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 |
|---|---|---|---|---|---|---|
Polynomial.coeff_divModByMonicAux_mem_span_pow_mul_span | Mathlib/Algebra/Polynomial/CoeffMem.lean | lemma coeff_divModByMonicAux_mem_span_pow_mul_span : ∀ (p q : S[X]) (hq : q.Monic) (i),
(p.divModByMonicAux hq).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧
(p.divModByMonicAux hq).2.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)
| p, q, hq, i => by
rw [divModByMonicAux]
have H₀ (i) : p.coef... | case neg
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
p q : S[X]
hq : q.Monic
i : ℕ
H₀ : ∀ (i : ℕ), p.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p)
hpq : ¬(q.degree ≤ p.degree ∧ p ≠ 0)
⊢ (0, p).1.coeff i ∈ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) ∧ (0, p).2.coeff i ∈ spanCoeffs(q) ... | simpa using H₀ _ | no goals | f188ae6d502b95a1 |
List.findM?_eq_findSomeM? | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Control.lean | theorem findM?_eq_findSomeM? [Monad m] [LawfulMonad m] (p : α → m Bool) (as : List α) :
as.findM? p = as.findSomeM? fun a => return if (← p a) then some a else none | case cons.e_a.h
m : Type → Type u_1
α : Type
inst✝¹ : Monad m
inst✝ : LawfulMonad m
p : α → m Bool
a : α
as : List α
ih :
findM? p as =
findSomeM?
(fun a => do
let __do_lift ← p a
pure (if __do_lift = true then some a else none))
as
b : Bool
⊢ (match b with
| true => pure (some a)
... | cases b <;> simp | no goals | ce5d2af180ea00d8 |
circleIntegrable_sub_zpow_iff | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | theorem circleIntegrable_sub_zpow_iff {c w : ℂ} {R : ℝ} {n : ℤ} :
CircleIntegrable (fun z => (z - w) ^ n) c R ↔ R = 0 ∨ 0 ≤ n ∨ w ∉ sphere c |R| | case mp.intro.intro
c w : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
hw : w ∈ sphere c |R|
⊢ ¬IntervalIntegrable (fun θ => (circleMap 0 R θ * I) • (circleMap c R θ - w) ^ n) volume 0 (2 * π) | rw [← image_circleMap_Ioc] at hw | case mp.intro.intro
c w : ℂ
R : ℝ
n : ℤ
hR : R ≠ 0
hn : n < 0
hw : w ∈ circleMap c R '' Ioc 0 (2 * π)
⊢ ¬IntervalIntegrable (fun θ => (circleMap 0 R θ * I) • (circleMap c R θ - w) ^ n) volume 0 (2 * π) | 8d16963f19b22485 |
AlgebraicGeometry.AffineTargetMorphismProperty.respectsIso_mk | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | theorem respectsIso_mk {P : AffineTargetMorphismProperty}
(h₁ : ∀ {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z) [IsAffine Z], P f → P (e.hom ≫ f))
(h₂ : ∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) [IsAffine Y],
P f → @P _ _ (f ≫ e.hom) (isAffine_of_isIso e.inv)) :
P.toProperty.RespectsIso | case hpostcomp.intro
P : AffineTargetMorphismProperty
h₁ : ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z], P f → P (e.hom ≫ f)
h₂ : ∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [inst : IsAffine Y], P f → P (f ≫ e.hom)
X Y Z : Scheme
e : Y ≅ Z
f : X ⟶ Y
a : IsAffine Y
h : P f
⊢ P.toProperty (f ≫ e.hom) | exact ⟨isAffine_of_isIso e.inv, h₂ e f h⟩ | no goals | a0abeb70d6935d67 |
TopologicalSpace.Closeds.coe_sup | Mathlib/Topology/Sets/Closeds.lean | theorem coe_sup (s t : Closeds α) : (↑(s ⊔ t) : Set α) = ↑s ∪ ↑t | α : Type u_2
inst✝ : TopologicalSpace α
s t : Closeds α
⊢ ↑(s ⊔ t) = ↑s ∪ ↑t | rfl | no goals | d62cbb5ccdecb25d |
Order.Ideal.isProper_of_not_mem | Mathlib/Order/Ideal.lean | theorem isProper_of_not_mem {I : Ideal P} {p : P} (nmem : p ∉ I) : IsProper I :=
⟨fun hp ↦ by
have := mem_univ p
rw [← hp] at this
exact nmem this⟩
| P : Type u_1
inst✝ : LE P
I : Ideal P
p : P
nmem : p ∉ I
hp : ↑I = univ
this : p ∈ univ
⊢ False | rw [← hp] at this | P : Type u_1
inst✝ : LE P
I : Ideal P
p : P
nmem : p ∉ I
hp : ↑I = univ
this : p ∈ ↑I
⊢ False | eedecb497d07f70f |
integral_sin_pow_aux | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_sin_pow_aux :
(∫ x in a..b, sin x ^ (n + 2)) =
(sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (↑n + 1) * ∫ x in a..b, sin x ^ n) -
(↑n + 1) * ∫ x in a..b, sin x ^ (n + 2) | a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
... | simp [H, h, sq] | a b : ℝ
n : ℕ
C : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
h : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)
hu : ∀ x ∈ [[a, b]], HasDerivAt (fun y => sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x
hv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x
H :
∫ (x : ℝ) in a..b, sin x ^ (n + 1) * sin x =
... | f362bf75d26c9d57 |
Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg | Mathlib/Analysis/MeanInequalities.lean | theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjExponent q) {A B : ℝ} (hA : 0 ≤ A)
(hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ HasSum (fun i => f i * g i) C | case intro.intro.intro.intro.intro.intro
ι : Type u
p q : ℝ
hpq : p.IsConjExponent q
f g : ι → ℝ≥0
A B : ℝ≥0
hf_sum : HasSum (fun a => f a ^ p) (A ^ p)
hg_sum : HasSum (fun a => g a ^ q) (B ^ q)
C : ℝ≥0
hC : C ≤ A * B
H : HasSum (fun i => f i * g i) C
⊢ HasSum (fun i => ↑(f i) * ↑(g i)) ↑C | norm_cast | no goals | 164a3a6315ccf271 |
WeierstrassCurve.exists_variableChange_of_char_three_of_j_eq_zero | Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean | private lemma exists_variableChange_of_char_three_of_j_eq_zero
[E.IsShortNF] [E'.IsShortNF] :
∃ C : VariableChange F, E.variableChange C = E' | F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
inst✝¹ : E.IsShortNF
inst✝ : E'.IsShortNF
ha₄ : E.a₄ ≠ 0
ha₄' : E'.a₄ ≠ 0
this : NeZero 4
u : F
hu : u ^ 4 = E.a₄ / E'.a₄
⊢ 2 ≤ 3 | norm_num | no goals | c130cdcad82e2167 |
CategoryTheory.MorphismProperty.colimitsOfShape_le_of_final | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | lemma colimitsOfShape_le_of_final {J' : Type*} [Category J'] (F : J ⥤ J') [F.Final] :
W.colimitsOfShape J' ≤ W.colimitsOfShape J | C : Type u
inst✝³ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝² : Category.{u_4, u_1} J
J' : Type u_2
inst✝¹ : Category.{u_3, u_2} J'
F : J ⥤ J'
inst✝ : F.Final
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X₁ X₂ : J' ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J' ... | have h₁' : IsColimit (c₁.whisker F) := (Functor.Final.isColimitWhiskerEquiv F c₁).symm h₁ | C : Type u
inst✝³ : Category.{v, u} C
W : MorphismProperty C
J : Type u_1
inst✝² : Category.{u_4, u_1} J
J' : Type u_2
inst✝¹ : Category.{u_3, u_2} J'
F : J ⥤ J'
inst✝ : F.Final
X✝ Y✝ : C
f✝ : X✝ ⟶ Y✝
X₁ X₂ : J' ⥤ C
c₁ : Cocone X₁
c₂ : Cocone X₂
h₁ : IsColimit c₁
h₂ : IsColimit c₂
f : X₁ ⟶ X₂
hf : W.functorCategory J' ... | 3cb77bf936c9d4d9 |
Orientation.linearIsometryEquiv_comp_rightAngleRotation | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | theorem linearIsometryEquiv_comp_rightAngleRotation (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x : E) : φ (J x) = J (φ x) | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
x : E
⊢ φ (o.rightAngleRotation x) = o.rightAngleRotation (φ x) | convert (o.rightAngleRotation_map φ (φ x)).symm | case h.e'_2.h.e'_6.h.e'_6
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
x : E
⊢ x = φ.symm (φ x)
case h.e'_3.h.e'_5.h.e'_5
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : In... | 92cca6cb1f7f9fef |
map_le_lineMap_iff_slope_le_slope | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | theorem map_le_lineMap_iff_slope_le_slope (hab : a < b) (h₀ : 0 < r) (h₁ : r < 1) :
f c ≤ lineMap (f a) (f b) r ↔ slope f a c ≤ slope f c b | k : Type u_1
E : Type u_2
inst✝³ : LinearOrderedField k
inst✝² : OrderedAddCommGroup E
inst✝¹ : Module k E
inst✝ : OrderedSMul k E
f : k → E
a b r : k
hab : a < b
h₀ : 0 < r
h₁ : r < 1
⊢ f ((lineMap a b) r) ≤ (lineMap (f a) (f b)) r ↔ slope f a ((lineMap a b) r) ≤ slope f ((lineMap a b) r) b | rw [map_le_lineMap_iff_slope_le_slope_left (mul_pos h₀ (sub_pos.2 hab)), ←
lineMap_slope_lineMap_slope_lineMap f a b r, right_le_lineMap_iff_le h₁] | no goals | 22cc763cce3393ca |
MeasureTheory.Measure.haar_singleton | Mathlib/MeasureTheory/Group/Measure.lean | theorem haar_singleton [IsTopologicalGroup G] [BorelSpace G] (g : G) : μ {g} = μ {(1 : G)} | case h.e'_2.h.e'_6
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsTopologicalGroup G
inst✝ : BorelSpace G
g : G
⊢ {g} = (fun h => g⁻¹ * h) ⁻¹' {1} | simp only [mul_one, preimage_mul_left_singleton, inv_inv] | no goals | f4a174aeac7f91b9 |
CoxeterSystem.isLeftInversion_simple_iff_isLeftDescent | Mathlib/GroupTheory/Coxeter/Inversion.lean | theorem isLeftInversion_simple_iff_isLeftDescent (w : W) (i : B) :
cs.IsLeftInversion w (s i) ↔ cs.IsLeftDescent w i | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
w : W
i : B
⊢ cs.IsLeftInversion w (cs.simple i) ↔ cs.IsLeftDescent w i | simp [IsLeftInversion, IsLeftDescent, cs.isReflection_simple i] | no goals | 99132206d5581048 |
exists_seq_of_forall_finset_exists | Mathlib/Data/Fintype/Basic.lean | theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
(h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) | α : Type u_4
P : α → Prop
r : α → α → Prop
h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
this : Nonempty α
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n) | choose! F hF using h | α : Type u_4
P : α → Prop
r : α → α → Prop
this : Nonempty α
F : Finset α → α
hF : ∀ (s : Finset α), (∀ x ∈ s, P x) → P (F s) ∧ ∀ x ∈ s, r x (F s)
⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n) | 84f3321d3efdbd70 |
Polynomial.coeff_hermite_explicit | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | theorem coeff_hermite_explicit :
∀ n k : ℕ, coeff (hermite (2 * n + k)) k = (-1) ^ n * (2 * n - 1)‼ * Nat.choose (2 * n + k) k
| 0, _ => by simp
| n + 1, 0 => by
convert coeff_hermite_succ_zero (2 * n + 1) using 1
-- Porting note: ring_nf did not solve the goal on line 165
rw [coeff_hermite_explicit... | case e_a.e_a
n✝ k✝ : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
n k : ℕ
⊢ (2 * (n + 1) - 1)‼ * (2 * (n + 1) + (k + 1)).choose (k + 1) =
(2 * (n + 1) - 1)‼ * (2 * (n + 1) + k).choose k + (2 * n - 1)‼ * ((2 * n + (k + 2)).choose (k + 2) * (k + 2)) | rw [(by rw [Nat.left_distrib, mul_one, Nat.add_one_sub_one] : 2 * (n + 1) - 1 = 2 * n + 1),
Nat.doubleFactorial_add_one, mul_comm (2 * n + 1)] | case e_a.e_a
n✝ k✝ : ℕ
hermite_explicit : ℕ → ℕ → ℤ := fun n k => (-1) ^ n * ↑(2 * n - 1)‼ * ↑((2 * n + k).choose k)
n k : ℕ
⊢ (2 * n - 1)‼ * (2 * n + 1) * (2 * (n + 1) + (k + 1)).choose (k + 1) =
(2 * n - 1)‼ * (2 * n + 1) * (2 * (n + 1) + k).choose k +
(2 * n - 1)‼ * ((2 * n + (k + 2)).choose (k + 2) * (k +... | 76b4bea6433d05b5 |
MeasureTheory.llr_smul_right | Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | lemma llr_smul_right [IsFiniteMeasure μ] [Measure.HaveLebesgueDecomposition μ ν]
(hμν : μ ≪ ν) (c : ℝ≥0∞) (hc : c ≠ 0) (hc_ne_top : c ≠ ∞) :
llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log c.toReal | case h.hx
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : μ.rnDeriv (c • ν) =ᶠ[ae ν] c⁻¹ • μ.rnDeriv ν
x : α
hx_eq : μ.rnDeriv (c • ν) x = (c⁻¹ • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top... | simp [hc, hc_ne_top] | no goals | f2c098759e8140a2 |
LucasLehmer.X.ext | Mathlib/NumberTheory/LucasLehmer.lean | theorem ext {x y : X q} (h₁ : x.1 = y.1) (h₂ : x.2 = y.2) : x = y | case mk.mk
q : ℕ+
fst✝¹ snd✝¹ fst✝ snd✝ : ZMod ↑q
h₁ : (fst✝¹, snd✝¹).1 = (fst✝, snd✝).1
h₂ : (fst✝¹, snd✝¹).2 = (fst✝, snd✝).2
⊢ (fst✝¹, snd✝¹) = (fst✝, snd✝) | congr | no goals | b7ee4edacbd6f2eb |
BoxIntegral.Box.subbox_induction_on | Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean | theorem subbox_induction_on {p : Box ι → Prop} (I : Box ι)
(H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J)
(H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ),
z ∈ Box.Icc J → Box.Icc J ⊆ U →
(∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) :
p I | ι : Type u_1
inst✝ : Fintype ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J
H_nhds :
∀ z ∈ Box.Icc I,
∃ U ∈ 𝓝[Box.Icc I] z,
∀ J ≤ I,
∀ (m : ℕ),
z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
⊢ p ... | refine subbox_induction_on' I (fun J hle hs => H_ind J hle fun J' h' => ?_) H_nhds | ι : Type u_1
inst✝ : Fintype ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J
H_nhds :
∀ z ∈ Box.Icc I,
∃ U ∈ 𝓝[Box.Icc I] z,
∀ J ≤ I,
∀ (m : ℕ),
z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
J : ... | 7bb7c3e429c684e0 |
HahnModule.add_smul | Mathlib/RingTheory/HahnSeries/Multiplication.lean | theorem add_smul [AddCommMonoid R] [SMulWithZero R V] {x y : HahnSeries Γ R}
{z : HahnModule Γ' R V} (h : ∀ (r s : R) (u : V), (r + s) • u = r • u + s • u) :
(x + y) • z = x • z + y • z | case h.h
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_5
inst✝⁶ : PartialOrder Γ
inst✝⁵ : PartialOrder Γ'
inst✝⁴ : VAdd Γ Γ'
inst✝³ : IsOrderedCancelVAdd Γ Γ'
inst✝² : AddCommMonoid V
inst✝¹ : AddCommMonoid R
inst✝ : SMulWithZero R V
x y : HahnSeries Γ R
z : HahnModule Γ' R V
h✝ : ∀ (r s : R) (u : V), (r + s) • u ... | rw [h.1, h.2, add_zero] | no goals | d6350673eea7c23e |
Subalgebra.LinearDisjoint.of_linearDisjoint_finite_left | Mathlib/RingTheory/LinearDisjoint.lean | theorem of_linearDisjoint_finite_left [Algebra.IsIntegral R A]
(H : ∀ A' : Subalgebra R S, A' ≤ A → [Module.Finite R A'] → A'.LinearDisjoint B) :
A.LinearDisjoint B | case intro.intro.intro.intro
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B... | obtain ⟨x', hx'⟩ := h (show x ∈ {x, y} by simp) | case intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmo... | 1d56e99b263dd6b5 |
Filter.inter_eventuallyEq_left | Mathlib/Order/Filter/Basic.lean | theorem inter_eventuallyEq_left {s t : Set α} {l : Filter α} :
(s ∩ t : Set α) =ᶠ[l] s ↔ ∀ᶠ x in l, x ∈ s → x ∈ t | α : Type u
s t : Set α
l : Filter α
⊢ s ∩ t =ᶠ[l] s ↔ ∀ᶠ (x : α) in l, x ∈ s → x ∈ t | simp only [eventuallyEq_set, mem_inter_iff, and_iff_left_iff_imp] | no goals | 2d1c75287f901022 |
Ideal.absNorm_bot | Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean | theorem absNorm_bot : absNorm (⊥ : Ideal S) = 0 | S : Type u_1
inst✝³ : CommRing S
inst✝² : Nontrivial S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
⊢ absNorm ⊥ = 0 | rw [← Ideal.zero_eq_bot, _root_.map_zero] | no goals | 73fd61bc09b59419 |
GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | theorem integral_cexp_neg_mul_sq_add_real_mul_I (hb : 0 < b.re) (c : ℝ) :
∫ x : ℝ, cexp (-b * (x + c * I) ^ 2) = (π / b) ^ (1 / 2 : ℂ) | b : ℂ
hb : 0 < b.re
c : ℝ
I₁ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
HI₁ : I₁ = fun T => ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)
I₂ : ℝ → ℂ := fun T => ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)
I₄ : ℝ → ℂ := fun T => ∫ (y : ℝ) in 0 ..c, cexp (-b * (↑T + ↑y * I) ^ 2)
I₅ : ℝ → ℂ := fu... | exact differentiable_pow 2 | no goals | 8f46f2d622ac75ef |
Real.Angle.cos_eq_iff_coe_eq_or_eq_neg | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | theorem cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} :
cos θ = cos ψ ↔ (θ : Angle) = ψ ∨ (θ : Angle) = -ψ | case mp
θ ψ : ℝ
Hcos : (∃ n, ↑n * π = (θ + ψ) / 2) ∨ ∃ n, ↑n * π = (θ - ψ) / 2
⊢ ↑θ = ↑ψ ∨ ↑θ = -↑ψ | rcases Hcos with (⟨n, hn⟩ | ⟨n, hn⟩) | case mp.inl.intro
θ ψ : ℝ
n : ℤ
hn : ↑n * π = (θ + ψ) / 2
⊢ ↑θ = ↑ψ ∨ ↑θ = -↑ψ
case mp.inr.intro
θ ψ : ℝ
n : ℤ
hn : ↑n * π = (θ - ψ) / 2
⊢ ↑θ = ↑ψ ∨ ↑θ = -↑ψ | 231d1921c276f456 |
ENNReal.rpow_le_one_of_one_le_of_neg | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | theorem rpow_le_one_of_one_le_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : z < 0) : x ^ z ≤ 1 | case coe
z : ℝ
hz : z < 0
x✝ : ℝ≥0
hx : 1 ≤ ↑x✝
⊢ ↑x✝ ^ z ≤ 1 | simp only [one_le_coe_iff, some_eq_coe] at hx | case coe
z : ℝ
hz : z < 0
x✝ : ℝ≥0
hx : 1 ≤ x✝
⊢ ↑x✝ ^ z ≤ 1 | 9fb379d2f486f39d |
ENNReal.exists_pos_sum_of_countable | Mathlib/Analysis/SpecificLimits/Basic.lean | theorem exists_pos_sum_of_countable {ε : ℝ≥0∞} (hε : ε ≠ 0) (ι) [Countable ι] :
∃ ε' : ι → ℝ≥0, (∀ i, 0 < ε' i) ∧ (∑' i, (ε' i : ℝ≥0∞)) < ε | case intro.intro
ε : ℝ≥0∞
hε : ε ≠ 0
ι : Type u_4
inst✝ : Countable ι
r : ℝ≥0∞
h0r : 0 < r
hrε : r < ε
⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∑' (i : ι), ↑(ε' i) < ε | rcases lt_iff_exists_coe.1 hrε with ⟨x, rfl, _⟩ | case intro.intro.intro.intro
ε : ℝ≥0∞
hε : ε ≠ 0
ι : Type u_4
inst✝ : Countable ι
x : ℝ≥0
right✝ : ↑x < ε
h0r : 0 < ↑x
hrε : ↑x < ε
⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∑' (i : ι), ↑(ε' i) < ε | 6474cb59d23a91c6 |
SameRay.trans | Mathlib/LinearAlgebra/Ray.lean | theorem trans (hxy : SameRay R x y) (hyz : SameRay R y z) (hy : y = 0 → x = 0 ∨ z = 0) :
SameRay R x z | case inr
R : Type u_1
inst✝² : StrictOrderedCommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x y z : M
hxy : SameRay R x y
hyz : SameRay R y z
hy : y = 0 → x = 0 ∨ z = 0
hx : x ≠ 0
⊢ SameRay R x z | rcases eq_or_ne z 0 with (rfl | hz) | case inr.inl
R : Type u_1
inst✝² : StrictOrderedCommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x y : M
hxy : SameRay R x y
hx : x ≠ 0
hyz : SameRay R y 0
hy : y = 0 → x = 0 ∨ 0 = 0
⊢ SameRay R x 0
case inr.inr
R : Type u_1
inst✝² : StrictOrderedCommSemiring R
M : Type u_2
inst✝¹ : AddCommMonoi... | 58153a221638fd7f |
IsLocalization.bot_lt_comap_prime | Mathlib/RingTheory/Localization/Ideal.lean | theorem bot_lt_comap_prime [IsDomain R] (hM : M ≤ R⁰) (p : Ideal S) [hpp : p.IsPrime]
(hp0 : p ≠ ⊥) : ⊥ < Ideal.comap (algebraMap R S) p | R : Type u_1
inst✝⁴ : CommRing R
M : Submonoid R
S : Type u_2
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalization M S
inst✝ : IsDomain R
hM : M ≤ R⁰
p : Ideal S
hpp : p.IsPrime
hp0 : p ≠ ⊥
this : IsDomain S
⊢ Ideal.comap (algebraMap R S) ⊥ < Ideal.comap (algebraMap R S) p | convert (orderIsoOfPrime M S).lt_iff_lt.mpr (show (⟨⊥, Ideal.bot_prime⟩ :
{ p : Ideal S // p.IsPrime }) < ⟨p, hpp⟩ from hp0.bot_lt) | no goals | 070619afe4075fe6 |
Set.SMulAntidiagonal.finite_of_isPWO | Mathlib/Data/Set/SMulAntidiagonal.lean | theorem finite_of_isPWO (hs : s.IsPWO) (ht : t.IsPWO) (a) : (smulAntidiagonal s t a).Finite | case refl
G : Type u_1
P : Type u_2
s : Set G
t : Set P
inst✝³ : PartialOrder G
inst✝² : PartialOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
hs : s.IsPWO
ht : t.IsPWO
a : P
h : (s.smulAntidiagonal t a).Infinite
h1 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x1 x2 => x1 ≤ x2)
h2 : (... | simp_all only [Order.Preimage, le_refl, Prod.forall, implies_true] | no goals | 7f1573a75cfc8a75 |
PowerSeries.X_dvd_iff | Mathlib/RingTheory/PowerSeries/Basic.lean | theorem X_dvd_iff {φ : R⟦X⟧} : (X : R⟦X⟧) ∣ φ ↔ constantCoeff R φ = 0 | R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ X ∣ φ ↔ (constantCoeff R) φ = 0 | rw [← pow_one (X : R⟦X⟧), X_pow_dvd_iff, ← coeff_zero_eq_constantCoeff_apply] | R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
⊢ (∀ m < 1, (coeff R m) φ = 0) ↔ (coeff R 0) φ = 0 | cf60b9c3c4ca0721 |
SetTheory.Game.small_setOf_birthday_lt | Mathlib/SetTheory/Game/Birthday.lean | theorem small_setOf_birthday_lt (o : Ordinal) : Small.{u} {x : Game.{u} // birthday x < o} | y : PGame
IH : ∀ k < Order.succ (birthday ⟦y⟧), Small.{u, u + 1} { x // x.birthday < k }
S : Set Game := ⋃ a ∈ Set.Iio (Order.succ (birthday ⟦y⟧)), {x | x.birthday < a}
H : Small.{u, u + 1} ↑S := small_biUnion (Set.Iio (Order.succ (birthday ⟦y⟧))) fun a h => {x | x.birthday < a}
f : Set ↑S × Set ↑S → Game :=
fun g =>... | simpa [S, hy'] using this | no goals | 942b1277300d8c20 |
Real.lt_tan | Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | theorem lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x | x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
⊢ x < tan x | have cos_pos {y : ℝ} (hy : y ∈ U) : 0 < cos y := by
exact cos_pos_of_mem_Ioo (Ico_subset_Ioo_left (neg_lt_zero.mpr half_pi_pos) hy) | x : ℝ
h1 : 0 < x
h2 : x < π / 2
U : Set ℝ := Ico 0 (π / 2)
intU : interior U = Ioo 0 (π / 2)
half_pi_pos : 0 < π / 2
cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y
⊢ x < tan x | beb6377e2b7c7c87 |
Profinite.exists_locallyConstant | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_locallyConstant {α : Type*} (hC : IsLimit C) (f : LocallyConstant C.pt α) :
∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _).hom | J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
S : DiscreteQuotient ↑C.pt.toTop := f.discreteQuotient
ff : Quotient S.toSetoid → α := ⇑f.lift
h✝ : IsEmpty (Quotient S.toSetoid)
this✝ : ∃ j, IsEmpty ↑(F.obj j).toTop
... | rw [this] | J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
S : DiscreteQuotient ↑C.pt.toTop := f.discreteQuotient
ff : Quotient S.toSetoid → α := ⇑f.lift
h✝ : IsEmpty (Quotient S.toSetoid)
this✝ : ∃ j, IsEmpty ↑(F.obj j).toTop
... | 72c7aeebc62ad404 |
ContDiffWithinAt.contDiffOn' | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem ContDiffWithinAt.contDiffOn' (hm : m ≤ n) (h' : m = ∞ → n = ω)
(h : ContDiffWithinAt 𝕜 n f s x) :
∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u) | case intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
m✝ n : WithTop ℕ∞
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ω
m : ℕ
hm ... | rw [inter_comm] at hut | case intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
m✝ n : WithTop ℕ∞
h : ContDiffWithinAt 𝕜 n f s x
hn : n ≠ ω
m : ℕ
hm ... | 252a1c681d4d0950 |
CochainComplex.mappingCone.triangleMapOfHomotopy_comm₃ | Mathlib/Algebra/Homology/HomotopyCategory/Pretriangulated.lean | @[reassoc]
lemma triangleMapOfHomotopy_comm₃ :
mapOfHomotopy H ≫ (triangle φ₂).mor₃ = (triangle φ₁).mor₃ ≫ a⟦1⟧' | C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K₁ L₁ K₂ L₂ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
⊢ mapOfHomotopy H ≫ (triangle φ₂).mor₃ = (triangle φ₁).mor₃ ≫ (CategoryTheory.shiftFunctor (CochainComplex C ... | ext p | case h
C : Type u_1
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K₁ L₁ K₂ L₂ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
p : ℤ
⊢ (mapOfHomotopy H ≫ (triangle φ₂).mor₃).f p =
((triangle φ₁).mor₃ ≫ (CategoryTheory.shiftFu... | 5e3487226dd990fa |
CategoryTheory.Limits.Sigma.eqToHom_comp_ι | Mathlib/CategoryTheory/Limits/Shapes/Products.lean | theorem Sigma.eqToHom_comp_ι {J : Type*} (f : J → C) [HasCoproduct f] {j j' : J} (w : j = j') :
eqToHom (by simp [w]) ≫ Sigma.ι f j' = Sigma.ι f j | case refl
C : Type u
inst✝¹ : Category.{v, u} C
J : Type u_1
f : J → C
inst✝ : HasCoproduct f
j : J
⊢ eqToHom ⋯ ≫ ι f j = ι f j | simp | no goals | 50733f72a97eb131 |
hasDerivAt_of_tendstoUniformlyOnFilter | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | theorem hasDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasDerivAt g (g' x) x | ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜)... | let G' z := (1 : 𝕜 →L[𝕜] 𝕜).smulRight (g' z) | ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜)... | 3364bb7f5ca1a8f4 |
PrimeSpectrum.toPiLocalization_surjective_of_discreteTopology | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | theorem toPiLocalization_surjective_of_discreteTopology :
Function.Surjective (toPiLocalization R) := fun x ↦ by
have (p : PrimeSpectrum R) : ∃ f, (basicOpen f : Set _) = {p} :=
have ⟨_, ⟨f, rfl⟩, hpf, hfp⟩ := isTopologicalBasis_basic_opens.isOpen_iff.mp
(isOpen_discrete {p}) p rfl
⟨f, hfp.antisymm ... | R : Type u
inst✝¹ : CommSemiring R
inst✝ : DiscreteTopology (PrimeSpectrum R)
x : PiLocalization R
f : PrimeSpectrum R → R
hf : ∀ (p : PrimeSpectrum R), ↑(basicOpen (f p)) = {p}
e : PrimeSpectrum R ≃ ↑(Set.range f) := Equiv.ofInjective f ⋯
a : ↑(Set.range f)
⊢ IsLocalization.Away (↑a) (Localization.Away ↑a) | infer_instance | no goals | 680284c76fd78d8c |
MeasureTheory.Martingale.condExp_stoppedValue_stopping_time_ae_eq_restrict_le | Mathlib/Probability/Martingale/OptionalSampling.lean | theorem condExp_stoppedValue_stopping_time_ae_eq_restrict_le (h : Martingale f ℱ μ)
(hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ) [SigmaFinite (μ.trim hσ.measurableSpace_le)]
(hτ_le : ∀ x, τ x ≤ i) :
μ[stoppedValue f τ|hσ.measurableSpace] =ᵐ[μ.restrict {x : Ω | τ x ≤ σ x}] stoppedValue f τ | case refine_1
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : Measurabl... | rw [hτ.measurableSet_inter_le_iff hσ, IsStoppingTime.measurableSet_min_iff hτ hσ] at ht | case refine_1
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : Measurabl... | 5957e7231ee5fbd1 |
contDiff_norm_rpow | Mathlib/Analysis/InnerProductSpace/NormPow.lean | theorem contDiff_norm_rpow {p : ℝ} (hp : 1 < p) : ContDiff ℝ 1 (fun x : E ↦ ‖x‖ ^ p) | case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
p : ℝ
hp : 1 < p
x : E
hx : x = 0
⊢ Filter.Tendsto (fderiv ℝ fun x => ‖x‖ ^ p) (𝓝 0) (𝓝 0) | rw [tendsto_zero_iff_norm_tendsto_zero] | case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
p : ℝ
hp : 1 < p
x : E
hx : x = 0
⊢ Filter.Tendsto (fun x => ‖fderiv ℝ (fun x => ‖x‖ ^ p) x‖) (𝓝 0) (𝓝 0) | fdf902c8d0df4633 |
NumberField.mixedEmbedding.fundamentalCone.integerSetToAssociates_eq_iff | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean | theorem integerSetToAssociates_eq_iff (a b : integerSet K) :
integerSetToAssociates K a = integerSetToAssociates K b ↔
∃ ζ : torsion K, ζ • a = b | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a b : ↑(integerSet K)
x✝ : ∃ ζ, ↑ζ • ↑a = ↑b
u : (𝓞 K)ˣ
property✝ : u ∈ torsion K
h : ↑⟨u, property✝⟩ • ↑a = ↑b
⊢ (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑(unitsNonZeroDivisorsEquiv.symm u)) * ↑a = ↑b | simpa using h | no goals | 33418b1081f2de4d |
CategoryTheory.colimitYonedaHomEquiv_π_apply | Mathlib/CategoryTheory/Limits/Indization/LocallySmall.lean | theorem colimitYonedaHomEquiv_π_apply (η : colimit (F ⋙ yoneda) ⟶ G) (i : Iᵒᵖ) :
limit.π (F.op ⋙ G) i (colimitYonedaHomEquiv F G η) =
η.app (op (F.obj i.unop)) ((colimit.ι (F ⋙ yoneda) i.unop).app _ (𝟙 _)) | C : Type u
inst✝⁴ : Category.{v, u} C
I : Type u₁
inst✝³ : Category.{v₁, u₁} I
inst✝² : HasColimitsOfShape I (Type v)
inst✝¹ : HasLimitsOfShape Iᵒᵖ (Type v)
inst✝ : HasLimitsOfShape Iᵒᵖ (Type (max u v))
F : I ⥤ C
G : Cᵒᵖ ⥤ Type v
η : colimit (F ⋙ yoneda) ⟶ G
i : Iᵒᵖ
this :
∀ (a : limit ((F.op ⋙ G) ⋙ uliftFunctor.{u, ... | simp | no goals | ab277699a6c63a6c |
CompositionSeries.mem_eraseLast | Mathlib/Order/JordanHolder.lean | theorem mem_eraseLast {s : CompositionSeries X} {x : X} (h : 0 < s.length) :
x ∈ s.eraseLast ↔ x ≠ s.last ∧ x ∈ s | case mp.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
⊢ (fun i => s.toFun ⟨↑i, ⋯⟩) i ≠ last s ∧ (fun i => s.toFun ⟨↑i, ⋯⟩) i ∈ range s.toFun | have hi : (i : ℕ) < s.length := by
conv_rhs => rw [← Nat.add_one_sub_one s.length, Nat.succ_sub h]
exact i.2 | case mp.intro
X : Type u
inst✝¹ : Lattice X
inst✝ : JordanHolderLattice X
s : CompositionSeries X
h : 0 < s.length
i : Fin (s.length - 1 + 1)
hi : ↑i < s.length
⊢ (fun i => s.toFun ⟨↑i, ⋯⟩) i ≠ last s ∧ (fun i => s.toFun ⟨↑i, ⋯⟩) i ∈ range s.toFun | 2ab0afadecbebacb |
ProbabilityTheory.Kernel.ae_kernel_lt_top | Mathlib/Probability/Kernel/Composition/CompProd.lean | theorem ae_kernel_lt_top (a : α) (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
∀ᵐ b ∂κ a, η (a, b) (Prod.mk b ⁻¹' s) < ∞ | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h2s : ((κ ⊗ₖ η) a) s ≠ ⊤
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
this : ∀ (b : β), (η (a, b)) ... | have h2t : (κ ⊗ₖ η) a t ≠ ∞ := by rwa [measure_toMeasurable] | α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h2s : ((κ ⊗ₖ η) a) s ≠ ⊤
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
this : ∀ (b : β), (η (a, b)) ... | 8ecbe6a27811f30f |
Matroid.map_isBasis_iff' | Mathlib/Data/Matroid/Map.lean | lemma map_isBasis_iff' {I X : Set β} {hf} :
(M.map f hf).IsBasis I X ↔ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ I = f '' I₀ ∧ X = f '' X₀ | case refine_1.intro.intro.intro.intro
α : Type u_1
β : Type u_2
f : α → β
M : Matroid α
hf : InjOn f M.E
I : Set α
hI : I ⊆ M.E
X : Set α
hX : X ⊆ M.E
h : M.IsBasis I X
⊢ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ f '' I = f '' I₀ ∧ f '' X = f '' X₀ | exact ⟨I, X, h, rfl, rfl⟩ | no goals | c192441893919a44 |
Array.find?_range'_eq_none | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Range.lean | theorem find?_range'_eq_none {s n : Nat} {p : Nat → Bool} :
(range' s n).find? p = none ↔ ∀ i, s ≤ i → i < s + n → !p i | s n : Nat
p : Nat → Bool
⊢ find? p (range' s n) = none ↔ ∀ (i : Nat), s ≤ i → i < s + n → (!p i) = true | rw [← List.toArray_range'] | s n : Nat
p : Nat → Bool
⊢ find? p (List.range' s n).toArray = none ↔ ∀ (i : Nat), s ≤ i → i < s + n → (!p i) = true | 1040f6ec7d7fe1a6 |
List.getLast_map | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem getLast_map (f : α → β) (l : List α) (h) :
getLast (map f l) h = f (getLast l (by simpa using h)) | α : Type ?u.133064
β : Type ?u.133073
f : α → β
l : List α
h : map f l ≠ []
⊢ l ≠ [] | simpa using h | no goals | 9abc0df8c509c2cd |
AddCircle.gcd_mul_addOrderOf_div_eq | Mathlib/Topology/Instances/AddCircle.lean | theorem gcd_mul_addOrderOf_div_eq {n : ℕ} (m : ℕ) (hn : 0 < n) :
m.gcd n * addOrderOf (↑(↑m / ↑n * p) : AddCircle p) = n | case h
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
p : 𝕜
hp : Fact (0 < p)
n m : ℕ
hn : 0 < n
⊢ IsOfFinAddOrder ↑(p / ↑n) | rwa [← addOrderOf_pos_iff, addOrderOf_period_div hn] | no goals | dbb8668096a7f28d |
CategoryTheory.Pretriangulated.Opposite.complete_distinguished_triangle_morphism | Mathlib/CategoryTheory/Triangulated/Opposite/Pretriangulated.lean | lemma complete_distinguished_triangle_morphism (T₁ T₂ : Triangle Cᵒᵖ)
(hT₁ : T₁ ∈ distinguishedTriangles C) (hT₂ : T₂ ∈ distinguishedTriangles C)
(a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁) :
∃ (c : T₁.obj₃ ⟶ T₂.obj₃), T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧
T₁.mor₃ ≫ a⟦1⟧' ... | case intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T₁ T₂ : Triangle Cᵒᵖ
hT₁ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁) ∈ Pretriangulated.distinguished... | dsimp at c hc₁ hc₂ | case intro.intro
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T₁ T₂ : Triangle Cᵒᵖ
hT₁ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁) ∈ Pretriangulated.distinguished... | 1f895fdca2611d63 |
catalan_eq_centralBinom_div | Mathlib/Combinatorics/Enumerative/Catalan.lean | theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) | case ind
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑m.centralBinom / (↑m + 1)
⊢ ↑(catalan d.succ) = ↑d.succ.centralBinom / (↑d.succ + 1) | simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul] | case ind
d : ℕ
hd : ∀ m ≤ d, ↑(catalan m) = ↑m.centralBinom / (↑m + 1)
⊢ ∑ x : Fin d.succ, ↑(catalan ↑x) * ↑(catalan (d - ↑x)) = ↑d.succ.centralBinom / (↑d.succ + 1) | 6311ac92f4458437 |
analyticAt_clog | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | theorem analyticAt_clog (m : z ∈ slitPlane) : AnalyticAt ℂ log z | z : ℂ
m : z ∈ slitPlane
⊢ AnalyticAt ℂ log z | rw [analyticAt_iff_eventually_differentiableAt] | z : ℂ
m : z ∈ slitPlane
⊢ ∀ᶠ (z : ℂ) in 𝓝 z, DifferentiableAt ℂ log z | 057972827dcb7615 |
PowerSeries.rescale_neg_one_X | Mathlib/RingTheory/PowerSeries/Basic.lean | theorem rescale_neg_one_X : rescale (-1 : A) X = -X | A : Type u_2
inst✝ : CommRing A
⊢ (rescale (-1)) X = -X | rw [rescale_X, map_neg, map_one, neg_one_mul] | no goals | 008008a1724770fe |
Polynomial.Gal.mul_splits_in_splittingField_of_mul | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | theorem mul_splits_in_splittingField_of_mul {p₁ q₁ p₂ q₂ : F[X]} (hq₁ : q₁ ≠ 0) (hq₂ : q₂ ≠ 0)
(h₁ : p₁.Splits (algebraMap F q₁.SplittingField))
(h₂ : p₂.Splits (algebraMap F q₂.SplittingField)) :
(p₁ * p₂).Splits (algebraMap F (q₁ * q₂).SplittingField) | case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : Splits (algebraMap F q₁.SplittingField) p₁
h₂ : Splits (algebraMap F q₂.SplittingField) p₂
⊢ Splits (algebraMap F (q₁ * q₂).SplittingField) p₁ | rw [←
(SplittingField.lift q₁
(splits_of_splits_of_dvd (algebraMap F (q₁ * q₂).SplittingField) (mul_ne_zero hq₁ hq₂)
(SplittingField.splits _) (dvd_mul_right q₁ q₂))).comp_algebraMap] | case hf
F : Type u_1
inst✝ : Field F
p₁ q₁ p₂ q₂ : F[X]
hq₁ : q₁ ≠ 0
hq₂ : q₂ ≠ 0
h₁ : Splits (algebraMap F q₁.SplittingField) p₁
h₂ : Splits (algebraMap F q₂.SplittingField) p₂
⊢ Splits ((↑(SplittingField.lift q₁ ⋯)).comp (algebraMap F q₁.SplittingField)) p₁ | 12f0bbd6ecef4d4b |
Nat.Prime.emultiplicity_choose' | Mathlib/Data/Nat/Multiplicity.lean | theorem emultiplicity_choose' {p n k b : ℕ} (hp : p.Prime) (hnb : log p (n + k) < b) :
emultiplicity p (choose (n + k) k) = #{i ∈ Ico 1 b | p ^ i ≤ k % p ^ i + n % p ^ i} | p n k b : ℕ
hp : Prime p
hnb : log p (n + k) < b
this : (n + k).choose k * k ! * n ! = (n + k)!
⊢ emultiplicity p ((n + k).choose k * k ! * n !) =
↑(#(filter (fun i => p ^ i ≤ k % p ^ i + n % p ^ i) (Ico 1 b))) + emultiplicity p (k ! * n !) | rw [this, hp.emultiplicity_factorial hnb, hp.emultiplicity_mul,
hp.emultiplicity_factorial ((log_mono_right (le_add_left k n)).trans_lt hnb),
hp.emultiplicity_factorial ((log_mono_right (le_add_left n k)).trans_lt
(add_comm n k ▸ hnb)), multiplicity_choose_aux hp (le_add_left k n)] | p n k b : ℕ
hp : Prime p
hnb : log p (n + k) < b
this : (n + k).choose k * k ! * n ! = (n + k)!
⊢ ↑(∑ i ∈ Ico 1 b, k / p ^ i + ∑ i ∈ Ico 1 b, (n + k - k) / p ^ i +
#(filter (fun i => p ^ i ≤ k % p ^ i + (n + k - k) % p ^ i) (Ico 1 b))) =
↑(#(filter (fun i => p ^ i ≤ k % p ^ i + n % p ^ i) (Ico 1 b))) +
... | ed202d3f67c3f305 |
tendsto_exp_div_rpow_atTop | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | theorem tendsto_exp_div_rpow_atTop (s : ℝ) : Tendsto (fun x : ℝ => exp x / x ^ s) atTop atTop | s : ℝ
⊢ Tendsto (fun x => rexp x / x ^ s) atTop atTop | obtain ⟨n, hn⟩ := archimedean_iff_nat_lt.1 Real.instArchimedean s | case intro
s : ℝ
n : ℕ
hn : s < ↑n
⊢ Tendsto (fun x => rexp x / x ^ s) atTop atTop | 2529d965a6e29a2f |
CompleteOrthogonalIdempotents.of_ker_isNilpotent_of_isMulCentral | Mathlib/RingTheory/Idempotents.lean | theorem CompleteOrthogonalIdempotents.of_ker_isNilpotent_of_isMulCentral
(h : ∀ x ∈ RingHom.ker f, IsNilpotent x)
(he : ∀ i, IsIdempotentElem (e i))
(he' : ∀ i, IsMulCentral (e i))
(he'' : CompleteOrthogonalIdempotents (f ∘ e)) :
CompleteOrthogonalIdempotents e | R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
e : I → R
inst✝ : Fintype I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
he : ∀ (i : I), IsIdempotentElem (e i)
he' : ∀ (i : I), IsMulCentral (e i)
he'' : CompleteOrthogonalIdempotents (⇑f ∘ e)
e' : I → R
h₁ : CompleteOrthogonalIdempotents e'
... | ext i | case h
R : Type u_1
S : Type u_2
inst✝² : Ring R
inst✝¹ : Ring S
f : R →+* S
I : Type u_3
e : I → R
inst✝ : Fintype I
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
he : ∀ (i : I), IsIdempotentElem (e i)
he' : ∀ (i : I), IsMulCentral (e i)
he'' : CompleteOrthogonalIdempotents (⇑f ∘ e)
e' : I → R
h₁ : CompleteOrthogonalIdempote... | d1848a670d0365bc |
hasFDerivAt_norm_rpow | Mathlib/Analysis/InnerProductSpace/NormPow.lean | theorem hasFDerivAt_norm_rpow (x : E) {p : ℝ} (hp : 1 < p) :
HasFDerivAt (fun x : E ↦ ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • innerSL ℝ x) x | case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
⊢ HasFDerivAt (fun x => ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x) x | simp only [hx, norm_zero, map_zero, smul_zero] | case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
x : E
p : ℝ
hp : 1 < p
hx : x = 0
⊢ HasFDerivAt (fun x => ‖x‖ ^ p) 0 0 | df712761f8e3d685 |
PartialHomeomorph.nhds_eq_comap_inf_principal | Mathlib/Topology/PartialHomeomorph.lean | theorem nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) :
𝓝 x = comap e (𝓝 (e x)) ⊓ 𝓟 e.source | X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : PartialHomeomorph X Y
x : X
hx : x ∈ e.source
⊢ 𝓝 x = comap (↑e) (𝓝 (↑e x)) ⊓ 𝓟 e.source | lift x to e.source using hx | case intro
X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : PartialHomeomorph X Y
x : { x // x ∈ e.source }
⊢ 𝓝 ↑x = comap (↑e) (𝓝 (↑e ↑x)) ⊓ 𝓟 e.source | ef44c68f626bbbb6 |
MeasureTheory.unifTight_of_subsingleton | Mathlib/MeasureTheory/Function/UnifTight.lean | theorem unifTight_of_subsingleton [Subsingleton ι] (hp_top : p ≠ ∞)
{f : ι → α → β} (hf : ∀ i, MemLp (f i) p μ) : UnifTight f p μ := fun ε hε ↦ by
by_cases hε_top : ε = ∞
· exact ⟨∅, by measurability, fun _ => hε_top.symm ▸ le_top⟩
by_cases hι : Nonempty ι
case neg => exact ⟨∅, (by measurability), fun i => ... | case pos
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
inst✝ : Subsingleton ι
hp_top : p ≠ ⊤
f : ι → α → β
hf : ∀ (i : ι), MemLp (f i) p μ
ε : ℝ≥0
hε : 0 < ε
hε_top : ↑ε = ⊤
⊢ ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε | exact ⟨∅, by measurability, fun _ => hε_top.symm ▸ le_top⟩ | no goals | 6576e3dc8dfe236e |
MeasureTheory.Measure.haar.mul_left_index_le | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem mul_left_index_le {K : Set G} (hK : IsCompact K) {V : Set G} (hV : (interior V).Nonempty)
(g : G) : index ((fun h => g * h) '' K) V ≤ index K V | case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V
h2s : s.card = index K V
g₁ g₂ : G
hg₂ :... | simp only [exists_prop, mem_iUnion, Finset.mem_map, Equiv.coe_mulRight,
exists_exists_and_eq_and, mem_preimage, Equiv.toEmbedding_apply] | case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : (interior V).Nonempty
g : G
s : Finset G
h1s : K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V
h2s : s.card = index K V
g₁ g₂ : G
hg₂ :... | d7d19e4b8c244509 |
FreeGroup.reduce_invRev | Mathlib/GroupTheory/FreeGroup/Reduce.lean | theorem reduce_invRev {w : List (α × Bool)} : reduce (invRev w) = invRev (reduce w) | case H
α : Type u_1
inst✝ : DecidableEq α
w : List (α × Bool)
⊢ Red (reduce (invRev w)) (invRev (reduce w)) | rw [← red_invRev_iff, invRev_invRev] | case H
α : Type u_1
inst✝ : DecidableEq α
w : List (α × Bool)
⊢ Red (invRev (reduce (invRev w))) (reduce w) | 931dc9c3210eb8cc |
IsConj.eq_of_left_mem_center | Mathlib/GroupTheory/Subgroup/Center.lean | theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h | M : Type u_2
inst✝ : Monoid M
g h : M
H : IsConj g h
Hg : g ∈ Set.center M
⊢ g = h | rcases H with ⟨u, hu⟩ | case intro
M : Type u_2
inst✝ : Monoid M
g h : M
Hg : g ∈ Set.center M
u : Mˣ
hu : SemiconjBy (↑u) g h
⊢ g = h | 7f39f38db800ef78 |
round_neg_two_inv | Mathlib/Algebra/Order/Round.lean | theorem round_neg_two_inv : round (-2⁻¹ : α) = 0 | α : Type u_2
inst✝¹ : LinearOrderedField α
inst✝ : FloorRing α
⊢ round (-2⁻¹) = 0 | simp only [round_eq, ← one_div, neg_add_cancel, floor_zero] | no goals | 421e31398e309b65 |
Nat.exists_most_significant_bit | Mathlib/Data/Nat/Bitwise.lean | theorem exists_most_significant_bit {n : ℕ} (h : n ≠ 0) :
∃ i, testBit n i = true ∧ ∀ j, i < j → testBit n j = false | case pos
b : Bool
n : ℕ
hn : n ≠ 0 → ∃ i, n.testBit i = true ∧ ∀ (j : ℕ), i < j → n.testBit j = false
h : bit b n ≠ 0
h' : n = 0
⊢ ∃ i, (bit b n).testBit i = true ∧ ∀ (j : ℕ), i < j → (bit b n).testBit j = false | subst h' | case pos
b : Bool
hn : 0 ≠ 0 → ∃ i, testBit 0 i = true ∧ ∀ (j : ℕ), i < j → testBit 0 j = false
h : bit b 0 ≠ 0
⊢ ∃ i, (bit b 0).testBit i = true ∧ ∀ (j : ℕ), i < j → (bit b 0).testBit j = false | 164bb1ef025f0b32 |
FormalMultilinearSeries.comp_coeff_one | Mathlib/Analysis/Analytic/Composition.lean | theorem comp_coeff_one (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
(v : Fin 1 → E) : (q.comp p) 1 v = q 1 fun _i => p 1 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... | refine q.congr (by simp) fun i hi1 hi2 => ?_ | 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... | 2ac1709b5f4213f3 |
FirstOrder.Language.HomClass.realize_term | Mathlib/ModelTheory/Semantics.lean | theorem HomClass.realize_term {F : Type*} [FunLike F M N] [HomClass L F M N]
(g : F) {t : L.Term α} {v : α → M} :
t.realize (g ∘ v) = g (t.realize v) | L : Language
M : Type w
N : Type u_1
inst✝³ : L.Structure M
inst✝² : L.Structure N
α : Type u'
F : Type u_4
inst✝¹ : FunLike F M N
inst✝ : L.HomClass F M N
g : F
t : L.Term α
v : α → M
⊢ Term.realize (⇑g ∘ v) t = g (Term.realize v t) | induction t | case var
L : Language
M : Type w
N : Type u_1
inst✝³ : L.Structure M
inst✝² : L.Structure N
α : Type u'
F : Type u_4
inst✝¹ : FunLike F M N
inst✝ : L.HomClass F M N
g : F
v : α → M
a✝ : α
⊢ Term.realize (⇑g ∘ v) (var a✝) = g (Term.realize v (var a✝))
case func
L : Language
M : Type w
N : Type u_1
inst✝³ : L.Structure ... | df81aa8676069e92 |
ContinuousMap.exists_extension_forall_mem_of_isClosedEmbedding | Mathlib/Topology/TietzeExtension.lean | theorem exists_extension_forall_mem_of_isClosedEmbedding (f : C(X, ℝ)) {t : Set ℝ} {e : X → Y}
[hs : OrdConnected t] (hf : ∀ x, f x ∈ t) (hne : t.Nonempty) (he : IsClosedEmbedding e) :
∃ g : C(Y, ℝ), (∀ y, g y ∈ t) ∧ g ∘ e = f | case intro.intro.refine_2.h
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : NormalSpace Y
f : C(X, ℝ)
t : Set ℝ
e : X → Y
hs : t.OrdConnected
hf : ∀ (x : X), f x ∈ t
hne : t.Nonempty
he : IsClosedEmbedding e
h : ℝ ≃o ↑(Ioo (-1) 1)
F : X →ᵇ ℝ := { toFun := Subtype.val ∘ ⇑h ∘ ⇑f,... | exact hgG.2 (congr_fun hGF _) | no goals | b5717c30218ed2f5 |
BoxIntegral.IntegrationParams.tendsto_embedBox_toFilteriUnion_top | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | theorem tendsto_embedBox_toFilteriUnion_top (l : IntegrationParams) (h : I ≤ J) :
Tendsto (TaggedPrepartition.embedBox I J h) (l.toFilteriUnion I ⊤)
(l.toFilteriUnion J (Prepartition.single J I h)) | ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
⊢ Tendsto (⇑(embedBox I J h)) (toFilteriUnion I ⊤) (toFilteriUnion J (Prepartition.single J I h)) | simp only [toFilteriUnion, tendsto_iSup] | ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
⊢ ∀ (i : ℝ≥0),
Tendsto (⇑(embedBox I J h)) (l.toFilterDistortioniUnion I i ⊤)
(⨆ c, l.toFilterDistortioniUnion J c (Prepartition.single J I h)) | 75f666dbd7c5bb0f |
Asymptotics.isLittleO_principal | Mathlib/Analysis/Asymptotics/Lemmas.lean | theorem isLittleO_principal {s : Set α} : f'' =o[𝓟 s] g' ↔ ∀ x ∈ s, f'' x = 0 | α : Type u_1
F' : Type u_7
E'' : Type u_9
inst✝¹ : SeminormedAddCommGroup F'
inst✝ : NormedAddCommGroup E''
g' : α → F'
f'' : α → E''
s : Set α
h : ∀ x ∈ s, f'' x = 0
⊢ (fun _x => 0) =ᶠ[𝓟 s] f'' | exact fun x hx ↦ (h x hx).symm | no goals | 9fb493bf550d3541 |
Multiset.count_map_eq_count' | Mathlib/Data/Multiset/Filter.lean | theorem count_map_eq_count' [DecidableEq β] (f : α → β) (s : Multiset α) (hf : Function.Injective f)
(x : α) : (s.map f).count (f x) = s.count x | case neg
α : Type u_1
β : Type v
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
s : Multiset α
hf : Injective f
x : α
H : x ∉ s
⊢ count (f x) (map f s) = count x s | rw [count_eq_zero_of_not_mem H, count_eq_zero, mem_map] | case neg
α : Type u_1
β : Type v
inst✝¹ : DecidableEq α
inst✝ : DecidableEq β
f : α → β
s : Multiset α
hf : Injective f
x : α
H : x ∉ s
⊢ ¬∃ a ∈ s, f a = f x | 617a7c8c055ce103 |
surjOn_Icc_of_monotone_surjective | Mathlib/Order/Interval/Set/SurjOn.lean | theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
{a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b)) | α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a b : α
hab : a ≤ b
p : β
hp : p ∈ Icc (f a) (f b)
⊢ p ∈ f '' Icc a b | rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp') | case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_surj : Surjective f
a b : α
hab : a ≤ b
hp : f a ∈ Icc (f a) (f b)
⊢ f a ∈ f '' Icc a b
case inr.inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
h_mono : Monotone f
h_s... | c609773ecb1a1f37 |
Nat.bit_mod_two_eq_zero_iff | Mathlib/Data/Nat/Bitwise.lean | lemma bit_mod_two_eq_zero_iff (a x) :
bit a x % 2 = 0 ↔ !a | a : Bool
x : ℕ
⊢ bit a x % 2 = 0 ↔ (!a) = true | simp | no goals | c6c9ec748d9d84d0 |
convexOn_iff_pairwise_pos | Mathlib/Analysis/Convex/Function.lean | theorem convexOn_iff_pairwise_pos {s : Set E} {f : E → β} :
ConvexOn 𝕜 s f ↔
Convex 𝕜 s ∧
s.Pairwise fun x y =>
∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y | 𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : OrderedAddCommMonoid β
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 β
s : Set E
f : E → β
⊢ (Convex 𝕜 s ∧
∀ ⦃x : E⦄,
x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a •... | refine
and_congr_right'
⟨fun h x hx y hy _ a b ha hb hab => h hx hy ha hb hab, fun h x hx y hy a b ha hb hab => ?_⟩ | 𝕜 : Type u_1
E : Type u_2
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : AddCommMonoid E
inst✝² : OrderedAddCommMonoid β
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 β
s : Set E
f : E → β
h : s.Pairwise fun x y => ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y
x : E
hx : x ∈ s
y : E
hy : ... | 24d415d43f469181 |
MeasureTheory.L1.edist_def | Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean | theorem edist_def (f g : α →₁[μ] β) : edist f g = ∫⁻ a, edist (f a) (g a) ∂μ | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ edist f g = ∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ | simp only [Lp.edist_def, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_enorm, Pi.sub_apply,
one_toReal, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false] | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : ↥(Lp β 1 μ)
⊢ (if 1 = ⊤ then eLpNormEssSup (↑↑f - ↑↑g) μ else ∫⁻ (a : α), ‖↑↑f a - ↑↑g a‖ₑ ∂μ) =
∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ | fd4f8e4e94f651b9 |
Polynomial.leadingCoeff_cubic | Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean | theorem leadingCoeff_cubic (ha : a ≠ 0) :
leadingCoeff (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d) = a | R : Type u
a b c d : R
inst✝ : Semiring R
ha : a ≠ 0
⊢ (C a * X ^ 3 + C b * X ^ 2 + C c * X + C d).leadingCoeff = a | rw [add_assoc, add_assoc, ← add_assoc (C b * X ^ 2), add_comm,
leadingCoeff_add_of_degree_lt <| degree_quadratic_lt_degree_C_mul_X_cb ha,
leadingCoeff_C_mul_X_pow] | no goals | e4540a9c25cdab0b |
UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto | Mathlib/Topology/UniformSpace/UniformConvergence.lean | theorem UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto
(hF : UniformCauchySeqOnFilter F p p')
(hF' : ∀ᶠ x : α in p', Tendsto (fun n => F n x) p (𝓝 (f x))) :
TendstoUniformlyOnFilter F f p p' | α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p : Filter ι
p' : Filter α
hF : UniformCauchySeqOnFilter F p p'
hF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))
⊢ TendstoUniformlyOnFilter F f p p' | rcases p.eq_or_neBot with rfl | _ | case inl
α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f : α → β
p' : Filter α
hF : UniformCauchySeqOnFilter F ⊥ p'
hF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) ⊥ (𝓝 (f x))
⊢ TendstoUniformlyOnFilter F f ⊥ p'
case inr
α : Type u
β : Type v
ι : Type x
inst✝ : UniformSpace β
F : ι → α → β
f ... | be7f1c945998c45a |
MeasureTheory.measurableSet_generateFrom_singleton_iff | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | theorem measurableSet_generateFrom_singleton_iff {s t : Set α} :
MeasurableSet[MeasurableSpace.generateFrom {s}] t ↔ t = ∅ ∨ t = s ∨ t = sᶜ ∨ t = univ | α : Type u_1
s : Set α
x : Set Prop
hT : True ∈ x
hF : False ∉ x
p : Prop
hp : p ∈ x
hpneg : False ↔ p
⊢ False ∈ x | convert hp | no goals | 39127bdfcef76ebf |
Real.invariant | Mathlib/NumberTheory/DiophantineApproximation/Basic.lean | theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋ * v) | case refine_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
h' : ξ - ↑u / ↑v < (↑v * (2 * ↑v - 1))⁻¹
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v | rw [Huv, ← sub_sub, sub_lt_iff_lt_add, self_sub_floor, Hv] at h' | case refine_2
ξ : ℝ
u v : ℤ
hv : 2 ≤ v
h : ContfracLegendre.Ass ξ u v
huv : u - ⌊ξ⌋ * v = 1
hv₀' : 0 < 2 * ↑v - 1
Hv : (↑v * (2 * ↑v - 1))⁻¹ + (↑v)⁻¹ = 2 / (2 * ↑v - 1)
Huv : ↑u / ↑v = ↑⌊ξ⌋ + (↑v)⁻¹
h' : fract ξ < 2 / (2 * ↑v - 1)
⊢ -(1 / 2) < (fract ξ)⁻¹ - ↑v | 8eb0aea68cc1bde9 |
Doset.doset_union_rightCoset | Mathlib/GroupTheory/DoubleCoset.lean | theorem doset_union_rightCoset (H K : Subgroup G) (a : G) :
⋃ k : K, op (a * k) • ↑H = doset a H K | case h.mpr.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a x✝ x : G
hx : x ∈ H
y : G
hy : y ∈ K
hxy : x✝ = x * a * y
⊢ ∃ i, x✝ * ((↑i)⁻¹ * a⁻¹) ∈ H | refine ⟨⟨y, hy⟩, ?_⟩ | case h.mpr.intro.intro.intro.intro
G : Type u_1
inst✝ : Group G
H K : Subgroup G
a x✝ x : G
hx : x ∈ H
y : G
hy : y ∈ K
hxy : x✝ = x * a * y
⊢ x✝ * ((↑⟨y, hy⟩)⁻¹ * a⁻¹) ∈ H | 1ad4289c4ad1f9f3 |
GenContFract.IntFractPair.coe_stream_nth_rat_eq | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | theorem coe_stream_nth_rat_eq (v_eq_q : v = (↑q : K)) (n : ℕ) :
((IntFractPair.stream q n).map (mapFr (↑)) : Option <| IntFractPair K) =
IntFractPair.stream v n | K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
b : ℤ
fr : ℚ
stream_q_nth_eq : IntFractPair.stream q n = some { b := b, fr := fr }
fr_ne_zero : ¬fr = 0
IH : some { b := b, fr := ↑fr } = IntFractPair.stream (↑q) n
⊢ (↑fr)⁻¹ = ↑fr⁻¹ | norm_cast | no goals | 3c20096a5980fe60 |
SimpleGraph.connected_iff_exists_forall_reachable | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma connected_iff_exists_forall_reachable : G.Connected ↔ ∃ v, ∀ w, G.Reachable v w | V : Type u
G : SimpleGraph V
⊢ G.Preconnected ∧ Nonempty V ↔ ∃ v, ∀ (w : V), G.Reachable v w | constructor | case mp
V : Type u
G : SimpleGraph V
⊢ G.Preconnected ∧ Nonempty V → ∃ v, ∀ (w : V), G.Reachable v w
case mpr
V : Type u
G : SimpleGraph V
⊢ (∃ v, ∀ (w : V), G.Reachable v w) → G.Preconnected ∧ Nonempty V | 1c445adc9e510c30 |
Real.Gamma_one_half_eq | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | theorem Real.Gamma_one_half_eq : Real.Gamma (1 / 2) = √π | case h.e'_2
⊢ ∫ (x : ℝ) in Ioi 0, (2 * x ^ (2 - 1)) • (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) =
∫ (a : ℝ) in Ioi 0, 2 * rexp (-1 * a ^ 2) | refine setIntegral_congr_fun measurableSet_Ioi fun x hx => ?_ | case h.e'_2
x : ℝ
hx : x ∈ Ioi 0
⊢ (2 * x ^ (2 - 1)) • (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) = 2 * rexp (-1 * x ^ 2) | a42e522f111124af |
QuasispectrumRestricts.cfc | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean | theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
(h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
NonUnitalContinuousFunctionalCalculus R p where
predicate_zero := h0
compactSpace_quasispectrum a | case hom_id
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousStar S
inst✝¹... | case hom_id => exact ((h a).mp ha).2.nonUnitalStarAlgHom_id <| cfcₙHom_id ((h a).mp ha).1 | case hom_map_spectrum
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²⁴ : Semifield R
inst✝²³ : StarRing R
inst✝²² : MetricSpace R
inst✝²¹ : IsTopologicalSemiring R
inst✝²⁰ : ContinuousStar R
inst✝¹⁹ : Field S
inst✝¹⁸ : StarRing S
inst✝¹⁷ : MetricSpace S
inst✝¹⁶ : IsTopologicalRing S
inst✝¹⁵ : ContinuousSta... | 3af5d542f22c6b37 |
GenContFract.coe_of_s_rat_eq | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | theorem coe_of_s_rat_eq (v_eq_q : v = (↑q : K)) :
((of q).s.map (Pair.map ((↑))) : Stream'.Seq <| Pair K) = (of v).s | case h.a
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
q : ℚ
v_eq_q : v = ↑q
n : ℕ
a✝ : Pair K
⊢ a✝ ∈ (Stream'.Seq.map (Pair.map Rat.cast) (of q).s).get? n ↔ a✝ ∈ Option.map (Pair.map Rat.cast) ((of q).s.get? n) | rfl | no goals | 68950bb07f32bd54 |
MeasureTheory.IsStoppingTime.add_const_nat | Mathlib/Probability/Process/Stopping.lean | theorem add_const_nat {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStoppingTime f τ) {i : ℕ} :
IsStoppingTime f fun ω => τ ω + i | case pos
Ω : Type u_1
m : MeasurableSpace Ω
f : Filtration ℕ m
τ : Ω → ℕ
hτ : IsStoppingTime f τ
i j : ℕ
hij : i ≤ j
⊢ MeasurableSet {ω | τ ω + i = j} | simp_rw [eq_comm, ← Nat.sub_eq_iff_eq_add hij, eq_comm] | case pos
Ω : Type u_1
m : MeasurableSpace Ω
f : Filtration ℕ m
τ : Ω → ℕ
hτ : IsStoppingTime f τ
i j : ℕ
hij : i ≤ j
⊢ MeasurableSet {ω | τ ω = j - i} | 089b9068779bbc69 |
Array.forIn_map | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Monadic.lean | theorem forIn_map [Monad m] [LawfulMonad m]
(l : Array α) (g : α → β) (f : β → γ → m (ForInStep γ)) :
forIn (l.map g) init f = forIn l init fun a y => f (g a) y | m : Type u_1 → Type u_2
α : Type u_3
β : Type u_4
γ : Type u_1
init : γ
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l : Array α
g : α → β
f : β → γ → m (ForInStep γ)
⊢ forIn (map g l) init f = forIn l init fun a y => f (g a) y | cases l | case mk
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_4
γ : Type u_1
init : γ
inst✝¹ : Monad m
inst✝ : LawfulMonad m
g : α → β
f : β → γ → m (ForInStep γ)
toList✝ : List α
⊢ forIn (map g { toList := toList✝ }) init f = forIn { toList := toList✝ } init fun a y => f (g a) y | 992467d3c065c582 |
Module.freeLocus_localization | Mathlib/RingTheory/Spectrum/Prime/FreeLocus.lean | lemma freeLocus_localization (S : Submonoid R) :
freeLocus (Localization S) (LocalizedModule S M) =
comap (algebraMap R _) ⁻¹' freeLocus R M | case h
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
⊢ p ∈ freeLocus (Localization S) (Localiz... | let Mₚ := LocalizedModule p'.primeCompl M | case h
R : Type uR
M : Type uM
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
S : Submonoid R
p : PrimeSpectrum (Localization S)
p' : Ideal R := Ideal.comap (algebraMap R (Localization S)) p.asIdeal
hp' : S ≤ p'.primeCompl
Rₚ : Type uR := Localization.AtPrime p'
Mₚ : Type (max uR uM) := LocalizedModule ... | ac4a766398c7061e |
Std.DHashMap.Internal.Raw₀.getKeyD_modify_self | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem getKeyD_modify_self (h : m.1.WF) [Inhabited α] {k fallback : α} {f : β k → β k} :
(m.modify k f).getKeyD k fallback = if m.contains k then k else fallback | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : LawfulBEq α
h : m.val.WF
inst✝ : Inhabited α
k fallback : α
f : β k → β k
⊢ (m.modify k f).getKeyD k fallback = if m.contains k = true then k else fallback | simp_to_model [modify] using List.getKeyD_modifyKey_self | no goals | 37c14462b0d2c3a7 |
CategoryTheory.Presheaf.isSheaf_iff_multiequalizer | Mathlib/CategoryTheory/Sites/Sheaf.lean | theorem isSheaf_iff_multiequalizer [∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)] :
IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequalizer P) | case refine_2
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
inst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
S : J.Cover X
⊢ IsIso (S.toMultiequalizer P) → Nonempty (IsLimit (S.multifork P)) | intro h | case refine_2
C : Type u₁
inst✝² : Category.{v₁, u₁} C
A : Type u₂
inst✝¹ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
inst✝ : ∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
X : C
S : J.Cover X
h : IsIso (S.toMultiequalizer P)
⊢ Nonempty (IsLimit (S.multifork P)) | 32f315e61f5eaf0d |
isZero_Ext_succ_of_projective | Mathlib/CategoryTheory/Abelian/Ext.lean | /-- If `X : C` is projective and `n : ℕ`, then `Ext^(n + 1) X Y ≅ 0` for any `Y`. -/
lemma isZero_Ext_succ_of_projective (X Y : C) [Projective X] (n : ℕ) :
IsZero (((Ext R C (n + 1)).obj (Opposite.op X)).obj Y) | R : Type u_1
inst✝⁵ : Ring R
C : Type u_2
inst✝⁴ : Category.{u_3, u_2} C
inst✝³ : Abelian C
inst✝² : Linear R C
inst✝¹ : EnoughProjectives C
X Y : C
inst✝ : Projective X
n : ℕ
⊢ 𝟙 (((linearYoneda R C).obj Y).obj (Opposite.op (((ChainComplex.single₀ C).obj X).X (n + 1)))) = 0 | ext (x : _ ⟶ _) | case hf.h
R : Type u_1
inst✝⁵ : Ring R
C : Type u_2
inst✝⁴ : Category.{u_3, u_2} C
inst✝³ : Abelian C
inst✝² : Linear R C
inst✝¹ : EnoughProjectives C
X Y : C
inst✝ : Projective X
n : ℕ
x : Opposite.unop (Opposite.op (((ChainComplex.single₀ C).obj X).X (n + 1))) ⟶ Y
⊢ (ModuleCat.Hom.hom (𝟙 (((linearYoneda R C).obj Y).... | 40abdca1318b58b3 |
ProbabilityTheory.Kernel.IsCondKernel.isProbabilityMeasure_ae | Mathlib/Probability/Kernel/Disintegration/Basic.lean | /-- A conditional kernel is almost everywhere a probability measure. -/
lemma IsCondKernel.isProbabilityMeasure_ae [IsFiniteKernel κ.fst] [κ.IsCondKernel κCond] (a : α) :
∀ᵐ b ∂(κ.fst a), IsProbabilityMeasure (κCond (a, b)) | case neg
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : 0 = κ
h_sfin : ¬IsSFiniteKernel κCond
⊢ ∀ᵐ (b : β) ∂κ.fst a, IsProbabilityMeasure (κCond... | simp [h.symm] | no goals | d615b2a173689899 |
Ordinal.card_opow_le_of_omega0_le_left | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem card_opow_le_of_omega0_le_left {a : Ordinal} (ha : ω ≤ a) (b : Ordinal) :
(a ^ b).card ≤ max a.card b.card | case h
a : Ordinal.{u_1}
ha : ω ≤ a
b✝ b : Ordinal.{u_1}
hb : b.IsLimit
IH : ∀ o' < b, (a ^ o').card ≤ a.card ⊔ o'.card
⊢ ∀ (i : ↑(Iio b)), (a ^ ↑i).card ≤ a.card ⊔ b.card | intro c | case h
a : Ordinal.{u_1}
ha : ω ≤ a
b✝ b : Ordinal.{u_1}
hb : b.IsLimit
IH : ∀ o' < b, (a ^ o').card ≤ a.card ⊔ o'.card
c : ↑(Iio b)
⊢ (a ^ ↑c).card ≤ a.card ⊔ b.card | ca96380056897695 |
MeasurableSet.exists_isOpen_symmDiff_lt | Mathlib/MeasureTheory/Measure/Regular.lean | theorem _root_.MeasurableSet.exists_isOpen_symmDiff_lt [InnerRegularCompactLTTop μ]
[IsLocallyFiniteMeasure μ] [R1Space α] [BorelSpace α]
{s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ U, IsOpen U ∧ μ U < ∞ ∧ μ (U ∆ s) < ε | α : Type u_1
inst✝⁵ : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : μ.InnerRegularCompactLTTop
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : R1Space α
inst✝ : BorelSpace α
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ⊤
ε : ℝ≥0∞
hε : ε ≠ 0
this : ε / 2 ≠ 0
K : Set α
hKs : K ⊆ s
hKco : IsCompact K
hKcl ... | exact ne_top_of_le_ne_top hμs (by gcongr) | no goals | 498e42d44efdd46a |
Real.tendsto_logb_atTop_of_base_lt_one | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot | b : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e : ℝ
⊢ ∃ i, ∀ (a : ℝ), i ≤ a → logb b a ≤ e | use 1 ⊔ b ^ e | case h
b : ℝ
b_pos : 0 < b
b_lt_one : b < 1
e : ℝ
⊢ ∀ (a : ℝ), 1 ⊔ b ^ e ≤ a → logb b a ≤ e | 294c4d8092b3c5ec |
EReal.right_distrib_of_nonneg | Mathlib/Data/Real/EReal.lean | lemma right_distrib_of_nonneg {a b c : EReal} (ha : 0 ≤ a) (hb : 0 ≤ b) :
(a + b) * c = a * c + b * c | case inr.inr.inr.inr.h_real.h_bot
b : EReal
hb : 0 ≤ b
b_pos : 0 < b
a✝ : ℝ
c_pos : 0 < ↑a✝
ha : 0 ≤ ⊥
a_pos : 0 < ⊥
⊢ False | exact not_lt_bot a_pos | no goals | 4e8a6473d4cbc325 |
isSimpleRing_iff_isField | Mathlib/RingTheory/SimpleRing/Field.lean | lemma isSimpleRing_iff_isField (A : Type*) [CommRing A] : IsSimpleRing A ↔ IsField A :=
⟨fun _ ↦ Subring.topEquiv.symm.toMulEquiv.isField _ <| by
rw [← Subring.center_eq_top A]; exact IsSimpleRing.isField_center A,
fun h ↦ letI := h.toField; inferInstance⟩
| A : Type u_1
inst✝ : CommRing A
x✝ : IsSimpleRing A
⊢ IsField ↥(Subring.center A) | exact IsSimpleRing.isField_center A | no goals | 5feac9988e67d3e2 |
MeasureTheory.lintegral_sub_le' | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_sub_le' (f g : α → ℝ≥0∞) (hf : AEMeasurable f μ) :
∫⁻ x, g x ∂μ - ∫⁻ x, f x ∂μ ≤ ∫⁻ x, g x - f x ∂μ | case pos
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ = ⊤
⊢ ∫⁻ (x : α), g x ∂μ ≤ ⊤ | exact le_top | no goals | 3c78cc5e3caf47ca |
Profinite.exists_locallyConstant_finite_aux | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _).hom | case intro.intro.e_f.h
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (Loca... | change ff a = _ | case intro.intro.e_f.h
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (Loca... | ad55d9835f0fad2e |
CompHausLike.sigmaComparison_eq_comp_isos | Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean | theorem sigmaComparison_eq_comp_isos : sigmaComparison X σ =
(X.mapIso (opCoproductIsoProduct'
(finiteCoproduct.isColimit.{u, u} (fun a ↦ of P (σ a)))
(productIsProduct fun x ↦ Opposite.op (of P (σ x))))).hom ≫
(PreservesProduct.iso X fun a ↦ ⟨of P (σ a)⟩).hom ≫
(Types.productIso.{u, max u w} fu... | case h.h
P : TopCat → Prop
inst✝⁶ : HasExplicitFiniteCoproducts P
X : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝⁵ : PreservesFiniteProducts X
α : Type u
inst✝⁴ : Finite α
σ : α → Type u
inst✝³ : (a : α) → TopologicalSpace (σ a)
inst✝² : ∀ (a : α), CompactSpace (σ a)
inst✝¹ : ∀ (a : α), T2Space (σ a)
inst✝ : ∀ (a : α), H... | have := congrFun (piComparison_comp_π X (fun a ↦ ⟨of P (σ a)⟩) a) | case h.h
P : TopCat → Prop
inst✝⁶ : HasExplicitFiniteCoproducts P
X : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst✝⁵ : PreservesFiniteProducts X
α : Type u
inst✝⁴ : Finite α
σ : α → Type u
inst✝³ : (a : α) → TopologicalSpace (σ a)
inst✝² : ∀ (a : α), CompactSpace (σ a)
inst✝¹ : ∀ (a : α), T2Space (σ a)
inst✝ : ∀ (a : α), H... | 6a810ee204114508 |
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