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