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.Re... | 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 ∈ ... | 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 : ... | 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 : ... | 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... | 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... | 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
f... | 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.i... | 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_... | 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... | 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 (er... | 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 ... | 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 ... | 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... | 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).fromG... | 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... | 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... | 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)))... | 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' : ModelWithCorn... | 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' : ModelWithCorn... | 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... | 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... | 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... | 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✝ : IsSca... | 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) = ... | 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 : ... | 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).P... | 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).P... | 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... | 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 (... | 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
hp... | 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 ..))
... | 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✝ →
(nextPowerOfT... | 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 ... | 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.mulS... | 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 a... | 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... | 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... | 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 r... | 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 : CommRi... | 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 : CommRi... | 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.... | 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 ∈ ... | 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... | 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... | 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, (∀ (... | 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, (∀ (... | 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 : IsDom... | 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 : IsDom... | 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
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
... | 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 + ... | 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 : ReducePostconditionInductio... | 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.toPl... | 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 ≫... | 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))) =
... | 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 ... | 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).Termi... | 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).Termi... | 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 : HasZeroM... | 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 : HasZeroM... | 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... | 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 ... | 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 → ... | 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... | 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... | 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).... | 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 :... | 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) aux... | 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) ... | 253d8e16fa48a5a4 |
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