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
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|---|---|---|---|---|---|---|
Nat.Partrec.Code.exists_code
|
Mathlib/Computability/PartrecCode.lean
|
theorem exists_code {f : ℕ →. ℕ} : Nat.Partrec f ↔ ∃ c : Code, eval c = f
|
case refine_2.intro.pair
cf cg : Code
pf : Partrec cf.eval
pg : Partrec cg.eval
⊢ Partrec (cf.pair cg).eval
|
exact pf.pair pg
|
no goals
|
ff5610342a211d54
|
Set.equitableOn_iff_exists_eq_eq_add_one
|
Mathlib/Data/Set/Equitable.lean
|
theorem equitableOn_iff_exists_eq_eq_add_one {s : Set α} {f : α → ℕ} :
s.EquitableOn f ↔ ∃ b, ∀ a ∈ s, f a = b ∨ f a = b + 1
|
α : Type u_1
s : Set α
f : α → ℕ
⊢ s.EquitableOn f ↔ ∃ b, ∀ a ∈ s, f a = b ∨ f a = b + 1
|
simp_rw [equitableOn_iff_exists_le_le_add_one, Nat.le_and_le_add_one_iff]
|
no goals
|
6809ba29f79d6a9a
|
ENNReal.inner_le_weight_mul_Lp_of_nonneg
|
Mathlib/Analysis/MeanInequalities.lean
|
/-- **Weighted Hölder inequality**. -/
lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0∞) :
∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
|
case neg
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ⊤) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ⊤
this :
↑(∑ i ∈ s, (w i).toNNReal * (f i).toNNReal) ≤
↑((∑ i ∈ s, (w i).toNNReal) ^ (1 - p⁻¹) * (∑ i ∈ s, (w i).toNNReal * (f i).toNNReal ^ p) ^ p⁻¹)
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
|
rw [coe_mul] at this
|
case neg
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ⊤) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ⊤
this :
↑(∑ i ∈ s, (w i).toNNReal * (f i).toNNReal) ≤
↑((∑ i ∈ s, (w i).toNNReal) ^ (1 - p⁻¹)) * ↑((∑ i ∈ s, (w i).toNNReal * (f i).toNNReal ^ p) ^ p⁻¹)
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
|
bb7d779f184c526a
|
PSet.rank_pair
|
Mathlib/SetTheory/ZFC/Rank.lean
|
theorem rank_pair (x y : PSet) : rank {x, y} = max (succ (rank x)) (succ (rank y))
|
x y : PSet.{u_1}
⊢ {x, y}.rank = succ x.rank ⊔ succ y.rank
|
simp
|
no goals
|
ccc567272ea68945
|
MeasureTheory.measurableSet_range_of_continuous_injective
|
Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
|
theorem measurableSet_range_of_continuous_injective {β : Type*} [TopologicalSpace γ]
[PolishSpace γ] [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [OpensMeasurableSpace β]
{f : γ → β} (f_cont : Continuous f) (f_inj : Injective f) :
MeasurableSet (range f)
|
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝¹ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n => ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
this : range f = ⋂ n, F n
E_meas : ∀ (s : ↑b), MeasurableSet (E s)
⊢ MeasurableSet (range f)
|
have F_meas : ∀ n, MeasurableSet (F n) := by
intro n
refine MeasurableSet.iUnion fun s => ?_
exact MeasurableSet.iUnion fun _ => E_meas _
|
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝¹ : UpgradedPolishSpace γ := upgradePolishSpace γ
b : Set (Set γ)
b_count : b.Countable
b_nonempty : ∅ ∉ b
hb : IsTopologicalBasis b
this✝ : Encodable ↑b
A : Type (max 0 u_3) := { p // Disjoint ↑p.1 ↑p.2 }
q : A → Set β
hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p
hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)
q_meas : ∀ (p : A), MeasurableSet (q p)
E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \ q ⟨(t, s), ⋯⟩
u : ℕ → ℝ
u_anti : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
F : ℕ → Set β := fun n => ⋃ s, ⋃ (_ : Bornology.IsBounded ↑s ∧ diam ↑s ≤ u n), E s
this : range f = ⋂ n, F n
E_meas : ∀ (s : ↑b), MeasurableSet (E s)
F_meas : ∀ (n : ℕ), MeasurableSet (F n)
⊢ MeasurableSet (range f)
|
b9138111f37aa037
|
Besicovitch.TauPackage.color_lt
|
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N
|
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
{p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
∀ k < N,
g k < i ∧
(closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
this✝ : p.index (G n) = Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
this : ∃ t, p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t
⊢ p.c (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t) ∉ p.iUnionUpTo (G n) ∧
p.R (G n) ≤ p.τ * p.r (Classical.epsilon fun t => p.c t ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r t)
|
exact Classical.epsilon_spec this
|
no goals
|
70d4d5d8aa8a7de2
|
CauchySeq.totallyBounded_range
|
Mathlib/Topology/UniformSpace/Cauchy.lean
|
theorem CauchySeq.totallyBounded_range {s : ℕ → α} (hs : CauchySeq s) :
TotallyBounded (range s)
|
case intro
α : Type u
uniformSpace : UniformSpace α
s : ℕ → α
hs : CauchySeq s
a : Set (α × α)
ha : a ∈ 𝓤 α
n : ℕ
hn : ∀ k ≥ n, ∀ l ≥ n, (s k, s l) ∈ a
⊢ ∃ t, t.Finite ∧ range s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ a}
|
refine ⟨s '' { k | k ≤ n }, (finite_le_nat _).image _, ?_⟩
|
case intro
α : Type u
uniformSpace : UniformSpace α
s : ℕ → α
hs : CauchySeq s
a : Set (α × α)
ha : a ∈ 𝓤 α
n : ℕ
hn : ∀ k ≥ n, ∀ l ≥ n, (s k, s l) ∈ a
⊢ range s ⊆ ⋃ y ∈ s '' {k | k ≤ n}, {x | (x, y) ∈ a}
|
3130ab10e1bf3e8f
|
ContinuousLinearMap.adjointAux_inner_left
|
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
theorem adjointAux_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪adjointAux A y, x⟫ = ⟪y, A x⟫
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : CompleteSpace E
A : E →L[𝕜] F
x : E
y : F
⊢ ⟪(adjointAux A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜
|
rw [adjointAux_apply, toDual_symm_apply, toSesqForm_apply_coe, coe_comp', innerSL_apply_coe,
Function.comp_apply]
|
no goals
|
fd8b218a8f1b3564
|
Ideal.IsLasker.minimal
|
Mathlib/RingTheory/Lasker.lean
|
lemma IsLasker.minimal [DecidableEq (Ideal R)] (h : IsLasker R) (I : Ideal R) :
∃ t : Finset (Ideal R), t.inf id = I ∧ (∀ ⦃J⦄, J ∈ t → J.IsPrimary) ∧
((t : Set (Ideal R)).Pairwise ((· ≠ ·) on radical)) ∧
(∀ ⦃J⦄, J ∈ t → ¬ (t.erase J).inf id ≤ J)
|
case intro.intro
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : DecidableEq (Ideal R)
h : IsLasker R
I : Ideal R
s : Finset (Ideal R)
hs : s.inf id = I
hs' : ∀ ⦃J : Ideal R⦄, J ∈ s → J.IsPrimary
⊢ ∃ t,
t.inf id = I ∧
(∀ ⦃J : Ideal R⦄, J ∈ t → J.IsPrimary) ∧
(↑t).Pairwise ((fun x1 x2 => x1 ≠ x2) on radical) ∧ ∀ ⦃J : Ideal R⦄, J ∈ t → ¬(t.erase J).inf id ≤ J
|
exact exists_minimal_isPrimary_decomposition_of_isPrimary_decomposition hs hs'
|
no goals
|
43556b91168d43a5
|
FirstOrder.Language.IsUltrahomogeneous.extend_embedding
|
Mathlib/ModelTheory/Fraisse.lean
|
theorem IsUltrahomogeneous.extend_embedding (M_homog : L.IsUltrahomogeneous M) {S : Type*}
[L.Structure S] (S_FG : FG L S) {T : Type*} [L.Structure T] [h : Nonempty (T ↪[L] M)]
(f : S ↪[L] M) (g : S ↪[L] T) :
∃ f' : T ↪[L] M, f = f'.comp g
|
case h.h
L : Language
M : Type w
inst✝² : L.Structure M
M_homog : L.IsUltrahomogeneous M
S : Type u_1
inst✝¹ : L.Structure S
S_FG : Structure.FG L S
T : Type u_2
inst✝ : L.Structure T
h : Nonempty (T ↪[L] M)
f : S ↪[L] M
g : S ↪[L] T
r : T ↪[L] M
s : S ↪[L] M := r.comp g
t : M ≃[L] M
eq : f.comp s.equivRange.symm.toEmbedding = t.toEmbedding.comp s.toHom.range.subtype
x : S
⊢ f x = (t.toEmbedding.comp s) x
|
have eq' := congr_fun (congr_arg DFunLike.coe eq) ⟨s x, Hom.mem_range.2 ⟨x, rfl⟩⟩
|
case h.h
L : Language
M : Type w
inst✝² : L.Structure M
M_homog : L.IsUltrahomogeneous M
S : Type u_1
inst✝¹ : L.Structure S
S_FG : Structure.FG L S
T : Type u_2
inst✝ : L.Structure T
h : Nonempty (T ↪[L] M)
f : S ↪[L] M
g : S ↪[L] T
r : T ↪[L] M
s : S ↪[L] M := r.comp g
t : M ≃[L] M
eq : f.comp s.equivRange.symm.toEmbedding = t.toEmbedding.comp s.toHom.range.subtype
x : S
eq' : (f.comp s.equivRange.symm.toEmbedding) ⟨s x, ⋯⟩ = (t.toEmbedding.comp s.toHom.range.subtype) ⟨s x, ⋯⟩
⊢ f x = (t.toEmbedding.comp s) x
|
207000e714ea7978
|
RightDerivMeasurableAux.differentiable_set_subset_D
|
Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
|
theorem differentiable_set_subset_D :
{ x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K } ⊆ D f K
|
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
⊢ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K} ⊆ D f K
|
intro x hx
|
F : Type u_1
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
K : Set F
x : ℝ
hx : x ∈ {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}
⊢ x ∈ D f K
|
02ae9336af5f07d4
|
TopCat.Presheaf.stalkSpecializes_stalkPushforward
|
Mathlib/Topology/Sheaves/Stalks.lean
|
theorem stalkSpecializes_stalkPushforward (f : X ⟶ Y) (F : X.Presheaf C) {x y : X} (h : x ⤳ y) :
(f _* F).stalkSpecializes (f.hom.map_specializes h) ≫ F.stalkPushforward _ f x =
F.stalkPushforward _ f y ≫ F.stalkSpecializes h
|
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds ((Hom.hom f) y))ᵒᵖ
⊢ colimit.ι
(((whiskeringLeft (OpenNhds ((Hom.hom f) y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion ((Hom.hom f) y)).op).obj
((pushforward C f).obj F))
j✝ ≫
((pushforward C f).obj F).stalkSpecializes ⋯ ≫
colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F) ≫
colimit.pre (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F)
(OpenNhds.map f x).op =
colimit.ι
(((whiskeringLeft (OpenNhds ((Hom.hom f) y))ᵒᵖ (Opens ↑Y)ᵒᵖ C).obj (OpenNhds.inclusion ((Hom.hom f) y)).op).obj
((pushforward C f).obj F))
j✝ ≫
(colim.map (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F) ≫
colimit.pre (((whiskeringLeft (OpenNhds y)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion y).op).obj F)
(OpenNhds.map f y).op) ≫
F.stalkSpecializes h
|
simp only [stalkSpecializes, colimit.ι_desc_assoc, colimit.ι_map_assoc, colimit.ι_pre,
Category.assoc, colimit.pre_desc, colimit.ι_desc]
|
case w
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y : TopCat
f : X ⟶ Y
F : Presheaf C X
x y : ↑X
h : x ⤳ y
j✝ : (OpenNhds ((Hom.hom f) y))ᵒᵖ
⊢ (whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f x).inv) F).app (op { obj := (unop j✝).obj, property := ⋯ }) ≫
colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion x).op).obj F)
((OpenNhds.map f x).op.obj (op { obj := (unop j✝).obj, property := ⋯ })) =
(whiskerRight (NatTrans.op (OpenNhds.inclusionMapIso f y).inv) F).app j✝ ≫
(Cocone.whisker (OpenNhds.map f y).op
{ pt := (stalkFunctor C x).toPrefunctor.1 F,
ι :=
{
app := fun U =>
colimit.ι ((OpenNhds.inclusion x).op ⋙ F) (op { obj := (unop U).obj, property := ⋯ }),
naturality := ⋯ } }).ι.app
j✝
|
bb98b71fe156acfc
|
IsCompact.elim_nhds_subcover_nhdsSet
|
Mathlib/Topology/Compactness/Compact.lean
|
lemma IsCompact.elim_nhds_subcover_nhdsSet (hs : IsCompact s) {U : X → Set X}
(hU : ∀ x ∈ s, U x ∈ 𝓝 x) : ∃ t : Finset X, (∀ x ∈ t, x ∈ s) ∧ (⋃ x ∈ t, U x) ∈ 𝓝ˢ s
|
X : Type u
inst✝ : TopologicalSpace X
s : Set X
hs : IsCompact s
U : X → Set X
hU : ∀ x ∈ s, U x ∈ 𝓝 x
t : Finset ↑s
ht : ⋃ x ∈ t, U ↑x ∈ 𝓝ˢ s
⊢ ⋃ x ∈ Finset.image Subtype.val t, U x ∈ 𝓝ˢ s
|
rwa [Finset.set_biUnion_finset_image]
|
no goals
|
cd4eb3db00104b0a
|
Array.ext
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Basic.lean
|
theorem ext (a b : Array α)
(h₁ : a.size = b.size)
(h₂ : (i : Nat) → (hi₁ : i < a.size) → (hi₂ : i < b.size) → a[i] = b[i])
: a = b
|
case mk.mk
α : Type u
toList✝¹ toList✝ : List α
h₁ : { toList := toList✝¹ }.size = { toList := toList✝ }.size
h₂ :
∀ (i : Nat) (hi₁ : i < { toList := toList✝¹ }.size) (hi₂ : i < { toList := toList✝ }.size),
getElem { toList := toList✝¹ } i hi₁ = getElem { toList := toList✝ } i hi₂
⊢ { toList := toList✝¹ } = { toList := toList✝ }
|
apply congrArg
|
case mk.mk.h
α : Type u
toList✝¹ toList✝ : List α
h₁ : { toList := toList✝¹ }.size = { toList := toList✝ }.size
h₂ :
∀ (i : Nat) (hi₁ : i < { toList := toList✝¹ }.size) (hi₂ : i < { toList := toList✝ }.size),
getElem { toList := toList✝¹ } i hi₁ = getElem { toList := toList✝ } i hi₂
⊢ toList✝¹ = toList✝
|
c533afa60c503a53
|
CategoryTheory.Comonad.ComonadicityInternal.comparisonAdjunction_unit_app
|
Mathlib/CategoryTheory/Monad/Comonadicity.lean
|
theorem comparisonAdjunction_unit_app
[∀ A : adj.toComonad.Coalgebra, HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))] (B : C) :
(comparisonAdjunction adj).unit.app B = limit.lift _ (unitFork adj B)
|
case h
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝ : ∀ (A : adj.toComonad.Coalgebra), HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))
B : C
⊢ equalizer.lift ((adj.homEquiv B (F.obj B)) (𝟙 (F.obj B))) ⋯ ≫
equalizer.ι (G.map ((comparison adj).obj B).a) (adj.unit.app (G.toPrefunctor.1 ((comparison adj).obj B).A)) =
equalizer.lift (adj.unit.app B) ⋯ ≫ equalizer.ι (G.map (F.map (adj.unit.app B))) (adj.unit.app (G.obj (F.obj B)))
|
simp [Adjunction.homEquiv_unit]
|
no goals
|
56285f6d00a0cc25
|
Ordinal.add_lt_add_iff_left'
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
theorem add_lt_add_iff_left' (a) {b c : Ordinal} : a + b < a + c ↔ b < c
|
a b c : Ordinal.{u_4}
⊢ a + b < a + c ↔ b < c
|
rw [← not_le, ← not_le, add_le_add_iff_left]
|
no goals
|
084361ab579da7fc
|
CStarModule.inner_mul_inner_swap_le
|
Mathlib/Analysis/CStarAlgebra/Module/Defs.lean
|
/-- The C⋆-algebra-valued Cauchy-Schwarz inequality for Hilbert C⋆-modules. -/
lemma inner_mul_inner_swap_le {x y : E} : ⟪y, x⟫ * ⟪x, y⟫ ≤ ‖x‖ ^ 2 • ⟪y, y⟫
|
case inr
A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul Aᵐᵒᵖ E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
h₁ :
∀ (a : A),
0 ≤
‖x‖ ^ 2 •> (star a * a) - ‖x‖ ^ 2 •> (⟪y, x⟫_A * a) - ‖x‖ ^ 2 •> (star a * ⟪x, y⟫_A) +
‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A
⊢ ⟪y, x⟫_A * ⟪x, y⟫_A ≤ ‖x‖ ^ 2 •> ⟪y, y⟫_A
|
specialize h₁ ⟪x, y⟫
|
case inr
A : Type u_1
E : Type u_2
inst✝⁷ : NonUnitalCStarAlgebra A
inst✝⁶ : PartialOrder A
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module ℂ E
inst✝³ : SMul Aᵐᵒᵖ E
inst✝² : Norm E
inst✝¹ : CStarModule A E
inst✝ : StarOrderedRing A
x y : E
h : x ≠ 0
h₁ :
0 ≤
‖x‖ ^ 2 •> (star ⟪x, y⟫_A * ⟪x, y⟫_A) - ‖x‖ ^ 2 •> (⟪y, x⟫_A * ⟪x, y⟫_A) - ‖x‖ ^ 2 •> (star ⟪x, y⟫_A * ⟪x, y⟫_A) +
‖x‖ ^ 2 •> ‖x‖ ^ 2 •> ⟪y, y⟫_A
⊢ ⟪y, x⟫_A * ⟪x, y⟫_A ≤ ‖x‖ ^ 2 •> ⟪y, y⟫_A
|
9539de1a07b6e267
|
Ordnode.Valid'.balanceL_aux
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem Valid'.balanceL_aux {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H₁ : size l = 0 → size r ≤ 1) (H₂ : 1 ≤ size l → 1 ≤ size r → size r ≤ delta * size l)
(H₃ : 2 * @size α l ≤ 9 * size r + 5 ∨ size l ≤ 3) : Valid' o₁ (@balanceL α l x r) o₂
|
case inl
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑x) r o₂
H₁ : l.size = 0 → r.size ≤ 1
H₂ : 1 ≤ l.size → 1 ≤ r.size → r.size ≤ delta * l.size
H₃ : 2 * l.size ≤ 9 * r.size + 5 ∨ l.size ≤ 3
r0 : r.size = 0
⊢ 2 * r.size ≤ 9 * l.size + 5
|
rw [r0]
|
case inl
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑x) r o₂
H₁ : l.size = 0 → r.size ≤ 1
H₂ : 1 ≤ l.size → 1 ≤ r.size → r.size ≤ delta * l.size
H₃ : 2 * l.size ≤ 9 * r.size + 5 ∨ l.size ≤ 3
r0 : r.size = 0
⊢ 2 * 0 ≤ 9 * l.size + 5
|
9c4e9afdd6227678
|
PMF.mem_support_bernoulli_iff
|
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
theorem mem_support_bernoulli_iff : b ∈ (bernoulli p h).support ↔ cond b (p ≠ 0) (p ≠ 1)
|
p : ℝ≥0∞
h : p ≤ 1
b : Bool
⊢ b ∈ (bernoulli p h).support ↔ bif b then p ≠ 0 else p ≠ 1
|
simp
|
no goals
|
2a4b3f2fb15ad932
|
AlgebraicGeometry.Scheme.IdealSheafData.vanishingIdeal_support
|
Mathlib/AlgebraicGeometry/IdealSheaf.lean
|
lemma vanishingIdeal_support {I : IdealSheafData X} :
vanishingIdeal I.support = I.radical
|
case ideal.h
X : Scheme
I : X.IdealSheafData
U : ↑X.affineOpens
⊢ (vanishingIdeal I.support).ideal U = I.radical.ideal U
|
dsimp
|
case ideal.h
X : Scheme
I : X.IdealSheafData
U : ↑X.affineOpens
⊢ PrimeSpectrum.vanishingIdeal (⇑(ConcreteCategory.hom (IsAffineOpen.fromSpec ⋯).base) ⁻¹' I.support) =
(I.ideal U).radical
|
636619f5fa70acf7
|
MeasureTheory.limsup_measure_closed_le_of_forall_tendsto_measure
|
Mathlib/MeasureTheory/Measure/Portmanteau.lean
|
/-- One implication of the portmanteau theorem:
Assuming that for all Borel sets E whose boundary ∂E carries no probability mass under a
candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
lemma limsup_measure_closed_le_of_forall_tendsto_measure
{Ω ι : Type*} {L : Filter ι} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
{μ : Measure Ω} [IsFiniteMeasure μ] {μs : ι → Measure Ω}
(h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 →
Tendsto (fun i ↦ μs i E) L (𝓝 (μ E)))
(F : Set Ω) (F_closed : IsClosed F) :
L.limsup (fun i ↦ μs i F) ≤ μ F
|
case inr
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i => (μs i) E) L (𝓝 (μ E))
F : Set Ω
F_closed : IsClosed F
h✝ : L.NeBot
ex : ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) F)) = 0
rs : ℕ → ℝ := Classical.choose ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), μ (frontier (Metric.thickening (rs n) F)) = 0
Fthicks_open : ∀ (n : ℕ), IsOpen (Metric.thickening (rs n) F)
key : ∀ (n : ℕ), Tendsto (fun i => (μs i) (Metric.thickening (rs n) F)) L (𝓝 (μ (Metric.thickening (rs n) F)))
⊢ limsup (fun i => (μs i) F) L ≤ μ F
|
apply ENNReal.le_of_forall_pos_le_add
|
case inr.h
Ω : Type u_2
ι : Type u_3
L : Filter ι
inst✝³ : MeasurableSpace Ω
inst✝² : PseudoEMetricSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
μs : ι → Measure Ω
h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i => (μs i) E) L (𝓝 (μ E))
F : Set Ω
F_closed : IsClosed F
h✝ : L.NeBot
ex : ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) F)) = 0
rs : ℕ → ℝ := Classical.choose ex
rs_lim : Tendsto rs atTop (𝓝 0)
rs_pos : ∀ (n : ℕ), 0 < rs n
rs_null : ∀ (n : ℕ), μ (frontier (Metric.thickening (rs n) F)) = 0
Fthicks_open : ∀ (n : ℕ), IsOpen (Metric.thickening (rs n) F)
key : ∀ (n : ℕ), Tendsto (fun i => (μs i) (Metric.thickening (rs n) F)) L (𝓝 (μ (Metric.thickening (rs n) F)))
⊢ ∀ (ε : ℝ≥0), 0 < ε → μ F < ⊤ → limsup (fun i => (μs i) F) L ≤ μ F + ↑ε
|
bb4384475908a757
|
List.prefix_of_prefix_length_le
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem prefix_of_prefix_length_le :
∀ {l₁ l₂ l₃ : List α}, l₁ <+: l₃ → l₂ <+: l₃ → length l₁ ≤ length l₂ → l₁ <+: l₂
| [], _, _, _, _, _ => nil_prefix
| _ :: _, b :: _, _, ⟨_, rfl⟩, ⟨_, e⟩, ll => by
injection e with _ e'; subst b
rcases prefix_of_prefix_length_le ⟨_, rfl⟩ ⟨_, e'⟩ (le_of_succ_le_succ ll) with ⟨r₃, rfl⟩
exact ⟨r₃, rfl⟩
|
α : Type u_1
head✝ : α
tail✝¹ : List α
b : α
tail✝ w✝¹ w✝ : List α
e : b :: tail✝ ++ w✝ = head✝ :: tail✝¹ ++ w✝¹
ll : (head✝ :: tail✝¹).length ≤ (b :: tail✝).length
⊢ head✝ :: tail✝¹ <+: b :: tail✝
|
injection e with _ e'
|
α : Type u_1
head✝ : α
tail✝¹ : List α
b : α
tail✝ w✝¹ w✝ : List α
ll : (head✝ :: tail✝¹).length ≤ (b :: tail✝).length
head_eq✝ : b = head✝
e' : tail✝.append w✝ = tail✝¹.append w✝¹
⊢ head✝ :: tail✝¹ <+: b :: tail✝
|
555eaf8606db38b1
|
fermatLastTheoremWith'_nat_int_tfae
|
Mathlib/NumberTheory/FLT/Basic.lean
|
lemma fermatLastTheoremWith'_nat_int_tfae (n : ℕ) :
TFAE [FermatLastTheoremFor n, FermatLastTheoremWith' ℕ n, FermatLastTheoremWith' ℤ n]
|
case pos
a b c : ℤ
ha : IsUnit a
hb : IsUnit b
hc : IsUnit c
tfae_2_iff_1 : FermatLastTheoremWith' ℕ 0 ↔ FermatLastTheoremFor 0
⊢ a ^ 0 + b ^ 0 ≠ c ^ 0
|
simp only [pow_zero, Int.reduceAdd, ne_eq, OfNat.ofNat_ne_one, not_false_eq_true]
|
no goals
|
6e7fe97333acee70
|
IsSepClosed.exists_root_C_mul_X_pow_add_C_mul_X_add_C
|
Mathlib/FieldTheory/IsSepClosed.lean
|
theorem exists_root_C_mul_X_pow_add_C_mul_X_add_C
[IsSepClosed k] {n : ℕ} (a b c : k) (hn : (n : k) = 0) (hn' : 2 ≤ n) (hb : b ≠ 0) :
∃ x, a * x ^ n + b * x + c = 0
|
case intro
k : Type u
inst✝¹ : Field k
inst✝ : IsSepClosed k
n : ℕ
a b c : k
hn : ↑n = 0
hn' : 2 ≤ n
hb : b ≠ 0
f : k[X] := C a * X ^ n + C b * X + C c
hdeg : f.degree ≠ 0
hsep : f.Separable
x : k
hx : f.IsRoot x
⊢ ∃ x, a * x ^ n + b * x + c = 0
|
exact ⟨x, by simpa [f] using hx⟩
|
no goals
|
1f3e118ba5e9f6ba
|
Int.tmod_one
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
|
theorem tmod_one (a : Int) : tmod a 1 = 0
|
a : Int
⊢ a.tmod 1 = 0
|
simp [tmod_def, Int.tdiv_one, Int.one_mul, Int.sub_self]
|
no goals
|
3d4a6d61a12f2090
|
lTensor.inverse_comp_lTensor
|
Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
|
lemma lTensor.inverse_comp_lTensor :
(lTensor.inverse Q hfg hg).comp (lTensor Q g) =
Submodule.mkQ (p := LinearMap.range (lTensor Q f))
|
R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : AddCommGroup P
inst✝⁴ : Module R M
inst✝³ : Module R N
inst✝² : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
Q : Type u_5
inst✝¹ : AddCommGroup Q
inst✝ : Module R Q
hfg : Exact ⇑f ⇑g
hg : Surjective ⇑g
⊢ inverse Q hfg hg ∘ₗ lTensor Q g = (range (lTensor Q f)).mkQ
|
rw [lTensor.inverse, lTensor.inverse_of_rightInverse_comp_lTensor]
|
no goals
|
c51f9dd14acb7afb
|
FiberPrebundle.continuous_totalSpaceMk
|
Mathlib/Topology/FiberBundle/Basic.lean
|
theorem continuous_totalSpaceMk (b : B) :
Continuous[_, a.totalSpaceTopology] (TotalSpace.mk b)
|
B : Type u_2
F : Type u_3
E : B → Type u_5
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
inst✝ : (x : B) → TopologicalSpace (E x)
a : FiberPrebundle F E
b : B
this : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
e : Trivialization F TotalSpace.proj := a.trivializationOfMemPretrivializationAtlas ⋯
⊢ Continuous (↑e.toPartialHomeomorph ∘ TotalSpace.mk b)
|
exact continuous_iff_le_induced.2 (a.totalSpaceMk_isInducing b).eq_induced.le
|
no goals
|
6e072313aca77777
|
Orientation.kahler_neg_orientation
|
Mathlib/Analysis/InnerProductSpace/TwoDim.lean
|
theorem kahler_neg_orientation (x y : E) : (-o).kahler x y = conj (o.kahler x y)
|
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)
x y : E
⊢ ((-o).kahler x) y = (starRingEnd ℂ) ((o.kahler x) y)
|
simp [kahler_apply_apply, Complex.conj_ofReal]
|
no goals
|
2f8880191c12dfa8
|
Order.PartialIso.exists_across
|
Mathlib/Order/CountableDenseLinearOrder.lean
|
theorem exists_across [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β]
(f : PartialIso α β) (a : α) :
∃ b : β, ∀ p ∈ f.val, cmp (Prod.fst p) a = cmp (Prod.snd p) b
|
case h.mk
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : DenselyOrdered β
inst✝² : NoMinOrder β
inst✝¹ : NoMaxOrder β
inst✝ : Nonempty β
f : PartialIso α β
a : α
h : ¬∃ b, (a, b) ∈ ↑f
this :
∀ x ∈ Finset.image Prod.snd (Finset.filter (fun p => p.1 < a) ↑f),
∀ y ∈ Finset.image Prod.snd (Finset.filter (fun p => a < p.1) ↑f), x < y
b : β
hb :
(∀ x ∈ Finset.image Prod.snd (Finset.filter (fun p => p.1 < a) ↑f), x < b) ∧
∀ y ∈ Finset.image Prod.snd (Finset.filter (fun p => a < p.1) ↑f), b < y
p1 : α
p2 : β
hp : (p1, p2) ∈ ↑f
⊢ cmp (p1, p2).1 a = cmp (p1, p2).2 b
|
have : p1 ≠ a := fun he ↦ h ⟨p2, he ▸ hp⟩
|
case h.mk
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : DenselyOrdered β
inst✝² : NoMinOrder β
inst✝¹ : NoMaxOrder β
inst✝ : Nonempty β
f : PartialIso α β
a : α
h : ¬∃ b, (a, b) ∈ ↑f
this✝ :
∀ x ∈ Finset.image Prod.snd (Finset.filter (fun p => p.1 < a) ↑f),
∀ y ∈ Finset.image Prod.snd (Finset.filter (fun p => a < p.1) ↑f), x < y
b : β
hb :
(∀ x ∈ Finset.image Prod.snd (Finset.filter (fun p => p.1 < a) ↑f), x < b) ∧
∀ y ∈ Finset.image Prod.snd (Finset.filter (fun p => a < p.1) ↑f), b < y
p1 : α
p2 : β
hp : (p1, p2) ∈ ↑f
this : p1 ≠ a
⊢ cmp (p1, p2).1 a = cmp (p1, p2).2 b
|
68389d355148de8a
|
Polynomial.degree_divX_lt
|
Mathlib/Algebra/Polynomial/Inductions.lean
|
theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree
|
R : Type u
inst✝ : Semiring R
p : R[X]
hp0 : p ≠ 0
this : Nontrivial R
h : p.degree ≤ 0
h' : C (p.coeff 0) ≠ 0
⊢ p.divX.degree < (p.divX * X + C (p.coeff 0)).degree
|
rw [eq_C_of_degree_le_zero h, divX_C, degree_zero, zero_mul, zero_add]
|
R : Type u
inst✝ : Semiring R
p : R[X]
hp0 : p ≠ 0
this : Nontrivial R
h : p.degree ≤ 0
h' : C (p.coeff 0) ≠ 0
⊢ ⊥ < (C ((C (p.coeff 0)).coeff 0)).degree
|
e85e1acbc2b7e23a
|
List.lex_eq_false_iff_exists
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lex.lean
|
theorem lex_eq_false_iff_exists [BEq α] [PartialEquivBEq α] (lt : α → α → Bool)
(lt_irrefl : ∀ x y, x == y → lt x y = false)
(lt_asymm : ∀ x y, lt x y = true → lt y x = false)
(lt_antisymm : ∀ x y, lt x y = false → lt y x = false → x == y) :
lex l₁ l₂ lt = false ↔
(l₂.isEqv (l₁.take l₂.length) (· == ·)) ∨
(∃ (i : Nat) (h₁ : i < l₁.length) (h₂ : i < l₂.length),
(∀ j, (hj : j < i) →
l₁[j]'(Nat.lt_trans hj h₁) == l₂[j]'(Nat.lt_trans hj h₂)) ∧ lt l₂[i] l₁[i])
|
α : Type u_1
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
lt : α → α → Bool
lt_irrefl : ∀ (x y : α), (x == y) = true → lt x y = false
lt_asymm : ∀ (x y : α), lt x y = true → lt y x = false
lt_antisymm : ∀ (x y : α), lt x y = false → lt y x = false → (x == y) = true
a : α
l₁ : List α
ih :
∀ {l₂ : List α},
l₁.lex l₂ lt = false ↔
(l₂.isEqv (take l₂.length l₁) fun x1 x2 => x1 == x2) = true ∨
∃ i h₁ h₂, (∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true) ∧ lt l₂[i] l₁[i] = true
b : α
l₂ : List α
hab : lt a b = false
h :
(a == b) = true →
(l₂.isEqv (take l₂.length l₁) fun x1 x2 => x1 == x2) = true ∨
∃ i h₁ h₂, (∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true) ∧ lt l₂[i] l₁[i] = true
eq : ¬(b == a) = true
hba : ¬lt b a = true
⊢ lt b a = false
|
simpa using hba
|
no goals
|
0c9240a9b47ee0a9
|
CategoryTheory.Functor.FullyFaithful.hasShift.map_add_hom_app
|
Mathlib/CategoryTheory/Shift/Basic.lean
|
@[simp]
lemma map_add_hom_app (a b : A) (X : C) :
F.map ((add hF s i a b).hom.app X) =
(i (a + b)).hom.app X ≫ (shiftFunctorAdd D a b).hom.app (F.obj X) ≫
((i a).inv.app X)⟦b⟧' ≫ (i b).inv.app ((s a).obj X)
|
C : Type u
A : Type u_1
inst✝³ : Category.{v, u} C
D : Type u_2
inst✝² : Category.{u_3, u_2} D
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : C ⥤ D
hF : F.FullyFaithful
s : A → C ⥤ C
i : (i : A) → s i ⋙ F ≅ F ⋙ shiftFunctor D i
a b : A
X : C
⊢ F.map
(hF.preimage
((i (a + b)).hom.app X ≫
(shiftFunctorAdd D a b).hom.app (F.obj X) ≫
𝟙 ((shiftFunctor D b).obj ((shiftFunctor D a).obj (F.obj X))) ≫
(shiftFunctor D b).map ((i a).inv.app X) ≫
𝟙 ((shiftFunctor D b).obj (F.obj ((s a).obj X))) ≫
(i b).inv.app ((s a).obj X) ≫ 𝟙 (F.obj ((s b).obj ((s a).obj X))))) =
(i (a + b)).hom.app X ≫
(shiftFunctorAdd D a b).hom.app (F.obj X) ≫ (shiftFunctor D b).map ((i a).inv.app X) ≫ (i b).inv.app ((s a).obj X)
|
simp
|
no goals
|
b2d90d624ce5ad6b
|
Grp.hom_inv_apply
|
Mathlib/Algebra/Category/Grp/Basic.lean
|
@[to_additive]
lemma hom_inv_apply {X Y : Grp} (e : X ≅ Y) (s : Y) : e.hom (e.inv s) = s
|
X Y : Grp
e : X ≅ Y
s : ↑Y
⊢ (ConcreteCategory.hom e.hom) ((ConcreteCategory.hom e.inv) s) = s
|
simp
|
no goals
|
47b4f7ea66325457
|
MvPowerSeries.constantCoeff_invOfUnit
|
Mathlib/RingTheory/MvPowerSeries/Inverse.lean
|
theorem constantCoeff_invOfUnit (φ : MvPowerSeries σ R) (u : Rˣ) :
constantCoeff σ R (invOfUnit φ u) = ↑u⁻¹
|
σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
⊢ (constantCoeff σ R) (φ.invOfUnit u) = ↑u⁻¹
|
classical
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]
|
no goals
|
c9d098c27bb1f219
|
Homeomorph.comp_isOpenMap_iff
|
Mathlib/Topology/Homeomorph.lean
|
theorem comp_isOpenMap_iff (h : X ≃ₜ Y) {f : Z → X} : IsOpenMap (h ∘ f) ↔ IsOpenMap f
|
X : Type u_1
Y : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
h : X ≃ₜ Y
f : Z → X
hf : IsOpenMap (⇑h ∘ f)
⊢ IsOpenMap (⇑h.symm ∘ ⇑h ∘ f)
|
exact h.symm.isOpenMap.comp hf
|
no goals
|
f372446ee3a1e3b6
|
mdifferentiableOn_iff
|
Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
|
theorem mdifferentiableOn_iff :
MDifferentiableOn I I' f s ↔
ContinuousOn f s ∧
∀ (x : M) (y : M'),
DifferentiableOn 𝕜 (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
((extChartAt I x).target ∩
(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source))
|
case mp.convert_2
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
s : Set M
inst✝¹ : IsManifold I 1 M
inst✝ : IsManifold I' 1 M'
x : M
y : M'
z : E
hz :
(z ∈ range ↑I ∧ ↑I.symm z ∈ (chartAt H x).target) ∧
↑(chartAt H x).symm (↑I.symm z) ∈ s ∧ f (↑(chartAt H x).symm (↑I.symm z)) ∈ (chartAt H' y).source
w : M := ↑(extChartAt I x).symm z
this : w ∈ s
h : MDifferentiableWithinAt I I' f s w
w1 : w ∈ (chartAt H x).source
w2 : f w ∈ (chartAt H' y).source
⊢ (extChartAt I x).target ∩ ↑(extChartAt I x).symm ⁻¹' (s ∩ f ⁻¹' (extChartAt I' y).source) ⊆
↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I
|
mfld_set_tac
|
no goals
|
32f42f1853f90b85
|
CategoryTheory.ComposableArrows.mk₄_surjective
|
Mathlib/CategoryTheory/ComposableArrows.lean
|
lemma mk₄_surjective (X : ComposableArrows C 4) :
∃ (X₀ X₁ X₂ X₃ X₄ : C) (f₀ : X₀ ⟶ X₁) (f₁ : X₁ ⟶ X₂) (f₂ : X₂ ⟶ X₃) (f₃ : X₃ ⟶ X₄),
X = mk₄ f₀ f₁ f₂ f₃ :=
⟨_, _, _, _, _, X.map' 0 1, X.map' 1 2, X.map' 2 3, X.map' 3 4,
ext₄ rfl rfl rfl rfl rfl (by simp) (by simp) (by simp) (by simp)⟩
|
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X : ComposableArrows C 4
⊢ X.map' 1 2 ⋯ ⋯ =
eqToHom ⋯ ≫ (mk₄ (X.map' 0 1 ⋯ ⋯) (X.map' 1 2 ⋯ ⋯) (X.map' 2 3 ⋯ ⋯) (X.map' 3 4 ⋯ ⋯)).map' 1 2 ⋯ ⋯ ≫ eqToHom ⋯
|
simp
|
no goals
|
c94b8d43b56d9139
|
Polynomial.support_integralNormalization
|
Mathlib/RingTheory/Polynomial/IntegralNormalization.lean
|
theorem support_integralNormalization {f : R[X]} :
(integralNormalization f).support = f.support
|
case neg.h
R : Type u
inst✝¹ : Semiring R
inst✝ : IsCancelMulZero R
f : R[X]
a✝ : Nontrivial R
this : IsDomain R
hf : ¬f = 0
i : ℕ
⊢ f.coeff i ≠ 0 → (if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)) ≠ 0
|
intro hfi
|
case neg.h
R : Type u
inst✝¹ : Semiring R
inst✝ : IsCancelMulZero R
f : R[X]
a✝ : Nontrivial R
this : IsDomain R
hf : ¬f = 0
i : ℕ
hfi : f.coeff i ≠ 0
⊢ (if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)) ≠ 0
|
9178f9ebeb3af80a
|
DFinsupp.addHom_ext
|
Mathlib/Data/DFinsupp/Ext.lean
|
theorem addHom_ext {γ : Type w} [AddZeroClass γ] ⦃f g : (Π₀ i, β i) →+ γ⦄
(H : ∀ (i : ι) (y : β i), f (single i y) = g (single i y)) : f = g
|
case intro.intro
ι : Type u
β : ι → Type v
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → AddZeroClass (β i)
γ : Type w
inst✝ : AddZeroClass γ
f g : (Π₀ (i : ι), β i) →+ γ
H : ∀ (i : ι) (y : β i), f (single i y) = g (single i y)
x : ι
y : β x
⊢ f (single x y) = g (single x y)
|
apply H
|
no goals
|
8563ef17426d7610
|
AlgebraicGeometry.AffineSpace.reindex_comp
|
Mathlib/AlgebraicGeometry/AffineSpace.lean
|
@[simp, reassoc]
lemma reindex_comp {n₁ n₂ n₃ : Type v} (i : n₁ → n₂) (j : n₂ → n₃) (S : Scheme.{max u v}) :
reindex (j ∘ i) S = reindex j S ≫ reindex i S
|
n₁ n₂ n₃ : Type v
i : n₁ → n₂
j : n₂ → n₃
S : Scheme
H₁ : reindex (j ∘ i) S ≫ 𝔸(n₁; S) ↘ S = (reindex j S ≫ reindex i S) ≫ 𝔸(n₁; S) ↘ S
k : n₁
⊢ coord S ((j ∘ i) k) = coord S (j (i k))
|
rfl
|
no goals
|
076b8b6bdaff2aa6
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.sat_of_insertRat
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean
|
theorem sat_of_insertRat {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (c : DefaultClause n) (p : PosFin n → Bool)
(pf : p ⊨ f) :
(insertRatUnits f (negate c)).2 = true → p ⊨ c
|
case intro.inl.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : f.assignments[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.ratUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
let update_res := List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate;
let_fun update_res_size := ⋯;
InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
h1 :
(List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] = f.assignments[↑⟨i.val, ⋯⟩]
h2 :
∀ (j : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
(List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j].fst.val ≠ ↑⟨i.val, ⋯⟩
hpos : hasAssignment true f.assignments[i.val] = true
hneg : hasAssignment false f.assignments[i.val] = true
⊢ p ⊨ c
|
have p_entails_i_true := hf.2.2 i true hpos p pf
|
case intro.inl.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
c : DefaultClause n
p : PosFin n → Bool
pf : p ⊨ f
insertUnit_fold_success : (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).2.snd = true
i : PosFin n
hboth : f.assignments[i.val] = both
i_in_bounds : i.val < f.assignments.size
h0 : InsertUnitInvariant f.assignments ⋯ f.ratUnits f.assignments ⋯
insertUnit_fold_satisfies_invariant :
let update_res := List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate;
let_fun update_res_size := ⋯;
InsertUnitInvariant f.assignments ⋯ update_res.fst update_res.snd.fst update_res_size
h1 :
(List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).snd.fst[↑⟨i.val, ⋯⟩] = f.assignments[↑⟨i.val, ⋯⟩]
h2 :
∀ (j : Fin (List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst.size),
(List.foldl insertUnit (f.ratUnits, f.assignments, false) c.negate).fst[j].fst.val ≠ ↑⟨i.val, ⋯⟩
hpos : hasAssignment true f.assignments[i.val] = true
hneg : hasAssignment false f.assignments[i.val] = true
p_entails_i_true : p ⊨ (i, true)
⊢ p ⊨ c
|
c0b439908684a182
|
Int.Cooper.resolve_left_dvd₂
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Cooper.lean
|
theorem resolve_left_dvd₂ (a c d p x : Int)
(h₁ : p ≤ a * x) (h₃ : d ∣ c * x + s) :
a * d ∣ c * resolve_left a c d p x + c * p + a * s
|
case mk.h
s a c d p x : Int
h₁ : p ≤ a * x
h₃ : d ∣ c * x + s
k' : Nat
w : a * x = p + ↑k'
⊢ a * d / ↑((a * d).gcd c) ∣ ↑(a.lcm (a * d / ↑((a * d).gcd c)))
|
exact Int.dvd_lcm_right
|
no goals
|
3a45046c1f4b368b
|
AlgebraicTopology.DoldKan.Γ₀.Obj.Termwise.mapMono_eq_zero
|
Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
|
theorem mapMono_eq_zero (i : Δ' ⟶ Δ) [Mono i] (h₁ : Δ ≠ Δ') (h₂ : ¬Isδ₀ i) : mapMono K i = 0
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h₁ : Δ ≠ Δ'
h₂ : ¬Isδ₀ i
⊢ (if h : Δ = Δ' then eqToHom ⋯ else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = 0
|
rw [Ne] at h₁
|
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K : ChainComplex C ℕ
Δ Δ' : SimplexCategory
i : Δ' ⟶ Δ
inst✝ : Mono i
h₁ : ¬Δ = Δ'
h₂ : ¬Isδ₀ i
⊢ (if h : Δ = Δ' then eqToHom ⋯ else if h : Isδ₀ i then K.d Δ.len Δ'.len else 0) = 0
|
b5d49891deb888d3
|
rieszContentAux_image_nonempty
|
Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean
|
theorem rieszContentAux_image_nonempty (K : Compacts X) :
(Λ '' { f : C_c(X, ℝ≥0) | ∀ x ∈ K, (1 : ℝ≥0) ≤ f x }).Nonempty
|
case intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hfeq1onK : EqOn (⇑f) 1 K.carrier
hfinicc : ∀ (x : X), f x ∈ Icc 0 1
⊢ {f | ∀ x ∈ K, 1 ≤ f x}.Nonempty
|
have hfHasCompactSupport : HasCompactSupport f :=
IsCompact.of_isClosed_subset hVcp (isClosed_tsupport f)
(Set.Subset.trans hsuppfsubV interior_subset)
|
case intro.intro.intro.intro.intro
X : Type u_1
inst✝² : TopologicalSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
K : Compacts X
V : Set X
hVcp : IsCompact V
hKsubintV : K.carrier ⊆ interior V
hIsCompact_closure_interior : IsCompact (closure (interior V))
f : C(X, ℝ)
hsuppfsubV : tsupport ⇑f ⊆ interior V
hfeq1onK : EqOn (⇑f) 1 K.carrier
hfinicc : ∀ (x : X), f x ∈ Icc 0 1
hfHasCompactSupport : HasCompactSupport ⇑f
⊢ {f | ∀ x ∈ K, 1 ≤ f x}.Nonempty
|
2a4990d43927e6b7
|
Compactum.cl_cl
|
Mathlib/Topology/Category/Compactum.lean
|
theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A
|
case intro
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
this : ∀ (T : fsu), ι T ⊆ C1 → (⋂₀ ι T).Nonempty
G : Ultrafilter (Ultrafilter X.A)
h1 : C1 ⊆ (↑G).sets
⊢ X.str F ∈ Compactum.cl A
|
use X.join G
|
case h
X : Compactum
A : Set X.A
F : Ultrafilter X.A
hF : F ∈ Compactum.basic (Compactum.cl A)
fsu : Type u_1 := Finset (Set (Ultrafilter X.A))
ssu : Type u_1 := Set (Set (Ultrafilter X.A))
ι : fsu → ssu := fun x => ↑x
C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}
AA : Set (Ultrafilter X.A) := {G | A ∈ G}
C1 : ssu := insert AA C0
C2 : Set (Set (Ultrafilter X.A)) := finiteInterClosure C1
claim1 : ∀ B ∈ C0, ∀ C ∈ C0, B ∩ C ∈ C0
claim2 : ∀ B ∈ C0, B.Nonempty
claim3 : ∀ B ∈ C0, (AA ∩ B).Nonempty
this : ∀ (T : fsu), ι T ⊆ C1 → (⋂₀ ι T).Nonempty
G : Ultrafilter (Ultrafilter X.A)
h1 : C1 ⊆ (↑G).sets
⊢ X.join G ∈ Compactum.basic A ∧ X.str (X.join G) = X.str F
|
1c30fb7238f2120d
|
EuclideanGeometry.Sphere.wbtw_secondInter
|
Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean
|
theorem Sphere.wbtw_secondInter {s : Sphere P} {p p' : P} (hp : p ∈ s)
(hp' : dist p' s.center ≤ s.radius) : Wbtw ℝ p p' (s.secondInter p (p' -ᵥ p))
|
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p
⊢ p ≠ s.secondInter p (p' -ᵥ p)
|
intro he
|
case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p p' : P
hp : p ∈ s
hp' : dist p' s.center ≤ s.radius
h : ¬p' = p
he : p = s.secondInter p (p' -ᵥ p)
⊢ False
|
d271772fa5acc4b9
|
HasDerivWithinAt.rpow_const
|
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
theorem HasDerivWithinAt.rpow_const (hf : HasDerivWithinAt f f' s x) (hx : f x ≠ 0 ∨ 1 ≤ p) :
HasDerivWithinAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) s x
|
case h.e'_9
f : ℝ → ℝ
f' x p : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hx : f x ≠ 0 ∨ 1 ≤ p
⊢ f' * p * f x ^ (p - 1) = p * f x ^ (p - 1) * f'
|
ring
|
no goals
|
78370223655791e7
|
ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq_on
|
Mathlib/NumberTheory/ArithmeticFunction.lean
|
theorem sum_eq_iff_sum_smul_moebius_eq_on [AddCommGroup R] {f g : ℕ → R}
(s : Set ℕ) (hs : ∀ m n, m ∣ n → n ∈ s → m ∈ s) :
(∀ n > 0, n ∈ s → (∑ i ∈ n.divisors, f i) = g n) ↔
∀ n > 0, n ∈ s → (∑ x ∈ n.divisorsAntidiagonal, μ x.fst • g x.snd) = f n
|
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n
F : ℕ → R := fun n => ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2
n : ℕ
hn : n > 0
hnP : n ∈ s
this : ∑ d ∈ n.divisors, F d = g n
⊢ ∑ i ∈ n.divisors, f i = g n
|
rw [← this, sum_congr rfl]
|
R : Type u_1
inst✝ : AddCommGroup R
f g : ℕ → R
s : Set ℕ
hs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s
h : ∀ n > 0, n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n
F : ℕ → R := fun n => ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2
n : ℕ
hn : n > 0
hnP : n ∈ s
this : ∑ d ∈ n.divisors, F d = g n
⊢ ∀ x ∈ n.divisors, f x = F x
|
806246a74af811fb
|
Profinite.NobelingProof.GoodProducts.spanFin
|
Mathlib/Topology/Category/Profinite/Nobeling.lean
|
theorem GoodProducts.spanFin [WellFoundedLT I] :
⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (· ∈ s))))
|
case intro.cons
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
s : Finset I
inst✝ : WellFoundedLT I
x : ↑(π C fun x => x ∈ s)
l : List I := Finset.sort (fun x1 x2 => x1 ≥ x2) s
a : I
as : List I
ih :
List.Chain' (fun x1 x2 => x1 > x2) as →
(List.map (fun i => if ↑x i = true then e (π C fun x => x ∈ s) i else 1 - e (π C fun x => x ∈ s) i) as).prod ∈
Submodule.span ℤ (Products.eval (π C fun x => x ∈ s) '' {m | ↑m ≤ as})
⊢ List.Chain' (fun x1 x2 => x1 > x2) (a :: as) →
(if ↑x a = true then e (π C fun x => x ∈ s) a else 1 - e (π C fun x => x ∈ s) a) *
(List.map (fun i => if ↑x i = true then e (π C fun x => x ∈ s) i else 1 - e (π C fun x => x ∈ s) i) as).prod ∈
Submodule.span ℤ (Products.eval (π C fun x => x ∈ s) '' {m | ↑m ≤ a :: as})
|
intro ha
|
case intro.cons
I : Type u
C : Set (I → Bool)
inst✝¹ : LinearOrder I
s : Finset I
inst✝ : WellFoundedLT I
x : ↑(π C fun x => x ∈ s)
l : List I := Finset.sort (fun x1 x2 => x1 ≥ x2) s
a : I
as : List I
ih :
List.Chain' (fun x1 x2 => x1 > x2) as →
(List.map (fun i => if ↑x i = true then e (π C fun x => x ∈ s) i else 1 - e (π C fun x => x ∈ s) i) as).prod ∈
Submodule.span ℤ (Products.eval (π C fun x => x ∈ s) '' {m | ↑m ≤ as})
ha : List.Chain' (fun x1 x2 => x1 > x2) (a :: as)
⊢ (if ↑x a = true then e (π C fun x => x ∈ s) a else 1 - e (π C fun x => x ∈ s) a) *
(List.map (fun i => if ↑x i = true then e (π C fun x => x ∈ s) i else 1 - e (π C fun x => x ∈ s) i) as).prod ∈
Submodule.span ℤ (Products.eval (π C fun x => x ∈ s) '' {m | ↑m ≤ a :: as})
|
84857795ad4a98fd
|
List.head_reverse
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem head_reverse {l : List α} (h : l.reverse ≠ []) :
l.reverse.head h = getLast l (by simp_all)
|
case neg
α : Type u_1
a : α
l : List α
ih : ∀ (h : l.reverse ≠ []), l.reverse.head h = l.getLast ⋯
h : (a :: l).reverse ≠ []
h' : ¬l = []
⊢ (l.reverse ++ [a]).head ⋯ = (a :: l).getLast ⋯
|
simp only [head_eq_iff_head?_eq_some, head?_reverse] at ih
|
case neg
α : Type u_1
a : α
l : List α
h : (a :: l).reverse ≠ []
h' : ¬l = []
ih : ∀ (h : l.reverse ≠ []), l.getLast? = some (l.getLast ⋯)
⊢ (l.reverse ++ [a]).head ⋯ = (a :: l).getLast ⋯
|
e5cad9770083e9e0
|
SimpleGraph.Walk.dropLast_concat
|
Mathlib/Combinatorics/SimpleGraph/Walk.lean
|
@[simp]
lemma dropLast_concat {t u v} (p : G.Walk u v) (h : G.Adj v t) :
(p.concat h).dropLast = p.copy rfl (by simp)
|
case cons.hp
V : Type u
G : SimpleGraph V
t u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
p_ih✝ : ∀ (h : G.Adj w✝ t), (p✝.concat h).dropLast = p✝.copy ⋯ ⋯
h : G.Adj w✝ t
⊢ ¬(p✝.concat h).Nil
|
simp [concat, nil_iff_length_eq]
|
no goals
|
d5401d0415d66680
|
IsCompact.finite_compact_cover
|
Mathlib/Topology/Separation/Basic.lean
|
theorem IsCompact.finite_compact_cover {s : Set X} (hs : IsCompact s) {ι : Type*}
(t : Finset ι) (U : ι → Set X) (hU : ∀ i ∈ t, IsOpen (U i)) (hsC : s ⊆ ⋃ i ∈ t, U i) :
∃ K : ι → Set X, (∀ i, IsCompact (K i)) ∧ (∀ i, K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i
|
case insert.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
ι : Type u_3
x : ι
t : Finset ι
hx : x ∉ t
ih :
∀ {s : Set X},
IsCompact s →
∀ (U : ι → Set X),
(∀ i ∈ t, IsOpen (U i)) →
s ⊆ ⋃ i ∈ t, U i → ∃ K, (∀ (i : ι), IsCompact (K i)) ∧ (∀ (i : ι), K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i
s : Set X
hs : IsCompact s
U : ι → Set X
hsC : s ⊆ U x ∪ ⋃ x ∈ t, U x
hU : IsOpen (U x) ∧ ∀ x ∈ t, IsOpen (U x)
hU' : ∀ i ∈ t, IsOpen (U i)
K₁ K₂ : Set X
h1K₁ : IsCompact K₁
h1K₂ : IsCompact K₂
h2K₁ : K₁ ⊆ U x
h2K₂ : K₂ ⊆ ⋃ i ∈ Membership.mem t.val, U i
hK : s = K₁ ∪ K₂
K : ι → Set X
h1K : ∀ (i : ι), IsCompact (K i)
h2K : ∀ (i : ι), K i ⊆ U i
h3K : K₂ = ⋃ i ∈ t, K i
i : ι
⊢ update K x K₁ i ⊆ U i
|
rcases eq_or_ne i x with rfl | hi
|
case insert.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.inl
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
ι : Type u_3
t : Finset ι
ih :
∀ {s : Set X},
IsCompact s →
∀ (U : ι → Set X),
(∀ i ∈ t, IsOpen (U i)) →
s ⊆ ⋃ i ∈ t, U i → ∃ K, (∀ (i : ι), IsCompact (K i)) ∧ (∀ (i : ι), K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i
s : Set X
hs : IsCompact s
U : ι → Set X
hU' : ∀ i ∈ t, IsOpen (U i)
K₁ K₂ : Set X
h1K₁ : IsCompact K₁
h1K₂ : IsCompact K₂
h2K₂ : K₂ ⊆ ⋃ i ∈ Membership.mem t.val, U i
hK : s = K₁ ∪ K₂
K : ι → Set X
h1K : ∀ (i : ι), IsCompact (K i)
h2K : ∀ (i : ι), K i ⊆ U i
h3K : K₂ = ⋃ i ∈ t, K i
i : ι
hx : i ∉ t
hsC : s ⊆ U i ∪ ⋃ x ∈ t, U x
hU : IsOpen (U i) ∧ ∀ x ∈ t, IsOpen (U x)
h2K₁ : K₁ ⊆ U i
⊢ update K i K₁ i ⊆ U i
case insert.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.inr
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : R1Space X
ι : Type u_3
x : ι
t : Finset ι
hx : x ∉ t
ih :
∀ {s : Set X},
IsCompact s →
∀ (U : ι → Set X),
(∀ i ∈ t, IsOpen (U i)) →
s ⊆ ⋃ i ∈ t, U i → ∃ K, (∀ (i : ι), IsCompact (K i)) ∧ (∀ (i : ι), K i ⊆ U i) ∧ s = ⋃ i ∈ t, K i
s : Set X
hs : IsCompact s
U : ι → Set X
hsC : s ⊆ U x ∪ ⋃ x ∈ t, U x
hU : IsOpen (U x) ∧ ∀ x ∈ t, IsOpen (U x)
hU' : ∀ i ∈ t, IsOpen (U i)
K₁ K₂ : Set X
h1K₁ : IsCompact K₁
h1K₂ : IsCompact K₂
h2K₁ : K₁ ⊆ U x
h2K₂ : K₂ ⊆ ⋃ i ∈ Membership.mem t.val, U i
hK : s = K₁ ∪ K₂
K : ι → Set X
h1K : ∀ (i : ι), IsCompact (K i)
h2K : ∀ (i : ι), K i ⊆ U i
h3K : K₂ = ⋃ i ∈ t, K i
i : ι
hi : i ≠ x
⊢ update K x K₁ i ⊆ U i
|
745e4d6bcd976884
|
AkraBazziRecurrence.GrowsPolynomially.eventually_zero_of_frequently_zero
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma eventually_zero_of_frequently_zero (hf : GrowsPolynomially f) (hf' : ∃ᶠ x in atTop, f x = 0) :
∀ᶠ x in atTop, f x = 0
|
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ x ⊔ 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ 0 ≤ -logb 2 (x / x₀)
|
rw [neg_nonneg]
|
f : ℝ → ℝ
hf✝ : GrowsPolynomially f
hf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0
c₁ : ℝ
hc₁_mem : c₁ > 0
c₂ : ℝ
hc₂_mem : c₂ > 0
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
x : ℝ
hx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y)
hx_pos : 0 < x
x₀ : ℝ
hx₀_ge : x₀ ≥ x ⊔ 1
hx₀ : f x₀ = 0
x₀_pos : 0 < x₀
hmain : ∀ (m : ℕ) (z : ℝ), x ≤ z → z ∈ Set.Icc (2 ^ (-↑m - 1) * x₀) (2 ^ (-↑m) * x₀) → f z = 0
⊢ logb 2 (x / x₀) ≤ 0
|
b2c5c4c5d5df705a
|
FDerivMeasurableAux.D_subset_differentiable_set
|
Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
|
theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K }
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
∀ (e p q : ℕ),
n e ≤ p →
n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := n e ⊔ n e'
⊢ 0 ≤ 1 / 2
|
norm_num
|
no goals
|
28c357cacce7671e
|
AffineBasis.convexHull_eq_nonneg_coord
|
Mathlib/Analysis/Convex/Combination.lean
|
theorem AffineBasis.convexHull_eq_nonneg_coord {ι : Type*} (b : AffineBasis ι R E) :
convexHull R (range b) = { x | ∀ i, 0 ≤ b.coord i x }
|
R : Type u_1
E : Type u_3
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
ι : Type u_8
b : AffineBasis ι R E
x : E
hx : x ∈ {x | ∀ (i : ι), 0 ≤ (b.coord i) x}
⊢ x ∈ ⊤
|
exact AffineSubspace.mem_top R E x
|
no goals
|
1fd8376dbf4dc8b4
|
CategoryTheory.Limits.SequentialProduct.functorMap_commSq
|
Mathlib/CategoryTheory/Limits/Shapes/SequentialProduct.lean
|
lemma functorMap_commSq {n m : ℕ} (h : ¬(m < n)) :
(Functor.ofOpSequence (functorMap f)).map (homOfLE (by omega : n ≤ m + 1)).op ≫ Pi.π _ m ≫
eqToHom (functorObj_eq_neg (by omega : ¬(m < n))) =
(Pi.π (fun i ↦ if _ : i < m + 1 then M i else N i) m) ≫
eqToHom (functorObj_eq_pos (by omega)) ≫ f m
|
C : Type u_1
M N : ℕ → C
inst✝¹ : Category.{u_2, u_1} C
f : (n : ℕ) → M n ⟶ N n
inst✝ : HasProductsOfShape ℕ C
n m : ℕ
h : ¬m + 1 < n
⊢ m + 1 ≤ m + 1 + 1
|
omega
|
no goals
|
916cb9b7a6b71e63
|
ProbabilityTheory.Kernel.compProd_apply_univ_le
|
Mathlib/Probability/Kernel/Composition/CompProd.lean
|
theorem compProd_apply_univ_le (κ : Kernel α β) (η : Kernel (α × β) γ) [IsFiniteKernel η] (a : α) :
(κ ⊗ₖ η) a Set.univ ≤ κ a Set.univ * IsFiniteKernel.bound η
|
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsFiniteKernel η
a : α
hκ : IsSFiniteKernel κ
Cη : ℝ≥0∞ := IsFiniteKernel.bound η
⊢ ∫⁻ (b : β), (η (a, b)) Set.univ ∂κ a ≤ (κ a) Set.univ * IsFiniteKernel.bound η
|
calc
∫⁻ b, η (a, b) Set.univ ∂κ a ≤ ∫⁻ _, Cη ∂κ a :=
lintegral_mono fun b => measure_le_bound η (a, b) Set.univ
_ = Cη * κ a Set.univ := MeasureTheory.lintegral_const Cη
_ = κ a Set.univ * Cη := mul_comm _ _
|
no goals
|
26504621268ddbe5
|
isPiSystem_Ixx
|
Mathlib/MeasureTheory/PiSystem.lean
|
theorem isPiSystem_Ixx {Ixx : α → α → Set α} {p : α → α → Prop}
(Hne : ∀ {a b}, (Ixx a b).Nonempty → p a b)
(Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (f : ι → α)
(g : ι' → α) : @IsPiSystem α { S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S }
|
α : Type u_1
ι : Sort u_3
ι' : Sort u_4
inst✝ : LinearOrder α
Ixx : α → α → Set α
p : α → α → Prop
Hne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b
Hi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (a₁ ⊔ a₂) (b₁ ⊓ b₂)
f : ι → α
g : ι' → α
⊢ IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S}
|
simpa only [exists_range_iff] using isPiSystem_Ixx_mem (@Hne) (@Hi) (range f) (range g)
|
no goals
|
e33d9e10c4765e58
|
Finset.pairwiseDisjoint_pair_insert
|
Mathlib/Data/Finset/Powerset.lean
|
lemma pairwiseDisjoint_pair_insert [DecidableEq α] {a : α} (ha : a ∉ s) :
(s.powerset : Set (Finset α)).PairwiseDisjoint fun t ↦ ({t, insert a t} : Set (Finset α))
|
α : Type u_1
s : Finset α
inst✝ : DecidableEq α
a : α
ha : a ∉ s
i : Finset α
hi : i ⊆ s
j : Finset α
hj : j ⊆ s
⊢ ({i, insert a i} ∩ {j, insert a j}).Nonempty → i = j
|
simp only [Set.Nonempty, Set.mem_inter_iff, Set.mem_insert_iff, Set.mem_singleton_iff,
exists_eq_or_imp, exists_eq_left, or_imp, imp_self, true_and]
|
α : Type u_1
s : Finset α
inst✝ : DecidableEq α
a : α
ha : a ∉ s
i : Finset α
hi : i ⊆ s
j : Finset α
hj : j ⊆ s
⊢ (i = insert a j → i = j) ∧ (insert a i = j → i = j) ∧ (insert a i = insert a j → i = j)
|
32b2fa687831b31a
|
preNormEDS_ofNat
|
Mathlib/NumberTheory/EllipticDivisibilitySequence.lean
|
@[simp]
lemma preNormEDS_ofNat (n : ℕ) : preNormEDS b c d n = preNormEDS' b c d n
|
R : Type u
inst✝ : CommRing R
b c d : R
n : ℕ
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n
|
by_cases hn : n = 0
|
case pos
R : Type u
inst✝ : CommRing R
b c d : R
n : ℕ
hn : n = 0
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n
case neg
R : Type u
inst✝ : CommRing R
b c d : R
n : ℕ
hn : ¬n = 0
⊢ preNormEDS b c d ↑n = preNormEDS' b c d n
|
38ff9fe8b90f874c
|
strictConcaveOn_log_Iio
|
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
theorem strictConcaveOn_log_Iio : StrictConcaveOn ℝ (Iio 0) log
|
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
hxy : x ≠ y
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
hx' : 0 < -x
hy' : 0 < -y
⊢ -x ≠ -y
|
contrapose! hxy
|
x : ℝ
hx : x < 0
y : ℝ
hy : y < 0
a b : ℝ
ha : 0 < a
hb : 0 < b
hab : a + b = 1
hx' : 0 < -x
hy' : 0 < -y
hxy : -x = -y
⊢ x = y
|
cfbfdbfd08dcbfbb
|
Ordinal.principal_mul_iff_mul_left_eq
|
Mathlib/SetTheory/Ordinal/Principal.lean
|
theorem principal_mul_iff_mul_left_eq : Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o
|
case refine_2.inr
o : Ordinal.{u}
h : ∀ (a : Ordinal.{u}), 0 < a → a < o → a * o = o
a b : Ordinal.{u}
hao : a < o
hbo : b < o
ha : 0 < a
⊢ (fun x1 x2 => x1 * x2) a b < a * o
|
exact (isNormal_mul_right ha).strictMono hbo
|
no goals
|
5ae376ccba37fa39
|
Array.pairwise_iff_getElem
|
Mathlib/.lake/packages/batteries/Batteries/Data/Array/Pairwise.lean
|
theorem pairwise_iff_getElem {as : Array α} : as.Pairwise R ↔
∀ (i j : Nat) (_ : i < as.size) (_ : j < as.size), i < j → R as[i] as[j]
|
α : Type u_1
R : α → α → Prop
as : Array α
⊢ List.Pairwise R as.toList ↔ ∀ (i j : Nat) (x : i < as.size) (x_1 : j < as.size), i < j → R as[i] as[j]
|
simp [List.pairwise_iff_getElem, length_toList]
|
no goals
|
74fb924004d3641a
|
List.replicate_eq_nil_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem replicate_eq_nil_iff {n : Nat} (a : α) : replicate n a = [] ↔ n = 0
|
α : Type u_1
n : Nat
a : α
⊢ replicate n a = [] ↔ n = 0
|
cases n <;> simp
|
no goals
|
9325182348b1caf8
|
CategoryTheory.toNerve₂.mk_naturality_δ1i
|
Mathlib/AlgebraicTopology/SimplicialSet/NerveAdjunction.lean
|
lemma toNerve₂.mk_naturality_δ1i (i : Fin 3) : toNerve₂.mk.naturalityProperty F (δ₂ i)
|
case h
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
i : Fin 3
x : X.obj (op { obj := [1 + 1], property := ⋯ })
⊢ mk.app F { obj := [1], property := ⋯ } (X.map (δ₂ i ⋯ ⋯).op x) =
(nerveFunctor₂.obj (Cat.of C)).map (δ₂ i ⋯ ⋯).op (mk.app F { obj := [1 + 1], property := ⋯ } x)
|
rw [toNerve₂.mk.app_one]
|
case h
C : Type u
inst✝ : SmallCategory C
X : SSet.Truncated 2
F : oneTruncation₂.obj X ⟶ ReflQuiv.of C
hyp : ∀ (φ : X.obj (op { obj := [2], property := ⋯ })), F.map (ev02₂ φ) = F.map (ev01₂ φ) ≫ F.map (ev12₂ φ)
i : Fin 3
x : X.obj (op { obj := [1 + 1], property := ⋯ })
⊢ ComposableArrows.mk₁ (F.map { edge := X.map (δ₂ i ⋯ ⋯).op x, src_eq := ⋯, tgt_eq := ⋯ }) =
(nerveFunctor₂.obj (Cat.of C)).map (δ₂ i ⋯ ⋯).op (mk.app F { obj := [1 + 1], property := ⋯ } x)
|
4128d2c817244193
|
Equiv.preimage_piEquivPiSubtypeProd_symm_pi
|
Mathlib/Logic/Equiv/Set.lean
|
theorem preimage_piEquivPiSubtypeProd_symm_pi {α : Type*} {β : α → Type*} (p : α → Prop)
[DecidablePred p] (s : ∀ i, Set (β i)) :
(piEquivPiSubtypeProd p β).symm ⁻¹' pi univ s =
(pi univ fun i : { i // p i } => s i) ×ˢ pi univ fun i : { i // ¬p i } => s i
|
α : Type u_1
β : α → Type u_2
p : α → Prop
inst✝ : DecidablePred p
s : (i : α) → Set (β i)
⊢ ⇑(piEquivPiSubtypeProd p β).symm ⁻¹' univ.pi s = (univ.pi fun i => s ↑i) ×ˢ univ.pi fun i => s ↑i
|
ext ⟨f, g⟩
|
case h.mk
α : Type u_1
β : α → Type u_2
p : α → Prop
inst✝ : DecidablePred p
s : (i : α) → Set (β i)
f : (i : { x // p x }) → β ↑i
g : (i : { x // ¬p x }) → β ↑i
⊢ (f, g) ∈ ⇑(piEquivPiSubtypeProd p β).symm ⁻¹' univ.pi s ↔ (f, g) ∈ (univ.pi fun i => s ↑i) ×ˢ univ.pi fun i => s ↑i
|
486cb77064f0cff1
|
LinearMap.rTensor_comp_apply
|
Mathlib/LinearAlgebra/TensorProduct/Basic.lean
|
theorem rTensor_comp_apply (x : N ⊗[R] M) :
(g.comp f).rTensor M x = (g.rTensor M) ((f.rTensor M) x)
|
R : Type u_1
inst✝⁸ : CommSemiring R
M : Type u_5
N : Type u_6
P : Type u_7
Q : Type u_8
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid N
inst✝⁵ : AddCommMonoid P
inst✝⁴ : AddCommMonoid Q
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module R Q
inst✝ : Module R P
g : P →ₗ[R] Q
f : N →ₗ[R] P
x : N ⊗[R] M
⊢ (⇑(rTensor M g) ∘ ⇑(rTensor M f)) x = (rTensor M g) ((rTensor M f) x)
|
rfl
|
no goals
|
03a0cdea438279f9
|
NumberField.mixedEmbedding.minkowskiBound_lt_top
|
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
|
theorem minkowskiBound_lt_top : minkowskiBound K I < ⊤
|
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ 2 ^ finrank ℝ (mixedSpace K) < ⊤
|
exact ENNReal.pow_lt_top (lt_top_iff_ne_top.mpr ENNReal.ofNat_ne_top) _
|
no goals
|
16fce13ef6ba90f5
|
List.head_append
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem head_append {l₁ l₂ : List α} (w : l₁ ++ l₂ ≠ []) :
head (l₁ ++ l₂) w =
if h : l₁.isEmpty then
head l₂ (by simp_all [isEmpty_iff])
else
head l₁ (by simp_all [isEmpty_iff])
|
case isTrue
α : Type u_1
l₁ l₂ : List α
w : l₁ ++ l₂ ≠ []
h✝ : l₁.isEmpty = true
h : l₁ = []
⊢ (l₁ ++ l₂).head w = l₂.head ⋯
|
subst h
|
case isTrue
α : Type u_1
l₂ : List α
w : [] ++ l₂ ≠ []
h : [].isEmpty = true
⊢ ([] ++ l₂).head w = l₂.head ⋯
|
ca0387d7a57b7ced
|
TopologicalSpace.Opens.isBasis_iff_cover
|
Mathlib/Topology/Sets/Opens.lean
|
theorem isBasis_iff_cover {B : Set (Opens α)} :
IsBasis B ↔ ∀ U : Opens α, ∃ Us, Us ⊆ B ∧ U = sSup Us
|
α : Type u_2
inst✝ : TopologicalSpace α
B : Set (Opens α)
⊢ IsBasis B ↔ ∀ (U : Opens α), ∃ Us ⊆ B, U = sSup Us
|
constructor
|
case mp
α : Type u_2
inst✝ : TopologicalSpace α
B : Set (Opens α)
⊢ IsBasis B → ∀ (U : Opens α), ∃ Us ⊆ B, U = sSup Us
case mpr
α : Type u_2
inst✝ : TopologicalSpace α
B : Set (Opens α)
⊢ (∀ (U : Opens α), ∃ Us ⊆ B, U = sSup Us) → IsBasis B
|
f8972d52577a55d8
|
QuasispectrumRestricts.of_subset_range_algebraMap
|
Mathlib/Algebra/Algebra/Quasispectrum.lean
|
theorem of_subset_range_algebraMap (hf : f.LeftInverse (algebraMap R S))
(h : quasispectrum S a ⊆ Set.range (algebraMap R S)) : QuasispectrumRestricts a f where
rightInvOn := fun s hs => by obtain ⟨r, rfl⟩ := h hs; rw [hf r]
left_inv := hf
|
case intro
R : Type u_3
S : Type u_4
A : Type u_5
inst✝⁵ : Semifield R
inst✝⁴ : Field S
inst✝³ : NonUnitalRing A
inst✝² : Module R A
inst✝¹ : Module S A
inst✝ : Algebra R S
a : A
f : S → R
hf : Function.LeftInverse f ⇑(algebraMap R S)
h : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)
r : R
hs : (algebraMap R S) r ∈ quasispectrum S a
⊢ (algebraMap R S) (f ((algebraMap R S) r)) = (algebraMap R S) r
|
rw [hf r]
|
no goals
|
99bdb357cf4096be
|
emultiplicity_ne_zero
|
Mathlib/RingTheory/Multiplicity.lean
|
theorem emultiplicity_ne_zero :
emultiplicity a b ≠ 0 ↔ a ∣ b
|
α : Type u_1
inst✝ : Monoid α
a b : α
⊢ emultiplicity a b ≠ 0 ↔ a ∣ b
|
simp [emultiplicity_eq_zero]
|
no goals
|
6b1f7985cafca7c9
|
BoxIntegral.HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO
|
Mathlib/Analysis/BoxIntegral/Basic.lean
|
theorem HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO (hl : l.bRiemann = false)
(B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : Set ℝⁿ) (hs : s.Countable)
(hlH : s.Nonempty → l.bHenstock = true)
(H₁ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > (0 : ℝ),
∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → x ∈ Box.Icc J →
(l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε)
(H₂ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I \ s, ∀ ε > (0 : ℝ),
∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → (l.bHenstock → x ∈ Box.Icc J) →
(l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε * B J) :
HasIntegral I l f vol (g I)
|
case intro.intro.intro.intro
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : s.Countable
hlH : s.Nonempty → l.bHenstock = true
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I ∩ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₁ c x ε) →
x ∈ Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I \ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₂ c x ε) →
(l.bHenstock = true → x ∈ Box.Icc J) →
(l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ i ∈ t, 2 ^ Fintype.card ι * εs i ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
⊢ ∃ ia,
(∀ (c : ℝ≥0), l.RCond (ia c)) ∧
∀ x ∈ {π | ∃ c, l.MemBaseSet I c (ia c) π ∧ π.IsPartition}, integralSum f vol x ∈ Metric.closedBall (g I) ε
|
set δ : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ) := fun c x => if x ∈ s then δ₁ c x (εs x) else (δ₂ c) x ε'
|
case intro.intro.intro.intro
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : s.Countable
hlH : s.Nonempty → l.bHenstock = true
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I ∩ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₁ c x ε) →
x ∈ Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I \ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₂ c x ε) →
(l.bHenstock = true → x ∈ Box.Icc J) →
(l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ i ∈ t, 2 ^ Fintype.card ι * εs i ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
δ : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => if x ∈ s then δ₁ c x (εs x) else δ₂ c x ε'
⊢ ∃ ia,
(∀ (c : ℝ≥0), l.RCond (ia c)) ∧
∀ x ∈ {π | ∃ c, l.MemBaseSet I c (ia c) π ∧ π.IsPartition}, integralSum f vol x ∈ Metric.closedBall (g I) ε
|
f7fbe41cacd6cc55
|
sdiff_sdiff
|
Mathlib/Order/Heyting/Basic.lean
|
theorem sdiff_sdiff (a b c : α) : (a \ b) \ c = a \ (b ⊔ c) :=
eq_of_forall_ge_iff fun d => by simp_rw [sdiff_le_iff, sup_assoc]
|
α : Type u_2
inst✝ : GeneralizedCoheytingAlgebra α
a b c d : α
⊢ (a \ b) \ c ≤ d ↔ a \ (b ⊔ c) ≤ d
|
simp_rw [sdiff_le_iff, sup_assoc]
|
no goals
|
46b739da6dee4d11
|
LinearMap.quotientInfEquivSupQuotient_symm_apply_eq_zero_iff
|
Mathlib/LinearAlgebra/Isomorphisms.lean
|
theorem quotientInfEquivSupQuotient_symm_apply_eq_zero_iff {p p' : Submodule R M} {x : ↥(p ⊔ p')} :
(quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = 0 ↔ (x : M) ∈ p' :=
(LinearEquiv.symm_apply_eq _).trans <| by simp
|
R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p p' : Submodule R M
x : ↥(p ⊔ p')
⊢ Submodule.Quotient.mk x = (quotientInfEquivSupQuotient p p') 0 ↔ ↑x ∈ p'
|
simp
|
no goals
|
d709be932a6916f6
|
List.count_filterMap
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Count.lean
|
theorem count_filterMap {α} [BEq β] (b : β) (f : α → Option β) (l : List α) :
count b (filterMap f l) = countP (fun a => f a == some b) l
|
β : Type u_1
α : Type u_2
inst✝ : BEq β
b : β
f : α → Option β
l : List α
⊢ count b (filterMap f l) = countP (fun a => f a == some b) l
|
rw [count_eq_countP, countP_filterMap]
|
β : Type u_1
α : Type u_2
inst✝ : BEq β
b : β
f : α → Option β
l : List α
⊢ countP (fun a => (Option.map (fun x => x == b) (f a)).getD false) l = countP (fun a => f a == some b) l
|
2667cf96d053912f
|
Matroid.IsCircuit.isBasis_iff_insert_eq
|
Mathlib/Data/Matroid/Circuit.lean
|
lemma IsCircuit.isBasis_iff_insert_eq (hC : M.IsCircuit C) :
M.IsBasis I C ↔ ∃ e ∈ C \ I, C = insert e I
|
case refine_1
α : Type u_1
M : Matroid α
C I : Set α
hC : M.IsCircuit C
x✝ : ∃ e ∈ C, I = C \ {e}
e : α
he : e ∈ C
hI : I = C \ {e}
⊢ C = insert e I
|
rw [hI, insert_diff_singleton, insert_eq_of_mem he]
|
no goals
|
26bb41495fb6fdb1
|
ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear
|
Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
|
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
{f : D → E} {g : D → F} {N : WithTop ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
(hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : n ≤ N) :
‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
‖B‖ * ∑ i ∈ Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
|
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
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
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : WithTop ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
‖B‖ *
∑ i ∈ Finset.range (n + 1),
↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
|
let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
|
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
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
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : WithTop ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
‖B‖ *
∑ i ∈ Finset.range (n + 1),
↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
|
c6fa542bdff00851
|
WeierstrassCurve.natDegree_coeff_ΨSq_ofNat
|
Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
|
private lemma natDegree_coeff_ΨSq_ofNat (n : ℕ) :
(W.ΨSq n).natDegree ≤ n ^ 2 - 1 ∧ (W.ΨSq n).coeff (n ^ 2 - 1) = (n ^ 2 : ℤ)
|
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n
h : ∀ {n : ℕ},
(W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧
(W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) :=
fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n
n : ℕ
⊢ n + 1 ≠ 0
|
omega
|
no goals
|
96420254b4130cb9
|
PartialHomeomorph.isLocalStructomorphWithinAt_iff
|
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff {G : StructureGroupoid H}
[ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H}
(hx : x ∈ f.source ∪ sᶜ) :
G.IsLocalStructomorphWithinAt (⇑f) s x ↔
x ∈ s → ∃ e : PartialHomeomorph H H,
e ∈ G ∧ e.source ⊆ f.source ∧ EqOn f (⇑e) (s ∩ e.source) ∧ x ∈ e.source
|
case mp.intro.intro.intro.refine_3
H : Type u_1
inst✝¹ : TopologicalSpace H
G : StructureGroupoid H
inst✝ : ClosedUnderRestriction G
f : PartialHomeomorph H H
s : Set H
x : H
hx : x ∈ f.source ∪ sᶜ
hf : G.IsLocalStructomorphWithinAt (↑f) s x
h2x : x ∈ s
e : PartialHomeomorph H H
he : e ∈ G
hfe : EqOn (↑f) (↑e.toPartialEquiv) (s ∩ e.source)
hxe : x ∈ e.source
⊢ x ∈ f.source
|
exact Or.resolve_right hx (not_not.mpr h2x)
|
no goals
|
8a619387a2fde1b4
|
Module.End.exists_hasEigenvalue_of_genEigenspace_eq_top
|
Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
|
theorem exists_hasEigenvalue_of_genEigenspace_eq_top [Nontrivial M] {f : End R M} (k : ℕ∞)
(hf : ⨆ μ, f.genEigenspace μ k = ⊤) :
∃ μ, f.HasEigenvalue μ
|
R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Nontrivial M
f : End R M
k : ℕ∞
hf : ⨆ μ, (f.genEigenspace μ) k = ⊤
⊢ ∃ μ, f.HasEigenvalue μ
|
suffices ∃ μ, f.HasUnifEigenvalue μ k by
peel this with μ hμ
exact HasUnifEigenvalue.lt zero_lt_one hμ
|
R : Type u_3
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Nontrivial M
f : End R M
k : ℕ∞
hf : ⨆ μ, (f.genEigenspace μ) k = ⊤
⊢ ∃ μ, f.HasUnifEigenvalue μ k
|
f778b3a875e36e4d
|
CategoryTheory.ComposableArrows.Precomp.map_comp
|
Mathlib/CategoryTheory/ComposableArrows.lean
|
lemma map_comp {i j k : Fin (n + 1 + 1)} (hij : i ≤ j) (hjk : j ≤ k) :
map F f i k (hij.trans hjk) = map F f i j hij ≫ map F f j k hjk
|
case mk.mk.mk.succ.succ.succ
C : Type u_1
inst✝ : Category.{u_2, u_1} C
n : ℕ
F : ComposableArrows C n
X : C
f : X ⟶ F.left
n✝ : ℕ
hi : n✝ + 1 < n + 1 + 1
j : ℕ
hj : j + 1 < n + 1 + 1
hij : ⟨n✝ + 1, hi⟩ ≤ ⟨j + 1, hj⟩
k : ℕ
hk : k + 1 < n + 1 + 1
hjk : ⟨j + 1, hj⟩ ≤ ⟨k + 1, hk⟩
⊢ F.map (homOfLE ⋯) = F.map (homOfLE ⋯) ≫ F.map (homOfLE ⋯)
|
rw [← F.map_comp, homOfLE_comp]
|
no goals
|
17b6c032c85bf5cd
|
Turing.tr_reaches
|
Mathlib/Computability/PostTuringMachine.lean
|
theorem tr_reaches {σ₁ σ₂ f₁ f₂} {tr : σ₁ → σ₂ → Prop} (H : Respects f₁ f₂ tr) {a₁ a₂}
(aa : tr a₁ a₂) {b₁} (ab : Reaches f₁ a₁ b₁) : ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂
|
case inr
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
a₂ : σ₂
aa : tr a₁ a₂
b₁ : σ₁
ab✝ : Reaches f₁ a₁ b₁
ab : TransGen (fun a b => b ∈ f₁ a) a₁ b₁
⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂
|
have ⟨b₂, bb, h⟩ := tr_reaches₁ H aa ab
|
case inr
σ₁ : Type u_1
σ₂ : Type u_2
f₁ : σ₁ → Option σ₁
f₂ : σ₂ → Option σ₂
tr : σ₁ → σ₂ → Prop
H : Respects f₁ f₂ tr
a₁ : σ₁
a₂ : σ₂
aa : tr a₁ a₂
b₁ : σ₁
ab✝ : Reaches f₁ a₁ b₁
ab : TransGen (fun a b => b ∈ f₁ a) a₁ b₁
b₂ : σ₂
bb : tr b₁ b₂
h : Reaches₁ f₂ a₂ b₂
⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂
|
9a1ddf3f5592df15
|
subset_piiUnionInter
|
Mathlib/MeasureTheory/PiSystem.lean
|
theorem subset_piiUnionInter {π : ι → Set (Set α)} {S : Set ι} {i : ι} (his : i ∈ S) :
π i ⊆ piiUnionInter π S
|
α : Type u_3
ι : Type u_4
π : ι → Set (Set α)
S : Set ι
i : ι
his : i ∈ S
h_ss : {i} ⊆ S
⊢ π i ⊆ π i ∪ {univ}
|
exact subset_union_left
|
no goals
|
c6ca9627dfa3c302
|
Multiset.map_set_pairwise
|
Mathlib/Data/Multiset/UnionInter.lean
|
theorem map_set_pairwise {f : α → β} {r : β → β → Prop} {m : Multiset α}
(h : { a | a ∈ m }.Pairwise fun a₁ a₂ => r (f a₁) (f a₂)) : { b | b ∈ m.map f }.Pairwise r :=
fun b₁ h₁ b₂ h₂ hn => by
obtain ⟨⟨a₁, H₁, rfl⟩, a₂, H₂, rfl⟩ := Multiset.mem_map.1 h₁, Multiset.mem_map.1 h₂
exact h H₁ H₂ (mt (congr_arg f) hn)
|
α : Type u_1
β : Type v
f : α → β
r : β → β → Prop
m : Multiset α
h : {a | a ∈ m}.Pairwise fun a₁ a₂ => r (f a₁) (f a₂)
b₁ : β
h₁ : b₁ ∈ {b | b ∈ map f m}
b₂ : β
h₂ : b₂ ∈ {b | b ∈ map f m}
hn : b₁ ≠ b₂
⊢ r b₁ b₂
|
obtain ⟨⟨a₁, H₁, rfl⟩, a₂, H₂, rfl⟩ := Multiset.mem_map.1 h₁, Multiset.mem_map.1 h₂
|
case intro.intro.intro.intro
α : Type u_1
β : Type v
f : α → β
r : β → β → Prop
m : Multiset α
h : {a | a ∈ m}.Pairwise fun a₁ a₂ => r (f a₁) (f a₂)
a₁ : α
H₁ : a₁ ∈ m
h₁ : f a₁ ∈ {b | b ∈ map f m}
a₂ : α
H₂ : a₂ ∈ m
h₂ : f a₂ ∈ {b | b ∈ map f m}
hn : f a₁ ≠ f a₂
⊢ r (f a₁) (f a₂)
|
e200b04d19b2c153
|
Set.image_subtype_val_Ixx_Ixi
|
Mathlib/Order/Interval/Set/Image.lean
|
private lemma image_subtype_val_Ixx_Ixi {p q r : α → α → Prop} {a b : α} (c : {x // p a x ∧ q x b})
(h : ∀ {x}, r c x → p a x) :
Subtype.val '' {y : {x // p a x ∧ q x b} | r c.1 y.1} = {y : α | r c.1 y ∧ q y b} :=
(Subtype.image_preimage_val {x | p a x ∧ q x b} {y | r c.1 y}).trans <| by
ext; simp +contextual [@and_comm (r _ _), h]
|
case h
α : Type u_1
p q r : α → α → Prop
a b : α
c : { x // p a x ∧ q x b }
h : ∀ {x : α}, r (↑c) x → p a x
x✝ : α
⊢ x✝ ∈ {x | p a x ∧ q x b} ∩ {y | r (↑c) y} ↔ x✝ ∈ {y | r (↑c) y ∧ q y b}
|
simp +contextual [@and_comm (r _ _), h]
|
no goals
|
84f3fa0604c44797
|
Matrix.det_diagonal
|
Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
|
theorem det_diagonal {d : n → R} : det (diagonal d) = ∏ i, d i
|
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
d : n → R
⊢ (diagonal d).det = ∏ i : n, d i
|
rw [det_apply']
|
n : Type u_2
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
d : n → R
⊢ ∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, diagonal d (σ i) i = ∏ i : n, d i
|
587194f644b4bcab
|
CategoryTheory.shiftFunctorCompIsoId_add'_inv_app
|
Mathlib/CategoryTheory/Shift/Basic.lean
|
lemma shiftFunctorCompIsoId_add'_inv_app :
(shiftFunctorCompIsoId C p' p hp).inv.app X =
(shiftFunctorCompIsoId C n' n hn).inv.app X ≫
(shiftFunctorCompIsoId C m' m hm).inv.app (X⟦n'⟧)⟦n⟧' ≫
(shiftFunctorAdd' C m n p h).inv.app (X⟦n'⟧⟦m'⟧) ≫
((shiftFunctorAdd' C n' m' p'
(by rw [← add_left_inj p, hp, ← h, add_assoc,
← add_assoc m', hm, zero_add, hn])).inv.app X)⟦p⟧'
|
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddGroup A
inst✝ : HasShift C A
X : C
m n p m' n' p' : A
hm : m' + m = 0
hn : n' + n = 0
hp : p' + p = 0
h : m + n = p
⊢ n' + m' + m = n'
|
rw [add_assoc, hm, add_zero]
|
no goals
|
1740b8167511ce6d
|
ContinuousMultilinearMap.norm_mkPiAlgebraFin
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
theorem norm_mkPiAlgebraFin [NormOneClass A] :
‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1
|
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
n : ℕ
A : Type u_1
inst✝² : SeminormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : NormOneClass A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1
|
cases n
|
case zero
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_1
inst✝² : SeminormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : NormOneClass A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = 1
case succ
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
A : Type u_1
inst✝² : SeminormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : NormOneClass A
n✝ : ℕ
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 (n✝ + 1) A‖ = 1
|
73fe76999a224379
|
contMDiffAt_iff_contMDiffAt_nhds
|
Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
|
theorem contMDiffAt_iff_contMDiffAt_nhds
[IsManifold I n M] [IsManifold I' n M'] (hn : n ≠ ∞) :
ContMDiffAt I I' n f x ↔ ∀ᶠ x' in 𝓝 x, ContMDiffAt I I' n f x'
|
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
x : M
n : WithTop ℕ∞
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
⊢ ContMDiffAt I I' n f x → ∀ᶠ (x' : M) in 𝓝 x, ContMDiffAt I I' n f x'
|
rw [contMDiffAt_iff_contMDiffOn_nhds hn]
|
𝕜 : Type u_1
inst✝¹² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁹ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁸ : TopologicalSpace M
inst✝⁷ : ChartedSpace H M
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁴ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝³ : TopologicalSpace M'
inst✝² : ChartedSpace H' M'
f : M → M'
x : M
n : WithTop ℕ∞
inst✝¹ : IsManifold I n M
inst✝ : IsManifold I' n M'
hn : n ≠ ∞
⊢ (∃ u ∈ 𝓝 x, ContMDiffOn I I' n f u) → ∀ᶠ (x' : M) in 𝓝 x, ContMDiffAt I I' n f x'
|
0a10d293b6a59353
|
HasFPowerSeriesWithinOnBall.iteratedFDerivWithin_eq_sum_of_completeSpace
|
Mathlib/Analysis/Analytic/IteratedFDeriv.lean
|
theorem HasFPowerSeriesWithinOnBall.iteratedFDerivWithin_eq_sum_of_completeSpace [CompleteSpace F]
(h : HasFPowerSeriesWithinOnBall f p s x r)
(hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (v : Fin n → E) :
iteratedFDerivWithin 𝕜 n f s x v = ∑ σ : Perm (Fin n), p n (fun i ↦ v (σ i))
|
case h's
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f p s x r
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
this : iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f (s ∩ EMetric.ball x r) x
⊢ s ∩ EMetric.ball x r ⊆ EMetric.ball x r
|
exact inter_subset_right
|
no goals
|
70511175dfe1fa7d
|
LieAlgebra.engel_isBot_of_isMin
|
Mathlib/Algebra/Lie/CartanExists.lean
|
/-- Let `L` be a Lie algebra of dimension `n` over a field `K` with at least `n` elements.
Given a Lie subalgebra `U` of `L`, and an element `x ∈ U` such that `U ≤ engel K x`.
Suppose that `engel K x` is minimal amongst the Engel subalgebras `engel K y` for `y ∈ U`.
Then `engel K x ≤ engel K y` for all `y ∈ U`.
Lemma 2 in [barnes1967]. -/
lemma engel_isBot_of_isMin (hLK : finrank K L ≤ #K) (U : LieSubalgebra K L)
(E : {engel K x | x ∈ U}) (hUle : U ≤ E) (hmin : IsMin E) :
IsBot E
|
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
let __src := engel K x;
{ toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K (↥E) x' u
ψ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K Q x' u
i : ℕ
hi : i < r
hi0 : i ≠ 0
hψ : constantCoeff ψ ≠ 0
s : Finset K
hs : r ≤ s.card
hsψ : ∀ α ∈ s, eval α (constantCoeff ψ) ≠ 0
α : K
hα : α ∈ s
⊢ ∀ (m : ↥E), ∃ n, ((toEnd K ↥U ↥E) (α • u + x') ^ n) m = 0
|
let v := α • u + x'
|
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : LieRing L
inst✝¹ : LieAlgebra K L
inst✝ : Module.Finite K L
hLK : ↑(finrank K L) ≤ #K
U : LieSubalgebra K L
x : L
hxU : x ∈ U
y : L
hyU : y ∈ U
Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩
Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩
hUle : U ≤ ↑Ex
hmin : ∀ E ≤ Ex, Ex ≤ E
E : LieSubmodule K (↥U) L :=
let __src := engel K x;
{ toSubmodule := __src.toSubmodule, lie_mem := ⋯ }
hx₀ : x ≠ 0
Q : Type u_2 := L ⧸ E
r : ℕ := finrank K ↥E
hr : r < finrank K L
x' : ↥U := ⟨x, hxU⟩
y' : ↥U := ⟨y, hyU⟩
u : ↥U := y' - x'
χ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K (↥E) x' u
ψ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K Q x' u
i : ℕ
hi : i < r
hi0 : i ≠ 0
hψ : constantCoeff ψ ≠ 0
s : Finset K
hs : r ≤ s.card
hsψ : ∀ α ∈ s, eval α (constantCoeff ψ) ≠ 0
α : K
hα : α ∈ s
v : ↥U := α • u + x'
⊢ ∀ (m : ↥E), ∃ n, ((toEnd K ↥U ↥E) (α • u + x') ^ n) m = 0
|
53ed38f54130aaf6
|
jacobiSum_mul_jacobiSum_inv
|
Mathlib/NumberTheory/JacobiSum/Basic.lean
|
/-- If `χ` and `φ` are multiplicative characters on a finite field `F` with values in another
field `F'` such that `χ`, `φ` and `χφ` are all nontrivial and `char F' ≠ char F`, then
`J(χ,φ) * J(χ⁻¹,φ⁻¹) = #F` (in `F'`). -/
lemma jacobiSum_mul_jacobiSum_inv (h : ringChar F' ≠ ringChar F) {χ φ : MulChar F F'} (hχ : χ ≠ 1)
(hφ : φ ≠ 1) (hχφ : χ * φ ≠ 1) :
jacobiSum χ φ * jacobiSum χ⁻¹ φ⁻¹ = Fintype.card F
|
case intro.intro.a
F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ringChar F' ≠ ringChar F
χ φ : MulChar F F'
hχ : χ ≠ 1
hφ : φ ≠ 1
hχφ : χ * φ ≠ 1
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h
FF' : Type u_2 := CyclotomicField ψ.n F'
χ' : MulChar F FF' := χ.ringHomComp (algebraMap F' FF')
φ' : MulChar F FF' := φ.ringHomComp (algebraMap F' FF')
hinj : Function.Injective ⇑(algebraMap F' FF')
Hχφ : χ' * φ' ≠ 1
Hχφ' : χ'⁻¹ * φ'⁻¹ ≠ 1
Hχ : χ' ≠ 1
Hφ : φ' ≠ 1
Hcard : ↑(Fintype.card F) ≠ 0
⊢ jacobiSum (χ.ringHomComp (algebraMap F' FF')) (φ.ringHomComp (algebraMap F' FF')) *
jacobiSum (χ⁻¹.ringHomComp (algebraMap F' FF')) (φ⁻¹.ringHomComp (algebraMap F' FF')) =
(algebraMap F' FF') ↑(Fintype.card F)
|
have H := (gaussSum_mul_gaussSum_eq_card Hχφ ψ.prim).trans_ne Hcard
|
case intro.intro.a
F : Type u_1
F' : Type u_2
inst✝² : Fintype F
inst✝¹ : Field F
inst✝ : Field F'
h : ringChar F' ≠ ringChar F
χ φ : MulChar F F'
hχ : χ ≠ 1
hφ : φ ≠ 1
hχφ : χ * φ ≠ 1
n : ℕ+
hp : Nat.Prime (ringChar F)
hc : Fintype.card F = ringChar F ^ ↑n
ψ : PrimitiveAddChar F F' := FiniteField.primitiveChar F F' h
FF' : Type u_2 := CyclotomicField ψ.n F'
χ' : MulChar F FF' := χ.ringHomComp (algebraMap F' FF')
φ' : MulChar F FF' := φ.ringHomComp (algebraMap F' FF')
hinj : Function.Injective ⇑(algebraMap F' FF')
Hχφ : χ' * φ' ≠ 1
Hχφ' : χ'⁻¹ * φ'⁻¹ ≠ 1
Hχ : χ' ≠ 1
Hφ : φ' ≠ 1
Hcard : ↑(Fintype.card F) ≠ 0
H : gaussSum (χ' * φ') ψ.char * gaussSum (χ' * φ')⁻¹ ψ.char⁻¹ ≠ 0
⊢ jacobiSum (χ.ringHomComp (algebraMap F' FF')) (φ.ringHomComp (algebraMap F' FF')) *
jacobiSum (χ⁻¹.ringHomComp (algebraMap F' FF')) (φ⁻¹.ringHomComp (algebraMap F' FF')) =
(algebraMap F' FF') ↑(Fintype.card F)
|
5b48f17367fbeedf
|
rpow_one_add_lt_one_add_mul_self
|
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
theorem rpow_one_add_lt_one_add_mul_self {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠ 0) {p : ℝ} (hp1 : 0 < p)
(hp2 : p < 1) : (1 + s) ^ p < 1 + p * s
|
case inr.a.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) * p < log (1 + p * s)
|
rw [← lt_div_iff₀ hp1, ← div_lt_div_right_of_neg hs']
|
case inr.a.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + p * s) / p / s < log (1 + s) / s
|
424bf16041482ff7
|
ExistsContDiffBumpBase.y_pos_of_mem_ball
|
Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
|
theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
(hx : x ∈ ball (0 : E) (1 + D)) : 0 < y D x
|
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
z : E := (D / (1 + D)) • x
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
y : E
hy : y ∈ ball z (D * (1 + D - ‖x‖) / (1 + D))
⊢ D * (1 + D - ‖x‖) / (1 + D) + D / (1 + D) * ‖x‖ ≤ D
|
simp only [div_le_iff₀ B, field_simps]
|
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
z : E := (D / (1 + D)) • x
hz : z = (D / (1 + D)) • x
B : 0 < 1 + D
y : E
hy : y ∈ ball z (D * (1 + D - ‖x‖) / (1 + D))
⊢ D * (1 + D - ‖x‖) + D * ‖x‖ ≤ D * (1 + D)
|
646f97b7cc0fbf51
|
FintypeCat.isSkeleton
|
Mathlib/CategoryTheory/FintypeCat.lean
|
/-- `Fintype.Skeleton` is a skeleton of `Fintype`. -/
lemma isSkeleton : IsSkeletonOf FintypeCat Skeleton Skeleton.incl where
skel := Skeleton.is_skeletal
eqv
|
⊢ Skeleton.incl.IsEquivalence
|
infer_instance
|
no goals
|
90fe647c92f21534
|
MeasureTheory.Measure.absolutelyContinuous_compProd_of_compProd
|
Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean
|
lemma absolutelyContinuous_compProd_of_compProd [SigmaFinite μ] [SigmaFinite ν]
(hκη : μ ⊗ₘ κ ≪ ν ⊗ₘ η) :
μ ⊗ₘ κ ≪ μ ⊗ₘ η
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hκη : μ ⊗ₘ κ ≪ μ.withDensity (ν.rnDeriv μ) ⊗ₘ η + ν.singularPart μ ⊗ₘ η
h : μ ⊗ₘ κ ≪ μ.withDensity (ν.rnDeriv μ) ⊗ₘ η
⊢ μ ⊗ₘ κ ≪ μ ⊗ₘ η
|
refine h.trans (AbsolutelyContinuous.compProd_left ?_ _)
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hκη : μ ⊗ₘ κ ≪ μ.withDensity (ν.rnDeriv μ) ⊗ₘ η + ν.singularPart μ ⊗ₘ η
h : μ ⊗ₘ κ ≪ μ.withDensity (ν.rnDeriv μ) ⊗ₘ η
⊢ μ.withDensity (ν.rnDeriv μ) ≪ μ
|
932f493f7cae303f
|
Set.preimage_const_mul_Ioi_or_Iio
|
Mathlib/Algebra/Order/Group/Pointwise/Interval.lean
|
lemma preimage_const_mul_Ioi_or_Iio (hb : a ≠ 0) {U V : Set α}
(hU : U ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a}) (hV : V = HMul.hMul a ⁻¹' U) :
V ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a}
|
case h.inl.h
α : Type u_1
inst✝ : LinearOrderedField α
a : α
hb✝ : a ≠ 0
U V : Set α
hV : V = HMul.hMul a ⁻¹' U
aU : α
haU : U = Ioi aU
hb : a < 0
⊢ HMul.hMul a ⁻¹' Ioi aU = Iio (a⁻¹ * aU)
|
rw [Set.preimage_const_mul_Ioi_of_neg _ hb, div_eq_inv_mul]
|
no goals
|
1f2cbf705c2a3cf3
|
List.takeWhile_filterMap
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean
|
theorem takeWhile_filterMap (f : α → Option β) (p : β → Bool) (l : List α) :
(l.filterMap f).takeWhile p = (l.takeWhile fun a => (f a).all p).filterMap f
|
case cons.h_2
α : Type u_1
β : Type u_2
f : α → Option β
p : β → Bool
x : α
xs : List α
ih : takeWhile p (filterMap f xs) = filterMap f (takeWhile (fun a => Option.all p (f a)) xs)
x✝ : Option β
b✝ : β
h : f x = some b✝
⊢ takeWhile p (b✝ :: filterMap f xs) = filterMap f (takeWhile (fun a => Option.all p (f a)) (x :: xs))
|
simp [takeWhile_cons, h, ih]
|
case cons.h_2
α : Type u_1
β : Type u_2
f : α → Option β
p : β → Bool
x : α
xs : List α
ih : takeWhile p (filterMap f xs) = filterMap f (takeWhile (fun a => Option.all p (f a)) xs)
x✝ : Option β
b✝ : β
h : f x = some b✝
⊢ (if p b✝ = true then b✝ :: filterMap f (takeWhile (fun a => Option.all p (f a)) xs) else []) =
filterMap f (if p b✝ = true then x :: takeWhile (fun a => Option.all p (f a)) xs else [])
|
d35ebb1fda16ec75
|
SeminormFamily.filter_eq_iInf
|
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
theorem filter_eq_iInf (p : SeminormFamily 𝕜 E ι) :
p.moduleFilterBasis.toFilterBasis.filter = ⨅ i, (𝓝 0).comap (p i)
|
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Nonempty ι
p : SeminormFamily 𝕜 E ι
⊢ AddGroupFilterBasis.toFilterBasis.filter = ⨅ i, comap (⇑(p i)) (𝓝 0)
|
refine le_antisymm (le_iInf fun i => ?_) ?_
|
case refine_1
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Nonempty ι
p : SeminormFamily 𝕜 E ι
i : ι
⊢ AddGroupFilterBasis.toFilterBasis.filter ≤ comap (⇑(p i)) (𝓝 0)
case refine_2
𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Nonempty ι
p : SeminormFamily 𝕜 E ι
⊢ ⨅ i, comap (⇑(p i)) (𝓝 0) ≤ AddGroupFilterBasis.toFilterBasis.filter
|
043221e62cb70beb
|
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