ↔ f =ᵐ[μ] 0 | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᶠ[ae μ] f
hfi : Integrable f μ
⊢ AEMeasurable (fun a => ENNReal.ofReal (f a)) μ | exact (ENNReal.measurable_ofReal.comp_aemeasurable hfi.1.aemeasurable) | no goals | 3f575e9bc56637ff |
Num.succ_ofInt' | Mathlib/Data/Num/Lemmas.lean | theorem succ_ofInt' : ∀ n, ZNum.ofInt' (n + 1) = ZNum.ofInt' n + 1
| (n : ℕ) => by
change ZNum.ofInt' (n + 1 : ℕ) = ZNum.ofInt' (n : ℕ) + 1
dsimp only [ZNum.ofInt', ZNum.ofInt']
rw [Num.ofNat'_succ, Num.add_one, toZNum_succ, ZNum.add_one]
| -[0+1] => by
change ZNum.ofInt' 0 = ZNum.ofInt' (-[0+1]) + ... | n : ℕ
⊢ ZNum.ofInt' ↑(n + 1) = ZNum.ofInt' ↑n + 1 | dsimp only [ZNum.ofInt', ZNum.ofInt'] | n : ℕ
⊢ (ofNat' (n + 1)).toZNum = (ofNat' n).toZNum + 1 | ca426ea9d61f4551 |
Complex.volume_ball | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | theorem Complex.volume_ball (a : ℂ) (r : ℝ) :
volume (Metric.ball a r) = .ofReal r ^ 2 * NNReal.pi | a : ℂ
r : ℝ
⊢ volume (Metric.ball a r) = ENNReal.ofReal r ^ 2 * ↑NNReal.pi | simp [InnerProductSpace.volume_ball_of_dim_even (k := 1) (by simp) a,
← NNReal.coe_real_pi, ofReal_coe_nnreal] | no goals | 08ef1cfdd4ddfd5b |
emultiplicity_le_emultiplicity_iff | Mathlib/RingTheory/Multiplicity.lean | theorem emultiplicity_le_emultiplicity_iff {c d : β} :
emultiplicity a b ≤ emultiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d | case mpr
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
⊢ emultiplicity a b ≤ emultiplicity c d | unfold emultiplicity | case mpr
α : Type u_1
β : Type u_2
inst✝¹ : Monoid α
inst✝ : Monoid β
a b : α
c d : β
h : ∀ (n : ℕ), a ^ n ∣ b → c ^ n ∣ d
⊢ (if h : FiniteMultiplicity a b then ↑(Nat.find h) else ⊤) ≤ if h : FiniteMultiplicity c d then ↑(Nat.find h) else ⊤ | bab270aee1ba305d |
spectrum.isUnit_one_sub_smul_of_lt_inv_radius | Mathlib/Analysis/Normed/Algebra/Spectrum.lean | theorem isUnit_one_sub_smul_of_lt_inv_radius {a : A} {z : 𝕜} (h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹) :
IsUnit (1 - z • a) | case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
⊢ IsUnit (1 - z • a) | suffices hu : IsUnit (u⁻¹ • (1 : A) - a) by
rwa [IsUnit.smul_sub_iff_sub_inv_smul, inv_inv u] at hu | case neg
𝕜 : Type u_1
A : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
a : A
z : 𝕜
h : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹
hz : ¬z = 0
u : 𝕜ˣ := Units.mk0 z hz
⊢ IsUnit (u⁻¹ • 1 - a) | 7c24ad5e48f2ddf5 |
LieModule.iterate_toEnd_mem_lowerCentralSeries₂ | Mathlib/Algebra/Lie/Nilpotent.lean | theorem iterate_toEnd_mem_lowerCentralSeries₂ (x y : L) (m : M) (k : ℕ) :
(toEnd R L M x ∘ₗ toEnd R L M y)^[k] m ∈
lowerCentralSeries R L M (2 * k) | R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
x y : L
m : M
k : ℕ
⊢ (⇑((toEnd R L M) x ∘ₗ (toEnd R L M) y))^[k] m ∈ lowerCentralSeries R L M (2 * k) | induction k with
| zero => simp
| succ k ih =>
have hk : 2 * k.succ = (2 * k + 1) + 1 := rfl
simp only [lowerCentralSeries_succ, Function.comp_apply, Function.iterate_succ', hk,
toEnd_apply_apply, LinearMap.coe_comp, toEnd_apply_apply]
refine LieSubmodule.lie_mem_lie (LieSubmodule.mem_top x) ?_
exact LieSub... | no goals | a700d4de77919e57 |
Real.log_lt_sub_one_of_pos | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | theorem log_lt_sub_one_of_pos (hx1 : 0 < x) (hx2 : x ≠ 1) : log x < x - 1 | x : ℝ
hx1 : 0 < x
hx2 : x ≠ 1
⊢ log x ≠ 0 | rwa [← log_one, log_injOn_pos.ne_iff hx1] | x : ℝ
hx1 : 0 < x
hx2 : x ≠ 1
⊢ 1 ∈ Ioi 0 | 9e3a3fb6c52026a5 |
CategoryTheory.IsDetecting.isIso_iff_of_mono | Mathlib/CategoryTheory/Generator/Basic.lean | lemma IsDetecting.isIso_iff_of_mono {𝒢 : Set C} (h𝒢 : IsDetecting 𝒢)
{X Y : C} (f : X ⟶ Y) [Mono f] :
IsIso f ↔ ∀ s ∈ 𝒢, Function.Surjective ((coyoneda.obj (op s)).map f) | case mp
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
𝒢 : Set C
h𝒢 : IsDetecting 𝒢
X Y : C
f : X ⟶ Y
inst✝ : Mono f
h : IsIso f
⊢ ∀ s ∈ 𝒢, Function.Surjective ((coyoneda.obj (op s)).map f) | rw [isIso_iff_yoneda_map_bijective] at h | case mp
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
𝒢 : Set C
h𝒢 : IsDetecting 𝒢
X Y : C
f : X ⟶ Y
inst✝ : Mono f
h : ∀ (T : C), Function.Bijective fun x => x ≫ f
⊢ ∀ s ∈ 𝒢, Function.Surjective ((coyoneda.obj (op s)).map f) | f84bf19dd3722ac1 |
Finsupp.linearCombination_linear_comp | Mathlib/LinearAlgebra/Finsupp/LinearCombination.lean | theorem linearCombination_linear_comp (f : M →ₗ[R] M') :
linearCombination R (f ∘ v) = f ∘ₗ linearCombination R v | α : Type u_1
M : Type u_2
R : Type u_5
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
M' : Type u_8
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
v : α → M
f : M →ₗ[R] M'
⊢ linearCombination R (⇑f ∘ v) = f ∘ₗ linearCombination R v | ext | case h.h
α : Type u_1
M : Type u_2
R : Type u_5
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
M' : Type u_8
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
v : α → M
f : M →ₗ[R] M'
a✝ : α
⊢ (linearCombination R (⇑f ∘ v) ∘ₗ lsingle a✝) 1 = ((f ∘ₗ linearCombination R v) ∘ₗ lsingle a✝) 1 | 9c925f59b92133d8 |
ContMDiffOn.contMDiffOn_tangentMapWithin | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | theorem ContMDiffOn.contMDiffOn_tangentMapWithin
(hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n)
(hs : UniqueMDiffOn I s) :
haveI : IsManifold I 1 M := .of_le (le_trans le_add_self hmn)
haveI : IsManifold I' 1 M' := .of_le (le_trans le_add_self hmn)
ContMDiffOn I.tangent I'.tangent m (tangentMapWith... | 𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
in... | exact ContMDiffWithinAt.clm_apply_of_inCoordinates hϕ hv hb₂ | no goals | cff72fb4de8037fe |
IsRelPrime.prod_left | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem IsRelPrime.prod_left : (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x | α : Type u_2
I : Type u_1
inst✝¹ : CommMonoid α
inst✝ : DecompositionMonoid α
x : α
s : I → α
t : Finset I
⊢ (∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x | classical
refine Finset.induction_on t (fun _ ↦ isRelPrime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2) | no goals | 21cdedf24c1fa426 |
mem_tangentCone_of_openSegment_subset | Mathlib/Analysis/Calculus/TangentCone.lean | theorem mem_tangentCone_of_openSegment_subset {s : Set G} {x y : G} (h : openSegment ℝ x y ⊆ s) :
y - x ∈ tangentConeAt ℝ s x | G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
s : Set G
x y : G
h : openSegment ℝ x y ⊆ s
n : ℕ
hn : n ≠ 0
⊢ x + (1 / 2) ^ n • (y - x) ∈ openSegment ℝ x y | rw [openSegment_eq_image] | G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
s : Set G
x y : G
h : openSegment ℝ x y ⊆ s
n : ℕ
hn : n ≠ 0
⊢ x + (1 / 2) ^ n • (y - x) ∈ (fun θ => (1 - θ) • x + θ • y) '' Ioo 0 1 | 6e94cb4690e67383 |
LieAlgebra.IsKilling.rootSpace_neg_nsmul_add_chainTop_of_le | Mathlib/Algebra/Lie/Weights/RootSystem.lean | lemma rootSpace_neg_nsmul_add_chainTop_of_le {n : ℕ} (hn : n ≤ chainLength α β) :
rootSpace H (- (n • α) + chainTop α β) ≠ ⊥ | case neg.intro.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L
n : ℕ
hn : n ≤ chainLength α ... | obtain ⟨h, e, f, isSl2, he, hf⟩ := exists_isSl2Triple_of_weight_isNonZero hα | case neg.intro.intro.intro.intro.intro.intro.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H : LieSubalgebra K L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H) L
α β : Weight K (↥H) L... | bc20a037735ef6ac |
Submonoid.LocalizationWithZeroMap.leftCancelMulZero_of_le_isLeftRegular | Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean | theorem leftCancelMulZero_of_le_isLeftRegular
(f : LocalizationWithZeroMap S N) [IsLeftCancelMulZero M]
(h : ∀ ⦃x⦄, x ∈ S → IsLeftRegular x) : IsLeftCancelMulZero N | M : Type u_1
inst✝² : CommMonoidWithZero M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoidWithZero N
f : S.LocalizationWithZeroMap N
inst✝ : IsLeftCancelMulZero M
h : ∀ ⦃x : M⦄, x ∈ S → IsLeftRegular x
fl : S.LocalizationMap N := f.toLocalizationMap
g : M →* N := f.toMap
a z w : N
ha : a ≠ 0
hazw : a * z = a * w
b : M... | rw [← map_mul g] | no goals | 395e8d90995bfe86 |
PartitionOfUnity.finsupport_subset_fintsupport | Mathlib/Topology/PartitionOfUnity.lean | theorem finsupport_subset_fintsupport : ρ.finsupport x₀ ⊆ ρ.fintsupport x₀ := fun i hi ↦ by
rw [ρ.mem_fintsupport_iff]
apply subset_closure
exact (ρ.mem_finsupport x₀).mp hi
| case a
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
i : ι
hi : i ∈ ρ.finsupport x₀
⊢ x₀ ∈ support ⇑(ρ i) | exact (ρ.mem_finsupport x₀).mp hi | no goals | 9ecb2bf735262740 |
Complex.norm_log_one_add_half_le_self | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | /-- For `‖z‖ ≤ 1/2`, the complex logarithm is bounded by `(3/2) * ‖z‖`. -/
lemma norm_log_one_add_half_le_self {z : ℂ} (hz : ‖z‖ ≤ 1/2) : ‖(log (1 + z))‖ ≤ (3/2) * ‖z‖ | case hab.c0
z : ℂ
hz : ‖z‖ ≤ 1 / 2
hz3 : (1 - ‖z‖)⁻¹ ≤ 2
⊢ 0 ≤ 1 - ‖z‖ | linarith | no goals | f3a688a3bed721f7 |
Ordinal.natCast_add_omega0 | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem natCast_add_omega0 (n : ℕ) : n + ω = ω | case intro.intro.intro
n : ℕ
a : Ordinal.{u_1}
ha : a < ↑n + ω
m : ℕ
hb' : ↑m < ω
hb : a ≤ ↑n + ↑m
⊢ ↑n + ↑m < ω | exact_mod_cast nat_lt_omega0 (n + m) | no goals | f552f4cf7a04b7d2 |
Valued.continuous_extension | Mathlib/Topology/Algebra/Valued/ValuedField.lean | theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) | case inr.intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀... | have : ∃ z₀ : K, ∃ y₀ ∈ V', ↑z₀ = y₀ * x₀ ∧ z₀ ≠ 0 := by
rcases Completion.denseRange_coe.mem_nhds nhds_right with ⟨z₀, y₀, y₀_in, H : y₀ * x₀ = z₀⟩
refine ⟨z₀, y₀, y₀_in, ⟨H.symm, ?_⟩⟩
rintro rfl
exact mul_ne_zero (ne_of_mem_of_not_mem y₀_in zeroV') h H | case inr.intro.intro.intro.intro.intro
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x₀ : hat K
h : x₀ ≠ 0
preimage_one : ⇑v ⁻¹' {1} ∈ 𝓝 1
V : Set (hat K)
V_in : V ∈ 𝓝 1
hV : ∀ (x : K), ↑x ∈ V → v x = 1
V' : Set (hat K)
V'_in : V' ∈ 𝓝 1
zeroV' : 0 ∉ V'
hV' : ∀... | af62a79ad478b7b6 |
PMF.apply_eq_one_iff | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | theorem apply_eq_one_iff (p : PMF α) (a : α) : p a = 1 ↔ p.support = {a} | α : Type u_1
p : PMF α
a : α
h : p a = 1
a' : α
ha' : a' ∈ p.support
ha : a' ∉ {a}
this : 0 < ∑' (b : α), if b = a then 0 else p b
⊢ (p a + ∑' (b : α), if b = a then 0 else p b) = (if a = a then p a else 0) + ∑' (b : α), if b = a then 0 else p b | rw [eq_self_iff_true, if_true] | no goals | ccbc613819a4be0f |
Real.smul_map_diagonal_volume_pi | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | theorem smul_map_diagonal_volume_pi [DecidableEq ι] {D : ι → ℝ} (h : det (diagonal D) ≠ 0) :
ENNReal.ofReal (abs (det (diagonal D))) • Measure.map (toLin' (diagonal D)) volume =
volume | ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
this : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i => s i) = univ.pi fun i => (fun x => D i * x) ⁻¹' s i
⊢ ∀ (i : ι), ofReal |D i| * volume ((fun x => D i * x) ⁻¹' s i) ... | intro i | ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
D : ι → ℝ
h : (Matrix.diagonal D).det ≠ 0
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
this : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i => s i) = univ.pi fun i => (fun x => D i * x) ⁻¹' s i
i : ι
⊢ ofReal |D i| * volume ((fun x => D i * x) ⁻¹' s i) = vol... | f356868f3a872858 |
Zsqrtd.nonneg_smul | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem nonneg_smul {a : ℤ√d} {n : ℕ} (ha : Nonneg a) : Nonneg ((n : ℤ√d) * a) | d : ℕ
a : ℤ√↑d
n : ℕ
ha✝ : a.Nonneg
x y : ℕ
ha : { re := -↑x, im := ↑y }.Nonneg
⊢ { re := ↑n * -↑x, im := ↑n * ↑y }.Nonneg | simpa using nonnegg_neg_pos.2 (sqLe_smul n <| nonnegg_neg_pos.1 ha) | no goals | ed3c83933b42a1bd |
LieModule.Weight.hasEigenvalueAt | Mathlib/Algebra/Lie/Weights/Basic.lean | lemma hasEigenvalueAt (χ : Weight R L M) (x : L) :
(toEnd R L M x).HasEigenvalue (χ x) | R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : LieRing.IsNilpotent L
χ : Weight R L M
x : L
⊢ ?m.173842 | simpa [genWeightSpaceOf, ← Module.End.iSup_genEigenspace_eq] using χ.genWeightSpaceOf_ne_bot x | no goals | e7ad90acf37962a9 |
Std.Sat.AIG.RefVec.ite.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/If.lean | theorem go_decl_eq (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (discr : Ref aig)
(lhs rhs : RefVec aig w) (s : RefVec aig curr) :
∀ (idx : Nat) (h1) (h2),
(go aig curr hcurr discr lhs rhs s).aig.decls[idx]'h2 = aig.decls[idx]'h1 | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
discr : aig.Ref
lhs rhs : aig.RefVec w
s : aig.RefVec curr
res : RefVecEntry α w
hgo :
(if hcurr : curr < w then
let input := { discr := discr, lhs := lhs.get curr hcurr, rhs := rhs.get curr hcurr };
let res... | dsimp only at hgo | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
discr : aig.Ref
lhs rhs : aig.RefVec w
s : aig.RefVec curr
res : RefVecEntry α w
hgo :
(if hcurr : curr < w then
go (aig.mkIfCached { discr := discr, lhs := lhs.get curr hcurr, rhs := rhs.get curr hcurr }).aig (cu... | adac79f2da711b55 |
MeasureTheory.Measure.dirac_apply | Mathlib/MeasureTheory/Measure/Dirac.lean | theorem dirac_apply [MeasurableSingletonClass α] (a : α) (s : Set α) :
dirac a s = s.indicator 1 a | α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∉ s
⊢ (dirac a) {a}ᶜ = 0 | simp [dirac_apply' _ (measurableSet_singleton _).compl] | no goals | ffeb750d32ef2676 |
Finset.pow_ssubset_pow_succ_of_pow_ne_closure | Mathlib/Geometry/Group/Growth/LinearLowerBound.lean | @[to_additive]
lemma pow_ssubset_pow_succ_of_pow_ne_closure (hX₁ : (1 : G) ∈ X) (hX : X.Nontrivial)
(hXclosure : (X ^ n : Set G) ≠ closure (X : Set G)) : X ^ n ⊂ X ^ (n + 1) | G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hXclosure : ↑X ^ n ≠ ↑(closure ↑X)
⊢ X ^ n ⊂ X ^ (n + 1) | obtain rfl | hn := eq_or_ne n 0 | case inl
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
hXclosure : ↑X ^ 0 ≠ ↑(closure ↑X)
⊢ X ^ 0 ⊂ X ^ (0 + 1)
case inr
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
n : ℕ
hX₁ : 1 ∈ X
hX : X.Nontrivial
hXclosure : ↑X ^ n ≠ ↑(closure ↑X)
hn : n ≠ 0
⊢ ... | baf8f0ea21c560bf |
WeierstrassCurve.Jacobian.nonsingular_iff_of_Z_ne_zero | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma nonsingular_iff_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
W.Nonsingular P ↔ W.Equation P ∧ (eval P W.polynomialX ≠ 0 ∨ eval P W.polynomialY ≠ 0) | F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ W.Nonsingular P ↔ W.Equation P ∧ ((eval P) W.polynomialX ≠ 0 ∨ (eval P) W.polynomialY ≠ 0) | rw [nonsingular_of_Z_ne_zero hPz, Affine.Nonsingular, ← equation_of_Z_ne_zero hPz,
← eval_polynomialX_of_Z_ne_zero hPz, div_ne_zero_iff, and_iff_left <| pow_ne_zero 4 hPz,
← eval_polynomialY_of_Z_ne_zero hPz, div_ne_zero_iff, and_iff_left <| pow_ne_zero 3 hPz] | no goals | 36d824e29353fb1b |
CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.exists_epi | Mathlib/CategoryTheory/Abelian/GrothendieckCategory/ModuleEmbedding/Opposite.lean | theorem exists_epi (X : D) : ∃ f : generator F ⟶ F.obj X, Epi f | C : Type u
inst✝³ : Category.{v, u} C
D : Type v
inst✝² : SmallCategory D
F : D ⥤ Cᵒᵖ
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
X : D
⊢ ∃ f, Epi f | refine ⟨Sigma.desc (Pi.single X (𝟙 _)) ≫ Sigma.desc (fun f => f), ?_⟩ | C : Type u
inst✝³ : Category.{v, u} C
D : Type v
inst✝² : SmallCategory D
F : D ⥤ Cᵒᵖ
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{v, v, u} C
X : D
⊢ Epi
(Sigma.desc
(Pi.single X
(𝟙 (∐ fun x => CategoryTheory.Abelian.IsGrothendieckAbelian.OppositeModuleEmbedding.projectiveSeparator C))) ≫
... | ba1692fa24a3cffa |
Euclidean.closedBall_eq_image | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | theorem closedBall_eq_image (x : E) (r : ℝ) :
closedBall x r = toEuclidean.symm '' Metric.closedBall (toEuclidean x) r | E : Type u_1
inst✝⁶ : AddCommGroup E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : T2Space E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : FiniteDimensional ℝ E
x : E
r : ℝ
⊢ closedBall x r = ⇑toEuclidean.symm '' Metric.closedBall (toEuclidean x) r | rw [toEuclidean.image_symm_eq_preimage, closedBall_eq_preimage] | no goals | 0001a621089f0d20 |
rotation_ne_conjLIE | Mathlib/Analysis/Complex/Isometry.lean | theorem rotation_ne_conjLIE (a : Circle) : rotation a ≠ conjLIE | a : Circle
⊢ rotation a ≠ conjLIE | intro h | a : Circle
h : rotation a = conjLIE
⊢ False | a62cbc42ea7f34f8 |
Nat.Linear.Certificate.of_combineHyps | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem Certificate.of_combineHyps (ctx : Context) (c : PolyCnstr) (cs : Certificate) (h : (combineHyps c cs).denote ctx → False) : c.denote ctx → cs.denote ctx | ctx : Context
c : PolyCnstr
cs✝ : Certificate
k : Nat
c' : ExprCnstr
cs : List (Nat × ExprCnstr)
h : PolyCnstr.denote ctx (combineHyps c ((k, c') :: cs)) → False
⊢ PolyCnstr.denote ctx c → denote ctx ((k, c') :: cs) | intro h₁ h₂ | ctx : Context
c : PolyCnstr
cs✝ : Certificate
k : Nat
c' : ExprCnstr
cs : List (Nat × ExprCnstr)
h : PolyCnstr.denote ctx (combineHyps c ((k, c') :: cs)) → False
h₁ : PolyCnstr.denote ctx c
h₂ : ExprCnstr.denote ctx c'
⊢ denote ctx cs | 31feb49d52d674e6 |
DirichletCharacter.LSeriesSummable_iff | Mathlib/NumberTheory/LSeries/Dirichlet.lean | /-- The L-series of a Dirichlet character mod `N > 0` converges absolutely at `s` if and only if
`re s > 1`. -/
lemma LSeriesSummable_iff {N : ℕ} (hN : N ≠ 0) (χ : DirichletCharacter ℂ N) {s : ℂ} :
LSeriesSummable ↗χ s ↔ 1 < s.re | N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
s : ℂ
⊢ LSeriesSummable (fun n => χ ↑n) s ↔ 1 < s.re | refine ⟨fun H ↦ ?_, LSeriesSummable_of_one_lt_re χ⟩ | N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
s : ℂ
H : LSeriesSummable (fun n => χ ↑n) s
⊢ 1 < s.re | def54cbe0d797bc5 |
Set.inter_indicator_mul | Mathlib/Algebra/GroupWithZero/Indicator.lean | lemma inter_indicator_mul (f g : ι → M₀) (i : ι) :
(s ∩ t).indicator (fun j ↦ f j * g j) i = s.indicator f i * t.indicator g i | ι : Type u_1
M₀ : Type u_4
inst✝ : MulZeroClass M₀
s t : Set ι
f g : ι → M₀
i : ι
⊢ (s ∩ t).indicator (fun j => f j * g j) i = s.indicator f i * t.indicator g i | rw [← Set.indicator_indicator] | ι : Type u_1
M₀ : Type u_4
inst✝ : MulZeroClass M₀
s t : Set ι
f g : ι → M₀
i : ι
⊢ s.indicator (t.indicator fun j => f j * g j) i = s.indicator f i * t.indicator g i | f1bcb382e3797e03 |
Multiset.sum_map_mul_right | Mathlib/Algebra/BigOperators/Ring/Multiset.lean | lemma sum_map_mul_right : sum (s.map fun i ↦ f i * a) = sum (s.map f) * a :=
Multiset.induction_on s (by simp) fun a s ih => by simp [ih, add_mul]
| ι : Type u_1
α : Type u_2
inst✝ : NonUnitalNonAssocSemiring α
a✝ : α
s✝ : Multiset ι
f : ι → α
a : ι
s : Multiset ι
ih : (map (fun i => f i * a✝) s).sum = (map f s).sum * a✝
⊢ (map (fun i => f i * a✝) (a ::ₘ s)).sum = (map f (a ::ₘ s)).sum * a✝ | simp [ih, add_mul] | no goals | 025acb62f462b207 |
SimpleGraph.IsTuranMaximal.nonempty_iso_turanGraph | Mathlib/Combinatorics/SimpleGraph/Turan.lean | theorem nonempty_iso_turanGraph :
Nonempty (G ≃g turanGraph (Fintype.card V) r) | case intro
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
r : ℕ
h : G.IsTuranMaximal r
zm : { x // x ∈ univ } ≃ Fin #univ
zp :
∀ (a b : { x // x ∈ univ }),
h.finpartition.part ↑a = h.finpartition.part ↑b ↔ ↑(zm a) % #h.finpartition.parts = ↑(zm b) % #h.finpartition.parts
⊢ Nonempty (... | use (Equiv.subtypeUnivEquiv mem_univ).symm.trans zm | case map_rel_iff'
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
r : ℕ
h : G.IsTuranMaximal r
zm : { x // x ∈ univ } ≃ Fin #univ
zp :
∀ (a b : { x // x ∈ univ }),
h.finpartition.part ↑a = h.finpartition.part ↑b ↔ ↑(zm a) % #h.finpartition.parts = ↑(zm b) % #h.finpartition.parts
⊢ ∀ {... | a368d21921cbec7c |
MeasureTheory.StronglyMeasurable.mono | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | theorem mono {m m' : MeasurableSpace α} [TopologicalSpace β]
(hf : StronglyMeasurable[m'] f) (h_mono : m' ≤ m) : StronglyMeasurable[m] f | α : Type u_1
β : Type u_2
f : α → β
m m' : MeasurableSpace α
inst✝ : TopologicalSpace β
hf : StronglyMeasurable f
h_mono : m' ≤ m
⊢ StronglyMeasurable f | let f_approx : ℕ → @SimpleFunc α m β := fun n =>
@SimpleFunc.mk α m β
(hf.approx n)
(fun x => h_mono _ (SimpleFunc.measurableSet_fiber' _ x))
(SimpleFunc.finite_range (hf.approx n)) | α : Type u_1
β : Type u_2
f : α → β
m m' : MeasurableSpace α
inst✝ : TopologicalSpace β
hf : StronglyMeasurable f
h_mono : m' ≤ m
f_approx : ℕ → α →ₛ β := fun n => { toFun := ⇑(hf.approx n), measurableSet_fiber' := ⋯, finite_range' := ⋯ }
⊢ StronglyMeasurable f | 9ffa2b32d9636e7e |
Ordinal.mul_add_div_mul | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem mul_add_div_mul {a c : Ordinal} (hc : c < a) (b d : Ordinal) :
(a * b + c) / (a * d) = b / d | case inr
a c : Ordinal.{u_4}
hc : c < a
b d : Ordinal.{u_4}
ha : a ≠ 0
hd : d ≠ 0
H : a * d ≠ 0
⊢ (a * b + c) / (a * d) = b / d | apply le_antisymm | case inr.a
a c : Ordinal.{u_4}
hc : c < a
b d : Ordinal.{u_4}
ha : a ≠ 0
hd : d ≠ 0
H : a * d ≠ 0
⊢ (a * b + c) / (a * d) ≤ b / d
case inr.a
a c : Ordinal.{u_4}
hc : c < a
b d : Ordinal.{u_4}
ha : a ≠ 0
hd : d ≠ 0
H : a * d ≠ 0
⊢ b / d ≤ (a * b + c) / (a * d) | a58e67332447199e |
Sym2.GameAdd.fix_eq | Mathlib/Order/GameAdd.lean | theorem GameAdd.fix_eq {C : α → α → Sort*} (hr : WellFounded rα)
(IH : ∀ a₁ b₁, (∀ a₂ b₂, Sym2.GameAdd rα s(a₂, b₂) s(a₁, b₁) → C a₂ b₂) → C a₁ b₁) (a b : α) :
GameAdd.fix hr IH a b = IH a b fun a' b' _ => GameAdd.fix hr IH a' b' | α : Type u_1
rα : α → α → Prop
C : α → α → Sort u_3
hr : WellFounded rα
IH : (a₁ b₁ : α) → ((a₂ b₂ : α) → GameAdd rα s(a₂, b₂) s(a₁, b₁) → C a₂ b₂) → C a₁ b₁
a b : α
⊢ fix hr IH a b = IH a b fun a' b' x => fix hr IH a' b' | dsimp [GameAdd.fix] | α : Type u_1
rα : α → α → Prop
C : α → α → Sort u_3
hr : WellFounded rα
IH : (a₁ b₁ : α) → ((a₂ b₂ : α) → GameAdd rα s(a₂, b₂) s(a₁, b₁) → C a₂ b₂) → C a₁ b₁
a b : α
⊢ ⋯.fix (fun x IH' => IH x.1 x.2 fun a' b' => IH' (a', b')) (a, b) =
IH a b fun a' b' x => ⋯.fix (fun x IH' => IH x.1 x.2 fun a' b' => IH' (a', b')) (... | 0a1156822e7dedaa |
hasCardinalLT_option_iff | Mathlib/SetTheory/Cardinal/HasCardinalLT.lean | lemma hasCardinalLT_option_iff (X : Type u) (κ : Cardinal.{w})
(hκ : Cardinal.aleph0 ≤ κ) :
HasCardinalLT (Option X) κ ↔ HasCardinalLT X κ | X : Type u
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
x✝ : HasCardinalLT X κ
⊢ HasCardinalLT PUnit.{1} Cardinal.aleph0 | rw [hasCardinalLT_aleph0_iff] | X : Type u
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
x✝ : HasCardinalLT X κ
⊢ Finite PUnit.{1} | db42753f48328993 |
Int.add_assoc | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean | theorem add_assoc : ∀ a b c : Int, a + b + c = a + (b + c)
| (m:Nat), (n:Nat), _ => aux1 ..
| Nat.cast m, b, Nat.cast k => by
rw [Int.add_comm, ← aux1, Int.add_comm k, aux1, Int.add_comm b]
| a, (n:Nat), (k:Nat) => by
rw [Int.add_comm, Int.add_comm a, ← aux1, Int.add_comm a, Int.add_comm k]
| -[_+1], -[... | m n k : Nat
⊢ -[m+1] + -[n+1] + -[k+1] = -[m+1] + (-[n+1] + -[k+1]) | simp [Nat.add_comm, Nat.add_left_comm, Nat.add_assoc] | no goals | 3685d74f433bd852 |
fermatLastTheoremWith'_polynomial | Mathlib/NumberTheory/FLT/Polynomial.lean | theorem fermatLastTheoremWith'_polynomial {n : ℕ} (hn : 3 ≤ n) (chn : (n : k) ≠ 0) :
FermatLastTheoremWith' k[X] n | case intro.intro.intro
k : Type u_1
inst✝ : Field k
n : ℕ
hn : 3 ≤ n
chn : ↑n ≠ 0
a b c a' b' : k[X]
d : k[X] := gcd a b
hb : d ≠ 0 ∧ b' ≠ 0
ha : d ≠ 0 ∧ a' ≠ 0
eq_a : a = a' * d
eq_b : b = b' * d
hd : d ≠ 0
c' : k[X]
heq : a' ^ n + b' ^ n = c' ^ n
hc : d ≠ 0 ∧ c' ≠ 0
eq_c : c = c' * d
⊢ IsUnit a' ∧ IsUnit b' ∧ IsUnit ... | suffices goal : a'.natDegree = 0 ∧ b'.natDegree = 0 ∧ c'.natDegree = 0 by
simp [natDegree_eq_zero] at goal
obtain ⟨⟨ca', ha'⟩, ⟨cb', hb'⟩, ⟨cc', hc'⟩⟩ := goal
rw [← ha', ← hb', ← hc']
rw [← ha', C_ne_zero] at ha
rw [← hb', C_ne_zero] at hb
rw [← hc', C_ne_zero] at hc
exact ⟨ha.right.isUnit_C, hb.right.isU... | case intro.intro.intro
k : Type u_1
inst✝ : Field k
n : ℕ
hn : 3 ≤ n
chn : ↑n ≠ 0
a b c a' b' : k[X]
d : k[X] := gcd a b
hb : d ≠ 0 ∧ b' ≠ 0
ha : d ≠ 0 ∧ a' ≠ 0
eq_a : a = a' * d
eq_b : b = b' * d
hd : d ≠ 0
c' : k[X]
heq : a' ^ n + b' ^ n = c' ^ n
hc : d ≠ 0 ∧ c' ≠ 0
eq_c : c = c' * d
⊢ a'.natDegree = 0 ∧ b'.natDegree... | 0b2f8f9d4d152b49 |
CategoryTheory.PreGaloisCategory.exists_hom_from_galois_of_fiber_nonempty | Mathlib/CategoryTheory/Galois/Decomposition.lean | /-- Any object with non-empty fiber admits a hom from a Galois object. -/
lemma exists_hom_from_galois_of_fiber_nonempty (X : C) (h : Nonempty (F.obj X)) :
∃ (A : C) (_ : A ⟶ X), IsGalois A | case intro
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X : C
x : (F.obj X).carrier
⊢ ∃ A x, IsGalois A | obtain ⟨A, f, a, h1, _⟩ := exists_hom_from_galois_of_fiber F X x | case intro.intro.intro.intro.intro
C : Type u₁
inst✝² : Category.{u₂, u₁} C
inst✝¹ : GaloisCategory C
F : C ⥤ FintypeCat
inst✝ : FiberFunctor F
X : C
x : (F.obj X).carrier
A : C
f : A ⟶ X
a : (F.obj A).carrier
h1 : IsGalois A
right✝ : F.map f a = x
⊢ ∃ A x, IsGalois A | c1e1ad914efa42c5 |
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