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
|
|---|---|---|---|---|---|---|
HomologicalComplex.HomologySequence.epi_homologyMap_τ₃
|
Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
|
lemma epi_homologyMap_τ₃ (i : ι)
(h₁ : Epi (homologyMap φ.τ₂ i))
(h₂ : ∀ j, c.Rel i j → Epi (homologyMap φ.τ₁ j))
(h₃ : ∀ j, c.Rel i j → Mono (homologyMap φ.τ₂ j)) :
Epi (homologyMap φ.τ₃ i)
|
case pos.intro.hR₁
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
j : ι
hij : c.Rel i j
⊢ ((δ₀Functor ⋙ δlastFunctor).obj (composableArrows₅ hS₁ i j hij)).Exact
|
exact (composableArrows₅_exact hS₁ i j hij).δ₀.δlast
|
no goals
|
92d458a0d828d6da
|
Filter.tendsto_lift'
|
Mathlib/Order/Filter/Lift.lean
|
theorem tendsto_lift' {m : γ → β} {l : Filter γ} :
Tendsto m l (f.lift' h) ↔ ∀ s ∈ f, ∀ᶠ a in l, m a ∈ h s
|
α : Type u_1
β : Type u_2
γ : Type u_3
f : Filter α
h : Set α → Set β
m : γ → β
l : Filter γ
⊢ Tendsto m l (f.lift' h) ↔ ∀ s ∈ f, ∀ᶠ (a : γ) in l, m a ∈ h s
|
simp only [Filter.lift', tendsto_lift, tendsto_principal, comp]
|
no goals
|
76aa4804ab3151d6
|
mem_affineSpan_iff_eq_affineCombination
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
theorem mem_affineSpan_iff_eq_affineCombination [Nontrivial k] {p1 : P} {p : ι → P} :
p1 ∈ affineSpan k (Set.range p) ↔
∃ (s : Finset ι) (w : ι → k), ∑ i ∈ s, w i = 1 ∧ p1 = s.affineCombination k p w
|
case mpr.intro.intro.intro
ι : Type u_1
k : Type u_2
V : Type u_3
P : Type u_4
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Nontrivial k
p : ι → P
s : Finset ι
w : ι → k
hw : ∑ i ∈ s, w i = 1
⊢ (Finset.affineCombination k s p) w ∈ affineSpan k (Set.range p)
|
exact affineCombination_mem_affineSpan hw p
|
no goals
|
0958d0945311e8cf
|
BitVec.eq_of_getLsbD_eq
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem eq_of_getLsbD_eq {x y : BitVec w}
(pred : ∀ i, i < w → x.getLsbD i = y.getLsbD i) : x = y
|
w : Nat
x y : BitVec w
pred : ∀ (i : Nat), i < w → x.getLsbD i = y.getLsbD i
⊢ x = y
|
apply eq_of_toNat_eq
|
case a
w : Nat
x y : BitVec w
pred : ∀ (i : Nat), i < w → x.getLsbD i = y.getLsbD i
⊢ x.toNat = y.toNat
|
c491b21788cdd007
|
UniformOnFun.uniformSpace_eq_inf_precomp_of_cover
|
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
theorem uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} (φ₁ : δ₁ → α) (φ₂ : δ₂ → α)
(𝔗₁ : Set (Set δ₁)) (𝔗₂ : Set (Set δ₂))
(h_image₁ : MapsTo (φ₁ '' ·) 𝔗₁ 𝔖) (h_image₂ : MapsTo (φ₂ '' ·) 𝔗₂ 𝔖)
(h_preimage₁ : MapsTo (φ₁ ⁻¹' ·) 𝔖 𝔗₁) (h_preimage₂ : MapsTo (φ₂ ⁻¹' ·) 𝔖 𝔗₂)
(h_cover : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂) :
𝒱(α, β, 𝔖, _) =
.comap (ofFun 𝔗₁ ∘ (· ∘ φ₁) ∘ toFun 𝔖) 𝒱(δ₁, β, 𝔗₁, _) ⊓
.comap (ofFun 𝔗₂ ∘ (· ∘ φ₂) ∘ toFun 𝔖) 𝒱(δ₂, β, 𝔗₂, _)
|
case refine_1
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
δ₁ : Type u_5
δ₂ : Type u_6
φ₁ : δ₁ → α
φ₂ : δ₂ → α
𝔗₁ : Set (Set δ₁)
𝔗₂ : Set (Set δ₂)
h_image₁ : MapsTo (fun x => φ₁ '' x) 𝔗₁ 𝔖
h_image₂ : MapsTo (fun x => φ₂ '' x) 𝔗₂ 𝔖
h_preimage₁ : MapsTo (fun x => φ₁ ⁻¹' x) 𝔖 𝔗₁
h_preimage₂ : MapsTo (fun x => φ₂ ⁻¹' x) 𝔖 𝔗₂
h_cover : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂
ψ₁ : (S : Set α) → ↑(φ₁ ⁻¹' S) → ↑S := fun S => S.restrictPreimage φ₁
ψ₂ : (S : Set α) → ↑(φ₂ ⁻¹' S) → ↑S := fun S => S.restrictPreimage φ₂
this :
∀ S ∈ 𝔖,
UniformFun.uniformSpace (↑S) β =
UniformSpace.comap (fun x => x ∘ ψ₁ S) (UniformFun.uniformSpace (↑(φ₁ ⁻¹' S)) β) ⊓
UniformSpace.comap (fun x => x ∘ ψ₂ S) (UniformFun.uniformSpace (↑(φ₂ ⁻¹' S)) β)
⊢ uniformSpace α β 𝔖 ≤ UniformSpace.comap (⇑(ofFun 𝔗₁) ∘ (fun x => x ∘ φ₁) ∘ ⇑(toFun 𝔖)) (uniformSpace δ₁ β 𝔗₁)
|
rw [← uniformContinuous_iff]
|
case refine_1
α : Type u_1
β : Type u_2
inst✝ : UniformSpace β
𝔖 : Set (Set α)
δ₁ : Type u_5
δ₂ : Type u_6
φ₁ : δ₁ → α
φ₂ : δ₂ → α
𝔗₁ : Set (Set δ₁)
𝔗₂ : Set (Set δ₂)
h_image₁ : MapsTo (fun x => φ₁ '' x) 𝔗₁ 𝔖
h_image₂ : MapsTo (fun x => φ₂ '' x) 𝔗₂ 𝔖
h_preimage₁ : MapsTo (fun x => φ₁ ⁻¹' x) 𝔖 𝔗₁
h_preimage₂ : MapsTo (fun x => φ₂ ⁻¹' x) 𝔖 𝔗₂
h_cover : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂
ψ₁ : (S : Set α) → ↑(φ₁ ⁻¹' S) → ↑S := fun S => S.restrictPreimage φ₁
ψ₂ : (S : Set α) → ↑(φ₂ ⁻¹' S) → ↑S := fun S => S.restrictPreimage φ₂
this :
∀ S ∈ 𝔖,
UniformFun.uniformSpace (↑S) β =
UniformSpace.comap (fun x => x ∘ ψ₁ S) (UniformFun.uniformSpace (↑(φ₁ ⁻¹' S)) β) ⊓
UniformSpace.comap (fun x => x ∘ ψ₂ S) (UniformFun.uniformSpace (↑(φ₂ ⁻¹' S)) β)
⊢ UniformContinuous (⇑(ofFun 𝔗₁) ∘ (fun x => x ∘ φ₁) ∘ ⇑(toFun 𝔖))
|
687e94ee2bcbcd2a
|
ContinuousMap.tendsto_concat
|
Mathlib/Topology/ContinuousMap/Interval.lean
|
theorem tendsto_concat {ι : Type*} {p : Filter ι} {F : ι → C(Icc a b, E)} {G : ι → C(Icc b c, E)}
(hfg : ∀ᶠ i in p, (F i) ⊤ = (G i) ⊥) (hfg' : f ⊤ = g ⊥)
(hf : Tendsto F p (𝓝 f)) (hg : Tendsto G p (𝓝 g)) :
Tendsto (fun i => concat (F i) (G i)) p (𝓝 (concat f g))
|
α : Type u_1
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
a b c : α
inst✝² : Fact (a ≤ b)
inst✝¹ : Fact (b ≤ c)
E : Type u_2
inst✝ : TopologicalSpace E
f : C(↑(Icc a b), E)
g : C(↑(Icc b c), E)
ι : Type u_3
p : Filter ι
F : ι → C(↑(Icc a b), E)
G : ι → C(↑(Icc b c), E)
hfg : ∀ᶠ (i : ι) in p, (F i) ⊤ = (G i) ⊥
hfg' : f ⊤ = g ⊥
hf :
∀ (K : Set ↑(Icc a b)),
IsCompact K → ∀ (U : Set E), IsOpen U → MapsTo (⇑f) K U → ∀ᶠ (a_4 : ι) in p, MapsTo (⇑(F a_4)) K U
hg :
∀ (K : Set ↑(Icc b c)), IsCompact K → ∀ (U : Set E), IsOpen U → MapsTo (⇑g) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(G a)) K U
K : Set ↑(Icc a c)
hK : IsCompact K
U : Set E
hU : IsOpen U
hfgU : MapsTo (⇑(f.concat g)) K U
h : b ∈ Icc a c
K₁ : Set ↑(Icc a b) := ⇑projIccCM '' (Subtype.val '' (K ∩ Iic ⟨b, h⟩))
K₂ : Set ↑(Icc b c) := ⇑projIccCM '' (Subtype.val '' (K ∩ Ici ⟨b, h⟩))
hK₁ : IsCompact K₁
hK₂ : IsCompact K₂
⊢ MapsTo (⇑f) K₁ U
|
rw [← concat_comp_IccInclusionLeft hfg']
|
α : Type u_1
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
a b c : α
inst✝² : Fact (a ≤ b)
inst✝¹ : Fact (b ≤ c)
E : Type u_2
inst✝ : TopologicalSpace E
f : C(↑(Icc a b), E)
g : C(↑(Icc b c), E)
ι : Type u_3
p : Filter ι
F : ι → C(↑(Icc a b), E)
G : ι → C(↑(Icc b c), E)
hfg : ∀ᶠ (i : ι) in p, (F i) ⊤ = (G i) ⊥
hfg' : f ⊤ = g ⊥
hf :
∀ (K : Set ↑(Icc a b)),
IsCompact K → ∀ (U : Set E), IsOpen U → MapsTo (⇑f) K U → ∀ᶠ (a_4 : ι) in p, MapsTo (⇑(F a_4)) K U
hg :
∀ (K : Set ↑(Icc b c)), IsCompact K → ∀ (U : Set E), IsOpen U → MapsTo (⇑g) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(G a)) K U
K : Set ↑(Icc a c)
hK : IsCompact K
U : Set E
hU : IsOpen U
hfgU : MapsTo (⇑(f.concat g)) K U
h : b ∈ Icc a c
K₁ : Set ↑(Icc a b) := ⇑projIccCM '' (Subtype.val '' (K ∩ Iic ⟨b, h⟩))
K₂ : Set ↑(Icc b c) := ⇑projIccCM '' (Subtype.val '' (K ∩ Ici ⟨b, h⟩))
hK₁ : IsCompact K₁
hK₂ : IsCompact K₂
⊢ MapsTo (⇑((f.concat g).comp IccInclusionLeft)) K₁ U
|
c35e3215284e9199
|
MeasureTheory.aecover_closedBall
|
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
AECover μ l (fun i ↦ Metric.closedBall x (r i)) where
measurableSet _ := Metric.isClosed_closedBall.measurableSet
ae_eventually_mem
|
α : Type u_1
ι : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝¹ : PseudoMetricSpace α
inst✝ : OpensMeasurableSpace α
x : α
r : ι → ℝ
hr : Tendsto r l atTop
⊢ ∀ᵐ (x_1 : α) ∂μ, ∀ᶠ (i : ι) in l, x_1 ∈ Metric.closedBall x (r i)
|
filter_upwards with y
|
case h
α : Type u_1
ι : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
l : Filter ι
inst✝¹ : PseudoMetricSpace α
inst✝ : OpensMeasurableSpace α
x : α
r : ι → ℝ
hr : Tendsto r l atTop
y : α
⊢ ∀ᶠ (i : ι) in l, y ∈ Metric.closedBall x (r i)
|
e1922ff051f90e65
|
ContinuousLinearMap.image_rayleigh_eq_image_rayleigh_sphere
|
Mathlib/Analysis/InnerProductSpace/Rayleigh.lean
|
theorem image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
rayleighQuotient T '' {0}ᶜ = rayleighQuotient T '' sphere 0 r
|
case h.mp.intro.intro.refine_2
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
r : ℝ
hr : 0 < r
a : ℝ
x : E
hx : x ≠ 0
hxT : T.rayleighQuotient x = a
this✝ : ‖x‖ ≠ 0
c : 𝕜 := ↑‖x‖⁻¹ * ↑r
this : c ≠ 0
⊢ T.rayleighQuotient x = a
|
exact hxT
|
no goals
|
bd99a26fc5847de3
|
AddCircle.addWellApproximable_ae_empty_or_univ
|
Mathlib/NumberTheory/WellApproximable.lean
|
theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
(∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x
|
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p => ↑(↑1 / ↑↑p * T)
⊢ ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
|
rintro ⟨p, hp⟩
|
case mk
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p => ↑(↑1 / ↑↑p * T)
p : ℕ
hp : Nat.Prime p
⊢ addOrderOf (u ⟨p, hp⟩) = ↑⟨p, hp⟩
|
aeca070d334a419e
|
isOpen_pi_iff
|
Mathlib/Topology/Constructions.lean
|
theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
IsOpen s ↔
∀ f, f ∈ s → ∃ (I : Finset ι) (u : ∀ a, Set (π a)),
(∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s
|
case refine_1.intro.intro.intro.refine_1
ι : Type u_5
π : ι → Type u_6
T : (i : ι) → TopologicalSpace (π i)
s : Set ((a : ι) → π a)
a : (a : ι) → π a
x✝ : a ∈ s
I : Finset ι
t : (i : ι) → Set (π i)
h1 : ∀ (i : ι), ∃ t_1 ⊆ t i, IsOpen t_1 ∧ a i ∈ t_1
h2 : (↑I).pi t ⊆ s
i : ι
hi : i ∈ I
⊢ IsOpen ⋯.choose ∧ a i ∈ ⋯.choose
|
exact (h1 i).choose_spec.2
|
no goals
|
aadda838155adaf1
|
Partrec.sumCasesOn_left
|
Mathlib/Computability/Partrec.lean
|
theorem sumCasesOn_left {f : α → β ⊕ γ} {g : α → β →. σ} {h : α → γ → σ} (hf : Computable f)
(hg : Partrec₂ g) (hh : Computable₂ h) :
@Partrec _ σ _ _ fun a => Sum.casesOn (f a) (g a) fun c => Part.some (h a c) :=
(sumCasesOn_right (sumCasesOn hf (sumInr.comp snd).to₂ (sumInl.comp snd).to₂) hh hg).of_eq
fun a => by cases f a <;> simp
|
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_4
inst✝³ : Primcodable α
inst✝² : Primcodable β
inst✝¹ : Primcodable γ
inst✝ : Primcodable σ
f : α → β ⊕ γ
g : α → β →. σ
h : α → γ → σ
hf : Computable f
hg : Partrec₂ g
hh : Computable₂ h
a : α
⊢ Sum.casesOn (Sum.casesOn (f a) (fun b => Sum.inr (a, b).2) fun b => Sum.inl (a, b).2) (fun b => Part.some (h a b))
(g a) =
Sum.casesOn (f a) (g a) fun c => Part.some (h a c)
|
cases f a <;> simp
|
no goals
|
ca9a891f7dbff9f0
|
NNReal.add_rpow_le_one_of_add_le_one
|
Mathlib/Analysis/MeanInequalitiesPow.lean
|
theorem add_rpow_le_one_of_add_le_one {p : ℝ} (a b : ℝ≥0) (hab : a + b ≤ 1) (hp1 : 1 ≤ p) :
a ^ p + b ^ p ≤ 1
|
p : ℝ
a b : ℝ≥0
hab : a + b ≤ 1
hp1 : 1 ≤ p
h_le_one : ∀ x ≤ 1, x ^ p ≤ x
ha : a ≤ 1
⊢ a ^ p + b ^ p ≤ 1
|
have hb : b ≤ 1 := (self_le_add_left b a).trans hab
|
p : ℝ
a b : ℝ≥0
hab : a + b ≤ 1
hp1 : 1 ≤ p
h_le_one : ∀ x ≤ 1, x ^ p ≤ x
ha : a ≤ 1
hb : b ≤ 1
⊢ a ^ p + b ^ p ≤ 1
|
21837d813a208442
|
Polynomial.Monic.mul_left_eq_zero_iff
|
Mathlib/Algebra/Polynomial/Monic.lean
|
theorem Monic.mul_left_eq_zero_iff (h : Monic p) {q : R[X]} : q * p = 0 ↔ q = 0
|
R : Type u
inst✝ : Semiring R
p : R[X]
h : p.Monic
q : R[X]
⊢ q * p = 0 ↔ q = 0
|
by_cases hq : q = 0 <;> simp [h.mul_left_ne_zero, hq]
|
no goals
|
5f9daa1e5ac6cc25
|
nodup_permsOfList
|
Mathlib/Data/Fintype/Perm.lean
|
theorem nodup_permsOfList : ∀ {l : List α}, l.Nodup → (permsOfList l).Nodup
| [], _ => by simp [permsOfList]
| a :: l, hl => by
have hl' : l.Nodup := hl.of_cons
have hln' : (permsOfList l).Nodup := nodup_permsOfList hl'
have hmeml : ∀ {f : Perm α}, f ∈ permsOfList l → f a = a := fun {f} hf =>
not_not.1 (mt (mem_of_mem_permsOfList hf _) (nodup_cons.1 hl).1)
rw [permsOfList, List.nodup_append, List.nodup_flatMap, pairwise_iff_getElem]
refine ⟨?_, ⟨⟨?_,?_ ⟩, ?_⟩⟩
· exact hln'
· exact fun _ _ => hln'.map fun _ _ => mul_left_cancel
· intros i j hi hj hij x hx₁ hx₂
let ⟨f, hf⟩ := List.mem_map.1 hx₁
let ⟨g, hg⟩ := List.mem_map.1 hx₂
have hix : x a = l[i]
|
α : Type u_1
inst✝ : DecidableEq α
a : α
l : List α
hl : (a :: l).Nodup
hl' : l.Nodup
⊢ (permsOfList (a :: l)).Nodup
|
have hln' : (permsOfList l).Nodup := nodup_permsOfList hl'
|
α : Type u_1
inst✝ : DecidableEq α
a : α
l : List α
hl : (a :: l).Nodup
hl' : l.Nodup
hln' : (permsOfList l).Nodup
⊢ (permsOfList (a :: l)).Nodup
|
1438bc15e355e5ea
|
Ordnode.size_erase_of_mem
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem size_erase_of_mem [DecidableRel (α := α) (· ≤ ·)] {x : α} {t a₁ a₂} (h : Valid' a₁ t a₂)
(h_mem : x ∈ t) : size (erase x t) = size t - 1
|
case node.lt.intro
α : Type u_1
inst✝¹ : Preorder α
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
x : α
size✝ : ℕ
t_l : Ordnode α
t_x : α
t_r : Ordnode α
a₁ : WithBot α
a₂ : WithTop α
h : Valid' a₁ (node size✝ t_l t_x t_r) a₂
h_mem : mem x t_l = true
t_ih_l : (erase x t_l).size = t_l.size - 1
t_l_valid : Valid' a₁ (erase x t_l) ↑t_x
t_l_size : Raised (erase x t_l).size t_l.size
⊢ 0 < t_l.size → t_l.size - 1 + t_r.size + 1 = (node (t_l.size + t_r.size + 1) t_l t_x t_r).size - 1
|
rcases t_l.size with - | t_l_size <;> intro h_pos_t_l_size
|
case node.lt.intro.zero
α : Type u_1
inst✝¹ : Preorder α
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
x : α
size✝ : ℕ
t_l : Ordnode α
t_x : α
t_r : Ordnode α
a₁ : WithBot α
a₂ : WithTop α
h : Valid' a₁ (node size✝ t_l t_x t_r) a₂
h_mem : mem x t_l = true
t_ih_l : (erase x t_l).size = t_l.size - 1
t_l_valid : Valid' a₁ (erase x t_l) ↑t_x
t_l_size : Raised (erase x t_l).size t_l.size
h_pos_t_l_size : 0 < 0
⊢ 0 - 1 + t_r.size + 1 = (node (0 + t_r.size + 1) t_l t_x t_r).size - 1
case node.lt.intro.succ
α : Type u_1
inst✝¹ : Preorder α
inst✝ : DecidableRel fun x1 x2 => x1 ≤ x2
x : α
size✝ : ℕ
t_l : Ordnode α
t_x : α
t_r : Ordnode α
a₁ : WithBot α
a₂ : WithTop α
h : Valid' a₁ (node size✝ t_l t_x t_r) a₂
h_mem : mem x t_l = true
t_ih_l : (erase x t_l).size = t_l.size - 1
t_l_valid : Valid' a₁ (erase x t_l) ↑t_x
t_l_size✝ : Raised (erase x t_l).size t_l.size
t_l_size : ℕ
h_pos_t_l_size : 0 < t_l_size + 1
⊢ t_l_size + 1 - 1 + t_r.size + 1 = (node (t_l_size + 1 + t_r.size + 1) t_l t_x t_r).size - 1
|
53892aba6b42beb8
|
Language.IsRegular.finite_range_leftQuotient
|
Mathlib/Computability/MyhillNerode.lean
|
theorem IsRegular.finite_range_leftQuotient (h : L.IsRegular) :
(Set.range L.leftQuotient).Finite
|
α : Type u
L : Language α
h : L.IsRegular
σ : Type
x : Fintype σ
M : DFA α σ
hM : M.accepts = L
⊢ (Set.range (M.acceptsFrom ∘ M.eval)).Finite
|
exact Set.finite_of_finite_preimage (Set.toFinite _)
(Set.range_comp_subset_range M.eval M.acceptsFrom)
|
no goals
|
d7d0b6cc243f4b64
|
Dense.eq_zero_of_inner_left
|
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
theorem eq_zero_of_inner_left (hK : Dense (K : Set E)) (h : ∀ v : K, ⟪x, v⟫ = 0) : x = 0
|
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
x : E
hK : Dense ↑K
h : ∀ (v : ↥K), ⟪x, ↑v⟫_𝕜 = 0
this : (fun x_1 => ⟪x, x_1⟫_𝕜) = 0
⊢ x = 0
|
simpa using congr_fun this x
|
no goals
|
af006bdd87720de1
|
Real.le_rpow_iff_log_le
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
theorem le_rpow_iff_log_le (hx : 0 < x) (hy : 0 < y) : x ≤ y ^ z ↔ log x ≤ z * log y
|
x y z : ℝ
hx : 0 < x
hy : 0 < y
⊢ x ≤ y ^ z ↔ log x ≤ z * log y
|
rw [← log_le_log_iff hx (rpow_pos_of_pos hy z), log_rpow hy]
|
no goals
|
3899b76360f158ae
|
List.findSome?_cons_of_isNone
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean
|
theorem findSome?_cons_of_isNone (l) (h : (f a).isNone) : findSome? f (a :: l) = findSome? f l
|
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → Option α✝
a : α✝¹
l : List α✝¹
h : (f a).isNone = true
⊢ findSome? f (a :: l) = findSome? f l
|
simp only [findSome?]
|
α✝¹ : Type u_1
α✝ : Type u_2
f : α✝¹ → Option α✝
a : α✝¹
l : List α✝¹
h : (f a).isNone = true
⊢ (match f a with
| some b => some b
| none => findSome? f l) =
findSome? f l
|
274529f5d3b93fb6
|
AlternatingMap.map_linearDependent
|
Mathlib/LinearAlgebra/Alternating/Basic.lean
|
theorem map_linearDependent {K : Type*} [Ring K] {M : Type*} [AddCommGroup M] [Module K M]
{N : Type*} [AddCommGroup N] [Module K N] [NoZeroSMulDivisors K N] (f : M [⋀^ι]→ₗ[K] N)
(v : ι → M) (h : ¬LinearIndependent K v) : f v = 0
|
case intro.intro.intro.intro.intro
ι : Type u_7
K : Type u_12
inst✝⁵ : Ring K
M : Type u_13
inst✝⁴ : AddCommGroup M
inst✝³ : Module K M
N : Type u_14
inst✝² : AddCommGroup N
inst✝¹ : Module K N
inst✝ : NoZeroSMulDivisors K N
f : M [⋀^ι]→ₗ[K] N
v : ι → M
h✝ : ¬LinearIndependent K v
s : Finset ι
g : ι → K
h : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
hz : g i ≠ 0
⊢ f v = 0
|
letI := Classical.decEq ι
|
case intro.intro.intro.intro.intro
ι : Type u_7
K : Type u_12
inst✝⁵ : Ring K
M : Type u_13
inst✝⁴ : AddCommGroup M
inst✝³ : Module K M
N : Type u_14
inst✝² : AddCommGroup N
inst✝¹ : Module K N
inst✝ : NoZeroSMulDivisors K N
f : M [⋀^ι]→ₗ[K] N
v : ι → M
h✝ : ¬LinearIndependent K v
s : Finset ι
g : ι → K
h : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
hz : g i ≠ 0
this : DecidableEq ι := Classical.decEq ι
⊢ f v = 0
|
e59a2621ba7004bf
|
PrimeSpectrum.isMax_iff
|
Mathlib/RingTheory/Spectrum/Prime/Basic.lean
|
/-- Also see `PrimeSpectrum.isClosed_singleton_iff_isMaximal` -/
lemma isMax_iff {x : PrimeSpectrum R} :
IsMax x ↔ x.asIdeal.IsMaximal
|
R : Type u
inst✝ : CommSemiring R
x : PrimeSpectrum R
hx : IsMax x
I : Ideal R
hI : x.asIdeal < I
e : ¬I = ⊤
⊢ False
|
obtain ⟨m, hm, hm'⟩ := Ideal.exists_le_maximal I e
|
case intro.intro
R : Type u
inst✝ : CommSemiring R
x : PrimeSpectrum R
hx : IsMax x
I : Ideal R
hI : x.asIdeal < I
e : ¬I = ⊤
m : Ideal R
hm : m.IsMaximal
hm' : I ≤ m
⊢ False
|
9dca787d9a1cc302
|
MeasureTheory.setIntegral_abs_condExp_le
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
|
theorem setIntegral_abs_condExp_le {s : Set α} (hs : MeasurableSet[m] s) (f : α → ℝ) :
∫ x in s, |(μ[f|m]) x| ∂μ ≤ ∫ x in s, |f x| ∂μ
|
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ ∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ
|
simp only [condExp_of_not_integrable hfint, Pi.zero_apply, abs_zero, integral_const,
Algebra.id.smul_eq_mul, mul_zero]
|
case neg
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hs : MeasurableSet s
f : α → ℝ
hnm : m ≤ m0
hfint : ¬Integrable f μ
⊢ 0 ≤ ∫ (x : α) in s, |f x| ∂μ
|
f75c894a42ad0f54
|
AlgebraicGeometry.Scheme.Pullback.gluedLift_p2
|
Mathlib/AlgebraicGeometry/Pullbacks.lean
|
theorem gluedLift_p2 : gluedLift 𝒰 f g s ≫ p2 𝒰 f g = s.snd
|
case h
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
s : PullbackCone f g
⊢ ∀ (b : (MultispanShape.prod (Cover.pullbackCover 𝒰 s.fst).gluedCover.J).R),
Multicoequalizer.π (Cover.pullbackCover 𝒰 s.fst).gluedCover.diagram b ≫
(Cover.pullbackCover 𝒰 s.fst).fromGlued ≫ gluedLift 𝒰 f g s ≫ p2 𝒰 f g =
Multicoequalizer.π (Cover.pullbackCover 𝒰 s.fst).gluedCover.diagram b ≫
(Cover.pullbackCover 𝒰 s.fst).fromGlued ≫ s.snd
|
intro b
|
case h
X Y Z : Scheme
𝒰 : X.OpenCover
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g
s : PullbackCone f g
b : (MultispanShape.prod (Cover.pullbackCover 𝒰 s.fst).gluedCover.J).R
⊢ Multicoequalizer.π (Cover.pullbackCover 𝒰 s.fst).gluedCover.diagram b ≫
(Cover.pullbackCover 𝒰 s.fst).fromGlued ≫ gluedLift 𝒰 f g s ≫ p2 𝒰 f g =
Multicoequalizer.π (Cover.pullbackCover 𝒰 s.fst).gluedCover.diagram b ≫
(Cover.pullbackCover 𝒰 s.fst).fromGlued ≫ s.snd
|
70923766d5abff15
|
translate_eq_domAddActMk_vadd
|
Mathlib/Algebra/Group/Translate.lean
|
lemma translate_eq_domAddActMk_vadd (a : G) (f : G → α) : τ a f = DomAddAct.mk (-a) +ᵥ f
|
α : Type u_2
G : Type u_5
inst✝ : AddCommGroup G
a : G
f : G → α
⊢ τ a f = DomAddAct.mk (-a) +ᵥ f
|
ext
|
case h
α : Type u_2
G : Type u_5
inst✝ : AddCommGroup G
a : G
f : G → α
x✝ : G
⊢ τ a f x✝ = (DomAddAct.mk (-a) +ᵥ f) x✝
|
a65c08f1373518e1
|
exists_lt_of_lt_ciSup'
|
Mathlib/Order/ConditionallyCompleteLattice/Indexed.lean
|
theorem exists_lt_of_lt_ciSup' {f : ι → α} {a : α} (h : a < ⨆ i, f i) : ∃ i, a < f i
|
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
a : α
h : ∀ (i : ι), f i ≤ a
⊢ ⨆ i, f i ≤ a
|
exact ciSup_le' h
|
no goals
|
713921a54fab7993
|
Subring.comap_map_eq
|
Mathlib/Algebra/Ring/Subring/Basic.lean
|
theorem comap_map_eq (f : R →+* S) (s : Subring R) :
(s.map f).comap f = s ⊔ closure (f ⁻¹' {0})
|
case a.intro.intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
s : Subring R
x y : R
hy : y ∈ s
hxy : x - y ∈ ⇑f ⁻¹' {0}
⊢ x ∈ s ⊔ closure (⇑f ⁻¹' {0})
|
rw [← closure_eq s, ← closure_union, ← add_sub_cancel y x]
|
case a.intro.intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
s : Subring R
x y : R
hy : y ∈ s
hxy : x - y ∈ ⇑f ⁻¹' {0}
⊢ y + (x - y) ∈ closure (↑s ∪ ⇑f ⁻¹' {0})
|
1888ddc34813b7fe
|
Multiset.singleton_eq_cons_iff
|
Mathlib/Data/Multiset/ZeroCons.lean
|
theorem singleton_eq_cons_iff {a b : α} (m : Multiset α) : {a} = b ::ₘ m ↔ a = b ∧ m = 0
|
α : Type u_1
a b : α
m : Multiset α
⊢ {a} = b ::ₘ m ↔ a = b ∧ m = 0
|
rw [← cons_zero, cons_eq_cons]
|
α : Type u_1
a b : α
m : Multiset α
⊢ (a = b ∧ 0 = m ∨ a ≠ b ∧ ∃ cs, 0 = b ::ₘ cs ∧ m = a ::ₘ cs) ↔ a = b ∧ m = 0
|
90ad7bc55698b478
|
FormalMultilinearSeries.compPartialSumTarget_tendsto_prod_atTop
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem compPartialSumTarget_tendsto_prod_atTop :
Tendsto (fun (p : ℕ × ℕ) => compPartialSumTarget 0 p.1 p.2) atTop atTop
|
case h'.mk
n : ℕ
c : Composition n
⊢ ∃ n_1, ⟨n, c⟩ ∈ compPartialSumTarget 0 n_1.1 n_1.2
|
simp only [mem_compPartialSumTarget_iff]
|
case h'.mk
n : ℕ
c : Composition n
⊢ ∃ n_1, 0 ≤ c.length ∧ c.length < n_1.1 ∧ ∀ (j : Fin c.length), c.blocksFun j < n_1.2
|
9e49e40e6afc4679
|
ProbabilityTheory.Kernel.iIndepSets.indepSets
|
Mathlib/Probability/Independence/Kernel.lean
|
theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} (h_indep : iIndepSets s κ μ) {i j : ι} (hij : i ≠ j) :
IndepSets (s i) (s j) κ μ
|
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
hf_m : ∀ x ∈ {i, j}, (if x = i then t₁ else t₂) ∈ s x
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t₁ ∩ t₂) = (κ a) t₁ * (κ a) t₂
|
have h1 : t₁ = ite (i = i) t₁ t₂ := by simp only [if_true, eq_self_iff_true]
|
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
hf_m : ∀ x ∈ {i, j}, (if x = i then t₁ else t₂) ∈ s x
h1 : t₁ = if i = i then t₁ else t₂
⊢ ∀ᵐ (a : α) ∂μ, (κ a) (t₁ ∩ t₂) = (κ a) t₁ * (κ a) t₂
|
f62ea0e10e361ba7
|
logEmbeddingQuot_injective
|
Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
|
theorem logEmbeddingQuot_injective :
Function.Injective (logEmbeddingQuot K)
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)
h :
(QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))
((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₁✝)) =
(QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))
((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₂✝))
⊢ a₁✝ = a₂✝
|
exact (EmbeddingLike.apply_eq_iff_eq _).mp <| (QuotientGroup.kerLift_injective _).eq_iff.mp h
|
no goals
|
bd3f79cf52e597d4
|
differentiableWithinAt_localInvariantProp
|
Mathlib/Geometry/Manifold/MFDeriv/Defs.lean
|
theorem differentiableWithinAt_localInvariantProp :
(contDiffGroupoid 1 I).LocalInvariantProp (contDiffGroupoid 1 I')
(DifferentiableWithinAtProp I I') :=
{ is_local
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) (↑I (↑e x))
|
have : I (e x) ∈ I.symm ⁻¹' e.target ∩ Set.range I := by simp only [hx, mfld_simps]
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
s : Set H
x : H
f : H → H'
e : PartialHomeomorph H H
he : e ∈ contDiffGroupoid 1 I
hx : x ∈ e.source
h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))
this✝ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))
this : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I
⊢ DifferentiableWithinAt 𝕜 (↑I' ∘ (f ∘ ↑e.symm) ∘ ↑I.symm) (↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I) (↑I (↑e x))
|
3d1405d37db22f10
|
CharP.intCast_injOn_Ico
|
Mathlib/Algebra/CharP/Basic.lean
|
lemma intCast_injOn_Ico [IsRightCancelAdd R] : InjOn (Int.cast : ℤ → R) (Ico 0 p)
|
case intro.intro.intro.intro
R : Type u_1
inst✝² : AddGroupWithOne R
p : ℕ
inst✝¹ : CharP R p
inst✝ : IsRightCancelAdd R
a b : ℕ
ha : a < p
hb : b < p
hab : ↑a = ↑b
⊢ a = b
|
exact natCast_injOn_Iio _ _ ha hb hab
|
no goals
|
bb28179ddf918e8c
|
List.set_eq_take_append_cons_drop
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean
|
theorem set_eq_take_append_cons_drop (l : List α) (n : Nat) (a : α) :
l.set n a = if n < l.length then l.take n ++ a :: l.drop (n + 1) else l
|
α : Type u_1
l : List α
n : Nat
a : α
⊢ l.set n a = if n < l.length then take n l ++ a :: drop (n + 1) l else l
|
split <;> rename_i h
|
case isTrue
α : Type u_1
l : List α
n : Nat
a : α
h : n < l.length
⊢ l.set n a = take n l ++ a :: drop (n + 1) l
case isFalse
α : Type u_1
l : List α
n : Nat
a : α
h : ¬n < l.length
⊢ l.set n a = l
|
9fb4c99baced165c
|
MatrixEquivTensor.invFun_add
|
Mathlib/RingTheory/MatrixAlgebra.lean
|
theorem invFun_add (M N : Matrix n n A) :
invFun n R A (M + N) = invFun n R A M + invFun n R A N
|
n : Type u_1
R : Type u_2
A : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Algebra R A
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M N : Matrix n n A
⊢ invFun n R A (M + N) = invFun n R A M + invFun n R A N
|
simp [invFun, add_tmul, Finset.sum_add_distrib]
|
no goals
|
abd2f857ec4e647f
|
lineDerivWithin_congr
|
Mathlib/Analysis/Calculus/LineDeriv/Basic.lean
|
theorem lineDerivWithin_congr (hs : EqOn f₁ f s) (hx : f₁ x = f x) :
lineDerivWithin 𝕜 f₁ s x v = lineDerivWithin 𝕜 f s x v :=
derivWithin_congr (fun _ hy ↦ hs hy) (by simpa using hx)
|
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
E : Type u_3
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
f f₁ : E → F
s : Set E
x v : E
hs : EqOn f₁ f s
hx : f₁ x = f x
⊢ f₁ (x + 0 • v) = f (x + 0 • v)
|
simpa using hx
|
no goals
|
5091d628b6956273
|
isInvertible_mfderiv_extChartAt
|
Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean
|
lemma isInvertible_mfderiv_extChartAt {y : M} (hy : y ∈ (extChartAt I x).source) :
(mfderiv I 𝓘(𝕜, E) (extChartAt I x) y).IsInvertible
|
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x y : M
hy : y ∈ (extChartAt I x).source
h'y : ↑(extChartAt I x) y ∈ (extChartAt I x).target
Z : (mfderiv I 𝓘(𝕜, E) (↑(extChartAt I x)) (↑(extChartAt I x).symm (↑(extChartAt I x) y))).IsInvertible
this : ↑(extChartAt I x).symm (↑(extChartAt I x) y) = y
⊢ (mfderiv I 𝓘(𝕜, E) (↑(extChartAt I x)) y).IsInvertible
|
rwa [this] at Z
|
no goals
|
a106f7bde02a2a5a
|
List.eq_replicate_or_eq_replicate_append_cons
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem eq_replicate_or_eq_replicate_append_cons {α : Type _} (l : List α) :
(l = []) ∨ (∃ n a, l = replicate n a ∧ 0 < n) ∨
(∃ n a b l', l = replicate n a ++ b :: l' ∧ 0 < n ∧ a ≠ b)
|
case neg.h
α : Type u_1
x : α
n : Nat
a : α
h : 0 < n
h' : ¬x = a
⊢ ∃ n_1 a_1 b l', x :: replicate n a = replicate n_1 a_1 ++ b :: l' ∧ 0 < n_1 ∧ a_1 ≠ b
|
refine ⟨1, x, a, replicate (n - 1) a, ?_, by decide, h'⟩
|
case neg.h
α : Type u_1
x : α
n : Nat
a : α
h : 0 < n
h' : ¬x = a
⊢ x :: replicate n a = replicate 1 x ++ a :: replicate (n - 1) a
|
1321b019df9b3ed1
|
MeasureTheory.IsProjectiveLimit.unique
|
Mathlib/MeasureTheory/Constructions/Projective.lean
|
theorem unique [∀ i, IsFiniteMeasure (P i)]
(hμ : IsProjectiveLimit μ P) (hν : IsProjectiveLimit ν P) :
μ = ν
|
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
μ ν : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)
hμ : IsProjectiveLimit μ P
hν : IsProjectiveLimit ν P
⊢ μ = ν
|
haveI : IsFiniteMeasure μ := hμ.isFiniteMeasure
|
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
μ ν : Measure ((i : ι) → α i)
inst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)
hμ : IsProjectiveLimit μ P
hν : IsProjectiveLimit ν P
this : IsFiniteMeasure μ
⊢ μ = ν
|
ef569de89d8ae529
|
PiLp.edist_comm
|
Mathlib/Analysis/Normed/Lp/PiLp.lean
|
theorem edist_comm (f g : PiLp p β) : edist f g = edist g f
|
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
f g : PiLp p β
⊢ edist f g = edist g f
|
rcases p.trichotomy with (rfl | rfl | h)
|
case inl
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
f g : PiLp 0 β
⊢ edist f g = edist g f
case inr.inl
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
f g : PiLp ⊤ β
⊢ edist f g = edist g f
case inr.inr
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
f g : PiLp p β
h : 0 < p.toReal
⊢ edist f g = edist g f
|
8220f242121bf124
|
Filter.EventuallyEq.iteratedDeriv_eq
|
Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean
|
lemma Filter.EventuallyEq.iteratedDeriv_eq (n : ℕ) {f g : 𝕜 → F} {x : 𝕜} (hfg : f =ᶠ[𝓝 x] g) :
iteratedDeriv n f x = iteratedDeriv n g x
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f g : 𝕜 → F
x : 𝕜
hfg : f =ᶠ[𝓝 x] g
⊢ iteratedDeriv n f x = iteratedDeriv n g x
|
simp only [← iteratedDerivWithin_univ, iteratedDerivWithin_eq_iteratedFDerivWithin]
|
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f g : 𝕜 → F
x : 𝕜
hfg : f =ᶠ[𝓝 x] g
⊢ ((iteratedFDerivWithin 𝕜 n f Set.univ x) fun x => 1) = (iteratedFDerivWithin 𝕜 n g Set.univ x) fun x => 1
|
714f8c22c6d22c1a
|
Cardinal.aleph_mul_aleph
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem aleph_mul_aleph (o₁ o₂ : Ordinal) : ℵ_ o₁ * ℵ_ o₂ = ℵ_ (max o₁ o₂)
|
o₁ o₂ : Ordinal.{u_1}
⊢ ℵ_ o₁ * ℵ_ o₂ = ℵ_ (o₁ ⊔ o₂)
|
rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), aleph_max]
|
no goals
|
3f107d0a8bd850a6
|
Algebra.IsPushout.symm
|
Mathlib/RingTheory/IsTensorProduct.lean
|
theorem Algebra.IsPushout.symm (h : Algebra.IsPushout R S R' S') : Algebra.IsPushout R R' S S'
|
R : Type u_1
S : Type v₃
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
R' : Type u_6
S' : Type u_7
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring S'
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R' S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : IsScalarTower R S S'
h : IsPushout R S R' S'
x✝ : Algebra R' (S ⊗[R] R') := TensorProduct.includeRight.toAlgebra
r : R'
x : R' ⊗[R] S
⊢ ⋯.equiv ((TensorProduct.comm R R' S) (r • x)) = r • ⋯.equiv ((TensorProduct.comm R R' S) x)
|
refine TensorProduct.induction_on x ?_ ?_ ?_
|
case refine_1
R : Type u_1
S : Type v₃
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
R' : Type u_6
S' : Type u_7
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring S'
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R' S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : IsScalarTower R S S'
h : IsPushout R S R' S'
x✝ : Algebra R' (S ⊗[R] R') := TensorProduct.includeRight.toAlgebra
r : R'
x : R' ⊗[R] S
⊢ ⋯.equiv ((TensorProduct.comm R R' S) (r • 0)) = r • ⋯.equiv ((TensorProduct.comm R R' S) 0)
case refine_2
R : Type u_1
S : Type v₃
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
R' : Type u_6
S' : Type u_7
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring S'
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R' S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : IsScalarTower R S S'
h : IsPushout R S R' S'
x✝ : Algebra R' (S ⊗[R] R') := TensorProduct.includeRight.toAlgebra
r : R'
x : R' ⊗[R] S
⊢ ∀ (x : R') (y : S),
⋯.equiv ((TensorProduct.comm R R' S) (r • x ⊗ₜ[R] y)) = r • ⋯.equiv ((TensorProduct.comm R R' S) (x ⊗ₜ[R] y))
case refine_3
R : Type u_1
S : Type v₃
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
R' : Type u_6
S' : Type u_7
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring S'
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R' S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : IsScalarTower R S S'
h : IsPushout R S R' S'
x✝ : Algebra R' (S ⊗[R] R') := TensorProduct.includeRight.toAlgebra
r : R'
x : R' ⊗[R] S
⊢ ∀ (x y : R' ⊗[R] S),
⋯.equiv ((TensorProduct.comm R R' S) (r • x)) = r • ⋯.equiv ((TensorProduct.comm R R' S) x) →
⋯.equiv ((TensorProduct.comm R R' S) (r • y)) = r • ⋯.equiv ((TensorProduct.comm R R' S) y) →
⋯.equiv ((TensorProduct.comm R R' S) (r • (x + y))) = r • ⋯.equiv ((TensorProduct.comm R R' S) (x + y))
|
ee0cd1db0e501e91
|
padicNorm.add_eq_max_of_ne
|
Mathlib/NumberTheory/Padics/PadicNorm.lean
|
theorem add_eq_max_of_ne {q r : ℚ} (hne : padicNorm p q ≠ padicNorm p r) :
padicNorm p (q + r) = max (padicNorm p q) (padicNorm p r)
|
case inr
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hne : padicNorm p q ≠ padicNorm p r
this :
∀ {p : ℕ} [hp : Fact (Nat.Prime p)] {q r : ℚ},
padicNorm p q ≠ padicNorm p r → padicNorm p r < padicNorm p q → padicNorm p (q + r) = padicNorm p q ⊔ padicNorm p r
hlt : ¬padicNorm p r < padicNorm p q
⊢ padicNorm p (r + q) = padicNorm p r ⊔ padicNorm p q
|
exact this hne.symm (hne.lt_or_lt.resolve_right hlt)
|
no goals
|
cd0b52cee2afa794
|
Complex.ofReal_cpow_of_nonpos
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
theorem ofReal_cpow_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℂ) :
(x : ℂ) ^ y = (-x : ℂ) ^ y * exp (π * I * y)
|
x : ℝ
hx : x ≤ 0
y : ℂ
⊢ ↑x ^ y = (-↑x) ^ y * cexp (↑π * I * y)
|
rcases hx.eq_or_lt with (rfl | hlt)
|
case inl
y : ℂ
hx : 0 ≤ 0
⊢ ↑0 ^ y = (-↑0) ^ y * cexp (↑π * I * y)
case inr
x : ℝ
hx : x ≤ 0
y : ℂ
hlt : x < 0
⊢ ↑x ^ y = (-↑x) ^ y * cexp (↑π * I * y)
|
206b85c721316ad4
|
div_eq_quo_add_sum_rem_div
|
Mathlib/Algebra/Polynomial/PartialFractions.lean
|
theorem div_eq_quo_add_sum_rem_div (f : R[X]) {ι : Type*} {g : ι → R[X]} {s : Finset ι}
(hg : ∀ i ∈ s, (g i).Monic) (hcop : Set.Pairwise ↑s fun i j => IsCoprime (g i) (g j)) :
∃ (q : R[X]) (r : ι → R[X]),
(∀ i ∈ s, (r i).degree < (g i).degree) ∧
((↑f : K) / ∏ i ∈ s, ↑(g i)) = ↑q + ∑ i ∈ s, (r i : K) / (g i : K)
|
case insert.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R : Type
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type
inst✝² : Field K
inst✝¹ : Algebra R[X] K
inst✝ : IsFractionRing R[X] K
ι : Type u_1
g : ι → R[X]
a : ι
b : Finset ι
hab : a ∉ b
Hind :
∀ (f : R[X]),
(∀ i ∈ b, (g i).Monic) →
((↑b).Pairwise fun i j => IsCoprime (g i) (g j)) →
∃ q r, (∀ i ∈ b, (r i).degree < (g i).degree) ∧ ↑f / ∏ i ∈ b, ↑(g i) = ↑q + ∑ i ∈ b, ↑(r i) / ↑(g i)
f : R[X]
hg : ∀ i ∈ insert a b, (g i).Monic
hcop : (↑(insert a b)).Pairwise fun i j => IsCoprime (g i) (g j)
q₀ r₁ r₂ : R[X]
hdeg₁ : r₁.degree < (g a).degree
left✝ : r₂.degree < (∏ i ∈ b, g i).degree
hf : ↑f / (↑(g a) * ↑(∏ i ∈ b, g i)) = ↑q₀ + ↑r₁ / ↑(g a) + ↑r₂ / ↑(∏ i ∈ b, g i)
q : R[X]
r : ι → R[X]
hrdeg : ∀ i ∈ b, (r i).degree < (g i).degree
IH : ↑?m.17923 / ∏ i ∈ b, ↑(g i) = ↑q + ∑ i ∈ b, ↑(r i) / ↑(g i)
⊢ ↑f / ∏ i ∈ insert a b, ↑(g i) = ↑(q₀ + q) + ∑ i ∈ insert a b, ↑((fun i => if i = a then r₁ else r i) i) / ↑(g i)
|
norm_cast at hf IH ⊢
|
case insert.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
R : Type
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
K : Type
inst✝² : Field K
inst✝¹ : Algebra R[X] K
inst✝ : IsFractionRing R[X] K
ι : Type u_1
g : ι → R[X]
a : ι
b : Finset ι
hab : a ∉ b
Hind :
∀ (f : R[X]),
(∀ i ∈ b, (g i).Monic) →
((↑b).Pairwise fun i j => IsCoprime (g i) (g j)) →
∃ q r, (∀ i ∈ b, (r i).degree < (g i).degree) ∧ ↑f / ∏ i ∈ b, ↑(g i) = ↑q + ∑ i ∈ b, ↑(r i) / ↑(g i)
f : R[X]
hg : ∀ i ∈ insert a b, (g i).Monic
hcop : (↑(insert a b)).Pairwise fun i j => IsCoprime (g i) (g j)
q₀ r₁ r₂ : R[X]
hdeg₁ : r₁.degree < (g a).degree
left✝ : r₂.degree < (∏ i ∈ b, g i).degree
hf✝ : ↑f / (↑(g a) * ↑(∏ i ∈ b, g i)) = ↑q₀ + ↑r₁ / ↑(g a) + ↑r₂ / ↑(∏ i ∈ b, g i)
q : R[X]
r : ι → R[X]
hrdeg : ∀ i ∈ b, (r i).degree < (g i).degree
hf : ↑f / ↑(g a * ∏ i ∈ b, g i) = ↑q₀ + ↑r₁ / ↑(g a) + ↑r₂ / ↑(∏ i ∈ b, g i)
IH : ↑?m.17923 / ↑(∏ i ∈ b, g i) = ↑q + ∑ i ∈ b, ↑(r i) / ↑(g i)
⊢ ↑f / ↑(∏ i ∈ insert a b, g i) = ↑(q₀ + q) + ∑ x ∈ insert a b, ↑(if x = a then r₁ else r x) / ↑(g x)
|
ab72339ad54ec2a6
|
FractionalIdeal.mem_zero_iff
|
Mathlib/RingTheory/FractionalIdeal/Basic.lean
|
theorem mem_zero_iff {x : P} : x ∈ (0 : FractionalIdeal S P) ↔ x = 0 :=
⟨fun ⟨x', x'_mem_zero, x'_eq_x⟩ => by
have x'_eq_zero : x' = 0 := x'_mem_zero
simp [x'_eq_x.symm, x'_eq_zero], fun hx => ⟨0, rfl, by simp [hx]⟩⟩
|
R : Type u_1
inst✝² : CommRing R
S : Submonoid R
P : Type u_2
inst✝¹ : CommRing P
inst✝ : Algebra R P
x : P
hx : x = 0
⊢ (Algebra.linearMap R P) 0 = x
|
simp [hx]
|
no goals
|
0682cefbc56573ad
|
Quantale.bot_mul
|
Mathlib/Algebra/Order/Quantale.lean
|
theorem bot_mul : ⊥ * x = ⊥
|
α : Type u_3
inst✝² : Semigroup α
inst✝¹ : CompleteLattice α
inst✝ : IsQuantale α
x : α
⊢ ⊥ * x = ⊥
|
rw [← sSup_empty, sSup_mul_distrib]
|
α : Type u_3
inst✝² : Semigroup α
inst✝¹ : CompleteLattice α
inst✝ : IsQuantale α
x : α
⊢ ⨆ y ∈ ∅, y * x = sSup ∅
|
ae1fb907b443c888
|
AffineMap.lineMap_apply_module
|
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
theorem lineMap_apply_module (p₀ p₁ : V1) (c : k) : lineMap p₀ p₁ c = (1 - c) • p₀ + c • p₁
|
k : Type u_1
V1 : Type u_2
inst✝² : Ring k
inst✝¹ : AddCommGroup V1
inst✝ : Module k V1
p₀ p₁ : V1
c : k
⊢ (lineMap p₀ p₁) c = (1 - c) • p₀ + c • p₁
|
simp [lineMap_apply_module', smul_sub, sub_smul]
|
k : Type u_1
V1 : Type u_2
inst✝² : Ring k
inst✝¹ : AddCommGroup V1
inst✝ : Module k V1
p₀ p₁ : V1
c : k
⊢ c • p₁ - c • p₀ + p₀ = p₀ - c • p₀ + c • p₁
|
37b76ea914db748b
|
Nat.primeFactors_one
|
Mathlib/Data/Nat/PrimeFin.lean
|
@[simp] lemma primeFactors_one : primeFactors 1 = ∅
|
case h
a✝ : ℕ
⊢ a✝ ∈ primeFactors 1 ↔ a✝ ∈ ∅
|
simpa using Prime.ne_one
|
no goals
|
674f6d87bcdf4ccf
|
MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported
|
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
theorem weightedHomogeneousSubmodule_eq_finsupp_supported (w : σ → M) (m : M) :
weightedHomogeneousSubmodule R w m = Finsupp.supported R R { d | weight w d = m }
|
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m : M
⊢ weightedHomogeneousSubmodule R w m = supported R R {d | (weight w) d = m}
|
ext x
|
case h
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w : σ → M
m : M
x : MvPolynomial σ R
⊢ x ∈ weightedHomogeneousSubmodule R w m ↔ x ∈ supported R R {d | (weight w) d = m}
|
1504e28071b2bf79
|
RingHom.finiteType_isStableUnderBaseChange
|
Mathlib/RingTheory/RingHom/FiniteType.lean
|
theorem finiteType_isStableUnderBaseChange : IsStableUnderBaseChange @FiniteType
|
case h₁
⊢ RespectsIso @FiniteType
|
exact finiteType_respectsIso
|
no goals
|
32090bc17de5e6dc
|
CategoryTheory.mateEquiv_vcomp
|
Mathlib/CategoryTheory/Adjunction/Mates.lean
|
theorem mateEquiv_vcomp (α : TwoSquare G₁ L₁ L₂ H₁) (β : TwoSquare G₂ L₂ L₃ H₂) :
(mateEquiv adj₁ adj₃) (α ≫ₕ β) = (mateEquiv adj₁ adj₂ α) ≫ᵥ (mateEquiv adj₂ adj₃ β)
|
case h
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
E : Type u₅
F : Type u₆
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} B
inst✝³ : Category.{v₃, u₃} C
inst✝² : Category.{v₄, u₄} D
inst✝¹ : Category.{v₅, u₅} E
inst✝ : Category.{v₆, u₆} F
G₁ : A ⥤ C
G₂ : C ⥤ E
H₁ : B ⥤ D
H₂ : D ⥤ F
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : E ⥤ F
R₃ : F ⥤ E
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : TwoSquare G₁ L₁ L₂ H₁
β : TwoSquare G₂ L₂ L₃ H₂
b : B
⊢ adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫
R₃.map (β.natTrans.app (G₁.obj (R₁.obj b))) ≫
R₃.map (H₂.map (α.natTrans.app (R₁.obj b))) ≫ R₃.map (H₂.map (H₁.map (adj₁.counit.app b))) =
adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫
R₃.map (β.app (G₁.obj (R₁.obj b))) ≫
(R₃.map (H₂.map (𝟙 (L₂.obj (G₁.obj (R₁.obj b))))) ≫ R₃.map (H₂.map (α.natTrans.app (R₁.obj b)))) ≫
R₃.map (H₂.map (H₁.map (adj₁.counit.app b)))
|
simp only [map_id, id_comp]
|
no goals
|
e571c91f29550453
|
seminormFromConst_seq_one
|
Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean
|
theorem seminormFromConst_seq_one (n : ℕ) (hn : 1 ≤ n) : seminormFromConst_seq c f 1 n = 1
|
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hc : f c ≠ 0
hpm : IsPowMul ⇑f
n : ℕ
hn : 1 ≤ n
⊢ seminormFromConst_seq c f 1 n = 1
|
simp only [seminormFromConst_seq]
|
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hc : f c ≠ 0
hpm : IsPowMul ⇑f
n : ℕ
hn : 1 ≤ n
⊢ f (1 * c ^ n) / f c ^ n = 1
|
cdcce5ae9812f7ce
|
SimplexCategory.eq_comp_δ_of_not_surjective'
|
Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean
|
theorem eq_comp_δ_of_not_surjective' {n : ℕ} {Δ : SimplexCategory} (θ : Δ ⟶ mk (n + 1))
(i : Fin (n + 2)) (hi : ∀ x, θ.toOrderHom x ≠ i) : ∃ θ' : Δ ⟶ mk n, θ = θ' ≫ δ i
|
case neg
n : ℕ
Δ : SimplexCategory
θ : Δ ⟶ ⦋n + 1⦌
i : Fin (n + 2)
hi : ∀ (x : Fin (Δ.len + 1)), (Hom.toOrderHom θ) x ≠ i
h : i < Fin.last (n + 1)
x : Fin (Δ.len + 1)
h' : i < (Hom.toOrderHom θ) x
⊢ (Hom.toOrderHom θ) x = i.succAbove (((Hom.toOrderHom θ) x).pred ⋯)
|
rw [Fin.succAbove_of_le_castSucc i _]
|
case neg
n : ℕ
Δ : SimplexCategory
θ : Δ ⟶ ⦋n + 1⦌
i : Fin (n + 2)
hi : ∀ (x : Fin (Δ.len + 1)), (Hom.toOrderHom θ) x ≠ i
h : i < Fin.last (n + 1)
x : Fin (Δ.len + 1)
h' : i < (Hom.toOrderHom θ) x
⊢ (Hom.toOrderHom θ) x = (((Hom.toOrderHom θ) x).pred ⋯).succ
case neg
n : ℕ
Δ : SimplexCategory
θ : Δ ⟶ ⦋n + 1⦌
i : Fin (n + 2)
hi : ∀ (x : Fin (Δ.len + 1)), (Hom.toOrderHom θ) x ≠ i
h : i < Fin.last (n + 1)
x : Fin (Δ.len + 1)
h' : i < (Hom.toOrderHom θ) x
⊢ i ≤ (((Hom.toOrderHom θ) x).pred ⋯).castSucc
|
4424d07c1d7e8eb1
|
Submonoid.leftInv_leftInv_eq
|
Mathlib/GroupTheory/Submonoid/Inverses.lean
|
theorem leftInv_leftInv_eq (hS : S ≤ IsUnit.submonoid M) : S.leftInv.leftInv = S
|
M : Type u_1
inst✝ : Monoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : M
hx : x ∈ S
⊢ x = ↑(IsUnit.unit ⋯)⁻¹⁻¹
|
rw [inv_inv (hS hx).unit]
|
M : Type u_1
inst✝ : Monoid M
S : Submonoid M
hS : S ≤ IsUnit.submonoid M
x : M
hx : x ∈ S
⊢ x = ↑(IsUnit.unit ⋯)
|
41ba8e94ab6aae76
|
Int.bitwise_diff
|
Mathlib/Data/Int/Bitwise.lean
|
theorem bitwise_diff : (bitwise fun a b => a && not b) = ldiff
|
case h.h.ofNat.negSucc
m n : ℕ
⊢ ↑(Nat.bitwise (fun x y => x && !!y) m n) = ↑(m &&& n)
|
congr
|
case h.h.ofNat.negSucc.e_a.e_f
m n : ℕ
⊢ (fun x y => x && !!y) = and
|
875421a4253732b9
|
NumberField.abs_discr_gt_two
|
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean
|
theorem abs_discr_gt_two (h : 1 < finrank ℚ K) : 2 < |discr K|
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Nat.succ 1 ≤ finrank ℚ K
⊢ 2 < |discr K|
|
rify
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Nat.succ 1 ≤ finrank ℚ K
⊢ 2 < |↑(discr K)|
|
33a91f7b5a5a4855
|
IsPrimitiveRoot.minpoly_dvd_expand
|
Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
|
theorem minpoly_dvd_expand {p : ℕ} (hdiv : ¬p ∣ n) :
minpoly ℤ μ ∣ expand ℤ p (minpoly ℤ (μ ^ p))
|
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hdiv : ¬p ∣ n
⊢ minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p))
|
rcases n.eq_zero_or_pos with (rfl | hpos)
|
case inl
K : Type u_1
inst✝² : CommRing K
μ : K
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
h : IsPrimitiveRoot μ 0
hdiv : ¬p ∣ 0
⊢ minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p))
case inr
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hdiv : ¬p ∣ n
hpos : n > 0
⊢ minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p))
|
765e9f715d5c1ba0
|
AkraBazziRecurrence.GrowsPolynomially.eventually_atTop_nonneg_or_nonpos
|
Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
|
lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
(∀ᶠ x in atTop, 0 ≤ f x) ∨ (∀ᶠ x in atTop, f x ≤ 0)
|
case h
f : ℝ → ℝ
hf : GrowsPolynomially f
c₁ : ℝ
left✝¹ : c₁ > 0
c₂ : ℝ
left✝ : c₂ > 0
h : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
heq : c₁ = c₂
c : ℝ
hc✝ : ∀ᶠ (x : ℝ) in atTop, f x = c
hpos : 0 ≤ c
a✝ : ℝ
hc : f a✝ = c
⊢ 0 ≤ f a✝
|
simpa only [hc]
|
no goals
|
69674f7264744f71
|
List.Perm.sym2
|
Mathlib/Data/List/Sym.lean
|
theorem Perm.sym2 {xs ys : List α} (h : xs ~ ys) :
xs.sym2 ~ ys.sym2
|
α : Type u_1
xs ys : List α
h : xs ~ ys
⊢ xs.sym2 ~ ys.sym2
|
induction h with
| nil => rfl
| cons x h ih =>
simp only [List.sym2, map_cons, cons_append, perm_cons]
exact (h.map _).append ih
| swap x y xs =>
simp only [List.sym2, map_cons, cons_append]
conv => enter [1,2,1]; rw [Sym2.eq_swap]
refine Perm.trans (Perm.swap ..) (Perm.trans (Perm.cons _ ?_) (Perm.swap ..))
simp only [← Multiset.coe_eq_coe, ← Multiset.cons_coe,
← Multiset.coe_add, ← Multiset.singleton_add]
simp only [add_assoc, add_left_comm]
| trans _ _ ih1 ih2 => exact ih1.trans ih2
|
no goals
|
1e06895f6b2940d7
|
Polynomial.degree_C_mul_of_isUnit
|
Mathlib/Algebra/Polynomial/Degree/Operations.lean
|
lemma degree_C_mul_of_isUnit (ha : IsUnit a) (p : R[X]) : (C a * p).degree = p.degree
|
R : Type u
a : R
inst✝ : Semiring R
ha : IsUnit a
p : R[X]
hp : p ≠ 0
a✝ : Nontrivial R
⊢ (C a * p).degree = p.degree
|
rw [degree_mul', degree_C ha.ne_zero]
|
R : Type u
a : R
inst✝ : Semiring R
ha : IsUnit a
p : R[X]
hp : p ≠ 0
a✝ : Nontrivial R
⊢ 0 + p.degree = p.degree
R : Type u
a : R
inst✝ : Semiring R
ha : IsUnit a
p : R[X]
hp : p ≠ 0
a✝ : Nontrivial R
⊢ (C a).leadingCoeff * p.leadingCoeff ≠ 0
|
3f26757d26394e37
|
Nat.isPowerOfTwo_nextPowerOfTwo
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Power2.lean
|
theorem isPowerOfTwo_nextPowerOfTwo (n : Nat) : n.nextPowerOfTwo.isPowerOfTwo
|
n : Nat
x✝ : (power : Nat) ×' (_ : power > 0) ×' power.isPowerOfTwo
a✝² :
∀ (y : (power : Nat) ×' (_ : power > 0) ×' power.isPowerOfTwo),
(invImage (fun x => PSigma.casesOn x fun power h₁ => PSigma.casesOn h₁ fun h₁ h₂ => n - power)
instWellFoundedRelationOfSizeOf).1
y x✝ →
(nextPowerOfTwo.go n y.1 ⋯).isPowerOfTwo
power : Nat
h₁✝ : (_ : power > 0) ×' power.isPowerOfTwo
a✝¹ :
∀ (y : (power : Nat) ×' (_ : power > 0) ×' power.isPowerOfTwo),
(invImage (fun x => PSigma.casesOn x fun power h₁ => PSigma.casesOn h₁ fun h₁ h₂ => n - power)
instWellFoundedRelationOfSizeOf).1
y ⟨power, h₁✝⟩ →
(nextPowerOfTwo.go n y.1 ⋯).isPowerOfTwo
h₁ : power > 0
h₂ : power.isPowerOfTwo
a✝ :
∀ (y : (power : Nat) ×' (_ : power > 0) ×' power.isPowerOfTwo),
(invImage (fun x => PSigma.casesOn x fun power h₁ => PSigma.casesOn h₁ fun h₁ h₂ => n - power)
instWellFoundedRelationOfSizeOf).1
y ⟨power, ⟨h₁, h₂⟩⟩ →
(nextPowerOfTwo.go n y.1 ⋯).isPowerOfTwo
h✝ : power < n
⊢ n - power * 2 < n - power
|
apply nextPowerOfTwo_dec <;> assumption
|
no goals
|
d389d9357b608488
|
MonoidHom.noncommPiCoprod_mulSingle
|
Mathlib/GroupTheory/NoncommPiCoprod.lean
|
theorem noncommPiCoprod_mulSingle [DecidableEq ι] (i : ι) (y : N i) :
noncommPiCoprod ϕ hcomm (Pi.mulSingle i y) = ϕ i y
|
M : Type u_1
inst✝³ : Monoid M
ι : Type u_2
inst✝² : Fintype ι
N : ι → Type u_3
inst✝¹ : (i : ι) → Monoid (N i)
ϕ : (i : ι) → N i →* M
hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)
inst✝ : DecidableEq ι
i : ι
y : N i
⊢ (ϕ i) (Pi.mulSingle i y i) * (Finset.univ.erase i).noncommProd (fun j => (ϕ j) (Pi.mulSingle i y j)) ⋯ = (ϕ i) y
|
rw [Pi.mulSingle_eq_same]
|
M : Type u_1
inst✝³ : Monoid M
ι : Type u_2
inst✝² : Fintype ι
N : ι → Type u_3
inst✝¹ : (i : ι) → Monoid (N i)
ϕ : (i : ι) → N i →* M
hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)
inst✝ : DecidableEq ι
i : ι
y : N i
⊢ (ϕ i) y * (Finset.univ.erase i).noncommProd (fun j => (ϕ j) (Pi.mulSingle i y j)) ⋯ = (ϕ i) y
|
df2c5930d6887abc
|
AddCircle.addWellApproximable_ae_empty_or_univ
|
Mathlib/NumberTheory/WellApproximable.lean
|
theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
(∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x
|
case neg.h.intro.inr.inl
T : ℝ
hT : Fact (0 < T)
δ : ℕ → ℝ
hδ : Tendsto δ atTop (𝓝 0)
this : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup Irreducible
μ : Measure 𝕊 := volume
u : Nat.Primes → 𝕊 := fun p => ↑(↑1 / ↑↑p * T)
hu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p
hu : Tendsto (addOrderOf ∘ u) atTop atTop
E : Set 𝕊 := addWellApproximable 𝕊 δ
X : ℕ → Set 𝕊 := fun n => approxAddOrderOf 𝕊 n (δ n)
A : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ ¬p ∣ n
B : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ∣ n ∧ ¬p * p ∣ n
C : ℕ → Set 𝕊 := fun p => blimsup X atTop fun n => 0 < n ∧ p ^ 2 ∣ n
hA₀ : ∀ (p : ℕ), MeasurableSet (A p)
hB₀ : ∀ (p : ℕ), MeasurableSet (B p)
hE₀ : NullMeasurableSet E μ
hE₁ : ∀ (p : ℕ), E = A p ∪ B p ∪ C p
hE₂ : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∧ B ↑p =ᶠ[ae μ] ∅ → E =ᶠ[ae μ] C ↑p
hA : ∀ (p : Nat.Primes), A ↑p =ᶠ[ae μ] ∅ ∨ A ↑p =ᶠ[ae μ] univ
hB : ∀ (p : Nat.Primes), B ↑p =ᶠ[ae μ] ∅ ∨ B ↑p =ᶠ[ae μ] univ
hC : ∀ (p : Nat.Primes), u p +ᵥ C ↑p = C ↑p
p : Nat.Primes
h✝¹ : ¬B ↑p =ᶠ[ae μ] ∅
h✝ : B ↑p =ᶠ[ae μ] ∅
⊢ A ↑p ∪ B ↑p ∪ C ↑p =ᶠ[ae volume] univ
|
contradiction
|
no goals
|
aeca070d334a419e
|
ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn
|
Mathlib/Topology/UniformSpace/CompactConvergence.lean
|
theorem tendsto_iff_forall_isCompact_tendstoUniformlyOn
{ι : Type u₃} {p : Filter ι} {F : ι → C(α, β)} {f} :
Tendsto F p (𝓝 f) ↔ ∀ K, IsCompact K → TendstoUniformlyOn (fun i a => F i a) f p K
|
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ (∀ (K : Set α), IsCompact K → ∀ (U : Set β), IsOpen U → MapsTo (⇑f) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U) ↔
∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a => (F i) a) (⇑f) p K
|
constructor
|
case mp
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ (∀ (K : Set α), IsCompact K → ∀ (U : Set β), IsOpen U → MapsTo (⇑f) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U) →
∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a => (F i) a) (⇑f) p K
case mpr
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ (∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a => (F i) a) (⇑f) p K) →
∀ (K : Set α), IsCompact K → ∀ (U : Set β), IsOpen U → MapsTo (⇑f) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U
|
db1a421b2350af85
|
RingHom.FinitePresentation.polynomial_induction
|
Mathlib/RingTheory/FinitePresentation.lean
|
/-- Induction principle for finitely presented ring homomorphisms.
For a property to hold for all finitely presented ring homs, it suffices for it to hold for
`Polynomial.C : R → R[X]`, surjective ring homs with finitely generated kernels, and to be closed
under composition.
Note that to state this conveniently for ring homs between rings of different universes, we carry
around two predicates `P` and `Q`, which should be "the same" apart from universes:
* `P`, for ring homs `(R : Type u) → (S : Type u)`.
* `Q`, for ring homs `(R : Type u) → (S : Type v)`.
-/
lemma polynomial_induction
(P : ∀ (R : Type u) [CommRing R] (S : Type u) [CommRing S], (R →+* S) → Prop)
(Q : ∀ (R : Type u) [CommRing R] (S : Type v) [CommRing S], (R →+* S) → Prop)
(polynomial : ∀ (R) [CommRing R], P R R[X] C)
(fg_ker : ∀ (R : Type u) [CommRing R] (S : Type v) [CommRing S] (f : R →+* S),
Surjective f → (ker f).FG → Q R S f)
(comp : ∀ (R) [CommRing R] (S) [CommRing S] (T) [CommRing T] (f : R →+* S) (g : S →+* T),
P R S f → Q S T g → Q R T (g.comp f))
{R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) (hf : f.FinitePresentation) :
Q R S f
|
case mk.intro.intro.intro
P : (R : Type u) → [inst : CommRing R] → (S : Type u) → [inst_1 : CommRing S] → (R →+* S) → Prop
Q : (R : Type u) → [inst : CommRing R] → (S : Type v) → [inst_1 : CommRing S] → (R →+* S) → Prop
polynomial : ∀ (R : Type u) [inst : CommRing R], P R R[X] C
fg_ker :
∀ (R : Type u) [inst : CommRing R] (S : Type v) [inst_1 : CommRing S] (f : R →+* S),
Surjective ⇑f → (ker f).FG → Q R S f
comp :
∀ (R : Type u) [inst : CommRing R] (S : Type u) [inst_1 : CommRing S] (T : Type v) [inst_2 : CommRing T] (f : R →+* S)
(g : S →+* T), P R S f → Q S T g → Q R T (g.comp f)
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
this✝ : Algebra R S := f.toAlgebra
n : ℕ
g : MvPolynomial (Fin n) R →ₐ[R] S
g' : MvPolynomial (Fin n) R →+* S
hg : Surjective ⇑g'
hg' : (ker g').FG
this : g'.comp MvPolynomial.C = f
⊢ Q R S f
|
subst this
|
case mk.intro.intro.intro
P : (R : Type u) → [inst : CommRing R] → (S : Type u) → [inst_1 : CommRing S] → (R →+* S) → Prop
Q : (R : Type u) → [inst : CommRing R] → (S : Type v) → [inst_1 : CommRing S] → (R →+* S) → Prop
polynomial : ∀ (R : Type u) [inst : CommRing R], P R R[X] C
fg_ker :
∀ (R : Type u) [inst : CommRing R] (S : Type v) [inst_1 : CommRing S] (f : R →+* S),
Surjective ⇑f → (ker f).FG → Q R S f
comp :
∀ (R : Type u) [inst : CommRing R] (S : Type u) [inst_1 : CommRing S] (T : Type v) [inst_2 : CommRing T] (f : R →+* S)
(g : S →+* T), P R S f → Q S T g → Q R T (g.comp f)
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℕ
g' : MvPolynomial (Fin n) R →+* S
hg : Surjective ⇑g'
hg' : (ker g').FG
this : Algebra R S := (g'.comp MvPolynomial.C).toAlgebra
g : MvPolynomial (Fin n) R →ₐ[R] S
⊢ Q R S (g'.comp MvPolynomial.C)
|
7a593dd7687244ec
|
finite_powers
|
Mathlib/GroupTheory/OrderOfElement.lean
|
@[to_additive (attr := simp)]
lemma finite_powers : (powers a : Set G).Finite ↔ IsOfFinOrder a
|
G : Type u_1
inst✝ : LeftCancelMonoid G
a : G
h : (↑(powers a)).Finite
⊢ IsOfFinOrder a
|
obtain ⟨m, n, hmn, ha⟩ := h.exists_lt_map_eq_of_forall_mem (f := fun n : ℕ ↦ a ^ n)
(fun n ↦ by simp [mem_powers_iff])
|
case intro.intro.intro
G : Type u_1
inst✝ : LeftCancelMonoid G
a : G
h : (↑(powers a)).Finite
m n : ℕ
hmn : m < n
ha : a ^ m = a ^ n
⊢ IsOfFinOrder a
|
2594fbf04d0aff1b
|
CategoryTheory.MonoidalCategory.leftUnitor_tensor_inv'
|
Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean
|
theorem leftUnitor_tensor_inv' (X Y : C) :
(λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom
|
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : MonoidalCategory C
X Y : C
⊢ (λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom
|
monoidal_coherence
|
no goals
|
22f4a25417a36d13
|
TopologicalSpace.Clopens.exists_finset_eq_sup_prod
|
Mathlib/Topology/ClopenBox.lean
|
theorem exists_finset_eq_sup_prod (W : Clopens (X × Y)) :
∃ (I : Finset (Clopens X × Clopens Y)), W = I.sup fun i ↦ i.1 ×ˢ i.2
|
case h
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : CompactSpace X
W : Clopens (X × Y)
U : X × Y → Clopens X
hxU : ∀ x ∈ W, x.1 ∈ U x
V : X × Y → Clopens Y
hxV : ∀ x ∈ W, x.2 ∈ V x
hUV : ∀ x ∈ W, U x ×ˢ V x ≤ W
I : Finset (X × Y)
hIW : ∀ x ∈ I, x ∈ W.carrier
hWI : W.carrier ⊆ ⋃ x ∈ I, ↑(U x) ×ˢ ↑(V x)
⊢ W = I.sup ((fun i => i.1 ×ˢ i.2) ∘ fun x => (U x, V x))
|
refine le_antisymm (fun x hx ↦ ?_) (Finset.sup_le fun x hx ↦ ?_)
|
case h.refine_1
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : CompactSpace X
W : Clopens (X × Y)
U : X × Y → Clopens X
hxU : ∀ x ∈ W, x.1 ∈ U x
V : X × Y → Clopens Y
hxV : ∀ x ∈ W, x.2 ∈ V x
hUV : ∀ x ∈ W, U x ×ˢ V x ≤ W
I : Finset (X × Y)
hIW : ∀ x ∈ I, x ∈ W.carrier
hWI : W.carrier ⊆ ⋃ x ∈ I, ↑(U x) ×ˢ ↑(V x)
x : X × Y
hx : x ∈ ↑W
⊢ x ∈ ↑(I.sup ((fun i => i.1 ×ˢ i.2) ∘ fun x => (U x, V x)))
case h.refine_2
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace Y
inst✝ : CompactSpace X
W : Clopens (X × Y)
U : X × Y → Clopens X
hxU : ∀ x ∈ W, x.1 ∈ U x
V : X × Y → Clopens Y
hxV : ∀ x ∈ W, x.2 ∈ V x
hUV : ∀ x ∈ W, U x ×ˢ V x ≤ W
I : Finset (X × Y)
hIW : ∀ x ∈ I, x ∈ W.carrier
hWI : W.carrier ⊆ ⋃ x ∈ I, ↑(U x) ×ˢ ↑(V x)
x : X × Y
hx : x ∈ I
⊢ ((fun i => i.1 ×ˢ i.2) ∘ fun x => (U x, V x)) x ≤ W
|
da4f184b61cb0200
|
List.set_set_perm'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/Perm.lean
|
theorem set_set_perm' {as : List α} {i j : Nat} (h₁ : i < as.length) (h₂ : i + j < as.length)
(hj : 0 < j) :
(as.set i as[i + j]).set (i + j) as[i] ~ as
|
α : Type u_1
as : List α
i j : Nat
h₁ : i < as.length
h₂ : i + j < as.length
hj : 0 < j
this : as = take i as ++ as[i] :: drop (i + 1) (take (i + j) as) ++ as[i + j] :: drop (i + j + 1) as
⊢ (as.set i as[i + j]).set (i + j) as[i] ~ as
|
conv => lhs; congr; congr; rw [this]
|
α : Type u_1
as : List α
i j : Nat
h₁ : i < as.length
h₂ : i + j < as.length
hj : 0 < j
this : as = take i as ++ as[i] :: drop (i + 1) (take (i + j) as) ++ as[i + j] :: drop (i + j + 1) as
⊢ ((take i as ++ as[i] :: drop (i + 1) (take (i + j) as) ++ as[i + j] :: drop (i + j + 1) as).set i as[i + j]).set
(i + j) as[i] ~
as
|
9b51820a9079599a
|
doublyStochastic_sum_perm_aux
|
Mathlib/Analysis/Convex/Birkhoff.lean
|
/--
If M is a scalar multiple of a doubly stochastic matrix, then it is a conical combination of
permutation matrices. This is most useful when M is a doubly stochastic matrix, in which case
the combination is convex.
This particular formulation is chosen to make the inductive step easier: we no longer need to
rescale each time a permutation matrix is subtracted.
-/
private lemma doublyStochastic_sum_perm_aux (M : Matrix n n R)
(s : R) (hs : 0 ≤ s)
(hM : ∃ M' ∈ doublyStochastic R n, M = s • M') :
∃ w : Equiv.Perm n → R, (∀ σ, 0 ≤ w σ) ∧ ∑ σ, w σ • σ.permMatrix R = M
|
case mk
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : LinearOrderedField R
h✝ : Nonempty n
d : ℕ
ih :
∀ m < d,
∀ (M : Matrix n n R) (s : R),
0 ≤ s →
(∃ M' ∈ doublyStochastic R n, M = s • M') →
#(filter (fun i => M i.1 i.2 ≠ 0) univ) = m →
∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ : Equiv.Perm n, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j : n, M i j = s) ∧ ∀ (j : n), ∑ i : n, M i j = s
hd : #(filter (fun i => M i.1 i.2 ≠ 0) univ) = d
hs'✝ : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
hs' : 0 ≤ s'
this : ∃ M' ∈ doublyStochastic R n, N = s' • M'
i' j' : n
hN' : ¬N i' j' = 0
hM' : M i' j' = 0
⊢ False
|
simp only [sub_apply, hM', smul_apply, PEquiv.toMatrix_apply, Equiv.toPEquiv_apply,
Option.mem_def, Option.some.injEq, smul_eq_mul, mul_ite, mul_one, mul_zero, zero_sub,
neg_eq_zero, ite_eq_right_iff, Classical.not_imp, N] at hN'
|
case mk
R : Type u_1
n : Type u_2
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : LinearOrderedField R
h✝ : Nonempty n
d : ℕ
ih :
∀ m < d,
∀ (M : Matrix n n R) (s : R),
0 ≤ s →
(∃ M' ∈ doublyStochastic R n, M = s • M') →
#(filter (fun i => M i.1 i.2 ≠ 0) univ) = m →
∃ w, (∀ (σ : Equiv.Perm n), 0 ≤ w σ) ∧ ∑ σ : Equiv.Perm n, w σ • Equiv.Perm.permMatrix R σ = M
M : Matrix n n R
s : R
hs : 0 ≤ s
hM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j : n, M i j = s) ∧ ∀ (j : n), ∑ i : n, M i j = s
hd : #(filter (fun i => M i.1 i.2 ≠ 0) univ) = d
hs'✝ : 0 < s
σ : Equiv.Perm n
hσ : ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0
i : n
hi : i ∈ univ
hi' : ∀ x' ∈ univ, M i (σ i) ≤ M x' (σ x')
N : Matrix n n R := M - M i (σ i) • Equiv.Perm.permMatrix R σ
hMi' : 0 < M i (σ i)
s' : R := s - M i (σ i)
hs' : 0 ≤ s'
this : ∃ M' ∈ doublyStochastic R n, N = s' • M'
i' j' : n
hM' : M i' j' = 0
hN' : σ i' = j' ∧ ¬M i (σ i) = 0
⊢ False
|
fe82d3c50ee8248f
|
exists_zpow_eq_one
|
Mathlib/GroupTheory/OrderOfElement.lean
|
theorem exists_zpow_eq_one (x : G) : ∃ (i : ℤ) (_ : i ≠ 0), x ^ (i : ℤ) = 1
|
case intro.intro
G : Type u_1
inst✝¹ : Group G
inst✝ : Finite G
x : G
w : ℕ
hw1 : w > 0
hw2 : IsPeriodicPt (fun x_1 => x * x_1) w 1
⊢ x ^ w = 1
|
exact (isPeriodicPt_mul_iff_pow_eq_one _).mp hw2
|
no goals
|
98741314826b3115
|
HahnSeries.embDomain_single
|
Mathlib/RingTheory/HahnSeries/Basic.lean
|
theorem embDomain_single {f : Γ ↪o Γ'} {g : Γ} {r : R} :
embDomain f (single g r) = single (f g) r
|
case pos
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
inst✝² : PartialOrder Γ
inst✝¹ : Zero R
inst✝ : PartialOrder Γ'
f : Γ ↪o Γ'
g : Γ
r : R
g' : Γ'
h : ¬g' = f g
hr : r = 0
⊢ g' ∉ ⇑f '' ((single g) r).support
|
simp [hr]
|
no goals
|
b834b49fe6448d2a
|
pow_sub_one_dvd_differentIdeal_aux
|
Mathlib/RingTheory/DedekindDomain/Different.lean
|
lemma pow_sub_one_dvd_differentIdeal_aux [IsFractionRing B L] [IsDedekindDomain A]
[NoZeroSMulDivisors A B] [Module.Finite A B]
{p : Ideal A} [p.IsMaximal] (P : Ideal B) {e : ℕ} (he : e ≠ 0) (hp : p ≠ ⊥)
(hP : P ^ e ∣ p.map (algebraMap A B)) : P ^ (e - 1) ∣ differentIdeal A B
|
case a
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : Algebra.IsSeparable K L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDedekindDomain B
inst✝⁴ : IsFractionRing B L
inst✝³ : IsDedekindDomain A
inst✝² : NoZeroSMulDivisors A B
inst✝¹ : Module.Finite A B
p : Ideal A
inst✝ : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP : P ^ e ∣ Ideal.map (algebraMap A B) p
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this✝ : Ideal.map (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, (intTrace A B) x ∈ p
⊢ (↑(P ^ (e - 1)))⁻¹⁻¹ = (↑a / ↑(Ideal.map (algebraMap A B) p))⁻¹
|
simp only [inv_inv, ha, FractionalIdeal.coeIdeal_mul, inv_div, ne_eq,
FractionalIdeal.coeIdeal_eq_zero, mul_div_assoc]
|
case a
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : FiniteDimensional K L
inst✝⁸ : Algebra.IsSeparable K L
inst✝⁷ : IsIntegralClosure B A L
inst✝⁶ : IsIntegrallyClosed A
inst✝⁵ : IsDedekindDomain B
inst✝⁴ : IsFractionRing B L
inst✝³ : IsDedekindDomain A
inst✝² : NoZeroSMulDivisors A B
inst✝¹ : Module.Finite A B
p : Ideal A
inst✝ : p.IsMaximal
P : Ideal B
e : ℕ
he : e ≠ 0
hp : p ≠ ⊥
hP : P ^ e ∣ Ideal.map (algebraMap A B) p
a : Ideal B
ha : Ideal.map (algebraMap A B) p = P ^ (e - 1) * a
hp' : ¬Ideal.map (algebraMap A B) p = ⊥
habot : a ≠ ⊥
hPbot : P ≠ ⊥
this✝ : Ideal.map (algebraMap A B) p ∣ a ^ e
this : ∀ x ∈ a, (intTrace A B) x ∈ p
⊢ ↑(P ^ (e - 1)) = ↑(P ^ (e - 1)) * (↑a / ↑a)
|
18a0d9e4648de8c5
|
IsPrimitiveRoot.minpoly_eq_pow
|
Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
|
theorem minpoly_eq_pow {p : ℕ} [hprime : Fact p.Prime] (hdiv : ¬p ∣ n) :
minpoly ℤ μ = minpoly ℤ (μ ^ p)
|
case refine_1
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
aux : Irreducible P ↔ Irreducible (map (Int.castRingHom ℚ) P)
⊢ ¬P ∣ Q
|
intro hdiv
|
case refine_1
n : ℕ
K : Type u_1
inst✝² : CommRing K
μ : K
h : IsPrimitiveRoot μ n
inst✝¹ : IsDomain K
inst✝ : CharZero K
p : ℕ
hprime : Fact (Nat.Prime p)
hdiv✝ : ¬p ∣ n
hn : ¬n = 0
hpos : 0 < n
P : ℤ[X] := minpoly ℤ μ
Q : ℤ[X] := minpoly ℤ (μ ^ p)
hdiff : ¬P = Q
Pmonic : P.Monic
Qmonic : Q.Monic
Pirr : Irreducible P
Qirr : Irreducible Q
PQprim : (P * Q).IsPrimitive
aux : Irreducible P ↔ Irreducible (map (Int.castRingHom ℚ) P)
hdiv : P ∣ Q
⊢ False
|
60585a9f1d93c365
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_fold_fn_preserves_induction_motive
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
|
theorem reduce_fold_fn_preserves_induction_motive {c_arr : Array (Literal (PosFin n))}
{assignment : Array Assignment}
(idx : Fin c_arr.size) (res : ReduceResult (PosFin n))
(ih : ReducePostconditionInductionMotive c_arr assignment idx.1 res) :
ReducePostconditionInductionMotive c_arr assignment (idx.1 + 1) (reduce_fold_fn assignment res c_arr[idx])
|
case right.h_4
n : Nat
c_arr : Array (Literal (PosFin n))
assignment : Array Assignment
idx : Fin c_arr.size
i : PosFin n
b : Bool
p : PosFin n → Bool
hp : p ⊨ assignment
j : Fin c_arr.size
j_lt_idx_add_one : ↑j < ↑idx + 1
p_entails_c_arr_j : p ⊨ c_arr[↑j]
acc✝ : ReduceResult (PosFin n)
ih : ReducePostconditionInductionMotive c_arr assignment (↑idx) reducedToNonunit
h : reducedToNonunit = reducedToUnit (i, b)
⊢ p ⊨ (i, b)
|
simp at h
|
no goals
|
681a60500ce41fce
|
CategoryTheory.GrothendieckTopology.toPlus_naturality
|
Mathlib/CategoryTheory/Sites/Plus.lean
|
theorem toPlus_naturality {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
η ≫ J.toPlus Q = J.toPlus _ ≫ J.plusMap η
|
case w.h
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
x✝ : Cᵒᵖ
⊢ (η ≫ J.toPlus Q).app x✝ = (J.toPlus P ≫ J.plusMap η).app x✝
|
dsimp [toPlus, plusMap]
|
case w.h
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
P Q : Cᵒᵖ ⥤ D
η : P ⟶ Q
x✝ : Cᵒᵖ
⊢ η.app x✝ ≫ ⊤.toMultiequalizer Q ≫ colimit.ι (J.diagram Q (unop x✝)) (op ⊤) =
(⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop x✝)) (op ⊤)) ≫ colimMap (J.diagramNatTrans η (unop x✝))
|
684ea2b09c5da0f6
|
fourierIntegral_half_period_translate
|
Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean
|
theorem fourierIntegral_half_period_translate {w : V} (hw : w ≠ 0) :
(∫ v : V, 𝐞 (-⟪v, w⟫) • f (v + i w)) = -∫ v : V, 𝐞 (-⟪v, w⟫) • f v
|
case h
E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
⊢ ‖w‖ ^ 2 ≠ 0
|
rwa [Ne, sq_eq_zero_iff, norm_eq_zero]
|
no goals
|
851451860b8c873e
|
Int.gcd_dvd_iff
|
Mathlib/Data/Int/GCD.lean
|
theorem gcd_dvd_iff {a b : ℤ} {n : ℕ} : gcd a b ∣ n ↔ ∃ x y : ℤ, ↑n = a * x + b * y
|
a b : ℤ
n : ℕ
⊢ a.gcd b ∣ n ↔ ∃ x y, ↑n = a * x + b * y
|
constructor
|
case mp
a b : ℤ
n : ℕ
⊢ a.gcd b ∣ n → ∃ x y, ↑n = a * x + b * y
case mpr
a b : ℤ
n : ℕ
⊢ (∃ x y, ↑n = a * x + b * y) → a.gcd b ∣ n
|
a5bf30f837d31f28
|
Array.getElem_extract_loop_ge_aux
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem getElem_extract_loop_ge_aux (as bs : Array α) (size start : Nat) (hge : i ≥ bs.size)
(h : i < (extract.loop as size start bs).size) : start + i - bs.size < as.size
|
case h
α : Type u_1
i : Nat
as bs : Array α
size start : Nat
hge : i ≥ bs.size
h : i < (extract.loop as size start bs).size
⊢ min size (as.size - start) ≤ as.size - start
|
exact Nat.min_le_right ..
|
no goals
|
be0a6fd7e95fa817
|
WeierstrassCurve.Jacobian.eval_polynomialY_of_Z_ne_zero
|
Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
|
lemma eval_polynomialY_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
eval P W.polynomialY / P z ^ 3 =
W.toAffine.polynomialY.evalEval (P x / P z ^ 2) (P y / P z ^ 3)
|
case a.a
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (eval P) W.polynomialY / P z ^ 3 + (W.a₁ * P x / P z ^ 2 * 1 + W.a₃ * 1) -
(Polynomial.evalEval (P x / P z ^ 2) (P y / P z ^ 3) (toAffine W).polynomialY +
(W.a₁ * P x / P z ^ 2 * (P z / P z) + W.a₃ * (P z ^ 3 / P z ^ 3))) =
0
|
rw [eval_polynomialY, Affine.evalEval_polynomialY]
|
case a.a
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (2 * P y + W.a₁ * P x * P z + W.a₃ * P z ^ 3) / P z ^ 3 + (W.a₁ * P x / P z ^ 2 * 1 + W.a₃ * 1) -
(2 * (P y / P z ^ 3) + (toAffine W).a₁ * (P x / P z ^ 2) + (toAffine W).a₃ +
(W.a₁ * P x / P z ^ 2 * (P z / P z) + W.a₃ * (P z ^ 3 / P z ^ 3))) =
0
|
ce8e9326e0978937
|
ENNReal.inv_rpow
|
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
theorem inv_rpow (x : ℝ≥0∞) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹
|
case inr.inr.inr
x : ℝ≥0∞
y : ℝ
hy : y < 0 ∨ 0 < y
h0 : x ≠ 0
h_top : x ≠ ⊤
⊢ x⁻¹ ^ y = (x ^ y)⁻¹
|
apply ENNReal.eq_inv_of_mul_eq_one_left
|
case inr.inr.inr.h
x : ℝ≥0∞
y : ℝ
hy : y < 0 ∨ 0 < y
h0 : x ≠ 0
h_top : x ≠ ⊤
⊢ x⁻¹ ^ y * x ^ y = 1
|
b19d8c607ccb3ad9
|
Surreal.Multiplication.mul_right_le_of_equiv
|
Mathlib/SetTheory/Surreal/Multiplication.lean
|
theorem mul_right_le_of_equiv (h₁ : x₁.Numeric) (h₂ : x₂.Numeric)
(h₁₂ : IH24 x₁ x₂ y) (h₂₁ : IH24 x₂ x₁ y) (he : x₁ ≈ x₂) : x₁ * y ≤ x₂ * y
|
case h₂.left
x₁ x₂ y : PGame
h₁ : x₁.Numeric
h₂ : x₂.Numeric
h₁₂ : IH24 x₁ x₂ y
h₂₁ : IH24 x₂ x₁ y
he : x₁ ≈ x₂
he' : -x₁ ≈ -x₂
i✝ : x₂.LeftMoves
j✝ : (-y).LeftMoves
⊢ ⟦x₂.mulOption (-y) i✝ j✝⟧ < ⟦x₁ * -y⟧
|
apply mulOption_lt_mul_of_equiv h₂ (ih24_neg h₂₁).2 (symm he)
|
no goals
|
e1812a149e6bada9
|
MeasureTheory.L2.norm_sq_eq_inner'
|
Mathlib/MeasureTheory/Function/L2Space.lean
|
theorem norm_sq_eq_inner' (f : α →₂[μ] E) : ‖f‖ ^ 2 = RCLike.re ⟪f, f⟫
|
α : Type u_1
E : Type u_2
𝕜 : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : ↥(Lp E 2 μ)
⊢ ENNReal.toReal 2 = 2
|
simp
|
no goals
|
629fb9c7e3401fd8
|
LieAlgebra.zeroRootSubalgebra_eq_of_is_cartan
|
Mathlib/Algebra/Lie/Weights/Cartan.lean
|
theorem zeroRootSubalgebra_eq_of_is_cartan (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
[IsNoetherian R L] : zeroRootSubalgebra R L H = H
|
R : Type u_1
L : Type u_2
inst✝⁴ : CommRing R
inst✝³ : LieRing L
inst✝² : LieAlgebra R L
H : LieSubalgebra R L
inst✝¹ : H.IsCartanSubalgebra
inst✝ : IsNoetherian R L
this : rootSpace H 0 ≤ H.toLieSubmodule
⊢ zeroRootSubalgebra R L H ≤ H
|
exact fun x hx => this hx
|
no goals
|
f049cf760aeeca8e
|
Multiset.prod_map_sum
|
Mathlib/Algebra/BigOperators/Ring/Multiset.lean
|
lemma prod_map_sum {s : Multiset (Multiset α)} :
prod (s.map sum) = sum ((Sections s).map prod) :=
Multiset.induction_on s (by simp) fun a s ih ↦ by
simp [ih, map_bind, sum_map_mul_left, sum_map_mul_right]
|
α : Type u_2
inst✝ : CommSemiring α
s : Multiset (Multiset α)
⊢ (map sum 0).prod = (map prod (Sections 0)).sum
|
simp
|
no goals
|
fb98ee7bf5561634
|
isOpen.dynEntourage
|
Mathlib/Dynamics/TopologicalEntropy/DynamicalEntourage.lean
|
lemma _root_.isOpen.dynEntourage [TopologicalSpace X] {T : X → X} (T_cont : Continuous T)
{U : Set (X × X)} (U_open : IsOpen U) (n : ℕ) :
IsOpen (dynEntourage T U n)
|
X : Type u_1
inst✝ : TopologicalSpace X
T : X → X
T_cont : Continuous T
U : Set (X × X)
U_open : IsOpen U
n : ℕ
k : ↑(Ico 0 n)
⊢ IsOpen ((map T T)^[↑k] ⁻¹' U)
|
exact U_open.preimage ((T_cont.prodMap T_cont).iterate k)
|
no goals
|
c6bf64c8430f1d4c
|
Matrix.isNilpotent_charpoly_sub_pow_of_isNilpotent
|
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
|
lemma isNilpotent_charpoly_sub_pow_of_isNilpotent (hM : IsNilpotent M) :
IsNilpotent (M.charpoly - X ^ (Fintype.card n))
|
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
hM : IsNilpotent M
a✝ : Nontrivial R
p : R[X] := M.charpolyRev
hp : p - 1 = X * (p /ₘ X)
⊢ IsNilpotent (M.charpoly - X ^ Fintype.card n)
|
have : IsNilpotent (p /ₘ X) :=
(Polynomial.isUnit_iff'.mp (isUnit_charpolyRev_of_isNilpotent hM)).2
|
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
hM : IsNilpotent M
a✝ : Nontrivial R
p : R[X] := M.charpolyRev
hp : p - 1 = X * (p /ₘ X)
this : IsNilpotent (p /ₘ X)
⊢ IsNilpotent (M.charpoly - X ^ Fintype.card n)
|
2fe27623d5fcf0fa
|
SimpContFract.determinant_aux
|
Mathlib/Algebra/ContinuedFractions/Determinant.lean
|
theorem determinant_aux (hyp : n = 0 ∨ ¬(↑s : GenContFract K).TerminatedAt (n - 1)) :
((↑s : GenContFract K).contsAux n).a * ((↑s : GenContFract K).contsAux (n + 1)).b -
((↑s : GenContFract K).contsAux n).b * ((↑s : GenContFract K).contsAux (n + 1)).a =
(-1) ^ n
|
K : Type u_1
inst✝ : Field K
s : SimpContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬(↑s).TerminatedAt (n + 1 - 1)
g : GenContFract K := ↑s
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH : n = 0 ∨ ¬(↑s).TerminatedAt (n - 1) → ppred_conts.a * pred_conts.b - ppred_conts.b * pred_conts.a = (-1) ^ n
ppred_conts_eq : ppred_conts = g.contsAux n
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
not_terminated_at_n : ¬g.TerminatedAt n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
this : ppA * pB - ppB * pA = (-1) ^ n
pow_succ_n : (-1) ^ (n + 1) = -1 * (-1) ^ n
⊢ pA * ppB - pB * ppA = (-1) ^ (n + 1)
|
rw [pow_succ_n, ← this]
|
K : Type u_1
inst✝ : Field K
s : SimpContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬(↑s).TerminatedAt (n + 1 - 1)
g : GenContFract K := ↑s
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH : n = 0 ∨ ¬(↑s).TerminatedAt (n - 1) → ppred_conts.a * pred_conts.b - ppred_conts.b * pred_conts.a = (-1) ^ n
ppred_conts_eq : ppred_conts = g.contsAux n
pA : K := pred_conts.a
pB : K := pred_conts.b
ppA : K := ppred_conts.a
ppB : K := ppred_conts.b
not_terminated_at_n : ¬g.TerminatedAt n
gp : Pair K
s_nth_eq : g.s.get? n = some gp
this : ppA * pB - ppB * pA = (-1) ^ n
pow_succ_n : (-1) ^ (n + 1) = -1 * (-1) ^ n
⊢ pA * ppB - pB * ppA = -1 * (ppA * pB - ppB * pA)
|
5478ed7427a774c4
|
Ideal.Filtration.submodule_closure_single
|
Mathlib/RingTheory/Filtration.lean
|
theorem submodule_closure_single :
AddSubmonoid.closure (⋃ i, single R i '' (F.N i : Set M)) = F.submodule.toAddSubmonoid
|
case a
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
f : PolynomialModule R M
hf : f ∈ F.submodule.toAddSubmonoid
⊢ Finsupp.sum f Finsupp.single ∈ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(F.N i))
|
apply AddSubmonoid.sum_mem _ _
|
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
I : Ideal R
F : I.Filtration M
f : PolynomialModule R M
hf : f ∈ F.submodule.toAddSubmonoid
⊢ ∀ c ∈ f.support, Finsupp.single c (f c) ∈ AddSubmonoid.closure (⋃ i, ⇑(single R i) '' ↑(F.N i))
|
bd336799b08b1c57
|
HomologicalComplex.mapBifunctor₂₃.d₁_eq_zero
|
Mathlib/Algebra/Homology/BifunctorAssociator.lean
|
lemma d₁_eq_zero (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) (h : ¬ c₁.Rel i₁ (c₁.next i₁)) :
d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j = 0
|
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{u_15, u_1} C₁
inst✝²¹ : Category.{u_17, u_2} C₂
inst✝²⁰ : Category.{u_16, u_5} C₃
inst✝¹⁹ : Category.{u_13, u_6} C₄
inst✝¹⁸ : Category.{u_14, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j : ι₄
h : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ d₁ F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ i₃ j = 0
|
dsimp [d₁]
|
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{u_15, u_1} C₁
inst✝²¹ : Category.{u_17, u_2} C₂
inst✝²⁰ : Category.{u_16, u_5} C₃
inst✝¹⁹ : Category.{u_13, u_6} C₄
inst✝¹⁸ : Category.{u_14, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j : ι₄
h : ¬c₁.Rel i₁ (c₁.next i₁)
⊢ c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) •
(F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫
ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ (c₁.next i₁) i₂ i₃ j =
0
|
27ea0ca2b0bbb349
|
Function.Embedding.setValue_eq
|
Mathlib/Logic/Embedding/Basic.lean
|
theorem setValue_eq {α β} (f : α ↪ β) (a : α) (b : β) [∀ a', Decidable (a' = a)]
[∀ a', Decidable (f a' = b)] : setValue f a b a = b
|
α : Sort u_1
β : Sort u_2
f : α ↪ β
a : α
b : β
inst✝¹ : (a' : α) → Decidable (a' = a)
inst✝ : (a' : α) → Decidable (f a' = b)
⊢ (f.setValue a b) a = b
|
simp [setValue]
|
no goals
|
116318157d574adc
|
LinearMap.quotientInfEquivSupQuotient_surjective
|
Mathlib/LinearAlgebra/Isomorphisms.lean
|
theorem quotientInfEquivSupQuotient_surjective (p p' : Submodule R M) :
Function.Surjective (quotientInfToSupQuotient p p')
|
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
⊢ Function.Surjective ⇑(quotientInfToSupQuotient p p')
|
rw [← range_eq_top, quotientInfToSupQuotient, range_liftQ, eq_top_iff']
|
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') ⧸ comap (p ⊔ p').subtype p'), x ∈ range (subToSupQuotient p p')
|
7b2f7728722e9e70
|
Projectivization.card
|
Mathlib/LinearAlgebra/Projectivization/Cardinality.lean
|
/-- Fraction free cardinality formula for the points of `ℙ k V` if `k` and `V` are finite
(for silly reasons the formula also holds when `k` and `V` are infinite).
See `Projectivization.card'` and `Projectivization.card''` for other spellings of the formula. -/
lemma card : Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
|
case inr
k : Type u_1
V : Type u_2
inst✝² : DivisionRing k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
a✝ : Nontrivial V
this✝ :
∀ (k : Type u_1) (V : Type u_2) [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V],
Nontrivial V → Finite k → Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
h : Infinite k
this : Infinite V
⊢ Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)
|
simp
|
no goals
|
1d5a95497a0b5f13
|
Rat.AbsoluteValue.equiv_real_of_unbounded
|
Mathlib/NumberTheory/Ostrowski.lean
|
theorem equiv_real_of_unbounded : f ≈ real
|
case intro.inr.inl
f : AbsoluteValue ℚ ℝ
notbdd : ¬∀ (n : ℕ), f ↑n ≤ 1
m : ℕ
hm : ¬f ↑m ≤ 1
oneltm : 1 < m
s : ℝ := logb (↑m) (f ↑m)
hs : s = logb (↑m) (f ↑m)
⊢ f ↑1 ^ s⁻¹ = real ↑1
|
simp
|
no goals
|
6f20338292d04472
|
MeasureTheory.setIntegral_gt_gt
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
theorem setIntegral_gt_gt {R : ℝ} {f : X → ℝ} (hR : 0 ≤ R)
(hfint : IntegrableOn f {x | ↑R < f x} μ) (hμ : μ {x | ↑R < f x} ≠ 0) :
(μ {x | ↑R < f x}).toReal * R < ∫ x in {x | ↑R < f x}, f x ∂μ
|
case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x} μ
⊢ 0 ≤ᶠ[ae (μ.restrict {x | R < f x})] fun a => f a - R
|
rw [Pi.zero_def, EventuallyLE, ae_restrict_iff₀]
|
case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x} μ
⊢ ∀ᵐ (x : X) ∂μ, x ∈ {x | R < f x} → 0 ≤ f x - R
case hf
X : Type u_1
mX : MeasurableSpace X
μ : Measure X
R : ℝ
f : X → ℝ
hR : 0 ≤ R
hfint : IntegrableOn f {x | R < f x} μ
hμ : μ {x | R < f x} ≠ 0
this : IntegrableOn (fun x => R) {x | R < f x} μ
⊢ NullMeasurableSet {x | 0 ≤ f x - R} (μ.restrict {x | R < f x})
|
d0b4505621d161b9
|
LieSubmodule.lcs_le_self
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
theorem lcs_le_self : N.lcs k ≤ N
|
case zero
R : Type u
L : Type v
M : Type w
inst✝⁵ : CommRing R
inst✝⁴ : LieRing L
inst✝³ : LieAlgebra R L
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : LieRingModule L M
k : ℕ
N : LieSubmodule R L M
⊢ lcs 0 N ≤ N
|
simp
|
no goals
|
cadb8f5db137edd1
|
IsOpen.continuous_piecewise_of_specializes
|
Mathlib/Topology/Inseparable.lean
|
theorem IsOpen.continuous_piecewise_of_specializes [DecidablePred (· ∈ s)] (hs : IsOpen s)
(hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g)
|
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s : Set X
f g : X → Y
inst✝ : DecidablePred fun x => x ∈ s
hs : IsOpen s
hf : Continuous f
hg : Continuous g
hspec : ∀ (x : X), f x ⤳ g x
this : ∀ (U : Set Y), IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U
U : Set Y
hU : IsOpen U
⊢ IsOpen (f ⁻¹' U ∩ s ∪ g ⁻¹' U)
|
exact hU.preimage hf |>.inter hs |>.union (hU.preimage hg)
|
no goals
|
2854524e4bf53082
|
CoxeterSystem.IsReduced.nodup_rightInvSeq
|
Mathlib/GroupTheory/Coxeter/Inversion.lean
|
theorem IsReduced.nodup_rightInvSeq {ω : List B} (rω : cs.IsReduced ω) : List.Nodup (ris ω)
|
case e_opt.e_a
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
rω : cs.IsReduced ω
j j' : ℕ
j_lt_j' : j < j'
j'_lt_length : j' < ω.length
dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1
t : W := (cs.rightInvSeq ω).getD j 1
h₁ : t = (cs.rightInvSeq ω).getD j 1
t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1
h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1
⊢ ω[j + 1 + (j' - 1 - (List.take j ω).length)]? = ω[j']?
|
congr
|
case e_opt.e_a.e_a
B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
rω : cs.IsReduced ω
j j' : ℕ
j_lt_j' : j < j'
j'_lt_length : j' < ω.length
dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1
t : W := (cs.rightInvSeq ω).getD j 1
h₁ : t = (cs.rightInvSeq ω).getD j 1
t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1
h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1
⊢ j + 1 + (j' - 1 - (List.take j ω).length) = j'
|
4f092e79e4ea2c15
|
LieSubalgebra.normalizer_eq_self_of_engel_le
|
Mathlib/Algebra/Lie/EngelSubalgebra.lean
|
/-- A Lie-subalgebra of an Artinian Lie algebra is self-normalizing
if it contains an Engel subalgebra.
See `LieSubalgebra.normalizer_engel` for a proof that Engel subalgebras are self-normalizing,
avoiding the Artinian condition. -/
lemma normalizer_eq_self_of_engel_le [IsArtinian R L]
(H : LieSubalgebra R L) (x : L) (h : engel R x ≤ H) :
normalizer H = H
|
case intro.a.h
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsArtinian R L
H : LieSubalgebra R L
x : L
h : engel R x ≤ H
N : LieSubalgebra R L := H.normalizer
aux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H
aux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N
dx : ↥N →ₗ[R] ↥N := LinearMap.restrict ((ad R L) x) aux₂
k : ℕ
hk : Codisjoint (LinearMap.ker (dx ^ (k + 1))) (LinearMap.range (dx ^ (k + 1)))
⊢ LinearMap.range (dx ^ (k + 1)) ≤ Submodule.comap N.subtype H.toSubmodule
|
rintro _ ⟨y, rfl⟩
|
case intro.a.h.intro
R : Type u_1
L : Type u_2
inst✝³ : CommRing R
inst✝² : LieRing L
inst✝¹ : LieAlgebra R L
inst✝ : IsArtinian R L
H : LieSubalgebra R L
x : L
h : engel R x ≤ H
N : LieSubalgebra R L := H.normalizer
aux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H
aux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N
dx : ↥N →ₗ[R] ↥N := LinearMap.restrict ((ad R L) x) aux₂
k : ℕ
hk : Codisjoint (LinearMap.ker (dx ^ (k + 1))) (LinearMap.range (dx ^ (k + 1)))
y : ↥N
⊢ (dx ^ (k + 1)) y ∈ Submodule.comap N.subtype H.toSubmodule
|
253d8e16fa48a5a4
|
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