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
|
|---|---|---|---|---|---|---|
BitVec.toInt_signExtend_of_lt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem toInt_signExtend_of_lt {x : BitVec w} (hv : w < v):
(x.signExtend v).toInt = x.toInt
|
w v : Nat
x : BitVec w
hv : w < v
⊢ (signExtend v x)[v - 1] = x.msb
|
simp [getElem_signExtend, Nat.le_sub_one_of_lt hv]
|
no goals
|
8d5f3d5abff36d51
|
Finset.toRight_cons_inr
|
Mathlib/Data/Finset/Sum.lean
|
@[simp] lemma toRight_cons_inr (hb) :
(cons (inr b) u hb).toRight = cons b u.toRight (by simpa)
|
α : Type u_1
β : Type u_2
b : β
u : Finset (α ⊕ β)
hb : inr b ∉ u
⊢ (cons (inr b) u hb).toRight = cons b u.toRight ⋯
|
ext y
|
case h
α : Type u_1
β : Type u_2
b : β
u : Finset (α ⊕ β)
hb : inr b ∉ u
y : β
⊢ y ∈ (cons (inr b) u hb).toRight ↔ y ∈ cons b u.toRight ⋯
|
8c6d972aff314ec7
|
FormalMultilinearSeries.ofScalars_radius_eq_top_of_tendsto
|
Mathlib/Analysis/Analytic/OfScalars.lean
|
theorem ofScalars_radius_eq_top_of_tendsto (hc : ∀ᶠ n in atTop, c n ≠ 0)
(hc' : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)) : (ofScalars E c).radius = ⊤
|
case neg.refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ Tendsto (fun n => ‖‖‖c (n + 1)‖ * ↑r' ^ (n + 1)‖‖ / ‖‖‖c n‖ * ↑r' ^ n‖‖) atTop (𝓝 0)
|
simp only [norm_norm]
|
case neg.refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedRing E
inst✝ : NormedAlgebra 𝕜 E
c : ℕ → 𝕜
hc : ∀ᶠ (n : ℕ) in atTop, c n ≠ 0
hc' : Tendsto (fun n => ‖c n.succ‖ / ‖c n‖) atTop (𝓝 0)
r' : ℝ≥0
hrz : ¬r' = 0
⊢ Tendsto (fun n => ‖‖c (n + 1)‖ * ↑r' ^ (n + 1)‖ / ‖‖c n‖ * ↑r' ^ n‖) atTop (𝓝 0)
|
fed995db04eea233
|
FintypeCat.uSwitchEquiv_symm_naturality
|
Mathlib/CategoryTheory/FintypeCat.lean
|
lemma uSwitchEquiv_symm_naturality {X Y : FintypeCat.{u}} (f : X ⟶ Y) (x : X) :
uSwitch.map f (X.uSwitchEquiv.symm x) = Y.uSwitchEquiv.symm (f x)
|
X Y : FintypeCat
f : X ⟶ Y
x : X.carrier
⊢ uSwitch.map f (X.uSwitchEquiv.symm x) = Y.uSwitchEquiv.symm (f x)
|
rw [← Equiv.apply_eq_iff_eq_symm_apply, ← uSwitchEquiv_naturality f,
Equiv.apply_symm_apply]
|
no goals
|
96883c09ded383e5
|
lp.norm_eq_zero_iff
|
Mathlib/Analysis/Normed/Lp/lpSpace.lean
|
theorem norm_eq_zero_iff {f : lp E p} : ‖f‖ = 0 ↔ f = 0
|
case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
h : ‖f‖ = 0
hp : 0 < p.toReal
hf : HasSum (fun i => ‖↑f i‖ ^ p.toReal) 0
⊢ f = 0
|
have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i => Real.rpow_nonneg (norm_nonneg _) _
|
case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : ↥(lp E p)
h : ‖f‖ = 0
hp : 0 < p.toReal
hf : HasSum (fun i => ‖↑f i‖ ^ p.toReal) 0
this : ∀ (i : α), 0 ≤ ‖↑f i‖ ^ p.toReal
⊢ f = 0
|
b5b6304c5b4b6762
|
RootPairing.infinite_of_linInd_coxeterWeight_four
|
Mathlib/LinearAlgebra/RootSystem/Reduced.lean
|
lemma infinite_of_linInd_coxeterWeight_four [NeZero (2 : R)] [NoZeroSMulDivisors ℤ M]
(hl : LinearIndependent R ![P.root i, P.root j]) (hc : P.coxeterWeight i j = 4) :
Infinite ι
|
case refine_3
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
i j : ι
inst✝¹ : NeZero 2
inst✝ : NoZeroSMulDivisors ℤ M
hc : P.coxeterWeight i j = 4
hl : ¬P.pairing j i • P.root i + -(2 • P.root j) = 0
⊢ (P.toLin.flip (P.coroot i)) (P.root j) • P.root i ≠ 2 • P.root j
|
rw [ne_eq, coroot_root_eq_pairing, ← sub_eq_zero, sub_eq_add_neg]
|
case refine_3
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
P : RootPairing ι R M N
i j : ι
inst✝¹ : NeZero 2
inst✝ : NoZeroSMulDivisors ℤ M
hc : P.coxeterWeight i j = 4
hl : ¬P.pairing j i • P.root i + -(2 • P.root j) = 0
⊢ ¬P.pairing j i • P.root i + -(2 • P.root j) = 0
|
6d8a6318d73cc39f
|
CategoryTheory.ShortComplex.HomologyMapData.comm
|
Mathlib/Algebra/Homology/ShortComplex/Homology.lean
|
@[reassoc]
lemma comm (h : HomologyMapData φ h₁ h₂) :
h.left.φH ≫ h₂.iso.hom = h₁.iso.hom ≫ h.right.φH
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
h : HomologyMapData φ h₁ h₂
⊢ h.left.φH ≫ h₂.iso.hom = h₁.iso.hom ≫ h.right.φH
|
simp only [← cancel_epi h₁.left.π, ← cancel_mono h₂.right.ι, assoc,
LeftHomologyMapData.commπ_assoc, HomologyData.comm, LeftHomologyMapData.commi_assoc,
RightHomologyMapData.commι, HomologyData.comm_assoc, RightHomologyMapData.commp]
|
no goals
|
63ba4f728922b48b
|
EuclideanDomain.mod_eq_zero
|
Mathlib/Algebra/EuclideanDomain/Basic.lean
|
theorem mod_eq_zero {a b : R} : a % b = 0 ↔ b ∣ a :=
⟨fun h => by
rw [← div_add_mod a b, h, add_zero]
exact dvd_mul_right _ _, fun ⟨c, e⟩ => by
rw [e, ← add_left_cancel_iff, div_add_mod, add_zero]
haveI := Classical.dec
by_cases b0 : b = 0
· simp only [b0, zero_mul]
· rw [mul_div_cancel_left₀ _ b0]⟩
|
case neg
R : Type u
inst✝ : EuclideanDomain R
a b : R
x✝ : b ∣ a
c : R
e : a = b * c
this : (p : Prop) → Decidable p
b0 : ¬b = 0
⊢ b * c = b * (b * c / b)
|
rw [mul_div_cancel_left₀ _ b0]
|
no goals
|
b42d0fe6e6a8faaf
|
GenContFract.contsAux_eq_contsAux_squashGCF_of_le
|
Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
|
theorem contsAux_eq_contsAux_squashGCF_of_le {m : ℕ} :
m ≤ n → contsAux g m = (squashGCF g n).contsAux m :=
Nat.strong_induction_on m
(by
clear m
intro m IH m_le_n
rcases m with - | m'
· rfl
· rcases n with - | n'
· exact (m'.not_succ_le_zero m_le_n).elim
-- 1 ≰ 0
· rcases m' with - | m''
· rfl
· -- get some inequalities to instantiate the IH for m'' and m'' + 1
have m'_lt_n : m'' + 1 < n' + 1 := m_le_n
have succ_m''th_contsAux_eq := IH (m'' + 1) (lt_add_one (m'' + 1)) m'_lt_n.le
have : m'' < m'' + 2 := lt_add_of_pos_right m'' zero_lt_two
have m''th_contsAux_eq := IH m'' this (le_trans this.le m_le_n)
have : (squashGCF g (n' + 1)).s.get? m'' = g.s.get? m'' :=
squashGCF_nth_of_lt (Nat.succ_lt_succ_iff.mp m'_lt_n)
simp [contsAux, succ_m''th_contsAux_eq, m''th_contsAux_eq, this])
|
case succ.succ.succ
K : Type u_1
g : GenContFract K
inst✝ : DivisionRing K
n' m'' : ℕ
IH : ∀ m < m'' + 1 + 1, m ≤ n' + 1 → g.contsAux m = (g.squashGCF (n' + 1)).contsAux m
m_le_n : m'' + 1 + 1 ≤ n' + 1
⊢ g.contsAux (m'' + 1 + 1) = (g.squashGCF (n' + 1)).contsAux (m'' + 1 + 1)
|
have m'_lt_n : m'' + 1 < n' + 1 := m_le_n
|
case succ.succ.succ
K : Type u_1
g : GenContFract K
inst✝ : DivisionRing K
n' m'' : ℕ
IH : ∀ m < m'' + 1 + 1, m ≤ n' + 1 → g.contsAux m = (g.squashGCF (n' + 1)).contsAux m
m_le_n : m'' + 1 + 1 ≤ n' + 1
m'_lt_n : m'' + 1 < n' + 1
⊢ g.contsAux (m'' + 1 + 1) = (g.squashGCF (n' + 1)).contsAux (m'' + 1 + 1)
|
cf127cafa12b15bc
|
SzemerediRegularity.edgeDensity_chunk_uniform
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
|
theorem edgeDensity_chunk_uniform [Nonempty α] (hPα : #P.parts * 16 ^ #P.parts ≤ card α)
(hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5) (hU : U ∈ P.parts) (hV : V ∈ P.parts) :
(G.edgeDensity U V : ℝ) ^ 2 - ε ^ 5 / ↑25 ≤
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts,
(G.edgeDensity ab.1 ab.2 : ℝ) ^ 2) / ↑16 ^ #P.parts
|
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
⊢ ↑(G.edgeDensity U V) ^ 2 - ε ^ 5 / 25 ≤
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2) ^ 2) / 16 ^ #P.parts
|
apply (edgeDensity_chunk_aux (hP := hP) hPα hPε hU hV).trans
|
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hU : U ∈ P.parts
hV : V ∈ P.parts
⊢ ((∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) / 16 ^ #P.parts) ^ 2 ≤
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2) ^ 2) / 16 ^ #P.parts
|
3385f6b5b19d88e6
|
Basis.le_span
|
Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
|
theorem Basis.le_span {J : Set M} (v : Basis ι R M) (hJ : span R J = ⊤) : #(range v) ≤ #J
|
R : Type u
M : Type v
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type w
inst✝ : RankCondition R
J : Set M
v : Basis ι R M
hJ : span R J = ⊤
this✝ : Nontrivial R
val✝ : Infinite ↑J
S : ↑J → Set ι := fun j => ↑(v.repr ↑j).support
S' : ↑J → Set M := fun j => ⇑v '' S j
hs : range ⇑v ⊆ ⋃ j, S' j
IJ : #↑J < #↑(range ⇑v)
this : #↑(⋃ j, S' j) < #↑(range ⇑v)
⊢ False
|
exact not_le_of_lt this ⟨Set.embeddingOfSubset _ _ hs⟩
|
no goals
|
1cc5b7e43db3a8f1
|
Array.foldl_append
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem foldl_append {β : Type _} (f : β → α → β) (b) (l l' : Array α) :
(l ++ l').foldl f b = l'.foldl f (l.foldl f b)
|
α : Type u_1
β : Type u_2
f : β → α → β
b : β
l l' : Array α
⊢ foldl f b (l ++ l') = foldl f (foldl f b l) l'
|
simp [foldl_eq_foldlM]
|
no goals
|
865e5e8a7f53e7b5
|
Basis.SmithNormalForm.toAddSubgroup_index_eq_pow_mul_prod
|
Mathlib/LinearAlgebra/FreeModule/Int.lean
|
/-- Given a submodule `N` in Smith normal form of a free `R`-module, its index as an additive
subgroup is an appropriate power of the cardinality of `R` multiplied by the product of the
indexes of the ideals generated by each basis vector. -/
lemma toAddSubgroup_index_eq_pow_mul_prod [Module R M] {N : Submodule R M}
(snf : Basis.SmithNormalForm N ι n) :
N.toAddSubgroup.index = Nat.card R ^ (Fintype.card ι - n) *
∏ i : Fin n, (Ideal.span {snf.a i}).toAddSubgroup.index
|
case convert_3
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN' : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
g : ι → R
h✝ : ∀ (i : ι), (if h : ∃ j, f j = i then a h.choose else 0) ∣ g i
i : ι
hj : ∃ j, f j = i
j : Fin n
h : j ≠ hj.choose
hinj : f j ≠ f hj.choose
⊢ (if i = f j then ⋯.choose • a j else 0) = 0
|
rw [hj.choose_spec] at hinj
|
case convert_3
ι : Type u_1
R : Type u_2
M : Type u_3
n : ℕ
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Fintype ι
inst✝ : Module R M
N : Submodule R M
bM : Basis ι R M
bN : Basis (Fin n) R ↥N
f : Fin n ↪ ι
a : Fin n → R
snf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)
N' : Submodule R (ι → R) := Submodule.map bM.equivFun N
hN' : N' = Submodule.map bM.equivFun N
bN' : Basis (Fin n) R ↥N' := bN.map (bM.equivFun.submoduleMap N)
snf' : ∀ (i : Fin n), ↑(bN' i) = Pi.single (f i) (a i)
hNN' : N.toAddSubgroup.index = N'.toAddSubgroup.index
g : ι → R
h✝ : ∀ (i : ι), (if h : ∃ j, f j = i then a h.choose else 0) ∣ g i
i : ι
hj : ∃ j, f j = i
j : Fin n
h : j ≠ hj.choose
hinj : f j ≠ i
⊢ (if i = f j then ⋯.choose • a j else 0) = 0
|
3d7a9cb7455a5c0f
|
List.findIdx?_go_eq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem findIdx?_go_eq {p : α → Bool} {xs : List α} {i : Nat} :
findIdx?.go p xs (i+1) = (findIdx?.go p xs 0).map fun k => k + (i + 1)
|
case cons.isFalse
α : Type u_1
p : α → Bool
head✝ : α
tail✝ : List α
i : Nat
tail_ih✝ :
∀ {i : Nat},
Option.map (fun i => i + 1) (findIdx?.go p tail✝ i) = Option.map (fun k => k + (i + 1)) (findIdx?.go p tail✝ 0)
h✝ : p head✝ = false
⊢ Option.map ((fun i => i + 1) ∘ fun k => k + (i + 1)) (findIdx?.go p tail✝ 0) =
Option.map ((fun k => k + (i + 1)) ∘ fun k => k + 1) (findIdx?.go p tail✝ 0)
|
congr
|
case cons.isFalse.e_f
α : Type u_1
p : α → Bool
head✝ : α
tail✝ : List α
i : Nat
tail_ih✝ :
∀ {i : Nat},
Option.map (fun i => i + 1) (findIdx?.go p tail✝ i) = Option.map (fun k => k + (i + 1)) (findIdx?.go p tail✝ 0)
h✝ : p head✝ = false
⊢ ((fun i => i + 1) ∘ fun k => k + (i + 1)) = (fun k => k + (i + 1)) ∘ fun k => k + 1
|
00b11065e9d67637
|
FermatLastTheoremForThreeGen.a_cube_b_cube_congr_one_or_neg_one
|
Mathlib/NumberTheory/FLT/Three.lean
|
/-- Given `S' : Solution'`, then `S'.a` and `S'.b` are both congruent to `1` modulo `λ ^ 4` or are
both congruent to `-1`. -/
lemma a_cube_b_cube_congr_one_or_neg_one :
λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1
|
case intro.inl.intro.inl.intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S' : Solution' hζ
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
z : 𝓞 K
hz : S'.c = λ * z
x : 𝓞 K
hx : S'.a ^ 3 - 1 = λ ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 - 1 = λ ^ 4 * y
⊢ False
|
replace hζ : IsPrimitiveRoot ζ ((3 : ℕ+) ^ 1) := by rwa [pow_one]
|
case intro.inl.intro.inl.intro
K : Type u_1
inst✝² : Field K
ζ : K
hζ✝ : IsPrimitiveRoot ζ ↑3
S' : Solution' hζ✝
inst✝¹ : NumberField K
inst✝ : IsCyclotomicExtension {3} ℚ K
z : 𝓞 K
hz : S'.c = (hζ✝.toInteger - 1) * z
x : 𝓞 K
hx : S'.a ^ 3 - 1 = (hζ✝.toInteger - 1) ^ 4 * x
y : 𝓞 K
hy : S'.b ^ 3 - 1 = (hζ✝.toInteger - 1) ^ 4 * y
hζ : IsPrimitiveRoot ζ (↑3 ^ 1)
⊢ False
|
7f0bf19b77a2dee2
|
Ideal.map_includeRight_eq
|
Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
|
/-- The ideal of `A ⊗[R] B` generated by `I` is the image of `A ⊗[R] I` -/
lemma Ideal.map_includeRight_eq (I : Ideal B) :
(I.map (Algebra.TensorProduct.includeRight : B →ₐ[R] A ⊗[R] B)).restrictScalars R
= LinearMap.range (LinearMap.lTensor A (Submodule.subtype (I.restrictScalars R)))
|
case a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
a : A
b : ↥(Submodule.restrictScalars R I)
this : a ⊗ₜ[R] ↑b = a ⊗ₜ[R] 1 * 1 ⊗ₜ[R] ↑b
⊢ 1 ⊗ₜ[R] ↑b ∈ map includeRight I
|
apply Ideal.mem_map_of_mem includeRight
|
case a
R : Type u_1
inst✝⁴ : CommSemiring R
A : Type u_2
B : Type u_3
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
I : Ideal B
a : A
b : ↥(Submodule.restrictScalars R I)
this : a ⊗ₜ[R] ↑b = a ⊗ₜ[R] 1 * 1 ⊗ₜ[R] ↑b
⊢ ↑b ∈ I
|
433dfbb2c63fd4cd
|
MeasureTheory.Measure.rnDeriv_add_right_of_mutuallySingular
|
Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
|
lemma rnDeriv_add_right_of_mutuallySingular {ν' : Measure α}
[SigmaFinite μ] [SigmaFinite ν] [SigmaFinite ν'] (hνν' : ν ⟂ₘ ν') :
μ.rnDeriv (ν + ν') =ᵐ[ν] μ.rnDeriv ν
|
α : Type u_1
m : MeasurableSpace α
μ ν ν' : Measure α
inst✝² : SigmaFinite μ
inst✝¹ : SigmaFinite ν
inst✝ : SigmaFinite ν'
hνν' : ν ⟂ₘ ν'
h_ac : ν ≪ ν + ν'
h₁ :
(μ.singularPart ν' + ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')]
(μ.singularPart ν').rnDeriv (ν + ν') + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv (ν + ν')
h₂ :
(μ.singularPart ν' + ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν =ᶠ[ae ν]
(μ.singularPart ν').rnDeriv ν + (ν'.withDensity (μ.rnDeriv ν')).rnDeriv ν
h₃ : (μ.singularPart ν').rnDeriv (ν + ν') =ᶠ[ae ν] (μ.singularPart ν').rnDeriv ν
⊢ ν'.withDensity (μ.rnDeriv ν') ⟂ₘ ν
|
exact hνν'.symm.withDensity
|
no goals
|
aa79bd6ae3fefa2b
|
IsLocalizedModule.exist_integer_multiples
|
Mathlib/Algebra/Module/LocalizedModule/Int.lean
|
theorem exist_integer_multiples {ι : Type*} (s : Finset ι) (g : ι → M') :
∃ b : S, ∀ i ∈ s, IsInteger f (b.val • g i)
|
case refine_2
R : Type u_1
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
M' : Type u_3
inst✝² : AddCommMonoid M'
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
ι : Type u_4
s : Finset ι
g : ι → M'
sec : ι → M × ↥S
hsec : ∀ (i : ι), (sec i).2 • g i = f (sec i).1
i : ι
hi : i ∈ s
⊢ (∏ j ∈ s.erase i, (sec j).2) • f (sec i).1 = (∏ i ∈ s, ↑(sec i).2) • g i
|
rw [← hsec, ← mul_smul, Submonoid.smul_def]
|
case refine_2
R : Type u_1
inst✝⁵ : CommSemiring R
S : Submonoid R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
M' : Type u_3
inst✝² : AddCommMonoid M'
inst✝¹ : Module R M'
f : M →ₗ[R] M'
inst✝ : IsLocalizedModule S f
ι : Type u_4
s : Finset ι
g : ι → M'
sec : ι → M × ↥S
hsec : ∀ (i : ι), (sec i).2 • g i = f (sec i).1
i : ι
hi : i ∈ s
⊢ ↑((∏ j ∈ s.erase i, (sec j).2) * (sec i).2) • g i = (∏ i ∈ s, ↑(sec i).2) • g i
|
0d28f5679ff37f99
|
Set.eq_insert_of_ncard_eq_succ
|
Mathlib/Data/Set/Card.lean
|
theorem eq_insert_of_ncard_eq_succ {n : ℕ} (h : s.ncard = n + 1) :
∃ a t, a ∉ t ∧ insert a t = s ∧ t.ncard = n
|
α : Type u_1
s : Set α
n : ℕ
h : s.ncard = n + 1
⊢ ∃ a t, a ∉ t ∧ insert a t = s ∧ t.ncard = n
|
have hsf := finite_of_ncard_pos (n.zero_lt_succ.trans_eq h.symm)
|
α : Type u_1
s : Set α
n : ℕ
h : s.ncard = n + 1
hsf : s.Finite
⊢ ∃ a t, a ∉ t ∧ insert a t = s ∧ t.ncard = n
|
a83d5923ba572b58
|
CategoryTheory.shiftFunctorAdd_zero_add_hom_app
|
Mathlib/CategoryTheory/Shift/Basic.lean
|
lemma shiftFunctorAdd_zero_add_hom_app (a : A) (X : C) :
(shiftFunctorAdd C 0 a).hom.app X =
eqToHom (by dsimp; rw [zero_add]) ≫ ((shiftFunctorZero C A).inv.app X)⟦a⟧'
|
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctorAdd C 0 a).hom.app X = eqToHom ⋯ ≫ (shiftFunctor C a).map ((shiftFunctorZero C A).inv.app X)
|
simp [← shiftFunctorAdd'_zero_add_hom_app, shiftFunctorAdd']
|
no goals
|
73e541f3a60f3c7f
|
tsub_tsub_tsub_cancel_right
|
Mathlib/Algebra/Order/Sub/Unbundled/Basic.lean
|
theorem tsub_tsub_tsub_cancel_right (h : c ≤ b) : a - c - (b - c) = a - b
|
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : AddLeftMono α
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c : α
h : c ≤ b
⊢ a - c - (b - c) = a - b
|
rw [tsub_tsub, add_tsub_cancel_of_le h]
|
no goals
|
6e29741ba1746be9
|
MvPowerSeries.map.isLocalHom
|
Mathlib/RingTheory/MvPowerSeries/Inverse.lean
|
theorem map.isLocalHom : IsLocalHom (map σ f) :=
⟨by
rintro φ ⟨ψ, h⟩
replace h := congr_arg (constantCoeff σ S) h
rw [constantCoeff_map] at h
have : IsUnit (constantCoeff σ S ↑ψ) := isUnit_constantCoeff _ ψ.isUnit
rw [h] at this
rcases isUnit_of_map_unit f _ this with ⟨c, hc⟩
exact isUnit_of_mul_eq_one φ (invOfUnit φ c) (mul_invOfUnit φ c hc.symm)⟩
|
case intro
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : (constantCoeff σ S) ↑ψ = f ((constantCoeff σ R) φ)
this : IsUnit ((constantCoeff σ S) ↑ψ)
⊢ IsUnit φ
|
rw [h] at this
|
case intro
σ : Type u_1
R : Type u_2
S : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
f : R →+* S
inst✝ : IsLocalHom f
φ : MvPowerSeries σ R
ψ : (MvPowerSeries σ S)ˣ
h : (constantCoeff σ S) ↑ψ = f ((constantCoeff σ R) φ)
this : IsUnit (f ((constantCoeff σ R) φ))
⊢ IsUnit φ
|
f085e8794e8de496
|
Topology.IsUpperSet.upperSet_le_upper
|
Mathlib/Topology/Order/UpperLowerSetTopology.lean
|
lemma upperSet_le_upper {t₁ t₂ : TopologicalSpace α} [@Topology.IsUpperSet α t₁ _]
[@Topology.IsUpper α t₂ _] : t₁ ≤ t₂ := fun s hs => by
rw [@isOpen_iff_isUpperSet α _ t₁]
exact IsUpper.isUpperSet_of_isOpen hs
|
α : Type u_1
inst✝² : Preorder α
t₁ t₂ : TopologicalSpace α
inst✝¹ : Topology.IsUpperSet α
inst✝ : IsUpper α
s : Set α
hs : IsOpen s
⊢ IsUpperSet s
|
exact IsUpper.isUpperSet_of_isOpen hs
|
no goals
|
acbfb9816d196487
|
CategoryTheory.Presheaf.isLocallyInjective_of_isLocallyInjective_of_isLocallySurjective
|
Mathlib/CategoryTheory/Sites/LocallySurjective.lean
|
lemma isLocallyInjective_of_isLocallyInjective_of_isLocallySurjective
{F₁ F₂ F₃ : Cᵒᵖ ⥤ A} (f₁ : F₁ ⟶ F₂) (f₂ : F₂ ⟶ F₃)
[IsLocallyInjective J (f₁ ≫ f₂)] [IsLocallySurjective J f₁] :
IsLocallyInjective J f₂ where
equalizerSieve_mem {X} x₁ x₂ h
|
case refine_2
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)
inst✝ : IsLocallySurjective J f₁
X : Cᵒᵖ
x₁ x₂ : ToType (F₂.obj X)
h : (ConcreteCategory.hom (f₂.app X)) x₁ = (ConcreteCategory.hom (f₂.app X)) x₂
S : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂
hS : S ∈ J (unop X)
T : ⦃Y : C⦄ → (f : Y ⟶ unop X) → S.arrows f → Sieve Y :=
fun Y f hf => equalizerSieve (localPreimage f₁ x₁ f ⋯) (localPreimage f₁ x₂ f ⋯)
Y : C
f : Y ⟶ unop X
hf : S.arrows f
⊢ Sieve.pullback f (Sieve.bind S.arrows T) ∈ J Y
|
apply J.superset_covering (Sieve.le_pullback_bind _ _ _ hf)
|
case refine_2
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁴ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝³ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝² : ConcreteCategory A FA
F₁ F₂ F₃ : Cᵒᵖ ⥤ A
f₁ : F₁ ⟶ F₂
f₂ : F₂ ⟶ F₃
inst✝¹ : IsLocallyInjective J (f₁ ≫ f₂)
inst✝ : IsLocallySurjective J f₁
X : Cᵒᵖ
x₁ x₂ : ToType (F₂.obj X)
h : (ConcreteCategory.hom (f₂.app X)) x₁ = (ConcreteCategory.hom (f₂.app X)) x₂
S : Sieve (unop X) := imageSieve f₁ x₁ ⊓ imageSieve f₁ x₂
hS : S ∈ J (unop X)
T : ⦃Y : C⦄ → (f : Y ⟶ unop X) → S.arrows f → Sieve Y :=
fun Y f hf => equalizerSieve (localPreimage f₁ x₁ f ⋯) (localPreimage f₁ x₂ f ⋯)
Y : C
f : Y ⟶ unop X
hf : S.arrows f
⊢ T f hf ∈ J Y
|
fb4794d97c93223f
|
Monotone.alternating_series_le_tendsto
|
Mathlib/Analysis/SpecificLimits/Normed.lean
|
theorem Monotone.alternating_series_le_tendsto
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfm : Monotone f) (k : ℕ) : ∑ i ∈ range (2 * k + 1), (-1) ^ i * f i ≤ l
|
E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
⊢ Monotone fun n => ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i
|
refine monotone_nat_of_le_succ (fun n ↦ ?_)
|
E : Type u_2
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k n : ℕ
⊢ ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i ≤ ∑ i ∈ Finset.range (2 * (n + 1) + 1), (-1) ^ i * f i
|
2a7dbce5730e93a1
|
Module.finitePresentation_of_ker
|
Mathlib/Algebra/Module/FinitePresentation.lean
|
lemma Module.finitePresentation_of_ker [Module.FinitePresentation R N]
(l : M →ₗ[R] N) (hl : Function.Surjective l) [Module.FinitePresentation R (LinearMap.ker l)] :
Module.FinitePresentation R M
|
case intro
R : Type u_1
M : Type u_3
N : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : FinitePresentation R N
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
inst✝ : FinitePresentation R ↥(LinearMap.ker l)
s : Finset M
hs : Submodule.span R ↑s = ⊤
⊢ FinitePresentation R M
|
refine ⟨s, hs, ?_⟩
|
case intro
R : Type u_1
M : Type u_3
N : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : FinitePresentation R N
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
inst✝ : FinitePresentation R ↥(LinearMap.ker l)
s : Finset M
hs : Submodule.span R ↑s = ⊤
⊢ (LinearMap.ker (linearCombination R Subtype.val)).FG
|
888cd3f368652bf6
|
LieModule.eventually_genWeightSpace_smul_add_eq_bot
|
Mathlib/Algebra/Lie/Weights/Chain.lean
|
lemma eventually_genWeightSpace_smul_add_eq_bot :
∀ᶠ (k : ℕ) in Filter.atTop, genWeightSpace M (k • χ₁ + χ₂) = ⊥
|
R : Type u_1
L : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
M : Type u_3
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : LieRing.IsNilpotent L
χ₁ χ₂ : L → R
inst✝² : NoZeroSMulDivisors ℤ R
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : IsNoetherian R M
hχ₁ : χ₁ ≠ 0
f : ℕ → L → R := fun k => k • χ₁ + χ₂
this : Function.Injective f
⊢ (f '' {x | ¬genWeightSpace M (x • χ₁ + χ₂) = ⊥}).Finite
|
apply (finite_genWeightSpace_ne_bot R L M).subset
|
R : Type u_1
L : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : LieAlgebra R L
M : Type u_3
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : LieRingModule L M
inst✝⁴ : LieModule R L M
inst✝³ : LieRing.IsNilpotent L
χ₁ χ₂ : L → R
inst✝² : NoZeroSMulDivisors ℤ R
inst✝¹ : NoZeroSMulDivisors R M
inst✝ : IsNoetherian R M
hχ₁ : χ₁ ≠ 0
f : ℕ → L → R := fun k => k • χ₁ + χ₂
this : Function.Injective f
⊢ f '' {x | ¬genWeightSpace M (x • χ₁ + χ₂) = ⊥} ⊆ {χ | genWeightSpace M χ ≠ ⊥}
|
ce52027cc0acda70
|
CPolynomialOn.fderiv
|
Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
|
theorem CPolynomialOn.fderiv (h : CPolynomialOn 𝕜 f s) :
CPolynomialOn 𝕜 (fderiv 𝕜 f) s
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
⊢ CPolynomialOn 𝕜 (_root_.fderiv 𝕜 f) s
|
intro y hy
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
h : CPolynomialOn 𝕜 f s
y : E
hy : y ∈ s
⊢ CPolynomialAt 𝕜 (_root_.fderiv 𝕜 f) y
|
fe76f93d74149110
|
CategoryTheory.NatTrans.CommShift.of_isIso
|
Mathlib/CategoryTheory/Shift/CommShift.lean
|
lemma of_isIso [IsIso τ] [NatTrans.CommShift τ A] :
NatTrans.CommShift (inv τ) A
|
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_7, u_1} C
inst✝⁷ : Category.{u_6, u_2} D
F₁ F₂ : C ⥤ D
τ : F₁ ⟶ F₂
A : Type u_5
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
inst✝⁴ : HasShift D A
inst✝³ : F₁.CommShift A
inst✝² : F₂.CommShift A
inst✝¹ : IsIso τ
inst✝ : CommShift τ A
⊢ CommShift (inv τ) A
|
haveI : NatTrans.CommShift (asIso τ).hom A := by assumption
|
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_7, u_1} C
inst✝⁷ : Category.{u_6, u_2} D
F₁ F₂ : C ⥤ D
τ : F₁ ⟶ F₂
A : Type u_5
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
inst✝⁴ : HasShift D A
inst✝³ : F₁.CommShift A
inst✝² : F₂.CommShift A
inst✝¹ : IsIso τ
inst✝ : CommShift τ A
this : CommShift (asIso τ).hom A
⊢ CommShift (inv τ) A
|
d35da6153719fb9d
|
ContinuousMap.sup_mem_closed_subalgebra
|
Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
|
theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A
|
case h.e'_4
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
h : IsClosed ↑A
f g : ↥A
⊢ A = A.topologicalClosure
|
apply SetLike.ext'
|
case h.e'_4.h
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
h : IsClosed ↑A
f g : ↥A
⊢ ↑A = ↑A.topologicalClosure
|
1e1c647738699fd7
|
AkraBazziRecurrence.eventually_atTop_sumTransform_ge
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma eventually_atTop_sumTransform_ge :
∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, c * g n ≤ sumTransform (p a b) g (r i n) n
|
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 > p a b + 1
⊢ 0 ≤ c₁
|
positivity
|
no goals
|
aff66f2e59f80837
|
RingHom.locally_iff_isLocalization
|
Mathlib/RingTheory/RingHom/Locally.lean
|
/-- In the definition of `Locally` we may replace `Localization.Away` with an arbitrary
algebra satisfying `IsLocalization.Away`. -/
lemma locally_iff_isLocalization (hP : RespectsIso P) (f : R →+* S) :
Locally P f ↔ ∃ (s : Finset S) (_ : Ideal.span (s : Set S) = ⊤),
∀ t ∈ s, ∀ (Sₜ : Type u) [CommRing Sₜ] [Algebra S Sₜ] [IsLocalization.Away t Sₜ],
P ((algebraMap S Sₜ).comp f)
|
case refine_1
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
hP : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
f : R →+* S
x✝³ : ∃ s, ∃ (_ : Ideal.span ↑s = ⊤), ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)
s : Finset S
hsone : Ideal.span ↑s = ⊤
hs : ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)
t : S
ht : t ∈ s
Sₜ : Type u
x✝² : CommRing Sₜ
x✝¹ : Algebra S Sₜ
x✝ : IsLocalization.Away t Sₜ
e : Localization.Away t ≃+* Sₜ := (IsLocalization.algEquiv (Submonoid.powers t) (Localization.Away t) Sₜ).toRingEquiv
⊢ P ((algebraMap S Sₜ).comp f)
|
have : algebraMap S Sₜ = e.toRingHom.comp (algebraMap S (Localization.Away t)) :=
RingHom.ext (fun x ↦ (AlgEquiv.commutes (IsLocalization.algEquiv _ _ _) _).symm)
|
case refine_1
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
hP : RespectsIso fun {R S} [CommRing R] [CommRing S] => P
f : R →+* S
x✝³ : ∃ s, ∃ (_ : Ideal.span ↑s = ⊤), ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)
s : Finset S
hsone : Ideal.span ↑s = ⊤
hs : ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)
t : S
ht : t ∈ s
Sₜ : Type u
x✝² : CommRing Sₜ
x✝¹ : Algebra S Sₜ
x✝ : IsLocalization.Away t Sₜ
e : Localization.Away t ≃+* Sₜ := (IsLocalization.algEquiv (Submonoid.powers t) (Localization.Away t) Sₜ).toRingEquiv
this : algebraMap S Sₜ = e.toRingHom.comp (algebraMap S (Localization.Away t))
⊢ P ((algebraMap S Sₜ).comp f)
|
48f562d83319536b
|
Ideal.span_pow_eq_top
|
Mathlib/RingTheory/Ideal/Basic.lean
|
theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
span ((fun (x : α) => x ^ n) '' s) = ⊤
|
case zero.inl
α : Type u_2
inst✝ : CommSemiring α
hs : span ∅ = ⊤
⊢ 1 ∈ span ((fun x => x ^ 0) '' ∅)
|
rw [Set.image_empty, hs]
|
case zero.inl
α : Type u_2
inst✝ : CommSemiring α
hs : span ∅ = ⊤
⊢ 1 ∈ ⊤
|
dd88b6d68cbdbab3
|
AddCircle.scaled_exp_map_periodic
|
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean
|
theorem scaled_exp_map_periodic : Function.Periodic (fun x => Circle.exp (2 * π / T * x)) T
|
T : ℝ
⊢ Periodic (fun x => Circle.exp (2 * π / T * x)) T
|
rcases eq_or_ne T 0 with (rfl | hT)
|
case inl
⊢ Periodic (fun x => Circle.exp (2 * π / 0 * x)) 0
case inr
T : ℝ
hT : T ≠ 0
⊢ Periodic (fun x => Circle.exp (2 * π / T * x)) T
|
27ec239a7d4a816a
|
Real.toNNReal_eq_nnnorm_of_nonneg
|
Mathlib/Analysis/Normed/Group/Basic.lean
|
theorem toNNReal_eq_nnnorm_of_nonneg (hr : 0 ≤ r) : r.toNNReal = ‖r‖₊
|
case a
r : ℝ
hr : 0 ≤ r
⊢ ↑⟨r, hr⟩ = ↑‖r‖₊
|
rw [coe_mk, coe_nnnorm r, Real.norm_eq_abs r, abs_of_nonneg hr]
|
no goals
|
edbd0a0d12ba3801
|
MulChar.IsQuadratic.pow_char
|
Mathlib/NumberTheory/MulChar/Basic.lean
|
theorem IsQuadratic.pow_char {χ : MulChar R R'} (hχ : χ.IsQuadratic) (p : ℕ) [hp : Fact p.Prime]
[CharP R' p] : χ ^ p = χ
|
case h.inr.inl
R : Type u_1
inst✝² : CommMonoid R
R' : Type u_2
inst✝¹ : CommRing R'
χ : MulChar R R'
hχ : χ.IsQuadratic
p : ℕ
hp : Fact (Nat.Prime p)
inst✝ : CharP R' p
x : Rˣ
hx : χ ↑x = 1
⊢ 1 ^ p = 1
|
rw [one_pow]
|
no goals
|
d7819318bd3ad273
|
Algebra.FormallyUnramified.ext_of_iInf
|
Mathlib/RingTheory/Unramified/Basic.lean
|
theorem ext_of_iInf [FormallyUnramified R A] (hI : ⨅ i, I ^ i = ⊥) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂
|
case pos
R : Type v
inst✝⁵ : CommRing R
A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
B : Type w
inst✝² : CommRing B
inst✝¹ : Algebra R B
I : Ideal B
inst✝ : FormallyUnramified R A
hI : ⨅ i, I ^ i = ⊥
g₁ g₂ : A →ₐ[R] B
H : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x)
i : ℕ
hi : i = 0
⊢ (Ideal.Quotient.mkₐ R (I ^ i)).comp g₁ = (Ideal.Quotient.mkₐ R (I ^ i)).comp g₂
|
ext x
|
case pos.H
R : Type v
inst✝⁵ : CommRing R
A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
B : Type w
inst✝² : CommRing B
inst✝¹ : Algebra R B
I : Ideal B
inst✝ : FormallyUnramified R A
hI : ⨅ i, I ^ i = ⊥
g₁ g₂ : A →ₐ[R] B
H : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk I) (g₂ x)
i : ℕ
hi : i = 0
x : A
⊢ ((Ideal.Quotient.mkₐ R (I ^ i)).comp g₁) x = ((Ideal.Quotient.mkₐ R (I ^ i)).comp g₂) x
|
18929210cec07ffd
|
MeasureTheory.pow_mul_meas_ge_le_eLpNorm
|
Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean
|
theorem pow_mul_meas_ge_le_eLpNorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
(ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }) ^ (1 / p.toReal) ≤ eLpNorm f p μ
|
case h₁
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α → E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
⊢ ε * μ {x | ε ≤ ↑‖f x‖₊ ^ p.toReal} ≤ ∫⁻ (x : α), ‖f x‖ₑ ^ p.toReal ∂μ
|
exact mul_meas_ge_le_lintegral₀ (hf.enorm.pow_const _) ε
|
no goals
|
2fe447c04e0f4a71
|
Nat.digits_add
|
Mathlib/Data/Nat/Digits.lean
|
theorem digits_add (b : ℕ) (h : 1 < b) (x y : ℕ) (hxb : x < b) (hxy : x ≠ 0 ∨ y ≠ 0) :
digits b (x + b * y) = x :: digits b y
|
case intro.succ.e_head
x b : ℕ
h : 1 < b + 2
hxb : x < b + 2
n✝ : ℕ
hxy : x ≠ 0 ∨ n✝ + 1 ≠ 0
⊢ (x + (b + 2) * (n✝ + 1)) % (b + 2) = x
|
simp [Nat.add_mod, mod_eq_of_lt hxb]
|
no goals
|
5707ca6647bd00ed
|
Stream.take_eq_takeTR
|
Mathlib/.lake/packages/batteries/Batteries/Data/Stream.lean
|
theorem take_eq_takeTR : @take = @takeTR
|
case h.h.h.h.h.snd
x✝⁴ : Type u_2
x✝³ : Type u_1
x✝² : Stream x✝⁴ x✝³
x✝¹ : x✝⁴
x✝ : Nat
⊢ (take x✝¹ x✝).snd = (takeTR x✝¹ x✝).snd
|
rw [snd_takeTR, snd_take_eq_drop]
|
no goals
|
cbdb14a1a0777cbf
|
List.mapIdx_reverse
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean
|
theorem mapIdx_reverse {l : List α} {f : Nat → α → β} :
l.reverse.mapIdx f = (mapIdx (fun i => f (l.length - 1 - i)) l).reverse
|
α : Type u_1
β : Type u_2
l : List α
f : Nat → α → β
i : Nat
⊢ (mapIdx (fun i => f (l.length - 1 - i)) l).reverse[i]? = Option.map (f i) l.reverse[i]?
|
by_cases h : i < l.length
|
case pos
α : Type u_1
β : Type u_2
l : List α
f : Nat → α → β
i : Nat
h : i < l.length
⊢ (mapIdx (fun i => f (l.length - 1 - i)) l).reverse[i]? = Option.map (f i) l.reverse[i]?
case neg
α : Type u_1
β : Type u_2
l : List α
f : Nat → α → β
i : Nat
h : ¬i < l.length
⊢ (mapIdx (fun i => f (l.length - 1 - i)) l).reverse[i]? = Option.map (f i) l.reverse[i]?
|
8086284ec8908f69
|
isMeagre_iff_countable_union_isNowhereDense
|
Mathlib/Topology/GDelta/Basic.lean
|
/-- A set is meagre iff it is contained in a countable union of nowhere dense sets. -/
lemma isMeagre_iff_countable_union_isNowhereDense {s : Set X} :
IsMeagre s ↔ ∃ S : Set (Set X), (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S
|
case refine_2
X : Type u_5
inst✝ : TopologicalSpace X
s : Set X
⊢ (∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S) →
∃ x, (∀ ⦃x_1 : Set X⦄, x_1 ∈ x → IsClosed x_1 ∧ IsNowhereDense x_1) ∧ (compl '' x).Countable ∧ s ⊆ ⋃₀ x
|
intro ⟨S, hS, hc, hsub⟩
|
case refine_2
X : Type u_5
inst✝ : TopologicalSpace X
s : Set X
S : Set (Set X)
hS : ∀ t ∈ S, IsNowhereDense t
hc : S.Countable
hsub : s ⊆ ⋃₀ S
⊢ ∃ x, (∀ ⦃x_1 : Set X⦄, x_1 ∈ x → IsClosed x_1 ∧ IsNowhereDense x_1) ∧ (compl '' x).Countable ∧ s ⊆ ⋃₀ x
|
bee0252956d68792
|
BooleanSubalgebra.map_symm_eq_iff_eq_map
|
Mathlib/Order/BooleanSubalgebra.lean
|
lemma map_symm_eq_iff_eq_map {M : BooleanSubalgebra β} {e : β ≃o α} :
L.map ↑e.symm = M ↔ L = M.map ↑e
|
α : Type u_2
β : Type u_3
inst✝¹ : BooleanAlgebra α
inst✝ : BooleanAlgebra β
L : BooleanSubalgebra α
M : BooleanSubalgebra β
e : β ≃o α
⊢ ↑(map { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ } L) = ↑M ↔
↑L = ↑(map { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ } M)
|
exact (Equiv.eq_image_iff_symm_image_eq _ _ _).symm
|
no goals
|
d4990c54f7f4a9cc
|
SimpleGraph.edgeDisjointTriangles_iff_mem_sym2_subsingleton
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
|
lemma edgeDisjointTriangles_iff_mem_sym2_subsingleton :
G.EdgeDisjointTriangles ↔
∀ ⦃e : Sym2 α⦄, ¬ e.IsDiag → {s ∈ G.cliqueSet 3 | e ∈ (s : Finset α).sym2}.Subsingleton
|
case h.mp.intro.intro.intro.intro.intro.intro.intro.intro.inr.inr.inl
α : Type u_1
G : SimpleGraph α
a b : α
hab : ¬a = b
d : α
hcd : G.Adj b d
hde : G.Adj a d
hce : G.Adj a b
⊢ ∃ c, G.Adj a c ∧ G.Adj b c ∧ {b, d, a} = {a, b, c}
|
exact ⟨d, by aesop⟩
|
no goals
|
dd5ab32164f7deff
|
groupCohomology.cochainsMap_f_map_mono
|
Mathlib/RepresentationTheory/GroupCohomology/Functoriality.lean
|
lemma cochainsMap_f_map_mono (hf : Function.Surjective f) [Mono φ] (i : ℕ) :
Mono ((cochainsMap f φ).f i)
|
k G H : Type u
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : Group H
A : Rep k H
B : Rep k G
f : G →* H
φ : (Action.res (ModuleCat k) f).obj A ⟶ B
hf : Function.Surjective ⇑f
inst✝ : Mono φ
i : ℕ
⊢ Mono ((cochainsMap f φ).f i)
|
simpa [ModuleCat.mono_iff_injective] using
((Rep.mono_iff_injective φ).1 inferInstance).comp_left.comp <|
LinearMap.funLeft_injective_of_surjective k A _ hf.comp_left
|
no goals
|
e4c1b9c4ec35a7ae
|
LinearMap.ker_eq_bot_range_liftQ_iff
|
Mathlib/Algebra/Exact.lean
|
lemma ker_eq_bot_range_liftQ_iff (h : range f ≤ ker g) :
ker ((range f).liftQ g h) = ⊥ ↔ ker g = range f
|
R : Type u_1
M : Type u_2
N : Type u_4
P : Type u_6
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
h : range f ≤ ker g
⊢ (∀ (x : N ⧸ range f), ((range f).liftQ g h) x = 0 ↔ x = 0) ↔ ∀ (x : N), g x = 0 ↔ ∃ y, f y = x
|
constructor
|
case mp
R : Type u_1
M : Type u_2
N : Type u_4
P : Type u_6
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
h : range f ≤ ker g
⊢ (∀ (x : N ⧸ range f), ((range f).liftQ g h) x = 0 ↔ x = 0) → ∀ (x : N), g x = 0 ↔ ∃ y, f y = x
case mpr
R : Type u_1
M : Type u_2
N : Type u_4
P : Type u_6
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup N
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N
g : N →ₗ[R] P
h : range f ≤ ker g
⊢ (∀ (x : N), g x = 0 ↔ ∃ y, f y = x) → ∀ (x : N ⧸ range f), ((range f).liftQ g h) x = 0 ↔ x = 0
|
d991d6866470e4a6
|
Set.encard_eq_three
|
Mathlib/Data/Set/Card.lean
|
theorem encard_eq_three {α : Type u_1} {s : Set α} :
encard s = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}
|
α : Type u_1
s : Set α
⊢ s.encard = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}
|
refine ⟨fun h ↦ ?_, fun ⟨x, y, z, hxy, hyz, hxz, hs⟩ ↦ ?_⟩
|
case refine_1
α : Type u_1
s : Set α
h : s.encard = 3
⊢ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}
case refine_2
α : Type u_1
s : Set α
x✝ : ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}
x y z : α
hxy : x ≠ y
hyz : x ≠ z
hxz : y ≠ z
hs : s = {x, y, z}
⊢ s.encard = 3
|
f3917f7e0cd671ea
|
CircleDeg1Lift.units_apply_inv_apply
|
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
theorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
f ((f⁻¹ : CircleDeg1Liftˣ) x) = x
|
f : CircleDeg1Liftˣ
x : ℝ
⊢ ↑f (↑f⁻¹ x) = x
|
simp only [← mul_apply, f.mul_inv, coe_one, id]
|
no goals
|
acac1f7bb9848a5c
|
HahnSeries.order_single_mul_of_isRegular
|
Mathlib/RingTheory/HahnSeries/Multiplication.lean
|
theorem order_single_mul_of_isRegular {g : Γ} {r : R} (hr : IsRegular r)
{x : HahnSeries Γ R} (hx : x ≠ 0) : (((single g) r) * x).order = g + x.order
|
R : Type u_3
Γ : Type u_6
inst✝¹ : LinearOrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
g : Γ
r : R
hr : IsRegular r
x : HahnSeries Γ R
hx : x ≠ 0
h✝ : Nontrivial R
⊢ ((single g) r).leadingCoeff * x.leadingCoeff ≠ 0
|
rwa [leadingCoeff_of_single, ne_eq, hr.left.mul_left_eq_zero_iff, leadingCoeff_eq_iff]
|
no goals
|
fd006c7f15ff647b
|
NumberField.InfinitePlace.ComplexEmbedding.exists_comp_symm_eq_of_comp_eq
|
Mathlib/NumberTheory/NumberField/Embeddings.lean
|
lemma ComplexEmbedding.exists_comp_symm_eq_of_comp_eq [IsGalois k K] (φ ψ : K →+* ℂ)
(h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)) :
∃ σ : K ≃ₐ[k] K, φ.comp σ.symm = ψ
|
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
⊢ ∃ σ, φ.comp ↑σ.symm = ψ
|
use (AlgHom.restrictNormal' ψ' K).symm
|
case h
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ ψ : K →+* ℂ
h : φ.comp (algebraMap k K) = ψ.comp (algebraMap k K)
this✝¹ : Algebra k ℂ := (φ.comp (algebraMap k K)).toAlgebra
this✝ : Algebra K ℂ := φ.toAlgebra
this : IsScalarTower k K ℂ
ψ' : K →ₐ[k] ℂ := { toRingHom := ψ, commutes' := ⋯ }
⊢ φ.comp ↑(ψ'.restrictNormal' K).symm.symm = ψ
|
912b9e2017d03fa6
|
bernoulliFourierCoeff_eq
|
Mathlib/NumberTheory/ZetaValues.lean
|
theorem bernoulliFourierCoeff_eq {k : ℕ} (hk : k ≠ 0) (n : ℤ) :
bernoulliFourierCoeff k n = -k ! / (2 * π * I * n) ^ k
|
case neg
k✝ : ℕ
hk✝ : k✝ ≠ 0
n : ℤ
hn : n ≠ 0
k : ℕ
hk : 1 ≤ k
h'k : bernoulliFourierCoeff k n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
h : ¬k + 1 = 1
⊢ -((↑k + 1) * ↑k !) / (2 * ↑π * I * ↑n * (2 * ↑π * I * ↑n) ^ k) =
-((↑k + 1) * ↑k !) / ((2 * ↑π * I * ↑n) ^ k * (2 * ↑π * I * ↑n))
|
ring_nf
|
no goals
|
71b5051efe4665bf
|
spectrum.hasFPowerSeriesOnBall_inverse_one_sub_smul
|
Mathlib/Analysis/Normed/Algebra/Spectrum.lean
|
theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [HasSummableGeomSeries A] (a : A) :
HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
(fun n => ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
{ r_le
|
𝕜 : Type u_1
A : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : HasSummableGeomSeries A
a : A
y : 𝕜
hy : y ∈ EMetric.ball 0 (↑‖a‖₊)⁻¹
h : ¬‖a‖₊ = 0
⊢ ‖y‖₊ < ‖a‖₊⁻¹
|
simpa only [← coe_inv h, mem_ball_zero_iff, Metric.emetric_ball_nnreal] using hy
|
no goals
|
040a66281ae63253
|
CategoryTheory.MonoidalCategory.tensorμ_tensorδ
|
Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean
|
@[reassoc (attr := simp)]
lemma tensorμ_tensorδ (X₁ X₂ Y₁ Y₂ : C) :
tensorμ X₁ X₂ Y₁ Y₂ ≫ tensorδ X₁ X₂ Y₁ Y₂ = 𝟙 _
|
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X₁ X₂ Y₁ Y₂ : C
⊢ tensorμ X₁ X₂ Y₁ Y₂ ≫ tensorδ X₁ X₂ Y₁ Y₂ = 𝟙 ((X₁ ⊗ X₂) ⊗ Y₁ ⊗ Y₂)
|
simp only [tensorμ, tensorδ, assoc, Iso.inv_hom_id_assoc,
← MonoidalCategory.whiskerLeft_comp_assoc, Iso.hom_inv_id_assoc,
hom_inv_whiskerRight_assoc, Iso.hom_inv_id, Iso.inv_hom_id,
MonoidalCategory.whiskerLeft_id, id_comp]
|
no goals
|
20f8ae0ec88ed79a
|
MeasureTheory.Measure.add_comp'
|
Mathlib/Probability/Kernel/Composition/MeasureComp.lean
|
/-- Same as `add_comp` except that it uses `⇑κ + ⇑η` instead of `⇑(κ + η)` in order to have
a simp-normal form on the left of the equality. -/
@[simp]
lemma add_comp' : (⇑κ + ⇑η) ∘ₘ μ = κ ∘ₘ μ + η ∘ₘ μ
|
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
⊢ (⇑κ + ⇑η) ∘ₘ μ = ⇑κ ∘ₘ μ + ⇑η ∘ₘ μ
|
rw [← Kernel.coe_add, add_comp]
|
no goals
|
843a18552080baea
|
MultilinearMap.mkPiRing_eq_iff
|
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
theorem mkPiRing_eq_iff [Fintype ι] {z₁ z₂ : M₂} :
MultilinearMap.mkPiRing R ι z₁ = MultilinearMap.mkPiRing R ι z₂ ↔ z₁ = z₂
|
R : Type uR
ι : Type uι
M₂ : Type v₂
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M₂
inst✝ : Fintype ι
z₁ z₂ : M₂
⊢ (∀ (x : ι → R), (∏ i : ι, x i) • z₁ = (∏ i : ι, x i) • z₂) ↔ z₁ = z₂
|
constructor <;> intro h
|
case mp
R : Type uR
ι : Type uι
M₂ : Type v₂
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M₂
inst✝ : Fintype ι
z₁ z₂ : M₂
h : ∀ (x : ι → R), (∏ i : ι, x i) • z₁ = (∏ i : ι, x i) • z₂
⊢ z₁ = z₂
case mpr
R : Type uR
ι : Type uι
M₂ : Type v₂
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M₂
inst✝ : Fintype ι
z₁ z₂ : M₂
h : z₁ = z₂
⊢ ∀ (x : ι → R), (∏ i : ι, x i) • z₁ = (∏ i : ι, x i) • z₂
|
e91bf17c3f81d970
|
Std.Sat.AIG.RefVec.ite.go_get_aux
|
Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/If.lean
|
theorem go_get_aux {w : Nat} (aig : AIG α) (curr : Nat) (hcurr : curr ≤ w) (discr : Ref aig)
(lhs rhs : RefVec aig w) (s : RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr) (hfoo),
(go aig curr hcurr discr lhs rhs s).vec.get idx (by omega)
=
(s.get idx hidx).cast hfoo
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
discr : aig.Ref
lhs rhs : aig.RefVec w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
⊢ ∀ (hfoo : aig.decls.size ≤ (go aig curr hcurr discr lhs rhs s).aig.decls.size),
(go aig curr hcurr discr lhs rhs s).vec.get idx ⋯ = (s.get idx hidx).cast hfoo
|
generalize hgo : go aig curr hcurr discr lhs rhs s = res
|
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ w
discr : aig.Ref
lhs rhs : aig.RefVec w
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
res : RefVecEntry α w
hgo : go aig curr hcurr discr lhs rhs s = res
⊢ ∀ (hfoo : aig.decls.size ≤ res.aig.decls.size), res.vec.get idx ⋯ = (s.get idx hidx).cast hfoo
|
c08e573c9b942d4a
|
LinearMap.IsProj.codRestrict_apply_cod
|
Mathlib/LinearAlgebra/Projection.lean
|
theorem codRestrict_apply_cod {f : M →ₗ[S] M} (h : IsProj m f) (x : m) : h.codRestrict x = x
|
case a
S : Type u_5
inst✝² : Semiring S
M : Type u_6
inst✝¹ : AddCommMonoid M
inst✝ : Module S M
m : Submodule S M
f : M →ₗ[S] M
h : IsProj m f
x : ↥m
⊢ f ↑x = ↑x
|
exact h.map_id x x.2
|
no goals
|
3e267d085658cef3
|
List.stronglyMeasurable_prod'
|
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
theorem _root_.List.stronglyMeasurable_prod' (l : List (α → M))
(hl : ∀ f ∈ l, StronglyMeasurable f) : StronglyMeasurable l.prod
|
α : Type u_1
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
l : List (α → M)
hl : ∀ f ∈ l, StronglyMeasurable f
⊢ StronglyMeasurable l.prod
|
induction' l with f l ihl
|
case nil
α : Type u_1
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
hl : ∀ f ∈ [], StronglyMeasurable f
⊢ StronglyMeasurable [].prod
case cons
α : Type u_1
M : Type u_5
inst✝² : Monoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousMul M
m : MeasurableSpace α
f : α → M
l : List (α → M)
ihl : (∀ f ∈ l, StronglyMeasurable f) → StronglyMeasurable l.prod
hl : ∀ f_1 ∈ f :: l, StronglyMeasurable f_1
⊢ StronglyMeasurable (f :: l).prod
|
281f39535a2f83b5
|
AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
/-- The main proof of the lower bound part of the Akra-Bazzi theorem. The factor
`1 + ε n` does not change the asymptotic order, but is needed for the induction step to go
through. -/
lemma smoothingFn_mul_asympBound_isBigO_T :
(fun (n : ℕ) => (1 + ε n) * asympBound g a b n) =O[atTop] T
|
case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
h_b_floor : 0 < ⌊b' * ↑n₀⌋₊
h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y
h_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y
h_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y
h_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)
h_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y
n₀_pos : 0 < n₀
h_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)
bound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))
h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y
h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y
h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)
h_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y
h_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty
⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖
|
set base_min : ℝ :=
(Finset.Ico (⌊b' * n₀⌋₊) n₀).inf' h_base_nonempty
(fun n => T n / ((1 + ε n) * asympBound g a b n)) with base_min_def
|
case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀
h_b_floor : 0 < ⌊b' * ↑n₀⌋₊
h_smoothing_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < 1 + ε ↑y
h_smoothing_pos' : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y
h_asympBound_pos : ∀ (y : ℕ), n₀ ≤ y → 0 < asympBound g a b y
h_asympBound_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < asympBound g a b (r i y)
h_asympBound_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < asympBound g a b y
n₀_pos : 0 < n₀
h_smoothing_r_pos : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), 0 < 1 + ε ↑(r i y)
bound2 : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b i ^ p a b * ↑y ^ p a b * (1 + ε ↑y) ≤ ↑(r i y) ^ p a b * (1 + ε ↑(r i y))
h_smoothingFn_floor : ∀ (y : ℕ), ⌊b' * ↑n₀⌋₊ ≤ y → 0 < 1 + ε ↑y
h_sumTransform : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), sumTransform (p a b) g (r i y) y ≤ c₁ * g ↑y
h_bi_le_r : ∀ (y : ℕ), n₀ ≤ y → ∀ (i : α), b (min_bi b) / 2 * ↑y ≤ ↑(r i y)
h_exp : ∀ (y : ℕ), n₀ ≤ y → ⌈rexp 1⌉₊ ≤ y
h_base_nonempty : (Ico ⌊b (min_bi b) / 2 * ↑n₀⌋₊ n₀).Nonempty
base_min : ℝ := (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)
base_min_def : base_min = (Ico ⌊b' * ↑n₀⌋₊ n₀).inf' h_base_nonempty fun n => T n / ((1 + ε ↑n) * asympBound g a b n)
⊢ ∃ c > 0, ∀ n ≥ n₀, c * ‖(1 + ε ↑n) * asympBound g a b n‖ ≤ ‖T n‖
|
82bb51116c3f899e
|
Set.Finite.image
|
Mathlib/Data/Set/Finite/Basic.lean
|
theorem Finite.image {s : Set α} (f : α → β) (hs : s.Finite) : (f '' s).Finite
|
α : Type u
β : Type v
s : Set α
f : α → β
hs : s.Finite
⊢ (f '' s).Finite
|
have := hs.to_subtype
|
α : Type u
β : Type v
s : Set α
f : α → β
hs : s.Finite
this : Finite ↑s
⊢ (f '' s).Finite
|
1d7b24a4e0931515
|
EMetric.isClosed_subsets_of_isClosed
|
Mathlib/Topology/MetricSpace/Closeds.lean
|
theorem isClosed_subsets_of_isClosed (hs : IsClosed s) :
IsClosed { t : Closeds α | (t : Set α) ⊆ s }
|
α : Type u
inst✝ : EMetricSpace α
s : Set α
hs : IsClosed s
t : Closeds α
ht : t ∈ closure {t | ↑t ⊆ s}
x : α
hx : x ∈ t
⊢ x ∈ closure s
|
refine mem_closure_iff.2 fun ε εpos => ?_
|
α : Type u
inst✝ : EMetricSpace α
s : Set α
hs : IsClosed s
t : Closeds α
ht : t ∈ closure {t | ↑t ⊆ s}
x : α
hx : x ∈ t
ε : ℝ≥0∞
εpos : ε > 0
⊢ ∃ y ∈ s, edist x y < ε
|
eed134e17fc5ce9d
|
Algebra.IsAlgebraic.lift_cardinalMk_le_sigma_polynomial
|
Mathlib/RingTheory/Algebraic/Cardinality.lean
|
theorem lift_cardinalMk_le_sigma_polynomial :
lift.{u} #L ≤ #(Σ p : R[X], { x : L // x ∈ p.aroots L })
|
R : Type u
inst✝⁵ : CommRing R
L : Type v
inst✝⁴ : CommRing L
inst✝³ : IsDomain L
inst✝² : Algebra R L
inst✝¹ : NoZeroSMulDivisors R L
inst✝ : Algebra.IsAlgebraic R L
x y : L
h :
(fun x =>
let p := Classical.indefiniteDescription (fun x_1 => x_1 ≠ 0 ∧ (Polynomial.aeval x) x_1 = 0) ⋯;
⟨↑p, ⟨x, ⋯⟩⟩)
x =
(fun x =>
let p := Classical.indefiniteDescription (fun x_1 => x_1 ≠ 0 ∧ (Polynomial.aeval x) x_1 = 0) ⋯;
⟨↑p, ⟨x, ⋯⟩⟩)
y
⊢ x = y
|
simp only [Set.coe_setOf, ne_eq, Set.mem_setOf_eq, Sigma.mk.inj_iff] at h
|
R : Type u
inst✝⁵ : CommRing R
L : Type v
inst✝⁴ : CommRing L
inst✝³ : IsDomain L
inst✝² : Algebra R L
inst✝¹ : NoZeroSMulDivisors R L
inst✝ : Algebra.IsAlgebraic R L
x y : L
h :
↑(Classical.indefiniteDescription (fun x_1 => ¬x_1 = 0 ∧ (Polynomial.aeval x) x_1 = 0) ⋯) =
↑(Classical.indefiniteDescription (fun x => ¬x = 0 ∧ (Polynomial.aeval y) x = 0) ⋯) ∧
HEq ⟨x, ⋯⟩ ⟨y, ⋯⟩
⊢ x = y
|
9dbc3398b59afa38
|
Types.Pushout.inl_rel'_inl_iff
|
Mathlib/CategoryTheory/Limits/Shapes/Types.lean
|
@[simp]
lemma inl_rel'_inl_iff (x₁ y₁ : X₁) :
Rel' f g (Sum.inl x₁) (Sum.inl y₁) ↔ x₁ = y₁ ∨
∃ (x₀ y₀ : S) (_ : g x₀ = g y₀), x₁ = f x₀ ∧ y₁ = f y₀
|
case mpr
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
x₁ y₁ : X₁
⊢ (x₁ = y₁ ∨ ∃ x₀ y₀, ∃ (_ : g x₀ = g y₀), x₁ = f x₀ ∧ y₁ = f y₀) → Rel' f g (Sum.inl x₁) (Sum.inl y₁)
|
rintro (rfl | ⟨_,_ , h, rfl, rfl⟩)
|
case mpr.inl
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
x₁ : X₁
⊢ Rel' f g (Sum.inl x₁) (Sum.inl x₁)
case mpr.inr.intro.intro.intro.intro
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
w✝¹ w✝ : S
h : g w✝¹ = g w✝
⊢ Rel' f g (Sum.inl (f w✝¹)) (Sum.inl (f w✝))
|
d928526eeaa2913f
|
BddOrd.id_apply
|
Mathlib/Order/Category/BddOrd.lean
|
lemma id_apply (X : BddOrd) (x : X) :
(𝟙 X : X ⟶ X) x = x
|
X : BddOrd
x : ↑X.toPartOrd
⊢ (ConcreteCategory.hom (𝟙 X)) x = x
|
simp
|
no goals
|
49293cb0f533256c
|
SimpleGraph.edist_bot
|
Mathlib/Combinatorics/SimpleGraph/Metric.lean
|
lemma edist_bot [DecidableEq V] : (⊥ : SimpleGraph V).edist u v = (if u = v then 0 else ⊤)
|
V : Type u_1
u v : V
inst✝ : DecidableEq V
⊢ ⊥.edist u v = if u = v then 0 else ⊤
|
by_cases h : u = v <;> simp [h, edist_bot_of_ne]
|
no goals
|
4f40291f70ad31c1
|
MvPolynomial.NewtonIdentities.pairMap_involutive
|
Mathlib/RingTheory/MvPolynomial/Symmetric/NewtonIdentities.lean
|
theorem pairMap_involutive : (pairMap σ).Involutive
|
σ : Type u_1
inst✝ : DecidableEq σ
⊢ Function.Involutive (MvPolynomial.NewtonIdentities.pairMap σ)
|
intro t
|
σ : Type u_1
inst✝ : DecidableEq σ
t : Finset σ × σ
⊢ MvPolynomial.NewtonIdentities.pairMap σ (MvPolynomial.NewtonIdentities.pairMap σ t) = t
|
d4c9b1d690e07836
|
Lean.Data.AC.Context.toList_nonEmpty
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean
|
theorem Context.toList_nonEmpty (e : Expr) : e.toList ≠ []
|
case op.cons
l r : Expr
ih₁ : l.toList ≠ []
rhs_ih✝ : r.toList ≠ []
head✝ : Nat
tail✝ : List Nat
h : l.toList = head✝ :: tail✝
⊢ ¬head✝ :: tail✝ ++ r.toList = []
|
simp [List.append]
|
no goals
|
c00518f72b14f35d
|
BitVec.toInt_abs_eq_ite
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem toInt_abs_eq_ite {x : BitVec w} :
x.abs.toInt =
if x = intMin w then (intMin w).toInt
else if x.msb then -x.toInt
else x.toInt
|
w : Nat
x : BitVec w
⊢ x.abs.toInt = if x = intMin w then (intMin w).toInt else if x.msb = true then -x.toInt else x.toInt
|
by_cases hx : x = intMin w
|
case pos
w : Nat
x : BitVec w
hx : x = intMin w
⊢ x.abs.toInt = if x = intMin w then (intMin w).toInt else if x.msb = true then -x.toInt else x.toInt
case neg
w : Nat
x : BitVec w
hx : ¬x = intMin w
⊢ x.abs.toInt = if x = intMin w then (intMin w).toInt else if x.msb = true then -x.toInt else x.toInt
|
27d1365952689b50
|
HomologicalComplex.d_comp_XIsoOfEq_hom
|
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
@[reassoc (attr := simp)]
lemma d_comp_XIsoOfEq_hom (K : HomologicalComplex V c) {p₂ p₃ : ι} (h : p₂ = p₃) (p₁ : ι) :
K.d p₁ p₂ ≫ (K.XIsoOfEq h).hom = K.d p₁ p₃
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K : HomologicalComplex V c
p₂ p₃ : ι
h : p₂ = p₃
p₁ : ι
⊢ K.d p₁ p₂ ≫ (K.XIsoOfEq h).hom = K.d p₁ p₃
|
subst h
|
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
K : HomologicalComplex V c
p₂ p₁ : ι
⊢ K.d p₁ p₂ ≫ (K.XIsoOfEq ⋯).hom = K.d p₁ p₂
|
5786182a83bb6e15
|
Lean.Data.AC.Context.evalList_append
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean
|
theorem Context.evalList_append
(ctx : Context α)
(l r : List Nat)
(h₁ : l ≠ [])
(h₂ : r ≠ [])
: evalList α ctx (l.append r) = ctx.op (evalList α ctx l) (evalList α ctx r)
|
case single
α : Sort u_1
ctx : Context α
r : List Nat
h₂ : r ≠ []
x : Nat
h₁ : [x] ≠ []
⊢ evalList α ctx ([x].append r) = ctx.op (evalList α ctx [x]) (evalList α ctx r)
|
cases r
|
case single.nil
α : Sort u_1
ctx : Context α
x : Nat
h₁ : [x] ≠ []
h₂ : [] ≠ []
⊢ evalList α ctx ([x].append []) = ctx.op (evalList α ctx [x]) (evalList α ctx [])
case single.cons
α : Sort u_1
ctx : Context α
x : Nat
h₁ : [x] ≠ []
head✝ : Nat
tail✝ : List Nat
h₂ : head✝ :: tail✝ ≠ []
⊢ evalList α ctx ([x].append (head✝ :: tail✝)) = ctx.op (evalList α ctx [x]) (evalList α ctx (head✝ :: tail✝))
|
9239bcb9323c7bb8
|
PadicSeq.add_eq_max_of_ne
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
theorem add_eq_max_of_ne {f g : PadicSeq p} (hfgne : f.norm ≠ g.norm) :
(f + g).norm = max f.norm g.norm
|
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfgne : f.norm ≠ g.norm
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : g ≈ 0
this✝ : (g - 0).LimZero
this : f + g ≈ f
h1 : (f + g).norm = f.norm
⊢ (f + g).norm = f.norm ⊔ g.norm
|
have h2 : g.norm = 0 := (norm_zero_iff _).2 hg
|
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfgne : f.norm ≠ g.norm
hfg : ¬f + g ≈ 0
hf : ¬f ≈ 0
hg : g ≈ 0
this✝ : (g - 0).LimZero
this : f + g ≈ f
h1 : (f + g).norm = f.norm
h2 : g.norm = 0
⊢ (f + g).norm = f.norm ⊔ g.norm
|
70e67c5122e56027
|
IsCyclic.exists_ofOrder_eq_natCard
|
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
@[to_additive]
lemma IsCyclic.exists_ofOrder_eq_natCard [h : IsCyclic α] : ∃ g : α, orderOf g = Nat.card α
|
case h
α : Type u_1
inst✝ : Group α
h : IsCyclic α
g : α
hg : ∀ (x : α), x ∈ zpowers g
⊢ orderOf g = Nat.card α
|
rw [← card_zpowers g, (eq_top_iff' (zpowers g)).mpr hg]
|
case h
α : Type u_1
inst✝ : Group α
h : IsCyclic α
g : α
hg : ∀ (x : α), x ∈ zpowers g
⊢ Nat.card ↥⊤ = Nat.card α
|
c40cde1ffa129f52
|
Real.summable_log_one_add_of_summable
|
Mathlib/Analysis/SpecialFunctions/Log/Summable.lean
|
lemma Real.summable_log_one_add_of_summable {f : ι → ℝ} (hf : Summable f) :
Summable (fun i : ι => log (1 + |f i|))
|
ι : Type u_1
f : ι → ℝ
hf : Summable f
⊢ Summable fun i => log (1 + |f i|)
|
have : Summable (fun n ↦ Complex.ofRealCLM (log (1 + |f n|))) := by
convert Complex.summable_log_one_add_of_summable (Complex.ofRealCLM.summable hf.norm) with x
rw [ofRealCLM_apply, ofReal_log (by positivity)]
simp only [ofReal_add, ofReal_one, norm_eq_abs, ofRealCLM_apply]
|
ι : Type u_1
f : ι → ℝ
hf : Summable f
this : Summable fun n => ofRealCLM (log (1 + |f n|))
⊢ Summable fun i => log (1 + |f i|)
|
4db2ca3d7bda0ab0
|
LieAlgebra.IsSemisimple.isSimple_of_isAtom
|
Mathlib/Algebra/Lie/Semisimple/Basic.lean
|
lemma isSimple_of_isAtom (I : LieIdeal R L) (hI : IsAtom I) : IsSimple R I where
non_abelian := IsSemisimple.non_abelian_of_isAtom I hI
eq_bot_or_eq_top
|
case intro.intro.intro.intro.intro.intro.a
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
y : ↥I
hy : y ∈ ↑↑J
a : L
ha : a ∈ I
b : L
hb : b ∈ sSup ({I' | IsAtom I'} \ {I})
⊢ ∃ a_1, ∃ (b : a_1 ∈ I), ⟨a_1, b⟩ ∈ J ∧ (↑I).subtype ⟨a_1, b⟩ = ⁅a, (↑I).subtype y⁆
|
erw [Submodule.coe_subtype]
|
case intro.intro.intro.intro.intro.intro.a
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsSemisimple R L
I : LieIdeal R L
hI : IsAtom I
J : LieIdeal R ↥I
y : ↥I
hy : y ∈ ↑↑J
a : L
ha : a ∈ I
b : L
hb : b ∈ sSup ({I' | IsAtom I'} \ {I})
⊢ ∃ a_1, ∃ (b : a_1 ∈ I), ⟨a_1, b⟩ ∈ J ∧ ↑⟨a_1, b⟩ = ⁅a, ↑y⁆
|
94384633946b96a7
|
Int.abs_sub_lt_one_of_floor_eq_floor
|
Mathlib/Algebra/Order/Floor.lean
|
theorem abs_sub_lt_one_of_floor_eq_floor {α : Type*} [LinearOrderedCommRing α] [FloorRing α]
{a b : α} (h : ⌊a⌋ = ⌊b⌋) : |a - b| < 1
|
α : Type u_4
inst✝¹ : LinearOrderedCommRing α
inst✝ : FloorRing α
a b : α
h : ⌊a⌋ = ⌊b⌋
this✝² : a < ↑⌊a⌋ + 1
this✝¹ : b < ↑⌊b⌋ + 1
this✝ : ↑⌊a⌋ = ↑⌊b⌋
this : ↑⌊a⌋ ≤ a
⊢ |a - b| < 1
|
have : (⌊b⌋ : α) ≤ b := floor_le b
|
α : Type u_4
inst✝¹ : LinearOrderedCommRing α
inst✝ : FloorRing α
a b : α
h : ⌊a⌋ = ⌊b⌋
this✝³ : a < ↑⌊a⌋ + 1
this✝² : b < ↑⌊b⌋ + 1
this✝¹ : ↑⌊a⌋ = ↑⌊b⌋
this✝ : ↑⌊a⌋ ≤ a
this : ↑⌊b⌋ ≤ b
⊢ |a - b| < 1
|
11281188035b59c7
|
IsIdempotentElem.mul_one_sub_self
|
Mathlib/Algebra/Ring/Idempotent.lean
|
@[simp]
lemma mul_one_sub_self (h : IsIdempotentElem a) : a * (1 - a) = 0
|
R : Type u_1
inst✝ : NonAssocRing R
a : R
h : IsIdempotentElem a
⊢ a * (1 - a) = 0
|
rw [mul_sub, mul_one, h.eq, sub_self]
|
no goals
|
b93c714e81a1fc1c
|
Module.exists_smul_eq_zero_and_mk_eq
|
Mathlib/Algebra/Module/PID.lean
|
theorem exists_smul_eq_zero_and_mk_eq {z : M} (hz : Module.IsTorsionBy R M (p ^ pOrder hM z))
{k : ℕ} (f : (R ⧸ R ∙ p ^ k) →ₗ[R] M ⧸ R ∙ z) :
∃ x : M, p ^ k • x = 0 ∧ Submodule.Quotient.mk (p := span R {z}) x = f 1
|
case h.e'_2.h.e'_6
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsPrincipalIdealRing R
M : Type v
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsDomain R
p : R
hp : Irreducible p
hM : IsTorsion' M ↥(Submonoid.powers p)
dec : (x : M) → Decidable (x = 0)
z : M
hz : IsTorsionBy R M (p ^ pOrder hM z)
k : ℕ
f : R ⧸ Submodule.span R {p ^ k} →ₗ[R] M ⧸ Submodule.span R {z}
f1 : ∃ a, Submodule.Quotient.mk a = f 1
⊢ p ^ k ∈ Submodule.span R {p ^ k}
|
exact Submodule.mem_span_singleton_self _
|
no goals
|
868eb814ce84c0fb
|
PreconnectedSpace.induction₂'
|
Mathlib/Topology/Connected/Clopen.lean
|
/-- In a preconnected space, given a transitive relation `P`, if `P x y` and `P y x` are true
for `y` close enough to `x`, then `P x y` holds for all `x, y`. This is a version of the fact
that, if an equivalence relation has open classes, then it has a single equivalence class. -/
lemma PreconnectedSpace.induction₂' [PreconnectedSpace α] (P : α → α → Prop)
(h : ∀ x, ∀ᶠ y in 𝓝 x, P x y ∧ P y x) (h' : Transitive P) (x y : α) :
P x y
|
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
P : α → α → Prop
h : ∀ (x : α), ∀ᶠ (y : α) in 𝓝 x, P x y ∧ P y x
h' : Transitive P
x y : α
⊢ P x y
|
let u := {z | P x z}
|
α : Type u
inst✝¹ : TopologicalSpace α
inst✝ : PreconnectedSpace α
P : α → α → Prop
h : ∀ (x : α), ∀ᶠ (y : α) in 𝓝 x, P x y ∧ P y x
h' : Transitive P
x y : α
u : Set α := {z | P x z}
⊢ P x y
|
97b52bfa4f7b2cfd
|
Vector.reverse_pmap
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Attach.lean
|
theorem reverse_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : Vector α n)
(H : ∀ (a : α), a ∈ xs → P a) :
(xs.pmap f H).reverse = xs.reverse.pmap f (fun a h => H a (by simpa using h))
|
α : Type u_1
β : Type u_2
n : Nat
P : α → Prop
f : (a : α) → P a → β
xs : Vector α n
H : ∀ (a : α), a ∈ xs → P a
⊢ (pmap f xs H).reverse = pmap f xs.reverse ⋯
|
rw [pmap_reverse]
|
no goals
|
2da462a799da24ae
|
ZMod.cast_neg_one
|
Mathlib/Data/ZMod/Basic.lean
|
theorem cast_neg_one {R : Type*} [Ring R] (n : ℕ) : cast (-1 : ZMod n) = (n - 1 : R)
|
case succ
R : Type u_1
inst✝ : Ring R
n : ℕ
⊢ (-1).cast = ↑(n + 1) - 1
|
rw [← natCast_val, val_neg_one, Nat.cast_succ, add_sub_cancel_right]
|
no goals
|
5d13c9152d6ff8e4
|
Polynomial.derivativeFinsupp_one
|
Mathlib/Algebra/Polynomial/Derivative.lean
|
theorem derivativeFinsupp_one : derivativeFinsupp (1 : R[X]) = .single 0 1
|
R : Type u
inst✝ : Semiring R
⊢ derivativeFinsupp 1 = Finsupp.single 0 1
|
simpa using derivativeFinsupp_C (1 : R)
|
no goals
|
8ebb754427a3b93b
|
OrthonormalBasis.orthonormal
|
Mathlib/Analysis/InnerProductSpace/PiL2.lean
|
theorem orthonormal (b : OrthonormalBasis ι 𝕜 E) : Orthonormal 𝕜 b
|
ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
⊢ ∀ (i j : ι), inner (b i) (b j) = if i = j then 1 else 0
|
intro i j
|
ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
i j : ι
⊢ inner (b i) (b j) = if i = j then 1 else 0
|
1605b523466c7150
|
MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv
|
Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
|
theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
(hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
{p : ℝ} (hp : Real.IsConjExponent (finrank ℝ E) p) :
∫⁻ x, ‖u x‖ₑ ^ p ∂μ ≤
lintegralPowLePowLIntegralFDerivConst μ p * (∫⁻ x, ‖fderiv ℝ u x‖ₑ ∂μ) ^ p
|
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).IsConjExponent p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝
this : (Measure.map (⇑e.symm) volume).IsAddHaarMeasure
hp : (↑#ι).IsConjExponent p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
⊢ Measurable fun x => ‖u x‖ₑ ^ p
|
borelize F
|
F : Type u_3
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
u : E → F
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p : ℝ
hp✝ : (↑(finrank ℝ E)).IsConjExponent p
C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p
ι : Type := Fin (finrank ℝ E)
hιcard : #ι = finrank ℝ E
this✝² : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝²
this : (Measure.map (⇑e.symm) volume).IsAddHaarMeasure
hp : (↑#ι).IsConjExponent p
h0p : 0 ≤ p
c : ℝ≥0 := μ.addHaarScalarFactor (Measure.map (⇑e.symm) volume)
hc : 0 < c
h2c : μ = c • Measure.map (⇑e.symm) volume
h3c : ↑c ≠ 0
h0C : C = c * ‖↑e.symm‖₊ ^ p * (c ^ p)⁻¹
hC : C * c ^ p = c * ‖↑e.symm‖₊ ^ p
v : (ι → ℝ) → F := u ∘ ⇑e.symm
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
this✝¹ : MeasurableSpace F := borel F
this✝ : BorelSpace F
⊢ Measurable fun x => ‖u x‖ₑ ^ p
|
7e7c86c82b1c72d8
|
PadicSeq.norm_nonarchimedean
|
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
theorem norm_nonarchimedean (f g : PadicSeq p) : (f + g).norm ≤ max f.norm g.norm
|
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : f ≈ 0
hfg' : f + g ≈ g
hcfg : (f + g).norm = g.norm
hcl : f.norm = 0
⊢ f.norm ⊔ g.norm = g.norm
|
rw [hcl]
|
p : ℕ
hp : Fact (Nat.Prime p)
f g : PadicSeq p
hfg : ¬f + g ≈ 0
hf : f ≈ 0
hfg' : f + g ≈ g
hcfg : (f + g).norm = g.norm
hcl : f.norm = 0
⊢ 0 ⊔ g.norm = g.norm
|
4a919ac1cede5d3f
|
MeasureTheory.lintegral_iSup
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
⊢ ∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ
|
set F := fun a : α => ⨆ n, f n a
|
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ≥0∞
hf : ∀ (n : ℕ), Measurable (f n)
h_mono : Monotone f
c : ℝ≥0 → ℝ≥0∞ := ofNNReal
F : α → ℝ≥0∞ := fun a => ⨆ n, f n a
⊢ lintegral μ F = ⨆ n, ∫⁻ (a : α), f n a ∂μ
|
e37d047cd3bb7874
|
PresentedMonoid.ext
|
Mathlib/Algebra/PresentedMonoid/Basic.lean
|
theorem ext {M : Type*} [Monoid M] (rels : FreeMonoid α → FreeMonoid α → Prop)
{φ ψ : PresentedMonoid rels →* M} (hx : ∀ (x : α), φ (.of rels x) = ψ (.of rels x)) :
φ = ψ
|
case h
α : Type u_2
M : Type u_3
inst✝ : Monoid M
rels : FreeMonoid α → FreeMonoid α → Prop
φ ψ : PresentedMonoid rels →* M
hx : ∀ (x : α), φ (of rels x) = ψ (of rels x)
x✝ : α
⊢ φ (of rels x✝) = (⇑ψ ∘ of rels) x✝
|
exact hx _
|
no goals
|
4e13ad6c6062fd98
|
antivary_inv_right₀
|
Mathlib/Algebra/Order/Monovary.lean
|
@[simp] lemma antivary_inv_right₀ (hg : StrongLT 0 g) : Antivary f g⁻¹ ↔ Monovary f g :=
forall_swap.trans <| forall₂_congr fun i j ↦ by simp [inv_lt_inv₀ (hg _) (hg _)]
|
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedSemifield α
inst✝ : LinearOrderedSemifield β
f : ι → α
g : ι → β
hg : StrongLT 0 g
i j : ι
⊢ g⁻¹ j < g⁻¹ i → f i ≤ f j ↔ g i < g j → f i ≤ f j
|
simp [inv_lt_inv₀ (hg _) (hg _)]
|
no goals
|
2da24a0e14981587
|
ZMod.val_neg_one
|
Mathlib/Data/ZMod/Basic.lean
|
theorem val_neg_one (n : ℕ) : (-1 : ZMod n.succ).val = n
|
case zero
⊢ ↑(-1) = 0
|
simp [Nat.mod_one]
|
no goals
|
93b3ed05716419f6
|
GromovHausdorff.candidates_lipschitz_aux
|
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
theorem candidates_lipschitz_aux (fA : f ∈ candidates X Y) :
f (x, y) - f (z, t) ≤ 2 * maxVar X Y * dist (x, y) (z, t) :=
calc
f (x, y) - f (z, t) ≤ f (x, t) + f (t, y) - f (z, t)
|
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
f : GromovHausdorff.ProdSpaceFun X Y
x y z t : X ⊕ Y
fA : f ∈ candidates X Y
⊢ f (x, z) + f (t, y) ≤ ↑(GromovHausdorff.maxVar X Y) * dist x z + ↑(GromovHausdorff.maxVar X Y) * dist t y
|
gcongr <;> apply candidates_dist_bound fA
|
no goals
|
1bc3e613166ef4b1
|
MvPolynomial.finSuccEquiv_eq
|
Mathlib/Algebra/MvPolynomial/Equiv.lean
|
theorem finSuccEquiv_eq :
(finSuccEquiv R n : MvPolynomial (Fin (n + 1)) R →+* Polynomial (MvPolynomial (Fin n) R)) =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R)) fun i : Fin (n + 1) =>
Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i
|
case hC.a
R : Type u
inst✝ : CommSemiring R
n : ℕ
i : R
⊢ ((↑(finSuccEquiv R n)).comp C) i =
((eval₂Hom (Polynomial.C.comp C) fun i => Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i).comp C) i
|
simp only [finSuccEquiv, optionEquivLeft_apply, aeval_C, AlgEquiv.coe_trans, RingHom.coe_coe,
coe_eval₂Hom, comp_apply, renameEquiv_apply, eval₂_C, RingHom.coe_comp, rename_C]
|
case hC.a
R : Type u
inst✝ : CommSemiring R
n : ℕ
i : R
⊢ (algebraMap R (MvPolynomial (Fin n) R)[X]) i = Polynomial.C (C i)
|
848c49066f7d034b
|
ContinuousMultilinearMap.changeOrigin_toFormalMultilinearSeries
|
Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
|
theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ι] :
continuousMultilinearCurryFin1 𝕜 (∀ i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
f.linearDeriv x
|
case h.inr.refine_3.h.e_6.h.h.inl
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
x : (i : ι) → E i
inst✝ : DecidableEq ι
y : (i : ι) → E i
h✝ : Nonempty ι
j : ι
⊢ Function.update x j (y j) j = if j = j then y j else x j
|
rw [Function.update_self, if_pos rfl]
|
no goals
|
d7dded01cf2a4e1b
|
Matrix.mulVec_injective_iff_isUnit
|
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
theorem mulVec_injective_iff_isUnit {A : Matrix m m K} :
Function.Injective A.mulVec ↔ IsUnit A
|
m : Type u
inst✝² : DecidableEq m
K : Type u_3
inst✝¹ : Field K
inst✝ : Fintype m
A : Matrix m m K
⊢ Function.Injective A.mulVec ↔ Function.Injective fun v => v ᵥ* Aᵀ
|
simp_rw [vecMul_transpose]
|
no goals
|
42648d76e6284bc4
|
Mathlib.Meta.NormNum.isInt_ediv
|
Mathlib/Tactic/NormNum/DivMod.lean
|
lemma isInt_ediv {a b q m a' : ℤ} {b' r : ℕ}
(ha : IsInt a a') (hb : IsNat b b')
(hm : q * b' = m) (h : r + m = a') (h₂ : Nat.blt r b' = true) :
IsInt (a / b) q := ⟨by
obtain ⟨⟨rfl⟩, ⟨rfl⟩⟩ := ha, hb
simp only [Nat.blt_eq] at h₂; simp only [← h, ← hm, Int.cast_id]
rw [Int.add_mul_ediv_right _ _ (Int.ofNat_ne_zero.2 ((Nat.zero_le ..).trans_lt h₂).ne')]
rw [Int.ediv_eq_zero_of_lt, zero_add] <;> [simp; simpa using h₂]⟩
|
a b q m a' : ℤ
b' r : ℕ
ha : IsInt a a'
hb : IsNat b b'
hm : q * ↑b' = m
h : ↑r + m = a'
h₂ : r.blt b' = true
⊢ a / b = ↑q
|
obtain ⟨⟨rfl⟩, ⟨rfl⟩⟩ := ha, hb
|
case mk.mk
a q m : ℤ
b' r : ℕ
hm : q * ↑b' = m
h₂ : r.blt b' = true
h : ↑r + m = a
⊢ a / ↑b' = ↑q
|
9ce47b0eb26c011f
|
Lake.BuildKey.eq_of_quickCmp
|
Mathlib/.lake/packages/lean4/src/lean/lake/Lake/Build/Key.lean
|
theorem eq_of_quickCmp {k k' : BuildKey} :
quickCmp k k' = Ordering.eq → k = k'
|
case customTarget.targetFacet
p t package✝ target✝ facet✝ : Name
⊢ (match customTarget p t with
| moduleFacet m f =>
match targetFacet package✝ target✝ facet✝ with
| moduleFacet m' f' =>
match m.quickCmp m' with
| Ordering.eq => f.quickCmp f'
| ord => ord
| x => Ordering.lt
| packageFacet p f =>
match targetFacet package✝ target✝ facet✝ with
| moduleFacet module facet => Ordering.gt
| packageFacet p' f' =>
match p.quickCmp p' with
| Ordering.eq => f.quickCmp f'
| ord => ord
| x => Ordering.lt
| targetFacet p t f =>
match targetFacet package✝ target✝ facet✝ with
| customTarget package target => Ordering.lt
| targetFacet p' t' f' =>
match p.quickCmp p' with
| Ordering.eq =>
match t.quickCmp t' with
| Ordering.eq => f.quickCmp f'
| ord => ord
| ord => ord
| x => Ordering.gt
| customTarget p t =>
match targetFacet package✝ target✝ facet✝ with
| customTarget p' t' =>
match p.quickCmp p' with
| Ordering.eq => t.quickCmp t'
| ord => ord
| x => Ordering.gt) =
Ordering.eq →
customTarget p t = targetFacet package✝ target✝ facet✝
|
intro
|
case customTarget.targetFacet
p t package✝ target✝ facet✝ : Name
a✝ :
(match customTarget p t with
| moduleFacet m f =>
match targetFacet package✝ target✝ facet✝ with
| moduleFacet m' f' =>
match m.quickCmp m' with
| Ordering.eq => f.quickCmp f'
| ord => ord
| x => Ordering.lt
| packageFacet p f =>
match targetFacet package✝ target✝ facet✝ with
| moduleFacet module facet => Ordering.gt
| packageFacet p' f' =>
match p.quickCmp p' with
| Ordering.eq => f.quickCmp f'
| ord => ord
| x => Ordering.lt
| targetFacet p t f =>
match targetFacet package✝ target✝ facet✝ with
| customTarget package target => Ordering.lt
| targetFacet p' t' f' =>
match p.quickCmp p' with
| Ordering.eq =>
match t.quickCmp t' with
| Ordering.eq => f.quickCmp f'
| ord => ord
| ord => ord
| x => Ordering.gt
| customTarget p t =>
match targetFacet package✝ target✝ facet✝ with
| customTarget p' t' =>
match p.quickCmp p' with
| Ordering.eq => t.quickCmp t'
| ord => ord
| x => Ordering.gt) =
Ordering.eq
⊢ customTarget p t = targetFacet package✝ target✝ facet✝
|
6e2940eeabc7d427
|
exists_gt_t2space
|
Mathlib/Topology/ShrinkingLemma.lean
|
theorem exists_gt_t2space (v : PartialRefinement u s (fun w => IsCompact (closure w)))
(hs : IsCompact s) (i : ι) (hi : i ∉ v.carrier) :
∃ v' : PartialRefinement u s (fun w => IsCompact (closure w)),
v < v' ∧ IsCompact (closure (v' i))
|
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w => IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
⊢ ∃ v', v < v' ∧ IsCompact (closure (v'.toFun i))
|
have : si ⊆ v i := by
intro x hx
have (j) (hj : j ≠ i) : x ∉ v j := by
rw [hsi] at hx
apply Set.not_mem_of_mem_compl
have hsi' : x ∈ (⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ) := Set.mem_of_mem_inter_right hx
rw [ne_eq] at hj
rw [Set.mem_iInter₂] at hsi'
exact hsi' j hj
obtain ⟨j, hj⟩ := Set.mem_iUnion.mp
(v.subset_iUnion (Set.mem_of_mem_inter_left hx))
obtain rfl : j = i := by
by_contra! h
exact this j h hj
exact hj
|
ι : Type u_1
X : Type u_2
inst✝² : TopologicalSpace X
u : ι → Set X
s : Set X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
v : PartialRefinement u s fun w => IsCompact (closure w)
hs : IsCompact s
i : ι
hi : i ∉ v.carrier
si : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ
hsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 = i), (v.toFun i_1)ᶜ
hsic : IsCompact si
this : si ⊆ v.toFun i
⊢ ∃ v', v < v' ∧ IsCompact (closure (v'.toFun i))
|
b61f3fede2ad34cb
|
Real.posLog_sub_posLog_inv
|
Mathlib/Analysis/SpecialFunctions/Log/PosLog.lean
|
theorem posLog_sub_posLog_inv {r : ℝ} : log⁺ r - log⁺ r⁻¹ = log r
|
case pos
r : ℝ
h : 0 ≤ log r
⊢ 0 ⊔ log r - 0 ⊔ -log r = log r
|
simp [h]
|
no goals
|
78a5076ba1e60175
|
Stream'.WSeq.destruct_ofSeq
|
Mathlib/Data/Seq/WSeq.lean
|
theorem destruct_ofSeq (s : Seq α) :
destruct (ofSeq s) = Computation.pure (s.head.map fun a => (a, ofSeq s.tail)) :=
destruct_eq_pure <| by
simp only [destruct, Seq.destruct, Option.map_eq_map, ofSeq, Computation.corec_eq, rmap,
Seq.head]
rw [show Seq.get? (some <$> s) 0 = some <$> Seq.get? s 0 by apply Seq.map_get?]
rcases Seq.get? s 0 with - | a
· rfl
dsimp only [(· <$> ·)]
simp [destruct]
|
α : Type u
s : Seq α
⊢ (↑s).destruct.destruct = Sum.inl (Option.map (fun a => (a, ↑s.tail)) s.head)
|
simp only [destruct, Seq.destruct, Option.map_eq_map, ofSeq, Computation.corec_eq, rmap,
Seq.head]
|
α : Type u
s : Seq α
⊢ (match
match Option.map (fun a' => (a', (some <$> s).tail)) ((some <$> s).get? 0) with
| none => Sum.inl none
| some (none, s') => Sum.inr s'
| some (some a, s') => Sum.inl (some (a, s')) with
| Sum.inl a => Sum.inl a
| Sum.inr b =>
Sum.inr
(Computation.corec
(fun s =>
match Option.map (fun a' => (a', Seq.tail s)) (Seq.get? s 0) with
| none => Sum.inl none
| some (none, s') => Sum.inr s'
| some (some a, s') => Sum.inl (some (a, s')))
b)) =
Sum.inl (Option.map (fun a => (a, some <$> s.tail)) (s.get? 0))
|
99deefc9eb9056e3
|
ZNum.cast_succ
|
Mathlib/Data/Num/Lemmas.lean
|
theorem cast_succ [AddGroupWithOne α] (n) : ((succ n : ZNum) : α) = n + 1
|
α : Type u_1
inst✝ : AddGroupWithOne α
n : ZNum
⊢ ↑n.succ = ↑n + 1
|
rw [← add_one, cast_add, cast_one]
|
no goals
|
1fc5017990152a38
|
VitaliFamily.withDensity_limRatioMeas_eq
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem withDensity_limRatioMeas_eq : μ.withDensity (v.limRatioMeas hρ) = ρ
|
case refine_1
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
s : Set α
hs : MeasurableSet s
⊢ Tendsto (fun t => ↑t ^ 2) (𝓝[>] 1) (𝓝 (1 ^ 2))
|
exact ENNReal.Tendsto.pow (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds)
|
no goals
|
c98b00e271c474ed
|
ArithmeticFunction.vonMangoldt.summable_residueClass_non_primes_div
|
Mathlib/NumberTheory/LSeries/PrimesInAP.lean
|
/-- The function `n ↦ Λ n / n`, restricted to non-primes in a residue class, is summable.
This is used to convert results on `ArithmeticFunction.vonMangoldt.residueClass` to results
on primes in an arithmetic progression. -/
lemma summable_residueClass_non_primes_div :
Summable fun n : ℕ ↦ (if n.Prime then 0 else residueClass a n) / n
|
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
⊢ Summable fun n => (if Nat.Prime n then 0 else residueClass a n) / ↑n
|
have hleF₀ (n : ℕ) : (if n.Prime then 0 else residueClass a n) / n ≤ F₀ n := by
refine div_le_div_of_nonneg_right ?_ n.cast_nonneg
split_ifs; exacts [le_rfl, residueClass_le a n]
|
q : ℕ
a : ZMod q
h₀ : ∀ (n : ℕ), 0 ≤ (if Nat.Prime n then 0 else residueClass a n) / ↑n
hleF₀ : ∀ (n : ℕ), (if Nat.Prime n then 0 else residueClass a n) / ↑n ≤ ArithmeticFunction.vonMangoldt.F₀ n
⊢ Summable fun n => (if Nat.Prime n then 0 else residueClass a n) / ↑n
|
6c8eee8a1fa8d5a9
|
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