name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
BitVec.toInt_concat | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem toInt_concat (x : BitVec w) (b : Bool) :
(concat x b).toInt = if w = 0 then -b.toInt else x.toInt * 2 + b.toInt | w : Nat
x : BitVec w
b : Bool
⊢ (if 2 * (x.toNat * 2 + b.toNat) < 2 ^ (w + 1) then ↑(x.toNat * 2 + b.toNat)
else ↑(x.toNat * 2 + b.toNat) - ↑(2 ^ (w + 1))) =
if w = 0 then -b.toInt else (if 2 * x.toNat < 2 ^ w then ↑x.toNat else ↑x.toNat - ↑(2 ^ w)) * 2 + b.toInt | cases w | case zero
b : Bool
x : BitVec 0
⊢ (if 2 * (x.toNat * 2 + b.toNat) < 2 ^ (0 + 1) then ↑(x.toNat * 2 + b.toNat)
else ↑(x.toNat * 2 + b.toNat) - ↑(2 ^ (0 + 1))) =
if 0 = 0 then -b.toInt else (if 2 * x.toNat < 2 ^ 0 then ↑x.toNat else ↑x.toNat - ↑(2 ^ 0)) * 2 + b.toInt
case succ
b : Bool
n✝ : Nat
x : BitVec (n✝ + ... | c7eb34aaf0a569ee |
hofer | Mathlib/Analysis/Hofer.lean | theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
{ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' | X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x'... | have key : ∀ n, d (u n) (u (n + 1)) ≤ ε / 2 ^ n ∧ 2 * ϕ (u n) < ϕ (u (n + 1)) := by
intro n
induction n using Nat.case_strong_induction_on with
| hz => simpa [u, ε_pos.le] using hu 0
| hi n IH =>
have A : d (u (n + 1)) x ≤ 2 * ε := by
rw [dist_comm]
let r := range (n + 1) calc
d (u 0) (u... | X : Type u_1
inst✝¹ : MetricSpace X
inst✝ : CompleteSpace X
x : X
ε : ℝ
ε_pos : 0 < ε
ϕ : X → ℝ
cont : Continuous ϕ
nonneg : ∀ (y : X), 0 ≤ ϕ y
reformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'
this : Nonempty X
F : ℕ → X → X
hF : ∀ (k : ℕ) (x' : X), d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ x'... | 7989b2596caedc2b |
Sbtw.angle₃₂₁_eq_pi | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | theorem _root_.Sbtw.angle₃₂₁_eq_pi {p₁ p₂ p₃ : P} (h : Sbtw ℝ p₁ p₂ p₃) : ∠ p₃ p₂ p₁ = π | 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
p₁ p₂ p₃ : P
h : Sbtw ℝ p₁ p₂ p₃
⊢ ∠ p₃ p₂ p₁ = π | rw [← h.angle₁₂₃_eq_pi, angle_comm] | no goals | dc1cfe5f4ae81779 |
Subring.map_comap_eq_self | Mathlib/Algebra/Ring/Subring/Basic.lean | theorem map_comap_eq_self
{f : R →+* S} {t : Subring S} (h : t ≤ f.range) : (t.comap f).map f = t | R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
t : Subring S
h : t ≤ f.range
⊢ map f (comap f t) = t | simpa only [inf_of_le_left h] using Subring.map_comap_eq f t | no goals | b52c232d3e7a9811 |
ComplexShape.Embedding.next_f | Mathlib/Algebra/Homology/Embedding/Boundary.lean | lemma next_f [e.IsTruncGE] {j k : ι} (hjk : c.next j = k) : c'.next (e.f j) = e.f k | case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsTruncGE
j k : ι
hjk : c.next j = k
hj : ¬c'.Rel (e.f j) (c'.next (e.f j))
⊢ ¬c.Rel j (c.next j) | intro hj' | case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsTruncGE
j k : ι
hjk : c.next j = k
hj : ¬c'.Rel (e.f j) (c'.next (e.f j))
hj' : c.Rel j (c.next j)
⊢ False | 4c4e3e70e2958597 |
BitVec.sshiftRightRec_zero_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem sshiftRightRec_zero_eq (x : BitVec w₁) (y : BitVec w₂) :
sshiftRightRec x y 0 = x.sshiftRight' (y &&& twoPow w₂ 0) | w₁ w₂ : Nat
x : BitVec w₁
y : BitVec w₂
⊢ x.sshiftRightRec y 0 = x.sshiftRight' (y &&& twoPow w₂ 0) | simp only [sshiftRightRec] | no goals | 9382154108edc327 |
MeromorphicAt.order_eq_top_iff | Mathlib/Analysis/Meromorphic/Order.lean | /-- The order of a meromorphic function `f` at a `z₀` is infinity iff `f` vanishes locally around
`z₀`. -/
lemma order_eq_top_iff {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) :
hf.order = ⊤ ↔ ∀ᶠ z in 𝓝[≠] x, f z = 0 | case neg.intro
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
m : ℕ
hm : ↑m = ⋯.order
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
⊢ ∃ᶠ (z : 𝕜) in 𝓝[≠] x, (z - x) ^ Exists.choose hf • f z = 0 | apply Eventually.frequently | case neg.intro.h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
x : 𝕜
hf : MeromorphicAt f x
m : ℕ
hm : ↑m = ⋯.order
h : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0
⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝[≠] x, (x_1 - x) ^ Exists.choose hf • f x_1 = 0 | 3c3d5a97dcd70e5d |
BitVec.getElem?_zero_ofBool | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem getElem?_zero_ofBool (b : Bool) : (ofBool b)[0]? = some b | b : Bool
⊢ (if b = true then 1#1 else 0#1)[0]? = some b | split <;> simp_all | no goals | e4c3f3fc6698a591 |
Topology.IsConstructible.image_of_isOpenEmbedding | Mathlib/Topology/Constructible.lean | @[stacks 09YD]
lemma IsConstructible.image_of_isOpenEmbedding (hfopen : IsOpenEmbedding f)
(hfcomp : IsRetrocompact (range f)) (hs : IsConstructible s) : IsConstructible (f '' s) | case union
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
s✝ : Set X
hfopen : IsOpenEmbedding f
hfcomp : IsRetrocompact (range f)
s : Set X
hs : IsConstructible s
t : Set X
ht : IsConstructible t
hs' : IsConstructible (f '' s)
ht' : IsConstructible (f '' t)
⊢ IsConstructible ... | exact hs'.union ht' | no goals | d2b2878af009198a |
MvQPF.Cofix.bisim_aux | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | theorem Cofix.bisim_aux {α : TypeVec n} (r : Cofix F α → Cofix F α → Prop) (h' : ∀ x, r x x)
(h : ∀ x y, r x y →
appendFun id (Quot.mk r) <$$> Cofix.dest x = appendFun id (Quot.mk r) <$$> Cofix.dest y) :
∀ x y, r x y → x = y | case a
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h' : ∀ (x : Cofix F α), r x x
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
x y : (P F).M α
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : (... | apply h' | no goals | 0af76fe163f43a26 |
irreducible_pow_sup_of_le | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem irreducible_pow_sup_of_le (hJ : Irreducible J) (n : ℕ) (hn : n ≤ emultiplicity J I) :
J ^ n ⊔ I = J ^ n | case neg
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : ↑n ≤ emultiplicity J I
hI : ¬I = ⊥
⊢ J ^ n ⊔ I = J ^ n | rw [irreducible_pow_sup hI hJ, min_eq_right] | case neg
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : ↑n ≤ emultiplicity J I
hI : ¬I = ⊥
⊢ n ≤ count J (normalizedFactors I) | c29a75c2640d6fa6 |
Turing.PartrecToTM2.tr_eval | Mathlib/Computability/TMToPartrec.lean | theorem tr_eval (c v) : eval (TM2.step tr) (init c v) = halt <$> Code.eval c v | case intro.intro.refine_2
c : Code
v : List ℕ
i : Cfg'
h₁ : TrCfg (stepNormal c Cont.halt v) i
h₂ : Reaches₁ (TM2.step tr) (init c v) i
x : TM2.Cfg (fun x => Γ') Λ' (Option Γ')
⊢ (∃ a ∈ c.eval v, halt a = x) → x ∈ eval (TM2.step tr) i | rintro ⟨v', hv, rfl⟩ | case intro.intro.refine_2.intro.intro
c : Code
v : List ℕ
i : Cfg'
h₁ : TrCfg (stepNormal c Cont.halt v) i
h₂ : Reaches₁ (TM2.step tr) (init c v) i
v' : List ℕ
hv : v' ∈ c.eval v
⊢ halt v' ∈ eval (TM2.step tr) i | 58e74fcd24b87e49 |
Vector.map_eq_flatMap | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem map_eq_flatMap {α β} (f : α → β) (l : Vector α n) :
map f l = (l.flatMap fun x => #v[f x]).cast (by simp) | case mk
α : Type u_1
β : Type u_2
f : α → β
l : Array α
⊢ map f { toArray := l, size_toArray := ⋯ } =
Vector.cast ⋯ ({ toArray := l, size_toArray := ⋯ }.flatMap fun x => { toArray := #[f x], size_toArray := ⋯ }) | simp [Array.map_eq_flatMap] | no goals | 4a06808d93e31f80 |
List.subperm_append_diff_self_of_count_le | Mathlib/.lake/packages/batteries/Batteries/Data/List/Perm.lean | theorem subperm_append_diff_self_of_count_le {l₁ l₂ : List α}
(h : ∀ x ∈ l₁, count x l₁ ≤ count x l₂) : l₁ ++ l₂.diff l₁ ~ l₂ | α : Type u_1
inst✝ : DecidableEq α
hd : α
tl : List α
IH : ∀ {l₂ : List α}, (∀ (x : α), x ∈ tl → count x tl ≤ count x l₂) → tl ++ l₂.diff tl ~ l₂
l₂ : List α
h : ∀ (x : α), x ∈ hd :: tl → count x (hd :: tl) ≤ count x l₂
⊢ 0 < count hd l₂ | exact Nat.lt_of_lt_of_le (count_pos_iff.mpr (.head _)) (h hd (.head _)) | no goals | 3d366c6e228bc2c2 |
ProbabilityTheory.Kernel.withDensity_rnDeriv_eq_zero_iff_apply_eq_zero | Mathlib/Probability/Kernel/RadonNikodym.lean | lemma withDensity_rnDeriv_eq_zero_iff_apply_eq_zero (κ η : Kernel α γ) [IsFiniteKernel κ]
[IsFiniteKernel η] (a : α) :
withDensity η (rnDeriv κ η) a = 0
↔ withDensity η (rnDeriv κ η) a (mutuallySingularSetSlice κ η a)ᶜ = 0 | α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
⊢ ((η.withDensity (κ.rnDeriv η)) a) univ = 0 ↔ ((η.withDensity (κ.rnDeriv η)) a) (κ.mutuallySingularSetSlice η a)ᶜ = 0 | have : univ = (mutuallySingularSetSlice κ η a) ∪ (mutuallySingularSetSlice κ η a)ᶜ := by simp | α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
κ η : Kernel α γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α
this : univ = κ.mutuallySingularSetSlice η a ∪ (κ.mutuallySingularSetSlice η a)ᶜ
⊢ ((η.withDensity (κ.rnDeriv η)) a) u... | 6395411a31dc6e00 |
AlgebraicTopology.DoldKan.identity_N₂_objectwise | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | theorem identity_N₂_objectwise (P : Karoubi (SimplicialObject C)) :
(N₂Γ₂.inv.app (N₂.obj P) : N₂.obj P ⟶ N₂.obj (Γ₂.obj (N₂.obj P))) ≫
N₂.map (Γ₂N₂.natTrans.app P) = 𝟙 (N₂.obj P) | C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
P : Karoubi (SimplicialObject C)
n : ℕ
⊢ (N₂Γ₂.inv.app (N₂.obj P)).f.f n =
PInfty.f n ≫ P.p.app (op ⦋n⦌) ≫ ((Γ₀.splitting (N₂.obj P).X).cofan (op ⦋n⦌)).inj (Splitting.IndexSet.id (op ⦋n⦌)) | simp only [N₂Γ₂_inv_app_f_f, N₂_obj_p_f, assoc] | no goals | f70bade0a77fed96 |
Polynomial.derivative_rootMultiplicity_of_root | Mathlib/Algebra/Polynomial/FieldDivision.lean | theorem derivative_rootMultiplicity_of_root [CharZero R] {p : R[X]} {t : R} (hpt : p.IsRoot t) :
p.derivative.rootMultiplicity t = p.rootMultiplicity t - 1 | R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : CharZero R
p : R[X]
t : R
hpt : p.IsRoot t
⊢ rootMultiplicity t (derivative p) = rootMultiplicity t p - 1 | by_cases h : p = 0 | case pos
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : CharZero R
p : R[X]
t : R
hpt : p.IsRoot t
h : p = 0
⊢ rootMultiplicity t (derivative p) = rootMultiplicity t p - 1
case neg
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : CharZero R
p : R[X]
t : R
hpt : p.IsRoot t
h : ¬p = 0
⊢ rootMultipl... | 793723e33e5d38d5 |
LinearLocallyFiniteOrder.isMax_of_succFn_le | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | theorem isMax_of_succFn_le [LocallyFiniteOrder ι] (i : ι) (hi : succFn i ≤ i) : IsMax i | ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i : ι
hi : succFn i ≤ i
j : ι
x✝ : i ≤ j
hij_lt : i < j
h_succFn_eq : succFn i = i
h_glb : IsGLB (↑(Finset.Ioc i j)) i
hi_mem : i ∈ Finset.Ioc i j
⊢ False | rw [Finset.mem_Ioc] at hi_mem | ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i : ι
hi : succFn i ≤ i
j : ι
x✝ : i ≤ j
hij_lt : i < j
h_succFn_eq : succFn i = i
h_glb : IsGLB (↑(Finset.Ioc i j)) i
hi_mem : i < i ∧ i ≤ j
⊢ False | 56626169ebf9d67b |
CompleteOrthogonalIdempotents.single | Mathlib/RingTheory/Idempotents.lean | lemma CompleteOrthogonalIdempotents.single {I : Type*} [Fintype I] [DecidableEq I]
(R : I → Type*) [∀ i, Semiring (R i)] :
CompleteOrthogonalIdempotents (Pi.single (f := R) · 1) | I : Type u_4
inst✝² : Fintype I
inst✝¹ : DecidableEq I
R : I → Type u_5
inst✝ : (i : I) → Semiring (R i)
⊢ Pairwise fun x1 x2 => Pi.single x1 1 * Pi.single x2 1 = 0 | intros i j hij | I : Type u_4
inst✝² : Fintype I
inst✝¹ : DecidableEq I
R : I → Type u_5
inst✝ : (i : I) → Semiring (R i)
i j : I
hij : i ≠ j
⊢ Pi.single i 1 * Pi.single j 1 = 0 | 4de28b9f4a72faae |
Equiv.Perm.isCycle_iff_exists_isCycleOn | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | theorem isCycle_iff_exists_isCycleOn :
f.IsCycle ↔ ∃ s : Set α, s.Nontrivial ∧ f.IsCycleOn s ∧ ∀ ⦃x⦄, ¬IsFixedPt f x → x ∈ s | case refine_1.intro
α : Type u_2
f : Perm α
a : α
ha : f a ≠ a ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle a y
⊢ {x | f x ≠ x}.Nontrivial | exact ⟨f a, f.injective.ne ha.1, a, ha.1, ha.1⟩ | no goals | 5c2cfe8fee87e6d7 |
IsPrimitiveRoot.coe_submonoidClass_iff | Mathlib/RingTheory/RootsOfUnity/PrimitiveRoots.lean | theorem coe_submonoidClass_iff {M B : Type*} [CommMonoid M] [SetLike B M] [SubmonoidClass B M]
{N : B} {ζ : N} : IsPrimitiveRoot (ζ : M) k ↔ IsPrimitiveRoot ζ k | k : ℕ
M : Type u_7
B : Type u_8
inst✝² : CommMonoid M
inst✝¹ : SetLike B M
inst✝ : SubmonoidClass B M
N : B
ζ : ↥N
⊢ IsPrimitiveRoot (↑ζ) k ↔ IsPrimitiveRoot ζ k | simp_rw [iff_def] | k : ℕ
M : Type u_7
B : Type u_8
inst✝² : CommMonoid M
inst✝¹ : SetLike B M
inst✝ : SubmonoidClass B M
N : B
ζ : ↥N
⊢ (↑ζ ^ k = 1 ∧ ∀ (l : ℕ), ↑ζ ^ l = 1 → k ∣ l) ↔ ζ ^ k = 1 ∧ ∀ (l : ℕ), ζ ^ l = 1 → k ∣ l | 00b8504c6ad4f89f |
HNNExtension.t_mul_of | Mathlib/GroupTheory/HNNExtension.lean | theorem t_mul_of (a : A) :
t * (of (a : G) : HNNExtension G A B φ) = of (φ a : G) * t :=
(Con.eq _).2 <| ConGen.Rel.of _ _ <| ⟨a, by simp⟩
| G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
a : ↥A
⊢ (fun x1 x2 => x1 * x2) (inr (ofAdd 1)) (inl ↑a) = inr (ofAdd 1) * inl ↑a ∧
(fun x1 x2 => x1 * x2) (inl ↑(φ a)) (inr (ofAdd 1)) = inl ↑(φ a) * inr (ofAdd 1) | simp | no goals | 35d09a862bd711d9 |
MeasureTheory.integral_llr_tilted_right | Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | lemma integral_llr_tilted_right [IsProbabilityMeasure μ] [SigmaFinite ν]
(hμν : μ ≪ ν) (hfμ : Integrable f μ) (hfν : Integrable (fun x ↦ exp (f x)) ν)
(h_int : Integrable (llr μ ν) μ) :
∫ x, llr μ (ν.tilted f) x ∂μ = ∫ x, llr μ ν x ∂μ - ∫ x, f x ∂μ + log (∫ x, exp (f x) ∂ν) | α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x => rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (x : α), -f x + log (∫ (x : α), rexp (f x) ∂ν) + llr μ ν x ∂μ =
-∫ (x : α), f x ∂μ + log (∫... | rw [← integral_neg, integral_add ?_ h_int] | α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x => rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a + log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
∫ (a : α), -f... | 618388524e9eb506 |
Polynomial.natDegree_monomial | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | theorem natDegree_monomial [DecidableEq R] (i : ℕ) (r : R) :
natDegree (monomial i r) = if r = 0 then 0 else i | case pos
R : Type u
inst✝¹ : Semiring R
inst✝ : DecidableEq R
i : ℕ
r : R
hr : r = 0
⊢ ((monomial i) r).natDegree = 0 | simp [hr] | no goals | 9d12030a7c0020fe |
sdiff_sup | Mathlib/Order/BooleanAlgebra.lean | theorem sdiff_sup : y \ (x ⊔ z) = y \ x ⊓ y \ z :=
sdiff_unique
(calc
y ⊓ (x ⊔ z) ⊔ y \ x ⊓ y \ z = (y ⊓ (x ⊔ z) ⊔ y \ x) ⊓ (y ⊓ (x ⊔ z) ⊔ y \ z) | α : Type u
x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ (y ⊓ z ⊔ (y ⊓ x ⊔ y \ x)) ⊓ (y ⊓ x ⊔ (y ⊓ z ⊔ y \ z)) = (y ⊓ z ⊔ y) ⊓ (y ⊓ x ⊔ y) | rw [sup_inf_sdiff, sup_inf_sdiff] | no goals | d036844b47839f50 |
CompleteSublattice.mem_iInf | Mathlib/Order/CompleteLattice/SetLike.lean | @[simp] lemma mem_iInf : x ∈ ⨅ i : I, f i ↔ ∀ i : I, x ∈ f i | X : Type u_1
L : CompleteSublattice (Set X)
I : Sort u_2
f : I → ↥L
x : X
⊢ x ∈ ⨅ i, f i ↔ ∀ (i : I), x ∈ f i | simp [← mem_subtype] | no goals | 3bd89b6287185cfa |
String.findAux_of_valid | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem findAux_of_valid (p) : ∀ l m r,
findAux ⟨l ++ m ++ r⟩ p ⟨utf8Len l + utf8Len m⟩ ⟨utf8Len l⟩ =
⟨utf8Len l + utf8Len (m.takeWhile (!p ·))⟩
| l, [], r => by unfold findAux List.takeWhile; simp
| l, c::m, r => by
unfold findAux List.takeWhile
rw [dif_pos (by exact Nat.lt_add_of_pos_right add_utf... | p : Char → Bool
l : List Char
c : Char
m r : List Char
⊢ { data := l ++ c :: m ++ r }.findAux p { byteIdx := utf8Len l + utf8Len (c :: m) } { byteIdx := utf8Len l } =
{ byteIdx := utf8Len l + utf8Len (List.takeWhile (fun x => !p x) (c :: m)) } | unfold findAux List.takeWhile | p : Char → Bool
l : List Char
c : Char
m r : List Char
⊢ (if h : { byteIdx := utf8Len l } < { byteIdx := utf8Len l + utf8Len (c :: m) } then
if p ({ data := l ++ c :: m ++ r }.get { byteIdx := utf8Len l }) = true then { byteIdx := utf8Len l }
else
let_fun this := ⋯;
{ data := l ++ c :: m ++ ... | d58119a3e133fdfa |
FiniteField.sum_subgroup_units_eq_zero | Mathlib/FieldTheory/Finite/Basic.lean | theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] (hg : G ≠ ⊥) :
∑ x : G, (x.val : K) = 0 | K : Type u_1
inst✝² : Ring K
inst✝¹ : NoZeroDivisors K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
hg : ∃ a, a ≠ 1
⊢ ∑ x : ↥G, ↑↑x = 0 | rcases hg with ⟨a, ha⟩ | case intro
K : Type u_1
inst✝² : Ring K
inst✝¹ : NoZeroDivisors K
G : Subgroup Kˣ
inst✝ : Fintype ↥G
a : ↥G
ha : a ≠ 1
⊢ ∑ x : ↥G, ↑↑x = 0 | e38b367708977b52 |
CategoryTheory.Idempotents.DoldKan.N₂_map_isoΓ₀_hom_app_f | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean | @[simp]
lemma N₂_map_isoΓ₀_hom_app_f (X : ChainComplex C ℕ) :
(N₂.map (isoΓ₀.hom.app X)).f = PInfty | case h
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : ChainComplex C ℕ
i✝ : ℕ
⊢ (N₂.map (isoΓ₀.hom.app X)).f.f i✝ = PInfty.f i✝ | apply comp_id | no goals | f45662f7b11eb378 |
ExistsContDiffBumpBase.y_eq_zero_of_not_mem_ball | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | theorem y_eq_zero_of_not_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (hx : x ∉ ball (0 : E) (1 + D)) :
y D x = 0 | 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
hx : x ∉ ball 0 (1 + D)
y : E
hy : y ∈ ball x D
⊢ φ y = 0 | simp only [φ, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero] | 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
hx : x ∉ ball 0 (1 + D)
y : E
hy : y ∈ ball x D
⊢ ‖y‖ ≤ 1 → False | 1521790313563a69 |
Finpartition.exists_equipartition_card_eq | Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean | theorem exists_equipartition_card_eq (hn : n ≠ 0) (hs : n ≤ #s) :
∃ P : Finpartition s, P.IsEquipartition ∧ #P.parts = n | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
n : ℕ
hn : 0 < n
hs : n ≤ #s
⊢ ∃ P, P.IsEquipartition ∧ #P.parts = n | have : (n - #s % n) * (#s / n) + #s % n * (#s / n + 1) = #s := by
rw [tsub_mul, mul_add, ← add_assoc,
tsub_add_cancel_of_le (Nat.mul_le_mul_right _ (mod_lt _ hn).le), mul_one, add_comm,
mod_add_div] | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
n : ℕ
hn : 0 < n
hs : n ≤ #s
this : (n - #s % n) * (#s / n) + #s % n * (#s / n + 1) = #s
⊢ ∃ P, P.IsEquipartition ∧ #P.parts = n | 2ac7403c622d01ca |
PartialHomeomorph.hasFPowerSeriesAt_symm | Mathlib/Analysis/Analytic/Inverse.lean | theorem PartialHomeomorph.hasFPowerSeriesAt_symm (f : PartialHomeomorph E F) {a : E}
{i : E ≃L[𝕜] F} (h0 : a ∈ f.source) {p : FormalMultilinearSeries 𝕜 E F}
(h : HasFPowerSeriesAt f p a) (hp : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
HasFPowerSeriesAt f.symm (p.leftInv i a) (f a) | 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 : PartialHomeomorph E F
a : E
i : E ≃L[𝕜] F
h0 : a ∈ f.source
p : FormalMultilinearSeries 𝕜 E F
h : HasFPowerSeriesAt (↑f) p a
hp... | filter_upwards [f.open_source.mem_nhds h0] with x hx using by simp [hx] | no goals | d8d26a141f5c0cc1 |
MeasureTheory.laverage_congr | Mathlib/MeasureTheory/Integral/Average.lean | theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
h : f =ᶠ[ae μ] g
⊢ ⨍⁻ (x : α), f x ∂μ = ⨍⁻ (x : α), g x ∂μ | simp only [laverage_eq, lintegral_congr_ae h] | no goals | d9af30b7d6d86991 |
sectionOfRetractionKerToTensorAux_prop | Mathlib/RingTheory/Smooth/Kaehler.lean | lemma sectionOfRetractionKerToTensorAux_prop (x y) (h : algebraMap P S x = algebraMap P S y) :
x - l (1 ⊗ₜ .D _ _ x) = y - l (1 ⊗ₜ .D _ _ y) | R P S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing P
inst✝² : CommRing S
inst✝¹ : Algebra R P
inst✝ : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
x y : P
h : (algebraMap P S) x = (algebraMap P S) y
⊢ x - y = ↑(l (1 ⊗ₜ[P] (D R P) (x - y))) | exact congr_arg Subtype.val (LinearMap.congr_fun hl.symm ⟨x - y, by simp [RingHom.mem_ker, h]⟩) | no goals | adb88c316fac4c11 |
le_min | Mathlib/Order/Defs/LinearOrder.lean | lemma le_min (h₁ : c ≤ a) (h₂ : c ≤ b) : c ≤ min a b | α : Type u_1
inst✝ : LinearOrder α
a b c : α
h₁ : c ≤ a
h₂ : c ≤ b
h : ¬a ≤ b
⊢ c ≤ min a b | simpa [min_def, if_neg h] using h₂ | no goals | 7383e3a33cc7f8e7 |
fermatLastTheoremWith_of_fermatLastTheoremWith_coprime | Mathlib/NumberTheory/FLT/Basic.lean | /-- To prove Fermat Last Theorem in any semiring that is a `NormalizedGCDMonoid` one can assume
that the `gcd` of `{a, b, c}` is `1`. -/
lemma fermatLastTheoremWith_of_fermatLastTheoremWith_coprime {n : ℕ} {R : Type*} [CommSemiring R]
[IsDomain R] [DecidableEq R] [NormalizedGCDMonoid R]
(hn : ∀ a b c : R, a ≠ 0... | n : ℕ
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : IsDomain R
inst✝¹ : DecidableEq R
inst✝ : NormalizedGCDMonoid R
hn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n
a b c : R
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
s : Finset R := {a, b, c}
⊢ False | let d := s.gcd id | n : ℕ
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : IsDomain R
inst✝¹ : DecidableEq R
inst✝ : NormalizedGCDMonoid R
hn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n
a b c : R
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
habc : a ^ n + b ^ n = c ^ n
s : Finset R := {a, b, c}
d : R := s.gcd id... | 94ffc8c4670030fa |
Matrix.vecMul_fromBlocks | Mathlib/Data/Matrix/Block.lean | theorem vecMul_fromBlocks [Fintype n] [Fintype o] [NonUnitalNonAssocSemiring α] (A : Matrix n l α)
(B : Matrix n m α) (C : Matrix o l α) (D : Matrix o m α) (x : n ⊕ o → α) :
x ᵥ* fromBlocks A B C D =
Sum.elim ((x ∘ Sum.inl) ᵥ* A + (x ∘ Sum.inr) ᵥ* C)
((x ∘ Sum.inl) ᵥ* B + (x ∘ Sum.inr) ᵥ* D) | l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type u_12
inst✝² : Fintype n
inst✝¹ : Fintype o
inst✝ : NonUnitalNonAssocSemiring α
A : Matrix n l α
B : Matrix n m α
C : Matrix o l α
D : Matrix o m α
x : n ⊕ o → α
⊢ x ᵥ* fromBlocks A B C D = Sum.elim (x ∘ Sum.inl ᵥ* A + x ∘ Sum.inr ᵥ* C) (x ∘ Sum.inl ᵥ* B + x ∘... | ext i | case h
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
α : Type u_12
inst✝² : Fintype n
inst✝¹ : Fintype o
inst✝ : NonUnitalNonAssocSemiring α
A : Matrix n l α
B : Matrix n m α
C : Matrix o l α
D : Matrix o m α
x : n ⊕ o → α
i : l ⊕ m
⊢ (x ᵥ* fromBlocks A B C D) i = Sum.elim (x ∘ Sum.inl ᵥ* A + x ∘ Sum.inr ᵥ* C) (x... | 94cbac95263c0568 |
Algebra.Etale.iff_exists_algEquiv_prod | Mathlib/RingTheory/Etale/Field.lean | theorem Algebra.Etale.iff_exists_algEquiv_prod :
Etale K A ↔
∃ (I : Type u) (_ : Finite I) (Ai : I → Type u) (_ : ∀ i, Field (Ai i))
(_ : ∀ i, Algebra K (Ai i)) (_ : A ≃ₐ[K] Π i, Ai i),
∀ i, Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i) | K A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ Etale K A ↔ ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i) | constructor | case mp
K A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ Etale K A → ∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ (i : I), Module.Finite K (Ai i) ∧ Algebra.IsSeparable K (Ai i)
case mpr
K A : Type u
inst✝² : Field K
inst✝¹ : CommRing A
inst✝ : Algebra K A
⊢ (∃ I, ∃ (_ : Finite I), ∃ Ai x x_1 x_2, ∀ ... | 06b608bc876f0b8c |
MeasureTheory.volume_sum_rpow_lt_one | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | theorem MeasureTheory.volume_sum_rpow_lt_one (hp : 1 ≤ p) :
volume {x : ι → ℝ | ∑ i, |x i| ^ p < 1} =
.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) | ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i : ι, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
⊢ volume {x | ∑ i : ι, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1)) | have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ofReal_one ▸ (ofReal_le_ofReal hp)) | ι : Type u_1
inst✝ : Fintype ι
p : ℝ
hp : 1 ≤ p
h₁ : 0 < p
h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i : ι, |x i| ^ p
eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)
this : Fact (1 ≤ ENNReal.ofReal p)
⊢ volume {x | ∑ i : ι, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1)) | 412f34b3848867a2 |
MeasureTheory.Measure.rnDeriv_smul_right_of_ne_top | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem rnDeriv_smul_right_of_ne_top (ν μ : Measure α) [IsFiniteMeasure ν]
[ν.HaveLebesgueDecomposition μ] {r : ℝ≥0∞} (hr : r ≠ 0) (hr_ne_top : r ≠ ∞) :
ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ | α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
⊢ r.toNNReal ≠ 0 | rw [ne_eq, ENNReal.toNNReal_eq_zero_iff] | α : Type u_1
m : MeasurableSpace α
ν μ : Measure α
inst✝¹ : IsFiniteMeasure ν
inst✝ : ν.HaveLebesgueDecomposition μ
r : ℝ≥0∞
hr : r ≠ 0
hr_ne_top : r ≠ ⊤
⊢ ¬(r = 0 ∨ r = ⊤) | dc22c3dfbd231803 |
UniformContinuousOn.cexp | Mathlib/Analysis/SpecialFunctions/Exp.lean | /-- The complex exponential function is uniformly continuous on left half planes. -/
lemma UniformContinuousOn.cexp (a : ℝ) : UniformContinuousOn exp {x : ℂ | x.re ≤ a} | a ε : ℝ
this✝ : ∀ (a : ℂ) (ε : ℝ), 0 < ε → ∀ᶠ (x : ℂ) in 𝓝 a, dist (cexp x) (cexp a) < ε
ha : 0 < ε / (2 * Real.exp a)
δ : ℝ
x : ℂ
a✝ : x.re ≤ a
y : ℂ
hy : y.re ≤ a
hxy : ‖x - y‖ < δ
h3 : ‖cexp (x - y) - 1‖ < ε / (2 * Real.exp a)
this : cexp x - cexp y = cexp y * (cexp (x - y) - 1)
hε : 0 < ε
hδ : 0 < δ ∧ ∀ ⦃y : ℂ⦄, ‖... | field_simp [mul_assoc] | no goals | e2639d45c5a634ab |
Matrix.IsDiag.fromBlocks_of_isSymm | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | theorem IsDiag.fromBlocks_of_isSymm [Zero α] {A : Matrix m m α} {C : Matrix n m α}
{D : Matrix n n α} (h : (A.fromBlocks 0 C D).IsSymm) (ha : A.IsDiag) (hd : D.IsDiag) :
(A.fromBlocks 0 C D).IsDiag | α : Type u_1
n : Type u_4
m : Type u_5
inst✝ : Zero α
A : Matrix m m α
C : Matrix n m α
D : Matrix n n α
h : (Matrix.fromBlocks A 0 C D).IsSymm
ha : A.IsDiag
hd : D.IsDiag
⊢ (Matrix.fromBlocks A 0 0ᵀ D).IsDiag | exact ha.fromBlocks hd | no goals | cbbb3240a7ebd823 |
MeasurableSpace.generateMeasurableRec_induction | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | theorem generateMeasurableRec_induction {s : Set (Set α)} {i : Ordinal} {t : Set α}
{p : Set α → Prop} (hs : ∀ t ∈ s, p t) (h0 : p ∅)
(hc : ∀ u, p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ)
(hn : ∀ f : ℕ → Set α,
(∀ n, p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)) :
... | α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t✝ : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
t : Set α
I... | unfold generateMeasurableRec | α : Type u
s : Set (Set α)
i : Ordinal.{u_1}
t✝ : Set α
p : Set α → Prop
hs : ∀ t ∈ s, p t
h0 : p ∅
hc : ∀ (u : Set α), p u → (∃ j < i, u ∈ generateMeasurableRec s j) → p uᶜ
hn : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n) ∧ ∃ j < i, f n ∈ generateMeasurableRec s j) → p (⋃ n, f n)
k✝ k : Ordinal.{u_1}
hk : k ≤ i
t : Set α
I... | 7a3cf976009de66a |
Batteries.HashMap.Imp.expand_WF | Mathlib/.lake/packages/batteries/Batteries/Data/HashMap/WF.lean | theorem expand_WF [BEq α] [Hashable α] {buckets : Buckets α β} (H : buckets.WF) :
(expand sz buckets).buckets.WF :=
go _ H.1 H.2 ⟨.mk' _, fun _ _ _ _ => by simp_all [Buckets.mk, List.mem_replicate]⟩
where
go (i) {source : Array (AssocList α β)}
(hs₁ : ∀ [LawfulHashable α] [PartialEquivBEq α], ∀ bucket ∈ s... | case refine_3
α : Type u_1
β : Type u_2
sz : Nat
inst✝¹ : BEq α
inst✝ : Hashable α
buckets : Buckets α β
H✝ : buckets.WF
i : Nat
source : Array (AssocList α β)
hs₁ :
∀ [inst : LawfulHashable α] [inst : PartialEquivBEq α] (bucket : AssocList α β),
bucket ∈ source.toList → List.Pairwise (fun a b => ¬(a.fst == b.fst... | let rank (k : α) := ((hash k).toUSize % USize.ofNat source.size).toNat | case refine_3
α : Type u_1
β : Type u_2
sz : Nat
inst✝¹ : BEq α
inst✝ : Hashable α
buckets : Buckets α β
H✝ : buckets.WF
i : Nat
source : Array (AssocList α β)
hs₁ :
∀ [inst : LawfulHashable α] [inst : PartialEquivBEq α] (bucket : AssocList α β),
bucket ∈ source.toList → List.Pairwise (fun a b => ¬(a.fst == b.fst... | e7bfb4067ae20a2d |
PNat.exists_prime_and_dvd | Mathlib/Data/PNat/Prime.lean | theorem exists_prime_and_dvd {n : ℕ+} (hn : n ≠ 1) : ∃ p : ℕ+, p.Prime ∧ p ∣ n | case intro
n : ℕ+
hn : n ≠ 1
p : ℕ
hp : Nat.Prime p ∧ p ∣ ↑n
⊢ Prime ⟨p, ⋯⟩ ∧ ↑⟨p, ⋯⟩ ∣ ↑n | apply hp | no goals | 0813ae5ed2031457 |
Associated.separable | Mathlib/FieldTheory/Separable.lean | theorem _root_.Associated.separable {f g : R[X]}
(ha : Associated f g) (h : f.Separable) : g.Separable | case intro.mk.intro.intro
R : Type u
inst✝ : CommSemiring R
f g u v : R[X]
h1 : u * v = 1
h2 : v * u = 1
ha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g
a b : R[X]
h : (a * f + b * derivative f) * (u * v) = 1 * 1
⊢ (a * v + b * derivative v) * g + b * v * derivative g = 1 | have h3 := congr(derivative $(h1)) | case intro.mk.intro.intro
R : Type u
inst✝ : CommSemiring R
f g u v : R[X]
h1 : u * v = 1
h2 : v * u = 1
ha : f * ↑{ val := u, inv := v, val_inv := h1, inv_val := h2 } = g
a b : R[X]
h : (a * f + b * derivative f) * (u * v) = 1 * 1
h3 : derivative (u * v) = derivative 1
⊢ (a * v + b * derivative v) * g + b * v * deriva... | b0038728ae30a917 |
Mon_ClassOfRepresentableBy_yonedaMonObjRepresentableBy | Mathlib/CategoryTheory/Monoidal/Yoneda.lean | lemma Mon_ClassOfRepresentableBy_yonedaMonObjRepresentableBy [Mon_Class X] :
Mon_ClassOfRepresentableBy X _ (yonedaMonObjRepresentableBy X) = ‹_› | case one
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : ChosenFiniteProducts C
X : C
inst✝ : Mon_Class X
⊢ toUnit (𝟙_ C) ≫ η = η | rw [toUnit_unique (toUnit _) (𝟙 _), Category.id_comp] | no goals | 4148e3493b7ae5c5 |
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 | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
⊢ ⨆ n, ∫⁻ (a : α), (eapprox f n) a ∂μ = ⨆ n, (eapprox f n).lintegral μ | congr | case e_s
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : Measurable f
⊢ (fun n => ∫⁻ (a : α), (eapprox f n) a ∂μ) = fun n => (eapprox f n).lintegral μ | 081ea2e4f2877c46 |
Path.truncate_range | Mathlib/Topology/Path.lean | theorem truncate_range {a b : X} (γ : Path a b) {t₀ t₁ : ℝ} :
range (γ.truncate t₀ t₁) ⊆ range γ | X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ℝ
⊢ range ⇑(γ.truncate t₀ t₁) ⊆ range γ.extend | simp only [range_subset_iff, SetCoe.exists, SetCoe.forall] | X : Type u_1
inst✝ : TopologicalSpace X
a b : X
γ : Path a b
t₀ t₁ : ℝ
⊢ ∀ (x : ℝ) (h : x ∈ I), (γ.truncate t₀ t₁) ⟨x, h⟩ ∈ range γ.extend | 3426c9c425a4aec7 |
Matrix.Pivot.listTransvecCol_mul_mul_listTransvecRow_last_row | Mathlib/LinearAlgebra/Matrix/Transvection.lean | theorem listTransvecCol_mul_mul_listTransvecRow_last_row (hM : M (inr unit) (inr unit) ≠ 0)
(i : Fin r) :
((listTransvecCol M).prod * M * (listTransvecRow M).prod) (inl i) (inr unit) = 0 | case hM
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
hM : M (inr ()) (inr ()) ≠ 0
i : Fin r
this : listTransvecCol M = listTransvecCol (M * (listTransvecRow M).prod)
⊢ (M * (listTransvecRow M).prod) (inr ()) (inr ()) ≠ 0 | simpa [mul_listTransvecRow_last_col] using hM | no goals | 798bc0c5bb9d9504 |
AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq | Mathlib/Analysis/Analytic/Uniqueness.lean | theorem eqOn_of_preconnected_of_eventuallyEq {f g : E → F} {U : Set E} (hf : AnalyticOnNhd 𝕜 f U)
(hg : AnalyticOnNhd 𝕜 g U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[𝓝 z₀] g) :
EqOn f g U | 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 g : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ U
hfg : f =ᶠ[𝓝 z₀] ... | have hfg' : f - g =ᶠ[𝓝 z₀] 0 := hfg.mono fun z h => by simp [h] | 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 g : E → F
U : Set E
hf : AnalyticOnNhd 𝕜 f U
hg : AnalyticOnNhd 𝕜 g U
hU : IsPreconnected U
z₀ : E
h₀ : z₀ ∈ U
hfg : f =ᶠ[𝓝 z₀] ... | 38cbed38c30a70a9 |
ENNReal.tendsto_atTop | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {f : β → ℝ≥0∞} {a : ℝ≥0∞}
(ha : a ≠ ∞) : Tendsto f atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, f n ∈ Icc (a - ε) (a + ε) :=
.trans (atTop_basis.tendsto_iff (hasBasis_nhds_of_ne_top ha)) (by simp only [true_and]; rfl)
| β : Type u_2
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → ℝ≥0∞
a : ℝ≥0∞
ha : a ≠ ⊤
⊢ (∀ (ib : ℝ≥0∞), 0 < ib → ∃ ia, ∀ x ∈ Ici ia, f x ∈ Icc (a - ib) (a + ib)) ↔
∀ ε > 0, ∃ N, ∀ n ≥ N, f n ∈ Icc (a - ε) (a + ε) | rfl | no goals | 6e16718bd5942c7b |
LaurentSeries.tendsto_valuation | Mathlib/RingTheory/LaurentSeries.lean | theorem tendsto_valuation (a : (idealX K).adicCompletion (RatFunc K)) :
Tendsto (Valued.v : RatFunc K → ℤₘ₀) (comap (↑) (𝓝 a)) (𝓝 (Valued.v a : ℤₘ₀)) | case h.left
K : Type u_2
inst✝ : Field K
a : adicCompletion (RatFunc K) (idealX K)
ψ : RatFunc K → ℤₘ₀ := ⇑Valued.v
hψ : ψ = ⇑Valued.v
this : ∀ (s : Set (adicCompletion (RatFunc K) (idealX K))), s ∈ 𝓝 0 ↔ ∃ γ, {x | Valued.v x < ↑γ} ⊆ s
ha : a = 0
S : Set ℤₘ₀
γ : ℤₘ₀
γ_ne_zero : γ ≠ 0
γ_le : Set.Iio γ ⊆ S
⊢ {t | Valued... | rw [ha, this] | case h.left
K : Type u_2
inst✝ : Field K
a : adicCompletion (RatFunc K) (idealX K)
ψ : RatFunc K → ℤₘ₀ := ⇑Valued.v
hψ : ψ = ⇑Valued.v
this : ∀ (s : Set (adicCompletion (RatFunc K) (idealX K))), s ∈ 𝓝 0 ↔ ∃ γ, {x | Valued.v x < ↑γ} ⊆ s
ha : a = 0
S : Set ℤₘ₀
γ : ℤₘ₀
γ_ne_zero : γ ≠ 0
γ_le : Set.Iio γ ⊆ S
⊢ ∃ γ_1, {x |... | 079b1bbdcfe0c224 |
ergodic_smul_of_denseRange_pow | Mathlib/Dynamics/Ergodic/Action/OfMinimal.lean | theorem ergodic_smul_of_denseRange_pow {M : Type*} [Monoid M] [TopologicalSpace M]
[MulAction M X] [ContinuousSMul M X] {g : M} (hg : DenseRange (g ^ · : ℕ → M))
(μ : Measure X) [IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicSMul M X μ] :
Ergodic (g • ·) μ | X : Type u_2
inst✝¹⁰ : TopologicalSpace X
inst✝⁹ : R1Space X
inst✝⁸ : MeasurableSpace X
inst✝⁷ : BorelSpace X
M : Type u_3
inst✝⁶ : Monoid M
inst✝⁵ : TopologicalSpace M
inst✝⁴ : MulAction M X
inst✝³ : ContinuousSMul M X
g : M
hg : DenseRange fun x => g ^ x
μ : Measure X
inst✝² : IsFiniteMeasure μ
inst✝¹ : μ.InnerRegula... | rw [mem_setOf, ← smul_iterate, preimage_iterate_eq, iterate_fixed hs] | no goals | 69e72ec408ea9393 |
SetTheory.PGame.sub_zero_eq_add_zero | Mathlib/SetTheory/Game/PGame.lean | theorem sub_zero_eq_add_zero (x : PGame) : x - 0 = x + 0 :=
show x + -0 = x + 0 by rw [neg_zero]
| x : PGame
⊢ x + -0 = x + 0 | rw [neg_zero] | no goals | 04a95978c7851b9b |
Real.aux₂ | Mathlib/NumberTheory/DiophantineApproximation/Basic.lean | theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v | ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
h : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1)
⊢ -1 < (u - ⌊ξ⌋ * v) * (2 * v - 1) | rw [← lt_sub_iff_add_lt, ← mul_assoc, ← sub_mul] at h | ξ : ℝ
u v : ℤ
hv : 2 ≤ v
hcop : IsCoprime u v
left✝ : v = 1 → -(1 / 2) < ξ - ↑u
hv₀ : 0 < ↑v
hv₀' : 0 < 2 * ↑v - 1
hv₁ : 0 < 2 * v - 1
h : -1 < (↑u - ξ * ↑v) * (2 * ↑v - 1)
⊢ -1 < (u - ⌊ξ⌋ * v) * (2 * v - 1) | b733bf52b4cb0f24 |
Nat.log_div_mul_self | Mathlib/Data/Nat/Log.lean | theorem log_div_mul_self (b n : ℕ) : log b (n / b * b) = log b n | b n : ℕ
hb : 1 < b
h : b ≤ n
⊢ 0 < b | omega | no goals | e0df15fc98715957 |
EReal.continuousAt_add | Mathlib/Topology/Instances/EReal/Lemmas.lean | theorem continuousAt_add {p : EReal × EReal} (h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥) (h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤) :
ContinuousAt (fun p : EReal × EReal => p.1 + p.2) p | case mk.h_real.h_real
a✝¹ a✝ : ℝ
h : (↑a✝¹, ↑a✝).1 ≠ ⊤ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊥
h' : (↑a✝¹, ↑a✝).1 ≠ ⊥ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (↑a✝¹, ↑a✝) | exact continuousAt_add_coe_coe _ _ | no goals | 4bd7a24b9f402842 |
ENNReal.add_rpow_le_rpow_add | Mathlib/Analysis/MeanInequalitiesPow.lean | theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p | case pos
p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
hp_pos : 0 < p
h_top : (a + b) ^ p = ⊤
⊢ a ^ p + b ^ p ≤ (a + b) ^ p | rw [h_top] | case pos
p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
hp_pos : 0 < p
h_top : (a + b) ^ p = ⊤
⊢ a ^ p + b ^ p ≤ ⊤ | da60e4c41d5e8c70 |
Filter.Tendsto.isCompact_insert_range_of_cocompact | Mathlib/Topology/Compactness/Compact.lean | theorem Tendsto.isCompact_insert_range_of_cocompact {f : X → Y} {y}
(hf : Tendsto f (cocompact X) (𝓝 y)) (hfc : Continuous f) : IsCompact (insert y (range f)) | case neg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
y : Y
hf : Tendsto f (cocompact X) (𝓝 y)
hfc : Continuous f
l : Filter Y
hne : l.NeBot
hle : l ≤ 𝓟 (insert y (range f))
hy : ¬ClusterPt y l
⊢ ∃ x ∈ insert y (range f), ClusterPt x l | simp only [clusterPt_iff, not_forall, ← not_disjoint_iff_nonempty_inter, not_not] at hy | case neg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
y : Y
hf : Tendsto f (cocompact X) (𝓝 y)
hfc : Continuous f
l : Filter Y
hne : l.NeBot
hle : l ≤ 𝓟 (insert y (range f))
hy : ∃ x, ∃ (_ : x ∈ 𝓝 y), ∃ x_1, ∃ (_ : x_1 ∈ l), Disjoint x x_1
⊢ ∃ x ∈ insert y (range f), Cluster... | bf4a62efeca4dca9 |
FiberPrebundle.inducing_totalSpaceMk_of_inducing_comp | Mathlib/Topology/FiberBundle/Basic.lean | theorem inducing_totalSpaceMk_of_inducing_comp (b : B)
(h : IsInducing (a.pretrivializationAt b ∘ TotalSpace.mk b)) :
@IsInducing _ _ _ 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
h :
IsInducing
((a.pretrivializationAt b).source.restrict ↑(a.pretrivializationAt b) ∘
codRestrict (TotalSpace.mk b) (a.pretrivializationAt... | exact (a.continuous_totalSpaceMk b).codRestrict (a.mem_pretrivializationAt_source b) | no goals | e6a89a880ec4de95 |
List.find?_flatten_eq_some_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Find.lean | theorem find?_flatten_eq_some_iff {xs : List (List α)} {p : α → Bool} {a : α} :
xs.flatten.find? p = some a ↔
p a ∧ ∃ as ys zs bs, xs = as ++ (ys ++ a :: zs) :: bs ∧
(∀ a ∈ as, ∀ x ∈ a, !p x) ∧ (∀ x ∈ ys, !p x) | case mp.intro.intro.intro.intro.inl.intro.intro.intro.intro
α : Type u_1
p : α → Bool
a : α
h : p a = true
zs : List α
as bs : List (List α)
h₂ : ∀ (a : α), a ∈ as.flatten → (!p a) = true
h₁ : bs.flatten = a :: zs
⊢ ∃ as_1 ys zs bs_1,
as ++ bs = as_1 ++ (ys ++ a :: zs) :: bs_1 ∧
(∀ (a : List α), a ∈ as_1 → ∀ ... | rw [flatten_eq_cons_iff] at h₁ | case mp.intro.intro.intro.intro.inl.intro.intro.intro.intro
α : Type u_1
p : α → Bool
a : α
h : p a = true
zs : List α
as bs : List (List α)
h₂ : ∀ (a : α), a ∈ as.flatten → (!p a) = true
h₁ : ∃ as bs_1 cs, bs = as ++ (a :: bs_1) :: cs ∧ (∀ (l : List α), l ∈ as → l = []) ∧ zs = bs_1 ++ cs.flatten
⊢ ∃ as_1 ys zs bs_1,
... | b3081438100b7db8 |
DivisorChain.element_of_chain_eq_pow_second_of_chain | Mathlib/RingTheory/ChainOfDivisors.lean | theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
(hr : r ∣ q) (hq : q ≠ 0) : ∃ i : Fin (n + 1), r = c 1 ^ (i : ℕ) | case a
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
⊢ ∀ b ∈ normalizedFactors r, b = c 1 | intro b hb | case a
M : Type u_1
inst✝¹ : CancelCommMonoidWithZero M
inst✝ : UniqueFactorizationMonoid M
q r : Associates M
n : ℕ
hn : n ≠ 0
c : Fin (n + 1) → Associates M
h₁ : StrictMono c
h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i
hr : r ∣ q
hq : q ≠ 0
i : ℕ := (normalizedFactors r).card
b : Associates M
hb : b ∈ normalizedF... | 671937a70cab0919 |
ENNReal.toNNReal_top_mul | Mathlib/Data/ENNReal/Real.lean | theorem toNNReal_top_mul (a : ℝ≥0∞) : ENNReal.toNNReal (∞ * a) = 0 | a : ℝ≥0∞
⊢ (⊤ * a).toNNReal = 0 | simp | no goals | 736ed26a2b92477a |
Submodule.mem_sup | Mathlib/LinearAlgebra/Span/Defs.lean | theorem mem_sup : x ∈ p ⊔ p' ↔ ∃ y ∈ p, ∃ z ∈ p', y + z = x :=
⟨fun h => by
rw [← span_eq p, ← span_eq p', ← span_union] at h
refine span_induction ?_ ?_ ?_ ?_ h
· rintro y (h | h)
· exact ⟨y, h, 0, by simp, by simp⟩
· exact ⟨0, by simp, y, h, by simp⟩
· exact ⟨0, by simp, 0, by simp⟩
... | case refine_2
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
p p' : Submodule R M
h : x ∈ span R (↑p ∪ ↑p')
⊢ ∃ y ∈ p, ∃ z ∈ p', y + z = 0 | exact ⟨0, by simp, 0, by simp⟩ | no goals | b19845d9667dca4f |
ADEInequality.lt_four | Mathlib/NumberTheory/ADEInequality.lean | theorem lt_four {q r : ℕ+} (hqr : q ≤ r) (H : 1 < sumInv {2, q, r}) : q < 4 | q r : ℕ+
hqr : q ≤ r
H : 1 < sumInv {2, q, r}
⊢ q < 4 | have h4 : (0 : ℚ) < 4 := by norm_num | q r : ℕ+
hqr : q ≤ r
H : 1 < sumInv {2, q, r}
h4 : 0 < 4
⊢ q < 4 | 5f44d66dcdd230c8 |
PreTilt.valAux_mul | Mathlib/RingTheory/Perfection.lean | theorem valAux_mul (f g : PreTilt O p) :
valAux K v O p (f * g) = valAux K v O p f * valAux K v O p g | case refine_2
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : ¬f = 0
hg : ¬g = 0
m n : ℕ
hm : (coeff (ModP O p) p (m ⊔ n)) f ≠ 0
hn : (coeff (ModP O p) p (m ⊔ n)) g ≠ ... | rw [← RingHom.map_pow, coeff_pow_p g] | case refine_2
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : ¬f = 0
hg : ¬g = 0
m n : ℕ
hm : (coeff (ModP O p) p (m ⊔ n)) f ≠ 0
hn : (coeff (ModP O p) p (m ⊔ n)) g ≠ ... | 3bd007ab1408e7f1 |
MeasureTheory.squareCylinders_eq_iUnion_image | Mathlib/MeasureTheory/Constructions/Cylinders.lean | theorem squareCylinders_eq_iUnion_image (C : ∀ i, Set (Set (α i))) :
squareCylinders C = ⋃ s : Finset ι, (fun t ↦ (s : Set ι).pi t) '' univ.pi C | case h
ι : Type u_2
α : ι → Type u_1
C : (i : ι) → Set (Set (α i))
f : Set ((i : ι) → α i)
⊢ f ∈ squareCylinders C ↔ f ∈ ⋃ s, (fun t => (↑s).pi t) '' univ.pi C | simp only [squareCylinders, mem_iUnion, mem_image, mem_univ_pi, exists_prop, mem_setOf_eq,
eq_comm (a := f)] | no goals | cc9933fc16d0a6b3 |
tprod_le_of_prod_le' | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | theorem tprod_le_of_prod_le' (ha₂ : 1 ≤ a₂) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : ∏' i, f i ≤ a₂ | case neg
ι : Type u_1
α : Type u_3
inst✝² : OrderedCommMonoid α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
f : ι → α
a₂ : α
ha₂ : 1 ≤ a₂
h : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂
hf : ¬Multipliable f
⊢ 1 ≤ a₂ | exact ha₂ | no goals | 3b302cfff58f9f80 |
CategoryTheory.MonoidalCategory.hom_inv_id_tensor | Mathlib/CategoryTheory/Monoidal/Category.lean | theorem hom_inv_id_tensor {V W X Y Z : C} (f : V ≅ W) (g : X ⟶ Y) (h : Y ⟶ Z) :
(f.hom ⊗ g) ≫ (f.inv ⊗ h) = (𝟙 V ⊗ g) ≫ (𝟙 V ⊗ h) | C : Type u
𝒞 : Category.{v, u} C
inst✝ : MonoidalCategory C
V W X Y Z : C
f : V ≅ W
g : X ⟶ Y
h : Y ⟶ Z
⊢ 𝟙 V ⊗ g ≫ h = (𝟙 V ⊗ g) ≫ (𝟙 V ⊗ h) | simp [id_tensorHom] | no goals | 2852a72e071acad2 |
ComplexShape.Embedding.liftExtend_homRestrict | Mathlib/Algebra/Homology/Embedding/HomEquiv.lean | @[simp]
lemma liftExtend_homRestrict (ψ : K ⟶ L.extend e) :
e.liftExtend (e.homRestrict ψ) (e.homRestrict_hasLift ψ) = ψ | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
⊢ e.liftExtend (e.homRestrict ψ) ⋯ = ψ | ext i' | case h
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{u_4, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i' : ι'
⊢ (e.liftExtend (e.homRestric... | 9240214e45cb80b5 |
MeasureTheory.SimpleFunc.const_lintegral_restrict | Mathlib/MeasureTheory/Function/SimpleFunc.lean | theorem const_lintegral_restrict (c : ℝ≥0∞) (s : Set α) :
(const α c).lintegral (μ.restrict s) = c * μ s | α : Type u_1
m : MeasurableSpace α
μ : Measure α
c : ℝ≥0∞
s : Set α
⊢ (const α c).lintegral (μ.restrict s) = c * μ s | rw [const_lintegral, Measure.restrict_apply MeasurableSet.univ, univ_inter] | no goals | b12a8c23f367c0d1 |
UnitAddTorus.mFourier_single | Mathlib/Analysis/Fourier/AddCircleMulti.lean | lemma mFourier_single [DecidableEq d] (z : d → AddCircle (1 : ℝ)) (i : d) :
mFourier (Pi.single i 1) z = fourier 1 (z i) | d : Type u_1
inst✝¹ : Fintype d
inst✝ : DecidableEq d
z : d → AddCircle 1
i : d
⊢ ∏ i_1 : d, (fourier (Pi.single i 1 i_1)) (z i_1) = (fourier 1) (z i) | have := Finset.prod_mul_prod_compl {i} (fun j ↦ fourier ((Pi.single i (1 : ℤ) : d → ℤ) j) (z j)) | d : Type u_1
inst✝¹ : Fintype d
inst✝ : DecidableEq d
z : d → AddCircle 1
i : d
this :
(∏ i_1 ∈ {i}, (fourier (Pi.single i 1 i_1)) (z i_1)) * ∏ i_1 ∈ {i}ᶜ, (fourier (Pi.single i 1 i_1)) (z i_1) =
∏ i_1 : d, (fourier (Pi.single i 1 i_1)) (z i_1)
⊢ ∏ i_1 : d, (fourier (Pi.single i 1 i_1)) (z i_1) = (fourier 1) (z i... | 629f9f2bd95cb4b4 |
PartialEquiv.trans_source' | Mathlib/Logic/Equiv/PartialEquiv.lean | theorem trans_source' : (e.trans e').source = e.source ∩ e ⁻¹' (e.target ∩ e'.source) | α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv β γ
⊢ (e.trans e').source = e.source ∩ ↑e ⁻¹' (e.target ∩ e'.source) | mfld_set_tac | no goals | 48dadd71f2dcfb98 |
NoetherNormalization.degreeOf_zero_t | Mathlib/RingTheory/NoetherNormalization.lean | private lemma degreeOf_zero_t {a : k} (ha : a ≠ 0) : ((T f) (monomial v a)).degreeOf 0 =
∑ i : Fin (n + 1), (r i) * v i | k : Type u_1
inst✝ : Field k
n : ℕ
f : MvPolynomial (Fin (n + 1)) k
v : Fin (n + 1) →₀ ℕ
a : k
ha : a ≠ 0
⊢ degreeOf 0 ((NoetherNormalization.T f) ((MvPolynomial.monomial v) a)) = ∑ i : Fin (n + 1), r i * v i | rw [← natDegree_finSuccEquiv, monomial_eq, Finsupp.prod_pow v fun a ↦ X a] | k : Type u_1
inst✝ : Field k
n : ℕ
f : MvPolynomial (Fin (n + 1)) k
v : Fin (n + 1) →₀ ℕ
a : k
ha : a ≠ 0
⊢ ((MvPolynomial.finSuccEquiv k n)
((NoetherNormalization.T f) (MvPolynomial.C a * ∏ a : Fin (n + 1), MvPolynomial.X a ^ v a))).natDegree =
∑ i : Fin (n + 1), r i * v i | c85c6fd1d2284785 |
Polynomial.exists_primitive_lcm_of_isPrimitive | Mathlib/RingTheory/Polynomial/Content.lean | theorem exists_primitive_lcm_of_isPrimitive {p q : R[X]} (hp : p.IsPrimitive) (hq : q.IsPrimitive) :
∃ r : R[X], r.IsPrimitive ∧ ∀ s : R[X], p ∣ s ∧ q ∣ s ↔ r ∣ s | case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
h : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h
rprim : r.IsPrimitive
pr : p ∣ r
qr... | have s0 : s ≠ 0 := by
contrapose! rs
simp [rs] | case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p q : R[X]
hp : p.IsPrimitive
hq : q.IsPrimitive
h : ∃ n r, r.natDegree = n ∧ r.IsPrimitive ∧ p ∣ r ∧ q ∣ r
r : R[X]
rdeg : r.natDegree = Nat.find h
rprim : r.IsPrimitive
pr : p ∣ r
qr... | 806b283d42e7cf3d |
MeasureTheory.Measure.measure_isHaarMeasure_eq_smul_of_isEverywherePos | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | theorem measure_isHaarMeasure_eq_smul_of_isEverywherePos [LocallyCompactSpace G]
(μ' μ : Measure G) [IsHaarMeasure μ] [IsHaarMeasure μ']
{s : Set G} (hs : MeasurableSet s) (h's : IsEverywherePos μ s) :
μ' s = haarScalarFactor μ' μ • μ s | case h.refine_3
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure ... | exact subset_biUnion_of_mem (u := id) ac | no goals | 05951a84047ceabf |
Algebra.traceMatrix_of_matrix_vecMul | Mathlib/RingTheory/Trace/Basic.lean | theorem traceMatrix_of_matrix_vecMul [Fintype κ] (b : κ → B) (P : Matrix κ κ A) :
traceMatrix A (b ᵥ* P.map (algebraMap A B)) = Pᵀ * traceMatrix A b * P | case a.e_f.h.e_f.h
κ : Type w
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : Fintype κ
b : κ → B
P : Matrix κ κ A
α β x y : κ
⊢ P y α * (trace A B) (b y * P x β • b x) = P y α * (traceMatrix A b y x * P x β) | simp [mul_comm] | no goals | 30a59ae6c9b52528 |
Order.height_eq_coe_add_one_iff | Mathlib/Order/KrullDimension.lean | lemma height_eq_coe_add_one_iff {x : α} {n : ℕ} :
height x = n + 1 ↔ height x < ⊤ ∧ (∃ y < x, height y = n) ∧ (∀ y < x, height y ≤ n) | α : Type u_1
inst✝ : Preorder α
x : α
n : ℕ
⊢ height x = ↑n + 1 ↔ height x < ⊤ ∧ (∃ y < x, height y = ↑n) ∧ ∀ y < x, height y ≤ ↑n | wlog hfin : height x < ⊤ | case inr
α : Type u_1
inst✝ : Preorder α
x : α
n : ℕ
this :
∀ {α : Type u_1} [inst : Preorder α] {x : α} {n : ℕ},
height x < ⊤ → (height x = ↑n + 1 ↔ height x < ⊤ ∧ (∃ y < x, height y = ↑n) ∧ ∀ y < x, height y ≤ ↑n)
hfin : ¬height x < ⊤
⊢ height x = ↑n + 1 ↔ height x < ⊤ ∧ (∃ y < x, height y = ↑n) ∧ ∀ y < x, heig... | e0f8bf729bbb5c14 |
MeasureTheory.pdf.IsUniform.mul_pdf_integrable | Mathlib/Probability/Distributions/Uniform.lean | theorem mul_pdf_integrable (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
Integrable fun x : ℝ => x * (pdf X ℙ volume x).toReal | case neg.right
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
hcs : IsCompact s
huX : IsUniform X s ℙ volume
hnt : ¬volume s = 0 ∧ ¬volume s = ⊤
this✝ : IsProbabilityMeasure ℙ
ind : ℝ → ℝ≥0∞ := (volume s)⁻¹ • 1
this : ∀ (x : ℝ), ‖x‖ₑ * s.indicator ind x = s.indicator (fun x => ‖x‖ₑ * ind x) x
⊢ ∫... | rw [lintegral_mul_const _ measurable_enorm] | case neg.right
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → ℝ
s : Set ℝ
hcs : IsCompact s
huX : IsUniform X s ℙ volume
hnt : ¬volume s = 0 ∧ ¬volume s = ⊤
this✝ : IsProbabilityMeasure ℙ
ind : ℝ → ℝ≥0∞ := (volume s)⁻¹ • 1
this : ∀ (x : ℝ), ‖x‖ₑ * s.indicator ind x = s.indicator (fun x => ‖x‖ₑ * ind x) x
⊢ (... | 41129caeb6bf5ff5 |
IsAdjoinRootMonic.coeff_apply_lt | Mathlib/RingTheory/IsAdjoinRoot.lean | theorem coeff_apply_lt (h : IsAdjoinRootMonic S f) (z : S) (i : ℕ) (hi : i < natDegree f) :
h.coeff z i = h.basis.repr z ⟨i, hi⟩ | R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : Ring S
f : R[X]
inst✝ : Algebra R S
h : IsAdjoinRootMonic S f
z : S
i : ℕ
hi : i < f.natDegree
⊢ { toFun := Polynomial.coeff, map_add' := ⋯ } (h.modByMonicHom z) i = (h.modByMonicHom z).coeff i | rfl | no goals | e78852ac90ccde5b |
MeasureTheory.Measure.haar.chaar_mono | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | theorem chaar_mono {K₀ : PositiveCompacts G} {K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂) :
chaar K₀ K₁ ≤ chaar K₀ K₂ | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' Ici 0 | apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤) | G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' Ici 0 | 33a7cc301dfbeb02 |
CategoryTheory.IsUniversalColimit.map_reflective | Mathlib/CategoryTheory/Limits/VanKampen.lean | theorem IsUniversalColimit.map_reflective
{Gl : C ⥤ D} {Gr : D ⥤ C} (adj : Gl ⊣ Gr) [Gr.Full] [Gr.Faithful]
{F : J ⥤ D} {c : Cocone (F ⋙ Gr)}
(H : IsUniversalColimit c)
[∀ X (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)]
[∀ X (f : X ⟶ Gl.obj c.pt), PreservesLimit (cospan (Gr.map ... | case refine_3
J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback ... | rw [← Category.assoc, Iso.comp_inv_eq] | case refine_3
J : Type v'
inst✝⁶ : Category.{u', v'} J
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_2
inst✝⁴ : Category.{u_3, u_2} D
Gl : C ⥤ D
Gr : D ⥤ C
adj : Gl ⊣ Gr
inst✝³ : Gr.Full
inst✝² : Gr.Faithful
F : J ⥤ D
c : Cocone (F ⋙ Gr)
H : IsUniversalColimit c
inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback ... | 0ba4821bd37956f9 |
JacobsonNoether.exists_separable_and_not_isCentral | Mathlib/FieldTheory/JacobsonNoether.lean | theorem exists_separable_and_not_isCentral (H : k ≠ (⊤ : Subring D)) :
∃ x : D, x ∉ k ∧ IsSeparable k x | D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : ... | exact hc | no goals | cf49e485bc09b885 |
mdifferentiableWithinAt_iff_of_mem_source' | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | theorem mdifferentiableWithinAt_iff_of_mem_source' {x' : M} {y : M'}
(hx : x' ∈ (chartAt H x).source) (hy : f x' ∈ (chartAt H' y).source) :
MDifferentiableWithinAt I I' f s x' ↔
ContinuousWithinAt f s x' ∧
DifferentiableWithinAt 𝕜 (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
((extChart... | 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... | set e := extChartAt I 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... | 9e7748b9367c9922 |
SimplexCategory.mkOfSucc_subinterval_eq | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | @[simp]
lemma mkOfSucc_subinterval_eq {n} (j l : ℕ) (hjl : j + l ≤ n) (i : Fin l) :
mkOfSucc i ≫ subinterval j l hjl =
mkOfSucc ⟨j + i.1, Nat.lt_of_lt_of_le (Nat.add_lt_add_left i.2 j) hjl⟩ | n j l : ℕ
hjl : j + l ≤ n
i✝ : Fin l
i : Fin (⦋1⦌.len + 1)
⊢ ↑(match 0 with
| 0 => i✝.castSucc
| 1 => i✝.succ) +
j =
↑(match 0 with
| 0 => ⟨j + ↑i✝, ⋯⟩
| 1 => ⟨j + ↑i✝ + 1, ⋯⟩) | rw [add_comm] | n j l : ℕ
hjl : j + l ≤ n
i✝ : Fin l
i : Fin (⦋1⦌.len + 1)
⊢ j +
↑(match 0 with
| 0 => i✝.castSucc
| 1 => i✝.succ) =
↑(match 0 with
| 0 => ⟨j + ↑i✝, ⋯⟩
| 1 => ⟨j + ↑i✝ + 1, ⋯⟩) | 34ac17f755ced5f6 |
lowerSemicontinuous_iff_isClosed_epigraph | Mathlib/Topology/Semicontinuous.lean | theorem lowerSemicontinuous_iff_isClosed_epigraph {f : α → γ} :
LowerSemicontinuous f ↔ IsClosed {p : α × γ | f p.1 ≤ p.2} | case mpr
α : Type u_1
inst✝⁴ : TopologicalSpace α
γ : Type u_3
inst✝³ : CompleteLinearOrder γ
inst✝² : DenselyOrdered γ
inst✝¹ : TopologicalSpace γ
inst✝ : OrderTopology γ
f : α → γ
⊢ IsClosed {p | f p.1 ≤ p.2} → ∀ (y : γ), IsClosed (f ⁻¹' Iic y) | exact fun hf y ↦ hf.preimage (Continuous.Prod.mk_left y) | no goals | 3832d193de0e5cf5 |
CategoryTheory.InjectiveResolution.rightDerivedToHomotopyCategory_app_eq | Mathlib/CategoryTheory/Abelian/RightDerived.lean | lemma InjectiveResolution.rightDerivedToHomotopyCategory_app_eq
{F G : C ⥤ D} [F.Additive] [G.Additive] (α : F ⟶ G) {X : C} (P : InjectiveResolution X) :
(NatTrans.rightDerivedToHomotopyCategory α).app X =
(P.isoRightDerivedToHomotopyCategoryObj F).hom ≫
(HomotopyCategory.quotient _ _).map
... | case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
β : (injectiveResolution X).cocomplex ⟶ P.cocomplex
hβ : (Hom... | dsimp | case intro
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{u_2, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasInjectiveResolutions C
inst✝² : Abelian D
F G : C ⥤ D
inst✝¹ : F.Additive
inst✝ : G.Additive
α : F ⟶ G
X : C
P : InjectiveResolution X
β : (injectiveResolution X).cocomplex ⟶ P.cocomplex
hβ : (Hom... | fb3a9807ff9e85f1 |
AddAction.IsBlock.of_addSubgroup_of_conjugate | Mathlib/GroupTheory/GroupAction/Blocks.lean | theorem _root_.AddAction.IsBlock.of_addSubgroup_of_conjugate
{G : Type*} [AddGroup G] {X : Type*} [AddAction G X] {B : Set X}
{H : AddSubgroup G} (hB : AddAction.IsBlock H B) (g : G) :
AddAction.IsBlock (H.map (AddAut.conj g).toMul.toAddMonoidHom) (g +ᵥ B) | case g
G : Type u_3
inst✝¹ : AddGroup G
X : Type u_4
inst✝ : AddAction G X
B : Set X
H : AddSubgroup G
hB : AddAction.IsBlock (↥H) B
g : G
h' : ↥(AddSubgroup.map (AddEquiv.toAddMonoidHom (Additive.toMul (AddAut.conj g))) H)
h : G
hH : h ∈ H
hh : g + h + -g = ↑h'
this : h' +ᵥ g +ᵥ B = g +ᵥ h +ᵥ B
⊢ Disjoint (⟨h, hH⟩ +ᵥ ... | exact Set.disjoint_image_of_injective (AddAction.injective g) | no goals | 847cf3b367a4f7f9 |
Submodule.localized₀_smul | Mathlib/Algebra/Module/LocalizedModule/Submodule.lean | lemma localized₀_smul (I : Submodule R R) : (I • M').localized₀ p f = I • M'.localized₀ p f | R : Type u_1
M : Type u_3
N : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : Module R M
inst✝¹ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M' : Submodule R M
I : Submodule R R
⊢ localized₀ p f (I • M') = I • localized₀ p f M' | apply le_antisymm | case a
R : Type u_1
M : Type u_3
N : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : Module R M
inst✝¹ : Module R N
p : Submonoid R
f : M →ₗ[R] N
inst✝ : IsLocalizedModule p f
M' : Submodule R M
I : Submodule R R
⊢ localized₀ p f (I • M') ≤ I • localized₀ p f M'
case a
R : T... | a5783bf24ee8d71d |
exists_norm_le_le_norm_sub_of_finset | Mathlib/Analysis/Normed/Module/FiniteDimension.lean | theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
(h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ | 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, ... | have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _ | 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
this : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed ↑F
⊢ ∃ ... | 3108e1e881ffd3aa |
hasSum_one_div_nat_pow_mul_sin | Mathlib/NumberTheory/ZetaValues.lean | theorem hasSum_one_div_nat_pow_mul_sin {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k + 1) * Real.sin (2 * π * n * x))
((-1 : ℝ) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
(Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1))... | case h.e'_3.h.e'_1.e_a
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
HasSum (fun n => 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
x✝ : ℕ
⊢ (cexp (2 * ↑π * I * -↑x✝ * ↑x) - cexp (2 ... | congr 4 | case h.e'_3.h.e'_1.e_a.e_a.e_a.e_a.e_z
k : ℕ
hk : k ≠ 0
x : ℝ
hx : x ∈ Icc 0 1
this :
HasSum (fun n => 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))
((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))
ofReal_two : ↑2 = 2
x✝ : ℕ
⊢ 2 * ↑π * I * -↑x✝ * ↑x = ... | 4ffa37bf4994910e |
ENNReal.pow_lt_top | Mathlib/Data/ENNReal/Order.lean | theorem pow_lt_top : a < ∞ → ∀ n : ℕ, a ^ n < ∞ | a : ℝ≥0∞
⊢ a < ⊤ → ∀ (n : ℕ), a ^ n < ⊤ | simpa only [lt_top_iff_ne_top] using pow_ne_top | no goals | 4063651c2567846c |
Polynomial.isNilpotent_iff | Mathlib/RingTheory/Polynomial/Nilpotent.lean | protected lemma isNilpotent_iff :
IsNilpotent P ↔ ∀ i, IsNilpotent (coeff P i) | case refine_2.succ
R : Type u_1
inst✝ : CommRing R
P p : R[X]
r : R
hp₀ : p.coeff 0 = 0
x✝ : r ≠ 0
hp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)
hpr : IsNilpotent (p + C r)
hr : IsNilpotent (C r)
i : ℕ
⊢ IsNilpotent ((p + C r).coeff (i + 1)) | simp only [coeff_add, coeff_C_succ, add_zero] | case refine_2.succ
R : Type u_1
inst✝ : CommRing R
P p : R[X]
r : R
hp₀ : p.coeff 0 = 0
x✝ : r ≠ 0
hp : IsNilpotent p → ∀ (i : ℕ), IsNilpotent (p.coeff i)
hpr : IsNilpotent (p + C r)
hr : IsNilpotent (C r)
i : ℕ
⊢ IsNilpotent (p.coeff (i + 1)) | 4d03fd07d02b3a60 |
CategoryTheory.Under.mapId_eq | Mathlib/CategoryTheory/Comma/Over/Basic.lean | theorem mapId_eq (Y : T) : map (𝟙 Y) = 𝟭 _ | case h_map
T : Type u₁
inst✝ : Category.{v₁, u₁} T
Y : T
x y : Under Y
u : x ⟶ y
⊢ (map (𝟙 Y)).map u = eqToHom ⋯ ≫ (𝟭 (Under Y)).map u ≫ eqToHom ⋯ | dsimp [Under, Under.map, Comma.mapLeft] | case h_map
T : Type u₁
inst✝ : Category.{v₁, u₁} T
Y : T
x y : Under Y
u : x ⟶ y
⊢ { left := 𝟙 x.left, right := u.right, w := ⋯ } = eqToHom ⋯ ≫ u ≫ eqToHom ⋯ | 67030982f590c6e3 |
Vector.lawfulBEq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem lawfulBEq_iff [BEq α] [NeZero n] : LawfulBEq (Vector α n) ↔ LawfulBEq α | case mpr.eq_of_beq.mk.mk
α : Type u_1
n✝ : Nat
inst✝¹ : BEq α
inst✝ : NeZero n✝
n : Nat
x✝ : n + 1 ≠ 0
h✝ : LawfulBEq α
a : Array α
ha : a.size = n + 1
b : Array α
hb : b.size = n + 1
h : ({ toArray := a, size_toArray := ha } == { toArray := b, size_toArray := hb }) = true
⊢ { toArray := a, size_toArray := ha } = { toA... | simp at h | case mpr.eq_of_beq.mk.mk
α : Type u_1
n✝ : Nat
inst✝¹ : BEq α
inst✝ : NeZero n✝
n : Nat
x✝ : n + 1 ≠ 0
h✝ : LawfulBEq α
a : Array α
ha : a.size = n + 1
b : Array α
hb : b.size = n + 1
h : (a == b) = true
⊢ { toArray := a, size_toArray := ha } = { toArray := b, size_toArray := hb } | fe7d41c34f58a0c2 |
MeasureTheory.lintegral_withDensity_eq_lintegral_mul_non_measurable | Mathlib/MeasureTheory/Measure/WithDensity.lean | theorem lintegral_withDensity_eq_lintegral_mul_non_measurable (μ : Measure α) {f : α → ℝ≥0∞}
(f_meas : Measurable f) (hf : ∀ᵐ x ∂μ, f x < ∞) (g : α → ℝ≥0∞) :
∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
x : α
⊢ (fun x => (f x)⁻¹ * i x) x ≤ g x | dsimp | α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
hf : ∀ᵐ (x : α) ∂μ, f x < ⊤
g i : α → ℝ≥0∞
i_meas : Measurable i
hi : i ≤ f * g
x : α
⊢ (f x)⁻¹ * i x ≤ g x | fdfc11b58110e190 |
eventuallyEq_prod | Mathlib/Topology/Algebra/Monoid.lean | theorem eventuallyEq_prod {X M : Type*} [CommMonoid M] {s : Finset ι} {l : Filter X}
{f g : ι → X → M} (hs : ∀ i ∈ s, f i =ᶠ[l] g i) : ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i | ι : Type u_1
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ᶠ (x : X) in l, ∀ i ∈ s, f i x = g i x
⊢ ∏ i ∈ s, f i =ᶠ[l] ∏ i ∈ s, g i | filter_upwards [hs] with x hx | case h
ι : Type u_1
X : Type u_6
M : Type u_7
inst✝ : CommMonoid M
s : Finset ι
l : Filter X
f g : ι → X → M
hs : ∀ᶠ (x : X) in l, ∀ i ∈ s, f i x = g i x
x : X
hx : ∀ i ∈ s, f i x = g i x
⊢ (∏ i ∈ s, f i) x = (∏ i ∈ s, g i) x | d4917e71c922a504 |
CategoryTheory.ShortComplex.LeftHomologyData.liftK_π_eq_zero_of_boundary | Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | @[reassoc]
lemma liftK_π_eq_zero_of_boundary (k : A ⟶ S.X₂) (x : A ⟶ S.X₁) (hx : k = x ≫ S.f) :
h.liftK k (by rw [hx, assoc, S.zero, comp_zero]) ≫ h.π = 0 | C : Type u_1
inst✝¹ : Category.{?u.12118, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A : C
k : A ⟶ S.X₂
x : A ⟶ S.X₁
hx : k = x ≫ S.f
⊢ k ≫ S.g = 0 | rw [hx, assoc, S.zero, comp_zero] | no goals | b39fbec65e9e3daa |
CategoryTheory.topology_eq_iff_same_sheaves | Mathlib/CategoryTheory/Sites/Closed.lean | theorem topology_eq_iff_same_sheaves {J₁ J₂ : GrothendieckTopology C} :
J₁ = J₂ ↔ ∀ P : Cᵒᵖ ⥤ Type max v u, Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P | case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ = J₂ | apply le_antisymm | case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ J₁ ≤ J₂
case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSh... | b7bd067a2a189db6 |
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