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 |
|---|---|---|---|---|---|---|
Module.End.independent_iInf_maxGenEigenspace_of_forall_mapsTo | Mathlib/LinearAlgebra/Eigenspace/Pi.lean | lemma independent_iInf_maxGenEigenspace_of_forall_mapsTo
(h : ∀ i j φ, MapsTo (f i) ((f j).maxGenEigenspace φ) ((f j).maxGenEigenspace φ)) :
iSupIndep fun χ : ι → R ↦ ⨅ i, (f i).maxGenEigenspace (χ i) | ι : Type u_1
R : Type u_2
M : Type u_4
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
f : ι → End R M
inst✝ : NoZeroSMulDivisors R M
h : ∀ (i j : ι) (φ : R), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)
l : ι
χ : ι → R
x : M
hx : ∀ (i : ι), x ∈ (f i).maxGenEigenspace (χ i)
⊢ ∀ ... | exact fun i ↦ h l i (χ i) (hx i) | no goals | 4dd72b81f13de433 |
Nat.Linear.Poly.denote_eq_cancelAux | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem Poly.denote_eq_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly)
(h : denote_eq ctx (r₁.reverse ++ m₁, r₂.reverse ++ m₂)) : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) | case neg
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
r₁ r₂ m₁✝ m₂✝ : Poly
k₁ : Nat
v₁ : Var
m₁ m₂ : List (Nat × Var)
hltv hgtv : ¬blt v₁ v₁ = true
h : denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) :: m... | apply ih | case neg.h
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
r₁ r₂ m₁✝ m₂✝ : Poly
k₁ : Nat
v₁ : Var
m₁ m₂ : List (Nat × Var)
hltv hgtv : ¬blt v₁ v₁ = true
h : denote_eq ctx (List.reverse r₁ ++ (k₁, v₁) ::... | 9bfcd466b3bdd638 |
PrimeSpectrum.iSup_basicOpen_eq_top_iff | Mathlib/RingTheory/Spectrum/Prime/Topology.lean | lemma iSup_basicOpen_eq_top_iff {ι : Type*} {f : ι → R} :
(⨆ i : ι, PrimeSpectrum.basicOpen (f i)) = ⊤ ↔ Ideal.span (Set.range f) = ⊤ | R : Type u
inst✝ : CommSemiring R
ι : Type u_1
f : ι → R
⊢ zeroLocus (⋃ i, {f i}) = Set.univᶜ ↔ zeroLocus ↑(Ideal.span (Set.range f)) = ∅ | simp only [Set.iUnion_singleton_eq_range, Set.compl_univ, PrimeSpectrum.zeroLocus_span] | no goals | f99a5f18aac9d319 |
BitVec.neg_eq_not_add | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem neg_eq_not_add (x : BitVec w) : -x = ~~~x + 1#w | case a.e_a
w : Nat
x : BitVec w
hx : x.toNat < 2 ^ w
⊢ 2 ^ w - x.toNat = 2 ^ w - 1 - x.toNat + 1 | rw [Nat.sub_sub, Nat.add_comm 1 x.toNat, ← Nat.sub_sub, Nat.sub_add_cancel (by omega)] | no goals | 0b1fe6f9c958eaa0 |
LinearMap.linearProjOfIsCompl_of_proj | Mathlib/LinearAlgebra/Projection.lean | theorem linearProjOfIsCompl_of_proj (f : E →ₗ[R] p) (hf : ∀ x : p, f x = x) :
p.linearProjOfIsCompl (ker f) (isCompl_of_proj hf) = f | case h.a.intro.intro
R : Type u_1
inst✝² : Ring R
E : Type u_2
inst✝¹ : AddCommGroup E
inst✝ : Module R E
p : Submodule R E
f : E →ₗ[R] ↥p
hf : ∀ (x : ↥p), f ↑x = x
x : ↥p
y : ↥(ker f)
this : ↑x + ↑y ∈ p ⊔ ker f
⊢ ↑((p.linearProjOfIsCompl (ker f) ⋯) (↑x + ↑y)) = ↑(f (↑x + ↑y)) | simp [hf] | no goals | 7c96f1fd31cda4bd |
Matrix.adjugate_transpose | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | theorem adjugate_transpose (A : Matrix n n α) : (adjugate A)ᵀ = adjugate Aᵀ | case pos.e_f.h
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
j : n
σ : Perm n
a✝ : σ ∈ univ
j' : n
this : σ j' = σ j ↔ j' = j
⊢ (if j' = j then Pi.single j 1 j' else A (σ j') j') = if j' = j then Pi.single (σ j) 1 (σ j') else A (σ j') j' | rw [← dite_eq_ite, ← dite_eq_ite] | case pos.e_f.h
n : Type v
α : Type w
inst✝² : DecidableEq n
inst✝¹ : Fintype n
inst✝ : CommRing α
A : Matrix n n α
j : n
σ : Perm n
a✝ : σ ∈ univ
j' : n
this : σ j' = σ j ↔ j' = j
⊢ (if x : j' = j then Pi.single j 1 j' else A (σ j') j') = if x : j' = j then Pi.single (σ j) 1 (σ j') else A (σ j') j' | 71e22f197b6f8782 |
MeasureTheory.measurableSet_range_of_continuous_injective | Mathlib/MeasureTheory/Constructions/Polish/Basic.lean | theorem measurableSet_range_of_continuous_injective {β : Type*} [TopologicalSpace γ]
[PolishSpace γ] [TopologicalSpace β] [T2Space β] [MeasurableSpace β] [OpensMeasurableSpace β]
{f : γ → β} (f_cont : Continuous f) (f_inj : Injective f) :
MeasurableSet (range f) | case intro.intro.intro.intro.intro.intro.h₂
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePo... | have I : ∀ m n, ((s m).1 ∩ (s n).1).Nonempty := by
intro m n
rw [← not_disjoint_iff_nonempty_inter]
by_contra! h
have A : x ∈ q ⟨(s m, s n), h⟩ \ q ⟨(s n, s m), h.symm⟩ :=
haveI := mem_iInter.1 (hxs m).2 (s n)
(mem_iInter.1 this h :)
have B : x ∈ q ⟨(s n, s m), h.symm⟩ \ q ⟨(s m, s n), h⟩ :=
haveI... | case intro.intro.intro.intro.intro.intro.h₂
γ : Type u_3
β : Type u_4
inst✝⁵ : TopologicalSpace γ
inst✝⁴ : PolishSpace γ
inst✝³ : TopologicalSpace β
inst✝² : T2Space β
inst✝¹ : MeasurableSpace β
inst✝ : OpensMeasurableSpace β
f : γ → β
f_cont : Continuous f
f_inj : Injective f
this✝ : UpgradedPolishSpace γ := upgradePo... | b9138111f37aa037 |
tprod_eq_tprod_primes_mul_tprod_primes_of_mulSupport_subset_prime_powers | Mathlib/NumberTheory/LSeries/PrimesInAP.lean | @[to_additive tsum_eq_tsum_primes_add_tsum_primes_of_support_subset_prime_powers]
lemma tprod_eq_tprod_primes_mul_tprod_primes_of_mulSupport_subset_prime_powers {f : ℕ → α}
(hfm : Multipliable f) (hf : Function.mulSupport f ⊆ {n | IsPrimePow n}) :
∏' n : ℕ, f n = (∏' p : Nat.Primes, f p) * ∏' (p : Nat.Primes) ... | α : Type u_1
inst✝⁴ : CommGroup α
inst✝³ : UniformSpace α
inst✝² : UniformGroup α
inst✝¹ : CompleteSpace α
inst✝ : T0Space α
f : ℕ → α
hfm : Multipliable f
hf : Function.mulSupport f ⊆ {n | IsPrimePow n}
hfs' : ∀ (p : Nat.Primes), Multipliable fun k => f (↑p ^ (k + 1))
⊢ ∏' (p : Nat.Primes), f ↑p * ∏' (k : ℕ), f (↑p ^ ... | exact tprod_mul (Multipliable.subtype hfm _) <|
Multipliable.prod (f := fun (pk : Nat.Primes × ℕ) ↦ f (pk.1 ^ (pk.2 + 2))) <|
hfm.comp_injective <| Subtype.val_injective |>.comp
Nat.Primes.prodNatEquiv.injective |>.comp <|
Function.Injective.prodMap (fun ⦃_ _⦄ a ↦ a) <| add_left_injective 1 | no goals | d92e126e2c6283c1 |
MeasureTheory.hitting_eq_sInf | Mathlib/Probability/Process/HittingTime.lean | theorem hitting_eq_sInf (ω : Ω) : hitting u s ⊥ ⊤ ω = sInf {i : ι | u i ω ∈ s} | Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : CompleteLattice ι
u : ι → Ω → β
s : Set β
ω : Ω
⊢ hitting u s ⊥ ⊤ ω = sInf {i | u i ω ∈ s} | simp only [hitting, Set.mem_Icc, bot_le, le_top, and_self_iff, exists_true_left, Set.Icc_bot,
Set.Iic_top, Set.univ_inter, ite_eq_left_iff, not_exists] | Ω : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : CompleteLattice ι
u : ι → Ω → β
s : Set β
ω : Ω
⊢ (∀ (x : ι), ¬(x ∈ Set.univ ∧ u x ω ∈ s)) → ⊤ = sInf {i | u i ω ∈ s} | 245ff96581a98bf4 |
Nat.factorizationLCMRight_dvd_right | Mathlib/Data/Nat/Factorization/Basic.lean | lemma factorizationLCMRight_dvd_right : factorizationLCMRight a b ∣ b | case inr.inr.h
a b : ℕ
ha : a ≠ 0
hb : b ≠ 0
i✝ : ℕ
a✝ : i✝ ∈ (a.lcm b).factorization.support
⊢ i✝ ^ 0 = 1 | rw [pow_zero] | no goals | 39bc287d3431f65d |
cfc_sum | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean | lemma cfc_sum {ι : Type*} (f : ι → R → R) (a : A) (s : Finset ι)
(hf : ∀ i ∈ s, ContinuousOn (f i) (spectrum R a) | case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
ι : Type u_3
f : ι ... | have hf' : ContinuousOn (∑ i : s, f i) (spectrum R a) := by
rw [sum_coe_sort s, hsum]
exact continuousOn_finset_sum s fun i hi => hf i hi | case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R p
ι : Type u_3
f : ι ... | d7c05d75018caec3 |
ProbabilityTheory.Kernel.compProd_zero_right | Mathlib/Probability/Kernel/Composition/CompProd.lean | @[simp]
lemma compProd_zero_right (κ : Kernel α β) (γ : Type*) {mγ : MeasurableSpace γ} :
κ ⊗ₖ (0 : Kernel (α × β) γ) = 0 | case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
γ : Type u_4
mγ : MeasurableSpace γ
h : ¬IsSFiniteKernel κ
⊢ κ ⊗ₖ 0 = 0 | rw [Kernel.compProd_of_not_isSFiniteKernel_left _ _ h] | no goals | 8fce4b1a2edb4728 |
Equiv.embeddingCongr_apply_trans | Mathlib/Logic/Embedding/Basic.lean | theorem embeddingCongr_apply_trans {α₁ β₁ γ₁ α₂ β₂ γ₂ : Sort*} (ea : α₁ ≃ α₂) (eb : β₁ ≃ β₂)
(ec : γ₁ ≃ γ₂) (f : α₁ ↪ β₁) (g : β₁ ↪ γ₁) :
Equiv.embeddingCongr ea ec (f.trans g) =
(Equiv.embeddingCongr ea eb f).trans (Equiv.embeddingCongr eb ec g) | α₁ : Sort u_1
β₁ : Sort u_2
γ₁ : Sort u_3
α₂ : Sort u_4
β₂ : Sort u_5
γ₂ : Sort u_6
ea : α₁ ≃ α₂
eb : β₁ ≃ β₂
ec : γ₁ ≃ γ₂
f : α₁ ↪ β₁
g : β₁ ↪ γ₁
⊢ (ea.embeddingCongr ec) (f.trans g) = ((ea.embeddingCongr eb) f).trans ((eb.embeddingCongr ec) g) | ext | case h
α₁ : Sort u_1
β₁ : Sort u_2
γ₁ : Sort u_3
α₂ : Sort u_4
β₂ : Sort u_5
γ₂ : Sort u_6
ea : α₁ ≃ α₂
eb : β₁ ≃ β₂
ec : γ₁ ≃ γ₂
f : α₁ ↪ β₁
g : β₁ ↪ γ₁
x✝ : α₂
⊢ ((ea.embeddingCongr ec) (f.trans g)) x✝ = (((ea.embeddingCongr eb) f).trans ((eb.embeddingCongr ec) g)) x✝ | 5d7256ac0b5549db |
Besicovitch.TauPackage.color_lt | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N | case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
{p.color ↑j}
color_i : p.color i = s... | simp only [G] | case inl
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
i : Ordinal.{u}
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
{p.color ↑j}
color_i : p.color i = s... | 70d4d5d8aa8a7de2 |
Nat.Primrec'.if_lt | Mathlib/Computability/Primrec.lean | theorem if_lt {n a b f g} (ha : @Primrec' n a) (hb : @Primrec' n b) (hf : @Primrec' n f)
(hg : @Primrec' n g) : @Primrec' n fun v => if a v < b v then f v else g v :=
(prec' (sub.comp₂ _ hb ha) hg (tail <| tail hf)).of_eq fun v => by
cases e : b v - a v
· simp [not_lt.2 (Nat.sub_eq_zero_iff_le.mp e)]
... | case succ
n : ℕ
a b f g : List.Vector ℕ n → ℕ
ha : Primrec' a
hb : Primrec' b
hf : Primrec' f
hg : Primrec' g
v : List.Vector ℕ n
n✝ : ℕ
e : b v - a v = n✝ + 1
⊢ Nat.rec (g v) (fun y IH => f (y ::ᵥ IH ::ᵥ v).tail.tail) (n✝ + 1) = if a v < b v then f v else g v | simp [Nat.lt_of_sub_eq_succ e] | no goals | 1dc1c946f29fea11 |
MeasureTheory.lintegral_eq_iSup_eapprox_lintegral | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_eq_iSup_eapprox_lintegral {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ a, f a ∂μ = ⨆ n, (eapprox f n).lintegral μ :=
calc
∫⁻ a, f a ∂μ = ∫⁻ a, ⨆ n, (eapprox f n : α → ℝ≥0∞) a ∂μ | case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
⊢ Monotone fun n => ⇑(eapprox f n) | intro i j h | case h_mono
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
i j : ℕ
h : i ≤ j
⊢ (fun n => ⇑(eapprox f n)) i ≤ (fun n => ⇑(eapprox f n)) j | 081ea2e4f2877c46 |
exists_idempotent_of_compact_t2_of_continuous_mul_left | Mathlib/Topology/Algebra/Semigroup.lean | theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m | case intro.intro.intro.intro
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun... | use m | case h
M : Type u_1
inst✝⁴ : Nonempty M
inst✝³ : Semigroup M
inst✝² : TopologicalSpace M
inst✝¹ : CompactSpace M
inst✝ : T2Space M
continuous_mul_left : ∀ (r : M), Continuous fun x => x * r
S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}
N : Set M
hN : Minimal (fun x => x ∈ S) N
N_close... | fc87b9dbd8289f3d |
preimage_connectedComponent_connected | Mathlib/Topology/Connected/Clopen.lean | theorem preimage_connectedComponent_connected
(connected_fibers : ∀ t : β, IsConnected (f ⁻¹' {t}))
(hcl : ∀ T : Set β, IsClosed T ↔ IsClosed (f ⁻¹' T)) (t : β) :
IsConnected (f ⁻¹' connectedComponent t) | case a.intro
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed v
hu... | constructor | case a.intro.left
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
u v : Set α
hu : IsClosed u
hv : IsClosed... | 4b0d93be45591aa1 |
Array.isEmpty_eq_false | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem isEmpty_eq_false {l : Array α} : l.isEmpty = false ↔ l ≠ #[] | α : Type u_1
l : Array α
⊢ l.isEmpty = false ↔ l ≠ #[] | cases l <;> simp | no goals | 7802ac57adbcfa1a |
ZFSet.omega_succ | Mathlib/SetTheory/ZFC/Basic.lean | theorem omega_succ {n} : n ∈ omega.{u} → insert n n ∈ omega.{u} :=
Quotient.inductionOn n fun x ⟨⟨n⟩, h⟩ =>
⟨⟨n + 1⟩,
ZFSet.exact <|
show insert (mk x) (mk x) = insert (mk <| ofNat n) (mk <| ofNat n) by
rw [ZFSet.sound h]
rfl⟩
| n✝ : ZFSet.{u}
x : PSet.{u}
x✝ : ⟦x⟧ ∈ omega
n : ℕ
h : x.Equiv (PSet.omega.Func { down := n })
⊢ insert (mk (PSet.omega.Func { down := n })) (mk (PSet.omega.Func { down := n })) =
insert (mk (ofNat n)) (mk (ofNat n)) | rfl | no goals | 2961ee9a9cf487d4 |
IsSimpleRing.isField_center | Mathlib/RingTheory/SimpleRing/Field.lean | lemma isField_center (A : Type*) [Ring A] [IsSimpleRing A] : IsField (Subring.center A) where
exists_pair_ne := ⟨0, 1, zero_ne_one⟩
mul_comm := mul_comm
mul_inv_cancel | case mk
A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 => x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
mem : 1 ∈ I
⊢ ∃ b, ⟨x, hx1✝⟩ * b = 1 | simp only [TwoSidedIdeal.mem_mk', Set.mem_range, I] at mem | case mk
A : Type u_1
inst✝¹ : Ring A
inst✝ : IsSimpleRing A
x : A
hx1✝ : x ∈ Subring.center A
hx1 : ∀ (g : A), g * x = x * g
hx2 : x ≠ 0
I : TwoSidedIdeal A := mk' (Set.range fun x_1 => x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯
mem : ∃ y, x * y = 1
⊢ ∃ b, ⟨x, hx1✝⟩ * b = 1 | 08d555f2a5a9e7b4 |
NumberField.mixedEmbedding.normAtPlace_negAt | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | theorem normAtPlace_negAt (x : mixedSpace K) (w : InfinitePlace K) :
normAtPlace w (negAt s x) = normAtPlace w x | case inl
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : InfinitePlace K
hw : w.IsReal
⊢ (normAtPlace w) ((negAt s) x) = (normAtPlace w) x | simp_rw [normAtPlace_apply_of_isReal hw, Real.norm_eq_abs, negAt_apply_abs_isReal] | no goals | 0325fa9e9c073963 |
UniformConvergenceCLM.uniformSpace_mono | Mathlib/Topology/Algebra/Module/StrongTopology.lean | theorem uniformSpace_mono [UniformSpace F] [UniformAddGroup F] (h : 𝔖₂ ⊆ 𝔖₁) :
instUniformSpace σ F 𝔖₁ ≤ instUniformSpace σ F 𝔖₂ | 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
𝔖₁ 𝔖₂ : Set (Set E)
inst✝¹ : UniformSpace F
inst✝ : UniformAddGroup F
h : 𝔖... | exact UniformSpace.comap_mono (UniformOnFun.mono (le_refl _) h) | no goals | d2d376aee2f9c3b0 |
ProbabilityTheory.setLIntegral_stieltjesOfMeasurableRat | Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean | lemma setLIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondKernelCDF f κ ν)
(a : α) (x : ℝ) {s : Set β} (hs : MeasurableSet s) :
∫⁻ b in s, ENNReal.ofReal (stieltjesOfMeasurableRat f hf.measurable (a, b) x) ∂(ν a)
= κ a (s ×ˢ Iic x) | case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMea... | have h_nonempty : Nonempty { r' : ℚ // x < ↑r' } := by
obtain ⟨r, hrx⟩ := exists_rat_gt x
exact ⟨⟨r, hrx⟩⟩ | case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMea... | 2b5f63b20150553c |
mem_posTangentConeAt_of_segment_subset | Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | theorem mem_posTangentConeAt_of_segment_subset (h : [x -[ℝ] x + y] ⊆ s) :
y ∈ posTangentConeAt s x | E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : [x-[ℝ]x + y] ⊆ s
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[>] 0, x + x_1 • y ∈ s | rw [eventually_nhdsWithin_iff] | E : Type u
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
x y : E
h : [x-[ℝ]x + y] ⊆ s
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝 0, x_1 ∈ Ioi 0 → x + x_1 • y ∈ s | 9cfc7f1a8c1fc7eb |
isUniformInducing_iff' | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | lemma isUniformInducing_iff' {f : α → β} :
IsUniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α | α : Type u
β : Type v
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
⊢ IsUniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α | rw [isUniformInducing_iff, UniformContinuous, tendsto_iff_comap, le_antisymm_iff, and_comm] | α : Type u
β : Type v
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
⊢ 𝓤 α ≤ comap (fun x => (f x.1, f x.2)) (𝓤 β) ∧ comap (fun x => (f x.1, f x.2)) (𝓤 β) ≤ 𝓤 α ↔
𝓤 α ≤ comap (fun x => (f x.1, f x.2)) (𝓤 β) ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α | 2cf0ebe6dcd6be5d |
CategoryTheory.RelCat.rel_iso_iff | Mathlib/CategoryTheory/Category/RelCat.lean | theorem rel_iso_iff {X Y : RelCat} (r : X ⟶ Y) :
IsIso (C := RelCat) r ↔ ∃ f : (Iso (C := Type u) X Y), graphFunctor.map f.hom = r | case h.left.h
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ y, r a y ∧ inv r y b) ↔ a = b
h2 : ∀ (a b : Y), (∃ y, inv r a y ∧ r y b) ↔ a = b
f : X → Y
hf : ∀ (x : X), r x (f x) ∧ inv r (f x) x
g : Y → X
hg : ∀ (x : Y), inv r x (g x) ∧ r (g x) x
x : X
⊢ (f ≫ g) x = 𝟙 X x | apply (h1 _ _).mp | case h.left.h
X Y : RelCat
r : X ⟶ Y
h : IsIso r
h1 : ∀ (a b : X), (∃ y, r a y ∧ inv r y b) ↔ a = b
h2 : ∀ (a b : Y), (∃ y, inv r a y ∧ r y b) ↔ a = b
f : X → Y
hf : ∀ (x : X), r x (f x) ∧ inv r (f x) x
g : Y → X
hg : ∀ (x : Y), inv r x (g x) ∧ r (g x) x
x : X
⊢ ∃ y, r ((f ≫ g) x) y ∧ inv r y (𝟙 X x) | a2d7764e00cdb24d |
max_aleph0_card_le_rank_fun_nat | Mathlib/LinearAlgebra/Dimension/ErdosKaplansky.lean | theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ → K) | case inr.intro.mk.intro.mk.intro
K : Type u
inst✝ : DivisionRing K
aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)
card_K : ℵ₀ < #K
this✝ : Module.rank K (ℕ → K) < #K
ιK : Type u
bK : Basis ιK K (ℕ → K)
L : Subfield K := Subfield.closure (Set.range fun i => bK i.1 i.2)
hLK : #↥L < #K
this : Module (↥L)ᵐᵒᵖ K := Module.compHom K ... | rw [Finsupp.linearCombination_apply, Finsupp.sum] at rep_e | case inr.intro.mk.intro.mk.intro
K : Type u
inst✝ : DivisionRing K
aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)
card_K : ℵ₀ < #K
this✝ : Module.rank K (ℕ → K) < #K
ιK : Type u
bK : Basis ιK K (ℕ → K)
L : Subfield K := Subfield.closure (Set.range fun i => bK i.1 i.2)
hLK : #↥L < #K
this : Module (↥L)ᵐᵒᵖ K := Module.compHom K ... | f7e0fbcdefa57fa3 |
HasDerivAt.lhopital_zero_nhds_left | Mathlib/Analysis/Calculus/LHopital.lean | theorem lhopital_zero_nhds_left (hff' : ∀ᶠ x in 𝓝[<] a, HasDerivAt f (f' x) x)
(hgg' : ∀ᶠ x in 𝓝[<] a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝[<] a, g' x ≠ 0)
(hfa : Tendsto f (𝓝[<] a) (𝓝 0)) (hga : Tendsto g (𝓝[<] a) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] a) l) :
Tendsto (fun x => ... | case intro.intro.intro.intro.intro.intro
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfa : Tendsto f (𝓝[<] a) (𝓝 0)
hga : Tendsto g (𝓝[<] a) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] a) l
s₁ : Set ℝ
hs₁ : s₁ ∈ 𝓝[<] a
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ 𝓝[<] a
hgg' : ∀ y ∈ s₂, HasDerivAt... | have hs : s ∈ 𝓝[<] a := inter_mem (inter_mem hs₁ hs₂) hs₃ | case intro.intro.intro.intro.intro.intro
a : ℝ
l : Filter ℝ
f f' g g' : ℝ → ℝ
hfa : Tendsto f (𝓝[<] a) (𝓝 0)
hga : Tendsto g (𝓝[<] a) (𝓝 0)
hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] a) l
s₁ : Set ℝ
hs₁ : s₁ ∈ 𝓝[<] a
hff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y
s₂ : Set ℝ
hs₂ : s₂ ∈ 𝓝[<] a
hgg' : ∀ y ∈ s₂, HasDerivAt... | da50ce2224ecb800 |
exists_seq_of_forall_finset_exists | Mathlib/Data/Fintype/Basic.lean | theorem exists_seq_of_forall_finset_exists {α : Type*} (P : α → Prop) (r : α → α → Prop)
(h : ∀ s : Finset α, (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y) :
∃ f : ℕ → α, (∀ n, P (f n)) ∧ ∀ m n, m < n → r (f m) (f n) | case intro
α : Type u_4
P : α → Prop
r : α → α → Prop
h : ∀ (s : Finset α), (∀ x ∈ s, P x) → ∃ y, P y ∧ ∀ x ∈ s, r x y
y : α
h✝ : P y ∧ ∀ x ∈ ∅, r x y
⊢ Nonempty α | exact ⟨y⟩ | no goals | 84f3321d3efdbd70 |
MeasureTheory.Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite | Mathlib/MeasureTheory/Measure/SeparableMeasure.lean | theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite [IsFiniteMeasure μ]
(h𝒜 : IsSetAlgebra 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) : μ.MeasureDense 𝒜 where
measurable s hs := hgen ▸ measurableSet_generateFrom hs
approx s ms | X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
this : MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (s ∆ t)).toReal < ε
⊢ μ s ≠ ⊤ → ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, μ (s ∆ t) < EN... | rintro - ε ε_pos | X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
inst✝ : IsFiniteMeasure μ
h𝒜 : IsSetAlgebra 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
this : MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (s ∆ t)).toReal < ε
ε : ℝ
ε_pos : 0 < ε
⊢ ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofR... | 92d6fe7992cc346b |
Orientation.oangle_add | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
o.oangle x y + o.oangle y z = o.oangle x z | case hy
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ (o.kahler y) z ≠ 0 | exact o.kahler_ne_zero hy hz | no goals | 9ab057dbb8d9890d |
MeasureTheory.FiniteMeasure.ext_of_forall_integral_eq | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | theorem ext_of_forall_integral_eq [HasOuterApproxClosed Ω] [BorelSpace Ω]
{μ ν : FiniteMeasure Ω} (h : ∀ (f : Ω →ᵇ ℝ), ∫ x, f x ∂μ = ∫ x, f x ∂ν) :
μ = ν | case h
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : BorelSpace Ω
μ ν : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), ∫ (x : Ω), f x ∂↑μ = ∫ (x : Ω), f x ∂↑ν
f : Ω →ᵇ ℝ≥0
⊢ (∫⁻ (x : Ω), ↑(f x) ∂↑μ).toReal = (∫⁻ (x : Ω), ↑(f x) ∂↑ν).toReal | rw [toReal_lintegral_coe_eq_integral f μ, toReal_lintegral_coe_eq_integral f ν] | case h
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : TopologicalSpace Ω
inst✝¹ : HasOuterApproxClosed Ω
inst✝ : BorelSpace Ω
μ ν : FiniteMeasure Ω
h : ∀ (f : Ω →ᵇ ℝ), ∫ (x : Ω), f x ∂↑μ = ∫ (x : Ω), f x ∂↑ν
f : Ω →ᵇ ℝ≥0
⊢ ∫ (x : Ω), ↑(f x) ∂↑μ = ∫ (x : Ω), ↑(f x) ∂↑ν | 623f02af03253cef |
cfcₙ_star | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean | lemma cfcₙ_star : cfcₙ (fun x ↦ star (f x)) a = star (cfcₙ f a) | case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalarTower R A A
... | obtain ⟨ha, hf, h0⟩ := h | case pos.intro.intro
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : Nontrivial R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : Module R A
inst✝¹ : IsScalar... | c2fb3c8e671683c4 |
MeasureTheory.Measure.QuasiMeasurePreserving.restrict | Mathlib/MeasureTheory/Measure/Restrict.lean | theorem QuasiMeasurePreserving.restrict {ν : Measure β} {f : α → β}
(hf : QuasiMeasurePreserving f μ ν) {t : Set β} (hmaps : MapsTo f s t) :
QuasiMeasurePreserving f (μ.restrict s) (ν.restrict t) where
measurable := hf.measurable
absolutelyContinuous | α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
⊢ (ν.restrict t) u = 0 → (map f (μ.restrict s)) u = 0 | suffices ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0 by simpa [hum, hf.measurable, hf.measurable hum] | α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
ν : Measure β
f : α → β
hf : QuasiMeasurePreserving f μ ν
t : Set β
hmaps : MapsTo f s t
u : Set β
hum : MeasurableSet u
⊢ ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0 | c96afaf9115a24a4 |
MeasureTheory.locallyIntegrable_map_homeomorph | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X ≃ₜ Y) {f : Y → E}
{μ : Measure X} : LocallyIntegrable f (Measure.map e μ) ↔ LocallyIntegrable (f ∘ e) μ | case refine_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ⇑e) μ
x : ... | refine ⟨e.symm ⁻¹' U, e.symm.continuous.continuousAt.preimage_mem_nhds hU, ?_⟩ | case refine_2.intro.intro
X : Type u_1
Y : Type u_2
E : Type u_3
inst✝⁶ : MeasurableSpace X
inst✝⁵ : TopologicalSpace X
inst✝⁴ : MeasurableSpace Y
inst✝³ : TopologicalSpace Y
inst✝² : NormedAddCommGroup E
inst✝¹ : BorelSpace X
inst✝ : BorelSpace Y
e : X ≃ₜ Y
f : Y → E
μ : Measure X
h : LocallyIntegrable (f ∘ ⇑e) μ
x : ... | cbcd4eb2baca93b2 |
Mathlib.Meta.Positivity.pos_of_isNat | Mathlib/Tactic/Positivity/Core.lean | lemma pos_of_isNat {n : ℕ} [OrderedSemiring A] [Nontrivial A]
(h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) | A : Type u_1
e : A
n : ℕ
inst✝¹ : OrderedSemiring A
inst✝ : Nontrivial A
h : NormNum.IsNat e n
w : Nat.ble 1 n = true
⊢ 0 < ↑n | apply Nat.cast_pos.2 | A : Type u_1
e : A
n : ℕ
inst✝¹ : OrderedSemiring A
inst✝ : Nontrivial A
h : NormNum.IsNat e n
w : Nat.ble 1 n = true
⊢ 0 < n | 4e81ca6912b2d16f |
CochainComplex.MappingConeCompHomotopyEquiv.hom_inv_id | Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean | @[reassoc (attr := simp)]
lemma hom_inv_id : hom f g ≫ inv f g = 𝟙 _ | case h
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
X₁ X₂ X₃ : CochainComplex C ℤ
f : X₁ ⟶ X₂
g : X₂ ⟶ X₃
n : ℤ
⊢ (hom f g ≫ inv f g).f n = (𝟙 (mappingCone g)).f n | simp [hom, inv, lift_desc_f _ _ _ _ _ _ _ n (n+1) rfl, ext_from_iff _ (n + 1) _ rfl] | no goals | 93e558e1fa54af44 |
IsLocalization.ideal_eq_iInf_comap_map_away | Mathlib/RingTheory/Localization/Ideal.lean | theorem ideal_eq_iInf_comap_map_away {S : Finset R} (hS : Ideal.span (α := R) S = ⊤) (I : Ideal R) :
I = ⨅ f ∈ S, (I.map (algebraMap R (Localization.Away f))).comap
(algebraMap R (Localization.Away f)) | case a.H
R : Type u_1
inst✝ : CommRing R
S : Finset R
hS : Ideal.span ↑S = ⊤
I : Ideal R
x : R
hx : x ∈ ⨅ f ∈ S, Ideal.comap (algebraMap R (Localization.Away f)) (Ideal.map (algebraMap R (Localization.Away f)) I)
⊢ ∀ (r : ↑↑S), ∃ n, ↑r ^ n • x ∈ I | rintro ⟨s, hs⟩ | case a.H.mk
R : Type u_1
inst✝ : CommRing R
S : Finset R
hS : Ideal.span ↑S = ⊤
I : Ideal R
x : R
hx : x ∈ ⨅ f ∈ S, Ideal.comap (algebraMap R (Localization.Away f)) (Ideal.map (algebraMap R (Localization.Away f)) I)
s : R
hs : s ∈ ↑S
⊢ ∃ n, ↑⟨s, hs⟩ ^ n • x ∈ I | 03519d2e121c9417 |
IntermediateField.exists_lt_finrank_of_infinite_dimensional | Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean | theorem exists_lt_finrank_of_infinite_dimensional
[Algebra.IsAlgebraic F E] (hnfd : ¬ FiniteDimensional F E) (n : ℕ) :
∃ L : IntermediateField F E, FiniteDimensional F L ∧ n < finrank F L | F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
n : ℕ
L : IntermediateField F E
fin : FiniteDimensional F ↥L
hn : n < finrank F ↥L
hnfd : ∀ (x : E), x ∈ L
⊢ FiniteDimensional F E | rw [show L = ⊤ from eq_top_iff.2 fun x _ ↦ hnfd x] at fin | F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
n : ℕ
L : IntermediateField F E
fin : FiniteDimensional F ↥⊤
hn : n < finrank F ↥L
hnfd : ∀ (x : E), x ∈ L
⊢ FiniteDimensional F E | e542ab6fcdb23a62 |
Ordinal.nmul_le_iff₃' | Mathlib/SetTheory/Ordinal/NaturalOps.lean | theorem nmul_le_iff₃' : a ⨳ (b ⨳ c) ≤ d ↔ ∀ a' < a, ∀ b' < b, ∀ c' < c,
a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') <
d ♯ a' ⨳ (b' ⨳ c) ♯ a' ⨳ (b ⨳ c') ♯ a ⨳ (b' ⨳ c') | case mpr
a b c d : Ordinal.{u}
h :
∀ a' < a,
∀ b' < b,
∀ c' < c,
toOrdinal
(toNatOrdinal (b ⨳ c ⨳ a') + toNatOrdinal (b' ⨳ c ⨳ a) + toNatOrdinal (b ⨳ c' ⨳ a) +
toNatOrdinal (b' ⨳ c' ⨳ a')) <
toOrdinal
(toNatOrdinal d + toNatOrdinal (b' ⨳ c ⨳ a') + toNa... | convert h c' hc a' ha b' hb using 1 <;> abel_nf | no goals | b189ce1ac63af5e2 |
padicNorm.nonarchimedean_aux | Mathlib/NumberTheory/Padics/PadicNorm.lean | theorem nonarchimedean_aux {q r : ℚ} (h : padicValRat p q ≤ padicValRat p r) :
padicNorm p (q + r) ≤ max (padicNorm p q) (padicNorm p r) :=
have hnqp : padicNorm p q ≥ 0 := padicNorm.nonneg _
have hnrp : padicNorm p r ≥ 0 := padicNorm.nonneg _
if hq : q = 0 then by simp [hq, max_eq_right hnrp, le_max_right]
... | p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
h : padicValRat p q ≤ padicValRat p r
hnqp : padicNorm p q ≥ 0
hnrp : padicNorm p r ≥ 0
hq : ¬q = 0
hr : ¬r = 0
hqr : ¬q + r = 0
⊢ ↑p ^ (-padicValRat p (q + r)) ≤ ↑p ^ (-padicValRat p q) ∨ ↑p ^ (-padicValRat p (q + r)) ≤ ↑p ^ (-padicValRat p r) | left | case h
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
h : padicValRat p q ≤ padicValRat p r
hnqp : padicNorm p q ≥ 0
hnrp : padicNorm p r ≥ 0
hq : ¬q = 0
hr : ¬r = 0
hqr : ¬q + r = 0
⊢ ↑p ^ (-padicValRat p (q + r)) ≤ ↑p ^ (-padicValRat p q) | ef1e6b3c3a9e971d |
MeasureTheory.withDensity_eq_iff | Mathlib/MeasureTheory/Function/AEEqOfLIntegral.lean | theorem withDensity_eq_iff {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
(hg : AEMeasurable g μ) (hfi : ∫⁻ x, f x ∂μ ≠ ∞) :
μ.withDensity f = μ.withDensity g ↔ f =ᵐ[μ] g :=
⟨fun hfg ↦ by
refine AEMeasurable.ae_eq_of_forall_setLIntegral_eq hf hg hfi ?_ fun s hs _ ↦ ?_
· rwa [← setLIntegral_univ, ← withDensi... | case refine_1
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hfg : μ.withDensity f = μ.withDensity g
⊢ ∫⁻ (x : α), g x ∂μ ≠ ⊤ | rwa [← setLIntegral_univ, ← withDensity_apply g MeasurableSet.univ, ← hfg,
withDensity_apply f MeasurableSet.univ, setLIntegral_univ] | no goals | d53e22828aab2f48 |
TopologicalSpace.IsTopologicalBasis.isQuotientMap | Mathlib/Topology/Bases.lean | theorem IsTopologicalBasis.isQuotientMap {V : Set (Set X)} (hV : IsTopologicalBasis V)
(h' : IsQuotientMap π) (h : IsOpenMap π) : IsTopologicalBasis (Set.image π '' V) | case h_nhds.intro.intro.intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_2
inst✝ : TopologicalSpace Y
π : X → Y
V : Set (Set X)
hV : IsTopologicalBasis V
h' : IsQuotientMap π
h : IsOpenMap π
U : Set Y
U_open : IsOpen U
x : X
y_in_U : π x ∈ U
W : Set X := π ⁻¹' U
x_in_W : x ∈ W
W_open : IsOpen W
Z : Set X... | have πZ_in_U : π '' Z ⊆ U := (Set.image_subset _ Z_in_W).trans (image_preimage_subset π U) | case h_nhds.intro.intro.intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
Y : Type u_2
inst✝ : TopologicalSpace Y
π : X → Y
V : Set (Set X)
hV : IsTopologicalBasis V
h' : IsQuotientMap π
h : IsOpenMap π
U : Set Y
U_open : IsOpen U
x : X
y_in_U : π x ∈ U
W : Set X := π ⁻¹' U
x_in_W : x ∈ W
W_open : IsOpen W
Z : Set X... | 0b2ca271bb3ece9e |
Cardinal.mk_preimage_of_injective_lift | Mathlib/SetTheory/Cardinal/Basic.lean | theorem mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β) (s : Set β)
(h : Injective f) : lift.{v} #(f ⁻¹' s) ≤ lift.{u} #s | case inj'.hf
α : Type u
β : Type v
f : α → β
s : Set β
h : Injective f
⊢ Injective fun x => f ↑x | exact h.comp Subtype.val_injective | no goals | 4897a23374911d36 |
Ordinal.eq_enumOrd | Mathlib/SetTheory/Ordinal/Enum.lean | theorem eq_enumOrd (f : Ordinal → Ordinal) (hs : ¬ BddAbove s) :
enumOrd s = f ↔ StrictMono f ∧ range f = s | case mp
s : Set Ordinal.{u}
hs : ¬BddAbove s
⊢ StrictMono (enumOrd s) ∧ range (enumOrd s) = s | exact ⟨enumOrd_strictMono hs, range_enumOrd hs⟩ | no goals | 7734d007ef2167c5 |
Pell.eq_of_xn_modEq_lem3 | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem eq_of_xn_modEq_lem3 {i n} (npos : 0 < n) :
∀ {j}, i < j → j ≤ 2 * n → j ≠ n → ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2) →
xn a1 i % xn a1 n < xn a1 j % xn a1 n
| 0, ij, _, _, _ => absurd ij (Nat.not_lt_zero _)
| j + 1, ij, j2n, jnn, ntriv =>
have lem2 : ∀ k > n, k ≤ 2 * n → (↑(xn a1 k % xn a1 n) : ℤ)... | a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n :=
lt_of_add_lt_add_right
(let_fun this :=
Eq.mpr (id (congrArg (fun _a => _a < n + k) (tsub_add_cancel_of_le k2n)... | rw [← Nat.mod_eq_of_lt (Nat.sub_lt (x_pos a1 n) (x_pos a1 (2 * n - k)))] | a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n :=
lt_of_add_lt_add_right
(let_fun this :=
Eq.mpr (id (congrArg (fun _a => _a < n + k) (tsub_add_cancel_of_le k2n)... | edca849e1f1b9de3 |
Subsemigroup.subsingleton_of_subsingleton | Mathlib/Algebra/Group/Subsemigroup/Defs.lean | theorem subsingleton_of_subsingleton [Subsingleton (Subsemigroup M)] : Subsingleton M | M : Type u_1
inst✝¹ : Mul M
inst✝ : Subsingleton (Subsemigroup M)
⊢ Subsingleton M | constructor | case allEq
M : Type u_1
inst✝¹ : Mul M
inst✝ : Subsingleton (Subsemigroup M)
⊢ ∀ (a b : M), a = b | 5e9123dac9d862c0 |
Nat.find_eq_iff | Mathlib/Data/Nat/Find.lean | lemma find_eq_iff (h : ∃ n : ℕ, p n) : Nat.find h = m ↔ p m ∧ ∀ n < m, ¬ p n | case mp
m : ℕ
p : ℕ → Prop
inst✝ : DecidablePred p
h : ∃ n, p n
⊢ Nat.find h = m → p m ∧ ∀ (n : ℕ), n < m → ¬p n | rintro rfl | case mp
p : ℕ → Prop
inst✝ : DecidablePred p
h : ∃ n, p n
⊢ p (Nat.find h) ∧ ∀ (n : ℕ), n < Nat.find h → ¬p n | 6e953eafe0710e9e |
AkraBazziRecurrence.base_nonempty | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma base_nonempty {n : ℕ} (hn : 0 < n) : (Finset.Ico (⌊b (min_bi b) / 2 * n⌋₊) n).Nonempty | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
n : ℕ
hn : 0 < n
b' : ℝ := b (min_bi b)
hb_pos : 0 < b'
⊢ ⌊b (min_bi b) / 2 * ↑n⌋₊ < n | exact_mod_cast calc ⌊b' / 2 * n⌋₊ ≤ b' / 2 * n := by exact Nat.floor_le (by positivity)
_ < 1 / 2 * n := by gcongr; exact R.b_lt_one (min_bi b)
_ ≤ 1 * n := by gcongr; norm_num
_ = n := by simp | no goals | 7e3cff39f102cb90 |
Set.PairwiseDisjoint.prod_left | Mathlib/Data/Set/Pairwise/Lattice.lean | theorem PairwiseDisjoint.prod_left {f : ι × ι' → α}
(hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i'))
(ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')) :
(s ×ˢ t : Set (ι × ι')).PairwiseDisjoint f | case mk.mk.inl
α : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝ : CompleteLattice α
s : Set ι
t : Set ι'
f : ι × ι' → α
hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')
ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')
i : ι
i' : ι'
hi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t
j' : ι'
hj : (i, j').1 ∈ s ∧ (i, j').2 ∈ t
h : (... | refine (ht hi.2 hj.2 <| (Prod.mk.inj_left _).ne_iff.1 h).mono ?_ ?_ | case mk.mk.inl.refine_1
α : Type u_1
ι : Type u_2
ι' : Type u_3
inst✝ : CompleteLattice α
s : Set ι
t : Set ι'
f : ι × ι' → α
hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')
ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')
i : ι
i' : ι'
hi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t
j' : ι'
hj : (i, j').1 ∈ s ∧ (i, j').2 ... | 34740219f815d1c5 |
CategoryTheory.IsPushout.isVanKampen_iff | Mathlib/CategoryTheory/Adhesive.lean | theorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :
H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w) | case mpr
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : IsVanKampenColimit (PushoutCocone.mk h i ⋯)
W' X' Y' Z' : C
f' : W' ⟶ X'
g' : W' ⟶ Y'
h' : X' ⟶ Z'
i' : Y' ⟶ Z'
αW : W' ⟶ W
αX : X' ⟶ X
αY : Y' ⟶ Y
αZ : Z' ⟶ Z
hf : IsPullback f' αW αX f
hg : IsP... | refine
Iff.trans ?_ ((H w.cocone ⟨by rintro (_ | _ | _); exacts [αW, αX, αY], ?_⟩ αZ ?_ ?_).trans ?_) | case mpr.refine_1
C : Type u
inst✝ : Category.{v, u} C
W X Y Z : C
f : W ⟶ X
g : W ⟶ Y
h : X ⟶ Z
i : Y ⟶ Z
H✝ : IsPushout f g h i
H : IsVanKampenColimit (PushoutCocone.mk h i ⋯)
W' X' Y' Z' : C
f' : W' ⟶ X'
g' : W' ⟶ Y'
h' : X' ⟶ Z'
i' : Y' ⟶ Z'
αW : W' ⟶ W
αX : X' ⟶ X
αY : Y' ⟶ Y
αZ : Z' ⟶ Z
hf : IsPullback f' αW αX f... | 03b3318bb2e454c2 |
CategoryTheory.Pretriangulated.isIso₂_of_isIso₁₃ | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | lemma isIso₂_of_isIso₁₃ {T T' : Triangle C} (φ : T ⟶ T') (hT : T ∈ distTriang C)
(hT' : T' ∈ distTriang C) (h₁ : IsIso φ.hom₁) (h₃ : IsIso φ.hom₃) : IsIso φ.hom₂ | C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono... | refine isIso_of_yoneda_map_bijective _ (fun A => ⟨?_, ?_⟩) | case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.ho... | 8a538ff25cabfee4 |
MeasureTheory.eq_of_cylinder_eq_of_subset | Mathlib/MeasureTheory/Constructions/Cylinders.lean | theorem eq_of_cylinder_eq_of_subset [h_nonempty : Nonempty (∀ i, α i)] {I J : Finset ι}
{S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (h_eq : cylinder I S = cylinder J T)
(hJI : J ⊆ I) :
S = Finset.restrict₂ hJI ⁻¹' T | ι : Type u_1
α : ι → Type u_2
h_nonempty : Nonempty ((i : ι) → α i)
I J : Finset ι
S : Set ((i : { x // x ∈ I }) → α ↑i)
T : Set ((i : { x // x ∈ J }) → α ↑i)
h_eq : cylinder I S = cylinder J T
hJI : J ⊆ I
⊢ S = Finset.restrict₂ hJI ⁻¹' T | rw [Set.ext_iff] at h_eq | ι : Type u_1
α : ι → Type u_2
h_nonempty : Nonempty ((i : ι) → α i)
I J : Finset ι
S : Set ((i : { x // x ∈ I }) → α ↑i)
T : Set ((i : { x // x ∈ J }) → α ↑i)
h_eq : ∀ (x : (i : ι) → α i), x ∈ cylinder I S ↔ x ∈ cylinder J T
hJI : J ⊆ I
⊢ S = Finset.restrict₂ hJI ⁻¹' T | 9294e60eab06ec06 |
Nat.add_le_of_le_sub | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean | theorem add_le_of_le_sub {a b c : Nat} (hle : b ≤ c) (h : a ≤ c - b) : a + b ≤ c | a b c : Nat
hle : b ≤ c
h : a ≤ c - b
d : Nat
hd : a + d = c - b
⊢ a + b + d = a + d + b | simp [Nat.add_comm, Nat.add_assoc, Nat.add_left_comm] | no goals | fa9a281b60141183 |
mellin_hasDerivAt_of_isBigO_rpow | Mathlib/Analysis/MellinTransform.lean | theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
{f : ℝ → E} {s : ℂ} (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a)))
(hs_top : s.re < a) (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
MellinConvergent (fun t => log t • f t) s ∧
HasDerivAt (mellin ... | E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E := fun z... | apply LocallyIntegrableOn.aestronglyMeasurable | case hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b : ℝ
f : ℝ → E
s : ℂ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s.re < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s.re
F : ℂ → ℝ → E := fun z t => ↑t ^ (z - 1) • f t
F' : ℂ → ℝ → E ... | 3ad5d9a87042b6f9 |
Nat.image_div_divisors_eq_divisors | Mathlib/NumberTheory/Divisors.lean | theorem image_div_divisors_eq_divisors (n : ℕ) :
image (fun x : ℕ => n / x) n.divisors = n.divisors | case pos
n : ℕ
hn : n = 0
⊢ image (fun x => n / x) n.divisors = n.divisors | simp [hn] | no goals | fa9501818042115c |
PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux | Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean | lemma PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks_aux
(p₁ p₂ : PrimeSpectrum (S ⊗[R] T))
(h : tensorProductTo R S T p₁ = tensorProductTo R S T p₂) :
p₁ ≤ p₂ | case intro.intro.intro.intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
p₁ p₂ : PrimeSpectrum (S ⊗[R] T)
h : tensorProductTo R S T p₁ = tensorProductTo R S T p₂
g : T →+* S ⊗[R] T :... | have h₁ : a ⊗ₜ[R] t ∈ p₁.asIdeal := e ▸ p₁.asIdeal.mul_mem_left (1 ⊗ₜ[R] (r • t)) hxp₁ | case intro.intro.intro.intro
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
hRT : (algebraMap R T).SurjectiveOnStalks
p₁ p₂ : PrimeSpectrum (S ⊗[R] T)
h : tensorProductTo R S T p₁ = tensorProductTo R S T p₂
g : T →+* S ⊗[R] T :... | c5c66534088770bc |
Polynomial.natDegree_le_iff_coeff_eq_zero | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | theorem natDegree_le_iff_coeff_eq_zero : p.natDegree ≤ n ↔ ∀ N : ℕ, n < N → p.coeff N = 0 | R : Type u
n : ℕ
inst✝ : Semiring R
p : R[X]
⊢ p.natDegree ≤ n ↔ ∀ (N : ℕ), n < N → p.coeff N = 0 | simp_rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero, Nat.cast_lt] | no goals | 887aff04220cb0c4 |
Submonoid.list_prod_mem | Mathlib/Algebra/Group/Submonoid/BigOperators.lean | theorem list_prod_mem {l : List M} (hl : ∀ x ∈ l, x ∈ s) : l.prod ∈ s | case intro
M : Type u_1
inst✝ : Monoid M
s : Submonoid M
l : List ↥s
⊢ (List.map Subtype.val l).prod ∈ s | rw [← coe_list_prod] | case intro
M : Type u_1
inst✝ : Monoid M
s : Submonoid M
l : List ↥s
⊢ ↑l.prod ∈ s | 664da354914202d6 |
IsSigmaCompact.of_isClosed_subset | Mathlib/Topology/Compactness/SigmaCompact.lean | /-- A closed subset of a σ-compact set is σ-compact. -/
lemma IsSigmaCompact.of_isClosed_subset {s t : Set X} (ht : IsSigmaCompact t)
(hs : IsClosed s) (h : s ⊆ t) : IsSigmaCompact s | case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsClosed s
h : s ⊆ t
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcov : ⋃ n, K n = t
⊢ IsSigmaCompact s | refine ⟨(fun n ↦ s ∩ (K n)), fun n ↦ (hcompact n).inter_left hs, ?_⟩ | case intro.intro
X : Type u_1
inst✝ : TopologicalSpace X
s t : Set X
hs : IsClosed s
h : s ⊆ t
K : ℕ → Set X
hcompact : ∀ (n : ℕ), IsCompact (K n)
hcov : ⋃ n, K n = t
⊢ ⋃ n, (fun n => s ∩ K n) n = s | a76d73e383c2e9e0 |
SimpleGraph.IsAlternating.sup_edge | Mathlib/Combinatorics/SimpleGraph/Matching.lean | lemma IsAlternating.sup_edge {u x : V} (halt : G.IsAlternating G') (hnadj : ¬G'.Adj u x)
(hu' : ∀ u', u' ≠ u → G.Adj x u' → G'.Adj x u')
(hx' : ∀ x', x' ≠ x → G.Adj x' u → G'.Adj x' u) : (G ⊔ edge u x).IsAlternating G' | case neg.inr.inl.inr.intro
V : Type u_1
G G' : SimpleGraph V
u x : V
halt : G.IsAlternating G'
hnadj : ¬G'.Adj u x
hu' : ∀ (u' : V), u' ≠ u → G.Adj x u' → G'.Adj x u'
hx' : ∀ (x' : V), x' ≠ x → G.Adj x' u → G'.Adj x' u
hadj : ¬G.Adj u x
v w w' : V
hww' : w ≠ w'
hr : (v = u ∧ w = x ∨ v = x ∧ w = u) ∧ v ≠ w
h1 : G.Adj v ... | aesop | no goals | a496c1ce5d983672 |
Polynomial.coeff_bdd_of_roots_le | Mathlib/Topology/Algebra/Polynomial.lean | theorem coeff_bdd_of_roots_le {B : ℝ} {d : ℕ} (f : F →+* K) {p : F[X]} (h1 : p.Monic)
(h2 : Splits f p) (h3 : p.natDegree ≤ d) (h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) (i : ℕ) :
‖(map f p).coeff i‖ ≤ max B 1 ^ d * d.choose (d / 2) | F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
B : ℝ
d : ℕ
f : F →+* K
p : F[X]
h1 : p.Monic
h2 : Splits f p
h3 : p.natDegree ≤ d
h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
i : ℕ
⊢ ‖(map f p).coeff i‖ ≤ (B ⊔ 1) ^ d * ↑(d.choose (d / 2)) | obtain hB | hB := le_or_lt 0 B | case inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
B : ℝ
d : ℕ
f : F →+* K
p : F[X]
h1 : p.Monic
h2 : Splits f p
h3 : p.natDegree ≤ d
h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
i : ℕ
hB : 0 ≤ B
⊢ ‖(map f p).coeff i‖ ≤ (B ⊔ 1) ^ d * ↑(d.choose (d / 2))
case inr
F : Type u_3
K : Type u_4
inst✝¹ : Comm... | 49af0bf207486cbc |
Batteries.RBNode.reverse_balance2 | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/WF.lean | theorem reverse_balance2 (l : RBNode α) (v : α) (r : RBNode α) :
(balance2 l v r).reverse = balance1 r.reverse v l.reverse | α : Type u_1
l : RBNode α
v : α
r : RBNode α
⊢ (l.balance2 v r).reverse = (r.reverse.balance1 v l.reverse).reverse.reverse | rw [reverse_balance1] | α : Type u_1
l : RBNode α
v : α
r : RBNode α
⊢ (l.balance2 v r).reverse = (l.reverse.reverse.balance2 v r.reverse.reverse).reverse | 6b5fe920356d4835 |
Nat.divisors_filter_squarefree | Mathlib/Data/Nat/Squarefree.lean | theorem divisors_filter_squarefree {n : ℕ} (h0 : n ≠ 0) :
{d ∈ n.divisors | Squarefree d}.val =
(UniqueFactorizationMonoid.normalizedFactors n).toFinset.powerset.val.map fun x =>
x.val.prod | case mpr.intro.intro
n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ (normalizedFactors n).dedup
hs0 : s.val.prod ≠ 0
⊢ Squarefree s.val.prod | have h :=
UniqueFactorizationMonoid.factors_unique irreducible_of_normalized_factor
(fun x hx =>
irreducible_of_normalized_factor x
(Multiset.mem_of_le (le_trans hs (Multiset.dedup_le _)) hx))
(prod_normalizedFactors hs0) | case mpr.intro.intro
n : ℕ
h0 : n ≠ 0
s : Finset ℕ
hs : s.val ≤ (normalizedFactors n).dedup
hs0 : s.val.prod ≠ 0
h : Multiset.Rel Associated (normalizedFactors s.val.prod) s.val
⊢ Squarefree s.val.prod | 4b75722f8636dcf2 |
CliffordAlgebra.toBaseChange_reverse | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | theorem toBaseChange_reverse (Q : QuadraticForm R V) (x : CliffordAlgebra (Q.baseChange A)) :
toBaseChange A Q (reverse x) =
TensorProduct.map LinearMap.id reverse (toBaseChange A Q x) | R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
⊢ (toBaseChange A Q) (reverse x) = (TensorProduct.map LinearMap.id reverse) ((toBa... | have := DFunLike.congr_fun (toBaseChange_comp_reverseOp A Q) x | R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
x : CliffordAlgebra (QuadraticForm.baseChange A Q)
this :
((AlgHom.op (toBaseChange A Q)).comp reverseOp) x =
((↑(Algebra.Tenso... | 1b433c5423f6e345 |
padicValInt.eq_zero_of_not_dvd | Mathlib/NumberTheory/Padics/PadicVal/Basic.lean | theorem eq_zero_of_not_dvd {z : ℤ} (h : ¬(p : ℤ) ∣ z) : padicValInt p z = 0 | case h.h
p : ℕ
z : ℤ
h : ¬↑p ∣ z
⊢ ¬p ∣ z.natAbs | rwa [← Int.ofNat_dvd_left] | no goals | 0c11c070559ea23d |
Vector.exists_mem_push | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem exists_mem_push {p : α → Prop} {a : α} {xs : Vector α n} :
(∃ x, ∃ _ : x ∈ xs.push a, p x) ↔ p a ∨ ∃ x, ∃ _ : x ∈ xs, p x | case mp.intro.intro.inl
α : Type u_1
n : Nat
p : α → Prop
a : α
xs : Vector α n
x : α
h' : p x
h : x ∈ xs
⊢ p a ∨ ∃ x, x ∈ xs ∧ p x | exact .inr ⟨x, h, h'⟩ | no goals | 110c8ee2b5790ab6 |
AffineIndependent.range | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem AffineIndependent.range {p : ι → P} (ha : AffineIndependent k p) :
AffineIndependent k (fun x => x : Set.range p → P) | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
f : ↑(Set.range p) → ι := fun x => Exists.choose ⋯
hf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x
⊢ AffineIndependent k fun x => ↑x | let fe : Set.range p ↪ ι := ⟨f, fun x₁ x₂ he => Subtype.ext (hf x₁ ▸ hf x₂ ▸ he ▸ rfl)⟩ | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
ha : AffineIndependent k p
f : ↑(Set.range p) → ι := fun x => Exists.choose ⋯
hf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x
fe : ↑(Set.range p) ↪ ι := { toFun := f, inj' := ⋯ }
⊢... | 74e562254d94f0aa |
Ideal.natAbs_det_equiv | Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean | theorem natAbs_det_equiv (I : Ideal S) {E : Type*} [EquivLike E S I] [AddEquivClass E S I] (e : E) :
Int.natAbs
(LinearMap.det
((Submodule.subtype I).restrictScalars ℤ ∘ₗ AddMonoidHom.toIntLinearMap (e : S →+ I))) =
Ideal.absNorm I | case pos
S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
E : Type u_2
e : E
inst✝¹ : EquivLike E S ↥⊥
inst✝ : AddEquivClass E S ↥⊥
⊢ (LinearMap.det (↑ℤ (Submodule.subtype ⊥) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm ⊥ | have : (1 : S) ≠ 0 := one_ne_zero | case pos
S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
E : Type u_2
e : E
inst✝¹ : EquivLike E S ↥⊥
inst✝ : AddEquivClass E S ↥⊥
this : 1 ≠ 0
⊢ (LinearMap.det (↑ℤ (Submodule.subtype ⊥) ∘ₗ (↑e).toIntLinearMap)).natAbs = absNorm ⊥ | 87e25ff10e007725 |
Polynomial.natDegree_expand | Mathlib/Algebra/Polynomial/Expand.lean | theorem natDegree_expand (p : ℕ) (f : R[X]) : (expand R p f).natDegree = f.natDegree * p | case neg.refine_2
R : Type u
inst✝ : CommSemiring R
p : ℕ
f : R[X]
hp : p > 0
hf : ¬f = 0
hf1 : (expand R p) f ≠ 0
⊢ f.leadingCoeff ≠ 0 | exact mt leadingCoeff_eq_zero.1 hf | no goals | eb7f41a4dec073d7 |
WittVector.ghostComponent_verschiebungFun | Mathlib/RingTheory/WittVector/Verschiebung.lean | theorem ghostComponent_verschiebungFun [hp : Fact p.Prime] (x : 𝕎 R) (n : ℕ) :
ghostComponent (n + 1) (verschiebungFun x) = p * ghostComponent n x | p : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x : 𝕎 R
n : ℕ
⊢ ∀ x_1 ∈ Finset.range (n + 1),
↑p ^ (x_1 + 1) * x.verschiebungFun.coeff (x_1 + 1) ^ p ^ (n + 1 - (x_1 + 1)) =
↑p * (↑p ^ x_1 * x.coeff x_1 ^ p ^ (n - x_1)) | rintro i - | p : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x : 𝕎 R
n i : ℕ
⊢ ↑p ^ (i + 1) * x.verschiebungFun.coeff (i + 1) ^ p ^ (n + 1 - (i + 1)) = ↑p * (↑p ^ i * x.coeff i ^ p ^ (n - i)) | ff1de095d2b9c02e |
Bimod.triangle_bimod | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem triangle_bimod {X Y Z : Mon_ C} (M : Bimod X Y) (N : Bimod Y Z) :
(associatorBimod M (regular Y) N).hom ≫ whiskerLeft M (leftUnitorBimod N).hom =
whiskerRight (rightUnitorBimod M).hom N | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N : Bimod Y Z
⊢ (α_ M.... | slice_lhs 2 3 => rw [← MonoidalCategory.whiskerLeft_comp, coequalizer.π_desc] | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N : Bimod Y Z
⊢ (α_ M.... | 4bcafaa16ddbb859 |
Orientation.oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | theorem oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle y (y - x) = Real.arctan (‖x‖ / ‖y‖) | V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
hs : (o.oangle y (y - x)).sign = 1
⊢ o.oangle y (y - x) = ↑(Real.arctan (‖x‖ / ‖y‖)) | rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_sub_eq_arctan_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h) (o.right_ne_zero_of_oangle_eq_pi_div_two h)] | no goals | f59a4087960daa85 |
MeasureTheory.VectorMeasure.restrict_le_restrict_iUnion | Mathlib/MeasureTheory/VectorMeasure/Basic.lean | theorem restrict_le_restrict_iUnion {f : ℕ → Set α} (hf₁ : ∀ n, MeasurableSet (f n))
(hf₂ : ∀ n, v ≤[f n] w) : v ≤[⋃ n, f n] w | case refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : TopologicalSpace M
inst✝¹ : OrderedAddCommMonoid M
inst✝ : OrderClosedTopology M
v w : VectorMeasure α M
f : ℕ → Set α
hf₁ : ∀ (n : ℕ), MeasurableSet (f n)
hf₂ : ∀ (n : ℕ), v ≤[f n] w
a : Set α
ha₁ : MeasurableSet a
ha₂ : a ⊆ ⋃ n, f n
ha₃ : ⋃ n, a ∩... | exact ha₁.inter (MeasurableSet.disjointed hf₁ n) | no goals | 2f1f18500f546fa0 |
CategoryTheory.LocalizerMorphism.IsLocalizedEquivalence.of_equivalence | Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean | /-- When the underlying functor `Φ.functor` of `Φ : LocalizerMorphism W₁ W₂` is
an equivalence of categories and that `W₁` and `W₂` essentially correspond to each
other via this equivalence, then `Φ` is a localized equivalence. -/
lemma IsLocalizedEquivalence.of_equivalence [Φ.functor.IsEquivalence]
(h : W₂ ≤ W₁.ma... | C₁ : Type u₁
C₂ : Type u₂
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝ : Φ.functor.IsEquivalence
h : W₂ ≤ W₁.map Φ.functor
⊢ (Φ.functor ⋙ W₂.Q).IsLocalization W₁ | refine Functor.IsLocalization.of_equivalence_source W₂.Q W₂ (Φ.functor ⋙ W₂.Q) W₁
(Functor.asEquivalence Φ.functor).symm ?_ (Φ.inverts W₂.Q)
((Functor.associator _ _ _).symm ≪≫ isoWhiskerRight ((Equivalence.unitIso _).symm) _ ≪≫
Functor.leftUnitor _) | C₁ : Type u₁
C₂ : Type u₂
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝ : Φ.functor.IsEquivalence
h : W₂ ≤ W₁.map Φ.functor
⊢ W₂ ≤ W₁.isoClosure.inverseImage Φ.functor.asEquivalence.symm.functor | 40b2369000dc33ca |
MeasureTheory.integral_tendsto_of_tendsto_of_monotone | Mathlib/MeasureTheory/Integral/Bochner.lean | /-- Monotone convergence theorem for real-valued functions and Bochner integrals -/
lemma integral_tendsto_of_tendsto_of_monotone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ}
(hf : ∀ n, Integrable (f n) μ) (hF : Integrable F μ) (h_mono : ∀ᵐ x ∂μ, Monotone fun n ↦ f n x)
(h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n ↦ f n... | case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x => f n x - f 0 x
hf'_nonneg : ∀ (i ... | refine h_cont.tendsto.comp ?_ | case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x => f n x - f 0 x
hf'_nonneg : ∀ (i ... | 34d96d2df2ee7826 |
EuclideanGeometry.inner_pos_or_eq_of_dist_le_radius | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | theorem inner_pos_or_eq_of_dist_le_radius {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
(hp₂ : dist p₂ s.center ≤ s.radius) : 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫ ∨ p₁ = p₂ | case neg.refine_2.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₁ p₂ : P
hp₁ : dist p₁ s.center = s.radius
hp₂ : dist p₂ s.center ≤ s.radius
h : ¬p₁ = p₂
hp₂' : ‖p₂ -ᵥ s.center‖ = ‖p₁ -ᵥ s.center‖
⊢ p₁ ≠ s.cent... | rintro rfl | case neg.refine_2.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₂ : P
hp₂ : dist p₂ s.center ≤ s.radius
hp₁ : dist s.center s.center = s.radius
h : ¬s.center = p₂
hp₂' : ‖p₂ -ᵥ s.center‖ = ‖s.center -ᵥ s.center... | 886356a6641013a5 |
Cycle.chain_iff_pairwise | Mathlib/Data/List/Cycle.lean | theorem chain_iff_pairwise [IsTrans α r] : Chain r s ↔ ∀ a ∈ s, ∀ b ∈ s, r a b :=
⟨by
induction' s with a l _
· exact fun _ b hb => (not_mem_nil _ hb).elim
intro hs b hb c hc
rw [Cycle.chain_coe_cons, List.chain_iff_pairwise] at hs
simp only [pairwise_append, pairwise_cons, mem_append, mem_singlet... | case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
inst✝ : IsTrans α r
a : α
l : List α
a✝ : Chain r ↑l → ∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b
b : α
hb : b ∈ ↑(a :: l)
c : α
hc : c ∈ ↑(a :: l)
hs : (∀ (a' : α), a' ∈ l ∨ a' = a → r a a') ∧ List.Pairwise r l ∧ ∀ a_1 ∈ l, r a_1 a
⊢ r b c | simp only [mem_coe_iff, mem_cons] at hb hc | case HI
α : Type u_1
r : α → α → Prop
s : Cycle α
inst✝ : IsTrans α r
a : α
l : List α
a✝ : Chain r ↑l → ∀ a ∈ ↑l, ∀ b ∈ ↑l, r a b
b c : α
hs : (∀ (a' : α), a' ∈ l ∨ a' = a → r a a') ∧ List.Pairwise r l ∧ ∀ a_1 ∈ l, r a_1 a
hb : b = a ∨ b ∈ l
hc : c = a ∨ c ∈ l
⊢ r b c | 402ba91df1a51b55 |
toAdd_multiset_sum | Mathlib/Algebra/BigOperators/Group/Finset/Defs.lean | theorem toAdd_multiset_sum (s : Multiset (Multiplicative α)) :
s.prod.toAdd = (s.map toAdd).sum | α : Type u_3
inst✝ : AddCommMonoid α
s : Multiset (Multiplicative α)
⊢ s.prod = s.sum | rfl | no goals | 14255307a4ae3d6c |
padicNorm.of_int | Mathlib/NumberTheory/Padics/PadicNorm.lean | theorem of_int (z : ℤ) : padicNorm p z ≤ 1 | p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
⊢ padicNorm p ↑z ≤ 1 | obtain rfl | hz := eq_or_ne z 0 | case inl
p : ℕ
hp : Fact (Nat.Prime p)
⊢ padicNorm p ↑0 ≤ 1
case inr
p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
hz : z ≠ 0
⊢ padicNorm p ↑z ≤ 1 | b29060c80fbff9ac |
Polynomial.natDegree_prod | Mathlib/Algebra/Polynomial/BigOperators.lean | theorem natDegree_prod (h : ∀ i ∈ s, f i ≠ 0) :
(∏ i ∈ s, f i).natDegree = ∑ i ∈ s, (f i).natDegree | R : Type u
ι : Type w
s : Finset ι
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f : ι → R[X]
h : ∀ i ∈ s, f i ≠ 0
a✝ : Nontrivial R
⊢ (∏ i ∈ s, f i).natDegree = ∑ i ∈ s, (f i).natDegree | apply natDegree_prod' | case h
R : Type u
ι : Type w
s : Finset ι
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
f : ι → R[X]
h : ∀ i ∈ s, f i ≠ 0
a✝ : Nontrivial R
⊢ ∏ i ∈ s, (f i).leadingCoeff ≠ 0 | c05c3285ef23f7a2 |
LieModule.traceForm_eq_sum_finrank_nsmul_mul | Mathlib/Algebra/Lie/TraceForm.lean | lemma traceForm_eq_sum_finrank_nsmul_mul (x y : L) :
traceForm K L M x y = ∑ χ : Weight K L M, finrank K (genWeightSpace M χ) • (χ x * χ y) | K : Type u_2
L : Type u_3
M : Type u_4
inst✝¹⁰ : LieRing L
inst✝⁹ : AddCommGroup M
inst✝⁸ : LieRingModule L M
inst✝⁷ : Field K
inst✝⁶ : LieAlgebra K L
inst✝⁵ : Module K M
inst✝⁴ : LieModule K L M
inst✝³ : FiniteDimensional K M
inst✝² : LieRing.IsNilpotent L
inst✝¹ : LinearWeights K L M
inst✝ : IsTriangularizable K L M
... | classical
have hds := DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top
(LieSubmodule.iSupIndep_iff_toSubmodule.mp <| iSupIndep_genWeightSpace' K L M)
(LieSubmodule.iSup_eq_top_iff_toSubmodule.mp <| iSup_genWeightSpace_eq_top' K L M)
simp_rw [traceForm_apply_apply, LinearMap.trace_eq_sum_trace_restrict hds... | no goals | 113a5a48d1ac7bbe |
reflection_sub | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem reflection_sub {v w : F} (h : ‖v‖ = ‖w‖) : reflection (ℝ ∙ (v - w))ᗮ v = w | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
v w : F
h : ‖v‖ = ‖w‖
R : F ≃ₗᵢ[ℝ] F := reflection (Submodule.span ℝ {v - w})ᗮ
⊢ R v + R v = w + w | have h₁ : R (v - w) = -(v - w) := reflection_orthogonalComplement_singleton_eq_neg (v - w) | F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
v w : F
h : ‖v‖ = ‖w‖
R : F ≃ₗᵢ[ℝ] F := reflection (Submodule.span ℝ {v - w})ᗮ
h₁ : R (v - w) = -(v - w)
⊢ R v + R v = w + w | 0665d6b9d7b08a87 |
Cardinal.lift_ord | Mathlib/SetTheory/Ordinal/Basic.lean | theorem lift_ord (c) : Ordinal.lift.{u,v} (ord c) = ord (lift.{u,v} c) | case refine_2
c : Cardinal.{v}
⊢ (lift.{u, v} c).ord ≤ Ordinal.lift.{u, v} c.ord | rw [ord_le, ← lift_card, card_ord] | no goals | 18b777ea3b4921fc |
Int.ediv_emod_unique' | Mathlib/Data/Int/Lemmas.lean | theorem ediv_emod_unique' {a b r q : Int} (h : b ≠ 0) :
a / b = q ∧ a % b = r ↔ r + b * q = a ∧ 0 ≤ r ∧ r < |b| | case mp
a b r q : ℤ
h : b ≠ 0
⊢ a / b = q ∧ a % b = r → r + b * q = a ∧ 0 ≤ r ∧ r < |b| | intro ⟨rfl, rfl⟩ | case mp
a b r q : ℤ
h : b ≠ 0
⊢ a % b + b * (a / b) = a ∧ 0 ≤ a % b ∧ a % b < |b| | 9b8e287beb8387b4 |
FirstOrder.Field.realize_eqZero | Mathlib/ModelTheory/Algebra/Field/CharP.lean | theorem realize_eqZero [CommRing K] [CompatibleRing K] (n : ℕ)
(v : Empty → K) : (Formula.Realize (eqZero n) v) ↔ ((n : K) = 0) | K : Type u_1
inst✝¹ : CommRing K
inst✝ : CompatibleRing K
n : ℕ
v : Empty → K
⊢ Formula.Realize (eqZero n) v ↔ ↑n = 0 | simp [eqZero, Term.realize] | no goals | ee78a6355f4127f0 |
IsOpen.analyticOn_iff_analyticOnNhd | Mathlib/Analysis/Analytic/Within.lean | /-- On open sets, `AnalyticOnNhd` and `AnalyticOn` coincide -/
lemma IsOpen.analyticOn_iff_analyticOnNhd {f : E → F} {s : Set E} (hs : IsOpen s) :
AnalyticOn 𝕜 f s ↔ AnalyticOnNhd 𝕜 f s | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
s : Set E
hs : IsOpen s
hf : AnalyticOn 𝕜 f s
x : E
m : x ∈ s
r : ℝ
r0 : r > 0
rs : Metric.ball x r ⊆ s
p : FormalMultilin... | positivity | no goals | 1b818455a3407b83 |
AdjoinRoot.aeval_algHom_eq_zero | Mathlib/RingTheory/AdjoinRoot.lean | theorem aeval_algHom_eq_zero (ϕ : AdjoinRoot f →ₐ[R] S) : aeval (ϕ (root f)) f = 0 | R : Type u
S : Type v
inst✝² : CommRing R
f : R[X]
inst✝¹ : CommRing S
inst✝ : Algebra R S
ϕ : AdjoinRoot f →ₐ[R] S
h : ϕ.comp (of f) = algebraMap R S
⊢ (aeval (ϕ (root f))) f = 0 | rw [aeval_def, ← h, ← RingHom.map_zero ϕ.toRingHom, ← eval₂_root f, hom_eval₂] | R : Type u
S : Type v
inst✝² : CommRing R
f : R[X]
inst✝¹ : CommRing S
inst✝ : Algebra R S
ϕ : AdjoinRoot f →ₐ[R] S
h : ϕ.comp (of f) = algebraMap R S
⊢ eval₂ (ϕ.comp (of f)) (ϕ (root f)) f = eval₂ (ϕ.comp (of f)) (ϕ.toRingHom (root f)) f | e222792cd0d376db |
WittVector.RecursionMain.succNthDefiningPoly_degree | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | theorem succNthDefiningPoly_degree [IsDomain k] (n : ℕ) (a₁ a₂ : 𝕎 k) (bs : Fin (n + 1) → k)
(ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
(succNthDefiningPoly p n a₁ a₂ bs).degree = p | p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝² : CommRing k
inst✝¹ : CharP k p
inst✝ : IsDomain k
n : ℕ
a₁ a₂ : 𝕎 k
bs : Fin (n + 1) → k
ha₁ : a₁.coeff 0 ≠ 0
ha₂ : a₂.coeff 0 ≠ 0
this : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1))).degree = ↑p
⊢ 1 < ↑p | exact mod_cast hp.out.one_lt | no goals | 913a0229c49bd7a8 |
Real.doublingGamma_add_one | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | theorem doublingGamma_add_one (s : ℝ) (hs : s ≠ 0) :
doublingGamma (s + 1) = s * doublingGamma s | s : ℝ
hs : s ≠ 0
⊢ Gamma (s / 2 + 1 / 2) * (s / 2 * Gamma (s / 2)) * (2 ^ (s - 1) * 2) / √π =
s * (Gamma (s / 2) * Gamma (s / 2 + 1 / 2) * 2 ^ (s - 1) / √π) | ring | no goals | c364431ec88e9229 |
Algebra.rank_adjoin_le | Mathlib/LinearAlgebra/FreeAlgebra.lean | theorem Algebra.rank_adjoin_le {R : Type u} {S : Type v} [CommRing R] [Ring S] [Algebra R S]
(s : Set S) : Module.rank R (adjoin R s) ≤ max #s ℵ₀ | R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
s : Set S
⊢ Module.rank R ↥(adjoin R s) ≤ #↑s ⊔ ℵ₀ | rw [adjoin_eq_range_freeAlgebra_lift] | R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
s : Set S
⊢ Module.rank R ↥((FreeAlgebra.lift R) Subtype.val).range ≤ #↑s ⊔ ℵ₀ | 066ae3e2b283e6be |
Fermat42.neg_of_minimal | Mathlib/NumberTheory/FLT/Four.lean | theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) | a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → c.natAbs ≤ c1.natAbs
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ a ^ 4 + b ^ 4 = (-c) ^ 2 | rw [heq] | a b c : ℤ
h2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → c.natAbs ≤ c1.natAbs
ha : a ≠ 0
hb : b ≠ 0
heq : a ^ 4 + b ^ 4 = c ^ 2
⊢ c ^ 2 = (-c) ^ 2 | 011e62322015e505 |
LaurentSeries.Cauchy.exists_lb_support | Mathlib/RingTheory/LaurentSeries.lean | theorem Cauchy.exists_lb_support {ℱ : Filter K⸨X⸩} (hℱ : Cauchy ℱ) :
∃ N, ∀ n, n < N → coeff hℱ n = 0 | case intro
K : Type u_2
inst✝ : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
x✝ : UniformSpace K := ⊥
N : ℤ
hN : ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0
n : ℤ
hn : n < N
⊢ Filter.map (fun f => f.coeff n) ℱ ≤ pure 0 | simp only [pure_zero, nonpos_iff] | case intro
K : Type u_2
inst✝ : Field K
ℱ : Filter K⸨X⸩
hℱ : Cauchy ℱ
x✝ : UniformSpace K := ⊥
N : ℤ
hN : ∀ᶠ (f : K⸨X⸩) in ℱ, ∀ n < N, f.coeff n = 0
n : ℤ
hn : n < N
⊢ 0 ∈ Filter.map (fun f => f.coeff n) ℱ | b58c34af0a5e0534 |
Real.qaryEntropy_strictAntiOn | Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean | /-- Qary entropy is strictly decreasing in the interval [1 - q⁻¹, 1]. -/
lemma qaryEntropy_strictAntiOn (qLe2 : 2 ≤ q) :
StrictAntiOn (qaryEntropy q) (Icc (1 - 1/q) 1) | case a
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
qpos : 0 < ↑q
⊢ ↑q - ↑q * p < 1 | have : (q : ℝ) - 1 < p * q := by
have tmp := mul_lt_mul_of_pos_right hp.1 qpos
simp at tmp
have : (q : ℝ) ≠ 0 := (ne_of_lt qpos).symm
have asdfasfd : (1 - (q : ℝ)⁻¹) * ↑q = q - 1 := by calc (1 - (q : ℝ)⁻¹) * ↑q
_ = q - (q : ℝ)⁻¹ * (q : ℝ) := by ring
_ = q - 1 := by simp_all only [ne_eq, isUnit_iff_ne_ze... | case a
q : ℕ
qLe2 : 2 ≤ q
p1 : ℝ
hp1 : p1 ∈ Icc (1 - 1 / ↑q) 1
p2 : ℝ
hp2 : p2 ∈ Icc (1 - 1 / ↑q) 1
p1le2 : p1 < p2
p : ℝ
this✝ : 2 ≤ ↑q
qinv_lt_1 : (↑q)⁻¹ < 1
zero_lt_1_sub_p : 0 < 1 - p
hp : 1 - (↑q)⁻¹ < p ∧ p < 1
qpos : 0 < ↑q
this : ↑q - 1 < p * ↑q
⊢ ↑q - ↑q * p < 1 | 37758adf34f3dcdd |
FDerivMeasurableAux.D_subset_differentiable_set | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } | case hy
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ... | positivity | no goals | 28c357cacce7671e |
intervalIntegral.intervalIntegrable_cpow' | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem intervalIntegrable_cpow' {r : ℂ} (h : -1 < r.re) :
IntervalIntegrable (fun x : ℝ => (x : ℂ) ^ r) volume a b | a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
c : ℝ
⊢ IntervalIntegrable (fun x => ↑x ^ r) volume 0 c | rcases le_total 0 c with (hc | hc) | case inl
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
c : ℝ
hc : 0 ≤ c
⊢ IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
case inr
a b : ℝ
r : ℂ
h : -1 < r.re
this : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => ↑x ^ r) volume 0 c
c : ℝ
hc : c ≤ 0
⊢ IntervalInt... | e33db8d106f7d67d |
Complex.arctan_tan | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | theorem arctan_tan {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re) (h₂ : z.re ≤ π / 2) :
arctan (tan z) = z | case hx₂
z : ℂ
h₀ : z ≠ ↑π / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ (2 * (I * z)).im ≤ π | norm_num | case hx₂
z : ℂ
h₀ : z ≠ ↑π / 2
h₁ : -(π / 2) < z.re
h₂ : z.re ≤ π / 2
h : cos z ≠ 0
⊢ 2 * z.re ≤ π | d18e3a0fad89d01b |
range_derivWithin_subset_closure_span_image | Mathlib/Analysis/Calculus/Deriv/Slope.lean | theorem range_derivWithin_subset_closure_span_image
(f : 𝕜 → F) {s t : Set 𝕜} (h : s ⊆ closure (s ∩ t)) :
range (derivWithin f s) ⊆ closure (Submodule.span 𝕜 (f '' t)) | case pos
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : (𝓝[s \ {x}] x).NeBot
H' : DifferentiableWithinAt 𝕜 f s x
I : (𝓝[(s ∩ t) \ {x}] x).NeBot
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x... | filter_upwards [self_mem_nhdsWithin] with y hy | case h
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : 𝕜 → F
s t : Set 𝕜
h : s ⊆ closure (s ∩ t)
x : 𝕜
H : (𝓝[s \ {x}] x).NeBot
H' : DifferentiableWithinAt 𝕜 f s x
I : (𝓝[(s ∩ t) \ {x}] x).NeBot
this : Tendsto (slope f x) (𝓝[(s ∩ t) \ {x}] x) ... | 81bfa572fe08868f |
mapClusterPt_self_zpow_atTop_pow | Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean | theorem mapClusterPt_self_zpow_atTop_pow (x : G) (m : ℤ) :
MapClusterPt (x ^ m) atTop (x ^ · : ℕ → G) | case intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
x : G
m : ℤ
y : G
hy : MapClusterPt y atTop fun x_1 => x ^ x_1
⊢ MapClusterPt (x ^ m) atTop fun x_1 => x ^ x_1 | have H : MapClusterPt (x ^ m) (atTop.curry atTop) ↿(fun a b ↦ x ^ (m + b - a)) := by
have : ContinuousAt (fun yz ↦ x ^ m * yz.2 / yz.1) (y, y) := by fun_prop
simpa only [comp_def, ← zpow_sub, ← zpow_add, div_eq_mul_inv, Prod.map, mul_inv_cancel_right]
using (hy.curry_prodMap hy).continuousAt_comp this | case intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : CompactSpace G
inst✝ : IsTopologicalGroup G
x : G
m : ℤ
y : G
hy : MapClusterPt y atTop fun x_1 => x ^ x_1
H : MapClusterPt (x ^ m) (atTop.curry atTop) ↿fun a b => x ^ (m + b - a)
⊢ MapClusterPt (x ^ m) atTop fun x_1 => x ^ x_1 | c67dd0c71ee8dd79 |
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