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 |
|---|---|---|---|---|---|---|
Finset.divConst_le_mulConst_sq | Mathlib/Combinatorics/Additive/DoublingConst.lean | /-- If `A` has small doubling, then it has small difference, with the constant squared.
This is a consequence of the Ruzsa triangle inequality. -/
@[to_additive
"If `A` has small doubling, then it has small difference, with the constant squared.
This is a consequence of the Ruzsa triangle inequality."]
lemma divConst... | case inl
G : Type u_1
inst✝¹ : CommGroup G
inst✝ : DecidableEq G
⊢ δₘ[∅] ≤ σₘ[∅] ^ 2 | simp | no goals | a3c2025e31fd48a2 |
Polynomial.cardPowDegree_isEuclidean | Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean | theorem cardPowDegree_isEuclidean : IsEuclidean (cardPowDegree : AbsoluteValue Fq[X] ℤ) :=
have card_pos : 0 < Fintype.card Fq := Fintype.card_pos_iff.mpr inferInstance
have pow_pos : ∀ n, 0 < (Fintype.card Fq : ℤ) ^ n := fun n =>
pow_pos (Int.natCast_pos.mpr card_pos) n
{ map_lt_map_iff' := fun {p q} => by
... | case neg
Fq : Type u_1
inst✝¹ : Field Fq
inst✝ : Fintype Fq
card_pos : 0 < Fintype.card Fq
pow_pos : ∀ (n : ℕ), 0 < ↑(Fintype.card Fq) ^ n
p q : Fq[X]
hp : p = 0
hq : ¬q = 0
⊢ 0 < ↑(Fintype.card Fq) ^ q.natDegree ↔ p.degree < q.degree | simp only [hp, hq, degree_zero, Ne, bot_lt_iff_ne_bot, degree_eq_bot, pow_pos,
not_false_iff] | no goals | 9b5952b60a5e1605 |
Vector.mapFinIdx_eq_append_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/MapIdx.lean | theorem mapFinIdx_eq_append_iff {l : Vector α (n + m)} {f : (i : Nat) → α → (h : i < n + m) → β}
{l₁ : Vector β n} {l₂ : Vector β m} :
l.mapFinIdx f = l₁ ++ l₂ ↔
∃ (l₁' : Vector α n) (l₂' : Vector α m), l = l₁' ++ l₂' ∧
l₁'.mapFinIdx (fun i a h => f i a (by omega)) = l₁ ∧
l₂'.mapFinIdx (fu... | case mk
α : Type u_1
n m : Nat
β : Type u_2
f : (i : Nat) → α → i < n + m → β
l₁ : Vector β n
l₂ : Vector β m
l : Array α
h : l.size = n + m
⊢ { toArray := l, size_toArray := h }.mapFinIdx f = l₁ ++ l₂ ↔
∃ l₁' l₂',
{ toArray := l, size_toArray := h } = l₁' ++ l₂' ∧
(l₁'.mapFinIdx fun i a h => f i a ⋯)... | rcases l₁ with ⟨l₁, rfl⟩ | case mk.mk
α : Type u_1
m : Nat
β : Type u_2
l₂ : Vector β m
l : Array α
l₁ : Array β
f : (i : Nat) → α → i < l₁.size + m → β
h : l.size = l₁.size + m
⊢ { toArray := l, size_toArray := h }.mapFinIdx f = { toArray := l₁, size_toArray := ⋯ } ++ l₂ ↔
∃ l₁' l₂',
{ toArray := l, size_toArray := h } = l₁' ++ l₂' ∧
... | 9b4b91865ad16511 |
DiffContOnCl.continuousOn_ball | Mathlib/Analysis/Calculus/DiffContOnCl.lean | theorem continuousOn_ball [NormedSpace ℝ E] {x : E} {r : ℝ} (h : DiffContOnCl 𝕜 f (ball x r)) :
ContinuousOn f (closedBall x r) | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
h : DiffContOnCl 𝕜 f (ball x r)
⊢ ContinuousOn f (closedBall x r) | rcases eq_or_ne r 0 with (rfl | hr) | case inl
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : NormedSpace ℝ E
x : E
h : DiffContOnCl 𝕜 f (ball x 0)
⊢ ContinuousOn f (closedBall x 0)
case inr
𝕜 : T... | 905015dd852d3f44 |
SimpleGraph.Walk.darts_nodup_of_support_nodup | Mathlib/Combinatorics/SimpleGraph/Walk.lean | theorem darts_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.Nodup) :
p.darts.Nodup | case cons
V : Type u
G : SimpleGraph V
u v u✝ v✝ w✝ : V
h✝ : G.Adj u✝ v✝
p' : G.Walk v✝ w✝
ih : p'.support.Nodup → p'.darts.Nodup
h : u✝ ∉ p'.support ∧ p'.support.Nodup
⊢ { toProd := (u✝, v✝), adj := h✝ } ∉ p'.darts ∧ p'.darts.Nodup | exact ⟨fun h' => h.1 (dart_fst_mem_support_of_mem_darts p' h'), ih h.2⟩ | no goals | f759df04312dd328 |
Multiset.union_le | Mathlib/Data/Multiset/UnionInter.lean | lemma union_le (h₁ : s ≤ u) (h₂ : t ≤ u) : s ∪ t ≤ u | α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
h₁ : s ≤ u
h₂ : t ≤ u
⊢ s ∪ t ≤ u ∪ t | exact union_le_union_right h₁ t | no goals | fa5e75449b7afd26 |
le_of_span_le_span | Mathlib/LinearAlgebra/LinearIndependent/Basic.lean | theorem le_of_span_le_span [Nontrivial R] {s t u : Set M} (hl : LinearIndepOn R id u)
(hsu : s ⊆ u) (htu : t ⊆ u) (hst : span R s ≤ span R t) : s ⊆ t | R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Nontrivial R
s t u : Set M
hl : LinearIndepOn R id u
hsu : s ⊆ u
htu : t ⊆ u
hst : span R s ≤ span R t
this : s ∪ t = t
⊢ s ⊆ s ∪ t | apply Set.subset_union_left | no goals | 9e4c1abdd9c87af8 |
HurwitzZeta.hasSum_int_cosKernel₀ | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | lemma hasSum_int_cosKernel₀ (a : ℝ) {t : ℝ} (ht : 0 < t) :
HasSum (fun n : ℤ ↦ if n = 0 then 0 else cexp (2 * π * I * a * n) * rexp (-π * n ^ 2 * t))
(↑(cosKernel a t) - 1) | a t : ℝ
ht : 0 < t
⊢ HasSum (fun n => if n = 0 then 0 else cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t)))
(↑(cosKernel (↑a) t) - 1) | simpa using hasSum_ite_sub_hasSum (hasSum_int_cosKernel a ht) 0 | no goals | 468001b4a32e1814 |
Std.DHashMap.Internal.exists_bucket' | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Model.lean | theorem exists_bucket' [BEq α] [Hashable α]
(self : Array (AssocList α β)) (i : USize) (hi : i.toNat < self.size) :
∃ l, Perm (self.toList.flatMap AssocList.toList) (self[i.toNat].toList ++ l) ∧
(∀ [LawfulHashable α], IsHashSelf self → ∀ k,
(mkIdx self.size (by omega) (hash k)).1.toNat = i.t... | case intro.intro.intro
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
self : Array (AssocList α β)
i : USize
hi : i.toNat < self.size
l : List ((a : α) × β a)
h₁ : (toListModel self).Perm (self[i.toNat].toList ++ l)
h₂ :
∀ [inst : LawfulHashable α],
IsHashSelf self → ∀ (k : α), (mkIdx self.size ⋯ (ha... | exact ⟨l, h₁, h₂⟩ | no goals | f84b72c0203fd6d7 |
Rat.lift_binop_eq | Mathlib/Data/Rat/Defs.lean | theorem lift_binop_eq (f : ℚ → ℚ → ℚ) (f₁ : ℤ → ℤ → ℤ → ℤ → ℤ) (f₂ : ℤ → ℤ → ℤ → ℤ → ℤ)
(fv :
∀ {n₁ d₁ h₁ c₁ n₂ d₂ h₂ c₂},
f ⟨n₁, d₁, h₁, c₁⟩ ⟨n₂, d₂, h₂, c₂⟩ = f₁ n₁ d₁ n₂ d₂ /. f₂ n₁ d₁ n₂ d₂)
(f0 : ∀ {n₁ d₁ n₂ d₂}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0) (a b c d : ℤ)
(b0 : b ≠ 0) (d0 : d ≠ 0... | case mk'.mk'
f : ℚ → ℚ → ℚ
f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ
fv :
∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0}
{c₂ : n₂.natAbs.Coprime d₂},
f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } =
f₁ n₁ (↑d₁)... | exact (divInt_eq_iff (f0 d₁0 d₂0) (f0 b0 d0)).2
(H ((divInt_eq_iff b0 d₁0).1 ha) ((divInt_eq_iff d0 d₂0).1 hc)) | no goals | f4e6dca8e0a02b67 |
List.dropLast_eq_take | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/TakeDrop.lean | theorem dropLast_eq_take (l : List α) : l.dropLast = l.take (l.length - 1) | α : Type u_1
l : List α
⊢ l.dropLast = take (l.length - 1) l | cases l with
| nil => simp [dropLast]
| cons x l =>
induction l generalizing x <;> simp_all [dropLast] | no goals | fe6ae036f9c7a9c2 |
List.Nodup.take_eq_filter_mem | Mathlib/Data/List/Nodup.lean | theorem Nodup.take_eq_filter_mem [DecidableEq α] :
∀ {l : List α} {n : ℕ} (_ : l.Nodup), l.take n = l.filter (l.take n).elem
| [], n, _ => by simp
| b::l, 0, _ => by simp
| b::l, n+1, hl => by
rw [take_succ_cons, Nodup.take_eq_filter_mem (Nodup.of_cons hl), filter_cons_of_pos (by simp)]
congr 1
re... | case e_tail
α : Type u
inst✝ : DecidableEq α
b : α
l : List α
n : ℕ
hl : (b :: l).Nodup
x : α
hx : x ∈ l
this : x ≠ b
⊢ elem x (take n l) = elem x (b :: List.filter (fun a => elem a (take n l)) l) | simp (config := {contextual := true}) [List.mem_filter, this, hx] | no goals | 11c2d2e7a90332e5 |
PowerSeries.invOneSubPow_val_eq_mk_sub_one_add_choose_of_pos | Mathlib/RingTheory/PowerSeries/WellKnown.lean | theorem invOneSubPow_val_eq_mk_sub_one_add_choose_of_pos (h : 0 < d) :
(invOneSubPow S d).val = (mk fun n => Nat.choose (d - 1 + n) (d - 1) : S⟦X⟧) | S : Type u_1
inst✝ : CommRing S
d : ℕ
h : 0 < d
⊢ ↑(invOneSubPow S d) = mk fun n => ↑((d - 1 + n).choose (d - 1)) | rw [← Nat.sub_one_add_one_eq_of_pos h, invOneSubPow, add_tsub_cancel_right] | no goals | 2588474e951d60fd |
ProbabilityTheory.IsRatStieltjesPoint.ite | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | lemma IsRatStieltjesPoint.ite {f g : α → ℚ → ℝ} {a : α} (p : α → Prop) [DecidablePred p]
(hf : p a → IsRatStieltjesPoint f a) (hg : ¬ p a → IsRatStieltjesPoint g a) :
IsRatStieltjesPoint (fun a ↦ if p a then f a else g a) a where
mono | case pos
α : Type u_1
f g : α → ℚ → ℝ
a : α
p : α → Prop
inst✝ : DecidablePred p
hf : p a → IsRatStieltjesPoint f a
hg : ¬p a → IsRatStieltjesPoint g a
h : p a
⊢ Monotone (f a)
case neg
α : Type u_1
f g : α → ℚ → ℝ
a : α
p : α → Prop
inst✝ : DecidablePred p
hf : p a → IsRatStieltjesPoint f a
hg : ¬p a → IsRatStieltjes... | exacts [(hf h).mono, (hg h).mono] | no goals | 07e5c46b6daf969c |
TopologicalSpace.IsOpenCover.comap | Mathlib/Topology/Sets/OpenCover.lean | /-- Pullback of a covering of `Y` by a continuous map `X → Y`, giving a covering of `X` with the
same index type. -/
lemma comap (hv : IsOpenCover v) (f : C(X, Y)) : IsOpenCover fun k ↦ (v k).comap f :=
by simp [IsOpenCover, ← preimage_iUnion, hv.iSup_set_eq_univ]
| κ : Type u_2
X : Type u_3
Y : Type u_4
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
v : κ → Opens Y
hv : IsOpenCover v
f : C(X, Y)
⊢ IsOpenCover fun k => (Opens.comap f) (v k) | simp [IsOpenCover, ← preimage_iUnion, hv.iSup_set_eq_univ] | no goals | 14e8d8ecea1e8679 |
Algebra.EssFiniteType.of_isLocalization | Mathlib/RingTheory/EssentialFiniteness.lean | lemma EssFiniteType.of_isLocalization (M : Submonoid R) [IsLocalization M S] :
EssFiniteType R S | case h
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
M : Submonoid R
inst✝ : IsLocalization M S
s : S
⊢ ∃ a, IsUnit ((algebraMap R S) a) ∧ ∃ y, (algebraMap R S) y = s * (algebraMap R S) a | obtain ⟨⟨x, t⟩, e⟩ := IsLocalization.surj M s | case h.intro.mk
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
M : Submonoid R
inst✝ : IsLocalization M S
s : S
x : R
t : ↥M
e : s * (algebraMap R S) ↑(x, t).2 = (algebraMap R S) (x, t).1
⊢ ∃ a, IsUnit ((algebraMap R S) a) ∧ ∃ y, (algebraMap R S) y = s * (algebraMap R S) a | e276ecc48164b1d8 |
Module.reflection_mul_reflection_zpow_apply_self | Mathlib/LinearAlgebra/Reflection.lean | /-- A formula for $(r_1 r_2)^m x$, where $m$ is an integer. This is the special case of
`Module.reflection_mul_reflection_zpow_apply` with $z = x$. -/
lemma reflection_mul_reflection_zpow_apply_self (m : ℤ)
(t : R := f y * g x - 2) (ht : t = f y * g x - 2 | case hp
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
t : optParam R (f y * g x - 2)
ht : autoParam (t = f y * g x - 2) _auto✝
S_eval_t_sub_two :
∀ (k : ℤ),
Polynomial.eval t (S R (k - 2)) = (f y * g x - 2) * Polynomial.ev... | rw [add_comm (m : ℤ) 1, zpow_one_add, LinearEquiv.mul_apply, LinearEquiv.mul_apply, ih] | case hp
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
x y : M
f g : Dual R M
hf : f x = 2
hg : g y = 2
t : optParam R (f y * g x - 2)
ht : autoParam (t = f y * g x - 2) _auto✝
S_eval_t_sub_two :
∀ (k : ℤ),
Polynomial.eval t (S R (k - 2)) = (f y * g x - 2) * Polynomial.ev... | f7647e25720205c5 |
existsUnique_isIdempotentElem_eq_of_ker_isNilpotent | Mathlib/RingTheory/Idempotents.lean | theorem existsUnique_isIdempotentElem_eq_of_ker_isNilpotent (h : ∀ x ∈ RingHom.ker f, IsNilpotent x)
(e : S) (he : e ∈ f.range) (he' : IsIdempotentElem e) :
∃! e' : R, IsIdempotentElem e' ∧ f e' = e | R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : Ring S
f : R →+* S
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e : S
he : e ∈ f.range
he' : IsIdempotentElem e
⊢ ∃! e', IsIdempotentElem e' ∧ f e' = e | obtain ⟨e', he₂, rfl⟩ := exists_isIdempotentElem_eq_of_ker_isNilpotent f h e he he' | case intro.intro
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : Ring S
f : R →+* S
h : ∀ x ∈ RingHom.ker f, IsNilpotent x
e' : R
he₂ : IsIdempotentElem e'
he : f e' ∈ f.range
he' : IsIdempotentElem (f e')
⊢ ∃! e'_1, IsIdempotentElem e'_1 ∧ f e'_1 = f e' | fa419631249a92f3 |
Turing.PartrecToTM2.head_main_ok | Mathlib/Computability/TMToPartrec.lean | theorem head_main_ok {q s L} {c d : List Γ'} :
Reaches₁ (TM2.step tr) ⟨some (head main q), s, K'.elim (trList L) [] c d⟩
⟨some q, none, K'.elim (trList [L.headI]) [] c d⟩ | case refine_2
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons
⊢ TransGen
(fun a b =>
(match a with
| { l := none, var := var, stk := stk } => none
| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =
some b... | rw [if_neg (show o ≠ some Γ'.consₗ by cases L <;> simp [o])] | case refine_2
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons
⊢ TransGen
(fun a b =>
(match a with
| { l := none, var := var, stk := stk } => none
| { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) =
some b... | 64264c0fb26f3257 |
Fin.dfoldl_succ_last | Mathlib/.lake/packages/batteries/Batteries/Data/Fin/Fold.lean | theorem dfoldl_succ_last (f : (i : Fin (n+1)) → α i.castSucc → α i.succ) (x) :
dfoldl (n+1) α f x = f (last n) (dfoldl n (α ∘ castSucc) (f ·.castSucc ·) x) | case succ
n : Nat
ih :
∀ {α : Fin (n + 1 + 1) → Type u_1} (f : (i : Fin (n + 1)) → α i.castSucc → α i.succ) (x : α 0),
dfoldl n (α ∘ succ) (fun x1 x2 => f x1.succ x2) (f 0 x) =
f (last n) (dfoldl n (α ∘ castSucc) (fun x1 x2 => f x1.castSucc x2) x)
α : Fin (n + 1 + 1 + 1) → Type u_1
f : (i : Fin (n + 1 + 1))... | congr | no goals | ee1a80e8ef7d6a6b |
OrderIso.convexOn_symm | Mathlib/Analysis/Convex/Function.lean | theorem OrderIso.convexOn_symm (f : α ≃o β) (hf : ConcaveOn 𝕜 univ f) :
ConvexOn 𝕜 univ f.symm | 𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : OrderedAddCommMonoid α
inst✝² : SMul 𝕜 α
inst✝¹ : OrderedAddCommMonoid β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : ConcaveOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
x' : α
hx'' : x = f x'
y' ... | simp | no goals | dbd840320ab5d4d8 |
fourierIntegral_half_period_translate | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | theorem fourierIntegral_half_period_translate {w : V} (hw : w ≠ 0) :
(∫ v : V, 𝐞 (-⟪v, w⟫) • f (v + i w)) = -∫ v : V, 𝐞 (-⟪v, w⟫) • f v | E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hiw : inner (i w) w = 1 / 2
this✝ : (fun v => 𝐞 (-inner v w) • f (v + i... | rw [this] | E : Type u_1
V : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
f : V → E
inst✝⁴ : NormedAddCommGroup V
inst✝³ : MeasurableSpace V
inst✝² : BorelSpace V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : FiniteDimensional ℝ V
w : V
hw : w ≠ 0
hiw : inner (i w) w = 1 / 2
this✝ : (fun v => 𝐞 (-inner v w) • f (v + i... | 851451860b8c873e |
Turing.PartrecToTM2.codeSupp'_supports | Mathlib/Computability/TMToPartrec.lean | theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S | case comp
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (f.comp g) k ⊆ S
⊢ Supports (codeSupp' (f.comp g) k) S | have H' := H | case comp
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H H' : codeSupp (f.comp g) k ⊆ S
⊢ Supports (codeSupp' (f.comp g) k) S | 5fb2278c84599fe0 |
Set.toFinset_nontrivial | Mathlib/Data/Fintype/Sets.lean | theorem toFinset_nontrivial [Fintype s] : s.toFinset.Nontrivial ↔ s.Nontrivial | α : Type u_1
s : Set α
inst✝ : Fintype ↑s
⊢ s.toFinset.Nontrivial ↔ s.Nontrivial | rw [Finset.Nontrivial, coe_toFinset] | no goals | 283303cb4bc1c7d2 |
TensorProduct.rTensor_injective_of_forall_vanishesTrivially | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean | theorem rTensor_injective_of_forall_vanishesTrivially
(hMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N},
∑ i, m i ⊗ₜ n i = (0 : M ⊗[R] N) → VanishesTrivially R m n)
(M' : Submodule R M) : Injective (rTensor N M'.subtype) | case intro.hmn
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i : Fin l, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n
M' : Submodule R M
s : Finset (↥M' × N)
hx : (rTensor... | simp only [map_sum, rTensor_tmul, coe_subtype] at hx | case intro.hmn
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
hMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i : Fin l, m i ⊗ₜ[R] n i = 0 → VanishesTrivially R m n
M' : Submodule R M
s : Finset (↥M' × N)
hx : ∑ x ∈ s,... | 3e74d15b86771baa |
closure_smul₀ | Mathlib/Topology/Algebra/ConstMulAction.lean | theorem closure_smul₀ {E} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E] [T1Space E]
[ContinuousConstSMul G₀ E] (c : G₀) (s : Set E) : closure (c • s) = c • closure s | case inl.inr
G₀ : Type u_4
inst✝⁵ : GroupWithZero G₀
E : Type u_5
inst✝⁴ : Zero E
inst✝³ : MulActionWithZero G₀ E
inst✝² : TopologicalSpace E
inst✝¹ : T1Space E
inst✝ : ContinuousConstSMul G₀ E
s : Set E
hs : s.Nonempty
⊢ closure 0 = 0 | exact closure_singleton | no goals | 4ccc2a8dac40d783 |
one_le_dite | Mathlib/Algebra/ZeroOne/Lemmas.lean | @[to_additive dite_nonneg]
lemma one_le_dite [LE α] (ha : ∀ h, 1 ≤ a h) (hb : ∀ h, 1 ≤ b h) : 1 ≤ dite p a b | case isTrue
α : Type u_1
inst✝² : One α
p : Prop
inst✝¹ : Decidable p
a : p → α
b : ¬p → α
inst✝ : LE α
ha : ∀ (h : p), 1 ≤ a h
hb : ∀ (h : ¬p), 1 ≤ b h
h✝ : p
⊢ 1 ≤ a h✝
case isFalse
α : Type u_1
inst✝² : One α
p : Prop
inst✝¹ : Decidable p
a : p → α
b : ¬p → α
inst✝ : LE α
ha : ∀ (h : p), 1 ≤ a h
hb : ∀ (h : ¬p), 1 ... | exacts [ha ‹_›, hb ‹_›] | no goals | 078a2f13959fa224 |
List.isEqv_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem isEqv_eq [DecidableEq α] {l₁ l₂ : List α} : l₁.isEqv l₂ (· == ·) = (l₁ = l₂) | case cons.cons
α : Type u_1
inst✝ : DecidableEq α
a : α
l₁ : List α
ih : ∀ {l₂ : List α}, ((l₁.isEqv l₂ fun x1 x2 => x1 == x2) = true) = (l₁ = l₂)
b : α
l₂ : List α
⊢ (((a :: l₁).isEqv (b :: l₂) fun x1 x2 => x1 == x2) = true) = (a :: l₁ = b :: l₂) | simp [isEqv, ih] | no goals | 6a9ca078710a7417 |
FreeGroup.injective_lift_of_ping_pong | Mathlib/GroupTheory/CoprodI.lean | theorem _root_.FreeGroup.injective_lift_of_ping_pong : Function.Injective (FreeGroup.lift a) | ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (... | have : FreeGroup.lift a =
(CoprodI.lift fun i => FreeGroup.lift fun _ => a i).comp
(@freeGroupEquivCoprodI ι).toMonoidHom := by
ext i
simp | ι : Type u_1
inst✝² : Nontrivial ι
G : Type u_1
inst✝¹ : Group G
a : ι → G
α : Type u_4
inst✝ : MulAction G α
X Y : ι → Set α
hXnonempty : ∀ (i : ι), (X i).Nonempty
hXdisj : Pairwise (Disjoint on X)
hYdisj : Pairwise (Disjoint on Y)
hXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)
hX : ∀ (i : ι), a i • (Y i)ᶜ ⊆ X i
hY : ∀ (... | 7d73ea3c881ec9f6 |
Zsqrtd.Nonneg.add | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem Nonneg.add {a b : ℤ√d} (ha : Nonneg a) (hb : Nonneg b) : Nonneg (a + b) | case intro.intro.inl.intro.intro.inr.inr.refine_1
d x y : ℕ
ha : { re := ↑x, im := ↑y }.Nonneg
z w : ℕ
hb : { re := -↑z, im := ↑w }.Nonneg
i : ℕ
h : ↑x + -↑z = -↑i
⊢ i ≤ z | exact Int.ofNat_le.1 (le_of_neg_le_neg (Int.le.intro x (by simp [add_comm, *]))) | no goals | 0f1aa1b9eeb4baf9 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUmod.denote_go_eq_divRec_r | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean | theorem denote_go_eq_divRec_r (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | case hq.htrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ =... | exact htrue | no goals | 123e2e165ae62131 |
List.Nodup.getElem_inj_iff | Mathlib/Data/List/Nodup.lean | theorem Nodup.getElem_inj_iff {l : List α} (h : Nodup l)
{i : Nat} {hi : i < l.length} {j : Nat} {hj : j < l.length} :
l[i] = l[j] ↔ i = j | α : Type u
l : List α
h : l.Nodup
i : ℕ
hi : i < l.length
j : ℕ
hj : j < l.length
⊢ l[i] = l[j] ↔ i = j | have := @Nodup.get_inj_iff _ _ h ⟨i, hi⟩ ⟨j, hj⟩ | α : Type u
l : List α
h : l.Nodup
i : ℕ
hi : i < l.length
j : ℕ
hj : j < l.length
this : l.get ⟨i, hi⟩ = l.get ⟨j, hj⟩ ↔ ⟨i, hi⟩ = ⟨j, hj⟩
⊢ l[i] = l[j] ↔ i = j | 53d6ce7c6656d7fb |
Multiset.inter_add_distrib | Mathlib/Data/Multiset/UnionInter.lean | lemma inter_add_distrib (s t u : Multiset α) : s ∩ t + u = (s + u) ∩ (t + u) | α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
h : s ∩ t + u ≠ (s + u) ∩ (t + u)
⊢ False | obtain ⟨a, ha⟩ := lt_iff_cons_le.1 <| h.lt_of_le <| le_inter
(Multiset.add_le_add_right inter_le_left) (Multiset.add_le_add_right inter_le_right) | case intro
α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
h : s ∩ t + u ≠ (s + u) ∩ (t + u)
a : α
ha : a ::ₘ (s ∩ t + u) ≤ (s + u) ∩ (t + u)
⊢ False | e2689f3d231690a7 |
MeasureTheory.integral_pos_iff_support_of_nonneg_ae | Mathlib/MeasureTheory/Integral/Bochner.lean | theorem integral_pos_iff_support_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : Integrable f μ) :
(0 < ∫ x, f x ∂μ) ↔ 0 < μ (Function.support f) | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᶠ[ae μ] f
hfi : Integrable f μ
⊢ 0 < ∫ (x : α), f x ∂μ ↔ 0 < μ (Function.support f) | simp_rw [(integral_nonneg_of_ae hf).lt_iff_ne, pos_iff_ne_zero, Ne, @eq_comm ℝ 0,
integral_eq_zero_iff_of_nonneg_ae hf hfi, Filter.EventuallyEq, ae_iff, Pi.zero_apply,
Function.support] | no goals | fb5673464a4e5426 |
MeasureTheory.ae_restrict_biUnion_iff | Mathlib/MeasureTheory/Measure/Restrict.lean | theorem ae_restrict_biUnion_iff (s : ι → Set α) {t : Set ι} (ht : t.Countable) (p : α → Prop) :
(∀ᵐ x ∂μ.restrict (⋃ i ∈ t, s i), p x) ↔ ∀ i ∈ t, ∀ᵐ x ∂μ.restrict (s i), p x | α : Type u_2
ι : Type u_6
m0 : MeasurableSpace α
μ : Measure α
s : ι → Set α
t : Set ι
ht : t.Countable
p : α → Prop
⊢ (∀ᵐ (x : α) ∂μ.restrict (⋃ i ∈ t, s i), p x) ↔ ∀ i ∈ t, ∀ᵐ (x : α) ∂μ.restrict (s i), p x | simp_rw [Filter.Eventually, ae_restrict_biUnion_eq s ht, mem_iSup] | no goals | 2fe83b76a301c64d |
Equiv.Perm.signBijAux_surj | Mathlib/GroupTheory/Perm/Sign.lean | theorem signBijAux_surj {n : ℕ} {f : Perm (Fin n)} :
∀ a ∈ finPairsLT n, ∃ b ∈ finPairsLT n, signBijAux f b = a :=
fun ⟨a₁, a₂⟩ ha =>
if hxa : f⁻¹ a₂ < f⁻¹ a₁ then
⟨⟨f⁻¹ a₁, f⁻¹ a₂⟩, mem_finPairsLT.2 hxa, by
dsimp [signBijAux]
rw [apply_inv_self, apply_inv_self, if_pos (mem_finPairsLT.1 ... | n : ℕ
f : Perm (Fin n)
x✝ : (_ : Fin n) × Fin n
a₁ a₂ : Fin n
ha : ⟨a₁, a₂⟩ ∈ finPairsLT n
hxa : f⁻¹ a₂ < f⁻¹ a₁
⊢ (if f (f⁻¹ a₂) < f (f⁻¹ a₁) then ⟨f (f⁻¹ a₁), f (f⁻¹ a₂)⟩ else ⟨f (f⁻¹ a₂), f (f⁻¹ a₁)⟩) = ⟨a₁, a₂⟩ | rw [apply_inv_self, apply_inv_self, if_pos (mem_finPairsLT.1 ha)] | no goals | 4716ef0c706297c6 |
SimpleGraph.dart_edge_eq_mk'_iff' | Mathlib/Combinatorics/SimpleGraph/Dart.lean | theorem dart_edge_eq_mk'_iff' :
∀ {d : G.Dart} {u v : V},
d.edge = s(u, v) ↔ d.fst = u ∧ d.snd = v ∨ d.fst = v ∧ d.snd = u | case mk.mk
V : Type u_1
G : SimpleGraph V
a b : V
h : G.Adj (a, b).1 (a, b).2
u v : V
⊢ { toProd := (a, b), adj := h }.edge = s(u, v) ↔
{ toProd := (a, b), adj := h }.toProd.1 = u ∧ { toProd := (a, b), adj := h }.toProd.2 = v ∨
{ toProd := (a, b), adj := h }.toProd.1 = v ∧ { toProd := (a, b), adj := h }.toPro... | rw [dart_edge_eq_mk'_iff] | case mk.mk
V : Type u_1
G : SimpleGraph V
a b : V
h : G.Adj (a, b).1 (a, b).2
u v : V
⊢ { toProd := (a, b), adj := h }.toProd = (u, v) ∨ { toProd := (a, b), adj := h }.toProd = (u, v).swap ↔
{ toProd := (a, b), adj := h }.toProd.1 = u ∧ { toProd := (a, b), adj := h }.toProd.2 = v ∨
{ toProd := (a, b), adj := ... | 704415a56cfd8f43 |
Module.End.injOn_genEigenspace | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | lemma injOn_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) (k : ℕ∞) :
InjOn (f.genEigenspace · k) {μ | f.genEigenspace μ k ≠ ⊥} | R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
k : ℕ∞
μ₁ : R
a✝ : μ₁ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}
μ₂ : R
hμ₂ : μ₂ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}
hμ₁₂ : (fun x => (f.genEigenspace x) k) μ₁ = (fun x => (f.genEigenspace x) k) μ₂
⊢ μ₁... | by_contra contra | R : Type v
M : Type w
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : NoZeroSMulDivisors R M
f : End R M
k : ℕ∞
μ₁ : R
a✝ : μ₁ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}
μ₂ : R
hμ₂ : μ₂ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}
hμ₁₂ : (fun x => (f.genEigenspace x) k) μ₁ = (fun x => (f.genEigenspace x) k) μ₂
cont... | a0c686e86c7fd170 |
Besicovitch.exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measure α)
[IsFiniteMeasure μ] (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ t : Set (α × ℝ), t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧
μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), close... | case intro.intro.intro.refine_2
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
... | intro p hp | case intro.intro.intro.refine_2
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
... | fb6f6078be13e9db |
LinearMap.eq_adjoint_iff_basis | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | theorem eq_adjoint_iff_basis {ι₁ : Type*} {ι₂ : Type*} (b₁ : Basis ι₁ 𝕜 E) (b₂ : Basis ι₂ 𝕜 F)
(A : E →ₗ[𝕜] F) (B : F →ₗ[𝕜] E) :
A = LinearMap.adjoint B ↔ ∀ (i₁ : ι₁) (i₂ : ι₂), ⟪A (b₁ i₁), b₂ i₂⟫ = ⟪b₁ i₁, B (b₂ i₂)⟫ | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
ι₁ : Type u_5
ι₂ : Type u_6
b₁ : Basis ι₁ 𝕜 E
b₂ : Basis ι₂ 𝕜 F
A : E →ₗ... | refine Basis.ext b₁ fun i₁ => ?_ | 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
ι₁ : Type u_5
ι₂ : Type u_6
b₁ : Basis ι₁ 𝕜 E
b₂ : Basis ι₂ 𝕜 F
A : E →ₗ... | 0f535bc4dd51830f |
Multiset.sub_eq_fold_erase | Mathlib/Data/Multiset/MapFold.lean | lemma sub_eq_fold_erase (s t : Multiset α) : s - t = foldl erase s t :=
Quotient.inductionOn₂ s t fun l₁ l₂ => by
show ofList (l₁.diff l₂) = foldl erase l₁ l₂
rw [diff_eq_foldl l₁ l₂]
symm
exact foldl_hom _ _ _ _ _ fun x y => rfl
| α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
l₁ l₂ : List α
⊢ foldl erase ↑l₁ ↑l₂ = ↑(List.foldl List.erase l₁ l₂) | exact foldl_hom _ _ _ _ _ fun x y => rfl | no goals | e85b123479925548 |
mul_eq_one_iff_of_one_le | Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | theorem mul_eq_one_iff_of_one_le [MulLeftMono α]
[MulRightMono α] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
a * b = 1 ↔ a = 1 ∧ b = 1 :=
Iff.intro
(fun hab : a * b = 1 =>
have : a ≤ 1 := hab ▸ le_mul_of_le_of_one_le le_rfl hb
have : a = 1 := le_antisymm this ha
have : b ≤ 1 := hab ▸ le_mul_o... | case intro
α : Type u_1
inst✝³ : MulOneClass α
inst✝² : PartialOrder α
inst✝¹ : MulLeftMono α
inst✝ : MulRightMono α
ha hb : 1 ≤ 1
⊢ 1 * 1 = 1 | rw [mul_one] | no goals | 257feea4dab48b2a |
tendsto_integral_peak_smul_of_integrable_of_tendsto | Mathlib/MeasureTheory/Integral/PeakFunction.lean | theorem tendsto_integral_peak_smul_of_integrable_of_tendsto
{t : Set α} (ht : MeasurableSet t) (h'ts : t ∈ 𝓝 x₀)
(h't : μ t ≠ ∞) (hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l uᶜ)
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂μ) l (𝓝 1))
(h'iφ : ∀ᶠ i ... | α : Type u_1
E : Type u_2
ι : Type u_3
hm : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace α
inst✝³ : BorelSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
g : α → E
l : Filter ι
x₀ : α
φ : ι → α → ℝ
a : E
inst✝ : CompleteSpace E
t : Set α
ht : MeasurableSet t
h'ts : t ∈ 𝓝 x₀
h't : μ t ≠ ⊤
hnφ... | simpa | no goals | 95790c5debc72743 |
AddCircle.lintegral_preimage | Mathlib/MeasureTheory/Integral/Periodic.lean | theorem lintegral_preimage (t : ℝ) (f : AddCircle T → ℝ≥0∞) :
(∫⁻ a in Ioc t (t + T), f a) = ∫⁻ b : AddCircle T, f b | case h
T : ℝ
hT : Fact (0 < T)
t : ℝ
f : AddCircle T → ℝ≥0∞
m : MeasurableSet (Ioc t (t + T))
this : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x => ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a
x : ↑(Ioc t (t + T))
⊢ ↑↑x = (QuotientAddGroup.mk ∘ Subtype.val) x | rfl | no goals | 0c520a241ff3aa69 |
eVariationOn.add_le_union | Mathlib/Topology/EMetricSpace/BoundedVariation.lean | theorem add_le_union (f : α → E) {s t : Set α} (h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) :
eVariationOn f s + eVariationOn f t ≤ eVariationOn f (s ∪ t) | α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈... | rw [tsub_eq_iff_eq_add_of_le] | α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s t : Set α
h : ∀ x ∈ s, ∀ y ∈ t, x ≤ y
hs : ¬s = ∅
this✝ : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ s }
ht : ¬t = ∅
this : Nonempty { u // Monotone u ∧ ∀ (i : ℕ), u i ∈ t }
n : ℕ
u : ℕ → α
hu : Monotone u
us : ∀ (i : ℕ), u i ∈... | 4878120f0018b659 |
BoxIntegral.IntegrationParams.tendsto_embedBox_toFilteriUnion_top | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | theorem tendsto_embedBox_toFilteriUnion_top (l : IntegrationParams) (h : I ≤ J) :
Tendsto (TaggedPrepartition.embedBox I J h) (l.toFilteriUnion I ⊤)
(l.toFilteriUnion J (Prepartition.single J I h)) | ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
⊢ Tendsto (⇑(embedBox I J h)) (l.toFilterDistortioniUnion I c ⊤)
(⨆ c, l.toFilterDistortioniUnion J c (Prepartition.single J I h)) | set π₀ := Prepartition.single J I h | ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
⊢ Tendsto (⇑(embedBox I J h)) (l.toFilterDistortioniUnion I c ⊤) (⨆ c, l.toFilterDistortioniUnion J c π₀) | 75f666dbd7c5bb0f |
List.Chain'.cons_of_le | Mathlib/Data/List/Chain.lean | theorem Chain'.cons_of_le [LinearOrder α] {a : α} {as m : List α}
(ha : List.Chain' (· > ·) (a :: as)) (hm : List.Chain' (· > ·) m) (hmas : m ≤ as) :
List.Chain' (· > ·) (a :: m) | case cons.cons.inl
α : Type u
inst✝ : LinearOrder α
a b : α
bs : List α
hm : Chain' (fun x1 x2 => x1 > x2) (b :: bs)
a' : α
as : List α
ha : a > a' ∧ Chain' (fun x1 x2 => x1 > x2) (a' :: as)
hmas : ¬a' :: as ≤ b :: bs
hh : a' < b
⊢ a' :: as ≤ b :: bs | apply le_of_lt | case cons.cons.inl.hab
α : Type u
inst✝ : LinearOrder α
a b : α
bs : List α
hm : Chain' (fun x1 x2 => x1 > x2) (b :: bs)
a' : α
as : List α
ha : a > a' ∧ Chain' (fun x1 x2 => x1 > x2) (a' :: as)
hmas : ¬a' :: as ≤ b :: bs
hh : a' < b
⊢ a' :: as < b :: bs | 9817e0c725681157 |
ContDiff.fourierPowSMulRight | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | lemma _root_.ContDiff.fourierPowSMulRight
{f : V → E} {k : WithTop ℕ∞} (hf : ContDiff ℝ k f) (n : ℕ) :
ContDiff ℝ k (fun v ↦ fourierPowSMulRight L f v n) | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
k : WithTop ℕ∞
hf : ContDiff ℝ k f
n : ℕ
⊢ ContDiff ℝ k fun v => VectorFourier.fourie... | simp_rw [fourierPowSMulRight_eq_comp] | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℂ E
V : Type u_2
W : Type u_3
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : NormedAddCommGroup W
inst✝ : NormedSpace ℝ W
L : V →L[ℝ] W →L[ℝ] ℝ
f : V → E
k : WithTop ℕ∞
hf : ContDiff ℝ k f
n : ℕ
⊢ ContDiff ℝ k fun v =>
(-(2 * ↑π * I)) ... | 9524ce0d7b23d0d7 |
FirstOrder.Language.Substructure.closure_withConstants_eq | Mathlib/ModelTheory/Substructures.lean | theorem closure_withConstants_eq :
closure (L[[A]]) s =
(closure L (A ∪ s)).withConstants ((A.subset_union_left).trans subset_closure) | L : Language
M : Type w
inst✝ : L.Structure M
A s : Set M
⊢ (L.lhomWithConstants ↑A).substructureReduct (((closure L).toFun (A ∪ s)).withConstants ⋯) ≤
(L.lhomWithConstants ↑A).substructureReduct ((closure (L[[↑A]])).toFun s) | simp only [subset_closure, reduct_withConstants, closure_le, LHom.coe_substructureReduct,
Set.union_subset_iff, and_true] | L : Language
M : Type w
inst✝ : L.Structure M
A s : Set M
⊢ A ⊆ ↑((closure (L[[↑A]])).toFun s) | a5a49fe6dc129c0d |
MeasureTheory.sigmaFinite_restrict_sigmaFiniteSetWRT' | Mathlib/MeasureTheory/Decomposition/Exhaustion.lean | lemma sigmaFinite_restrict_sigmaFiniteSetWRT' (μ ν : Measure α) [IsFiniteMeasure ν] :
SigmaFinite (μ.restrict (μ.sigmaFiniteSetWRT' ν)) | α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
fun p =>
(μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ℕ := Nat.pairE... | refine ⟨fun n ↦ f (e n), fun _ ↦ by simp, fun n ↦ ?_, ?_⟩ | case refine_1
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : IsFiniteMeasure ν
this : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))
f : ℕ × ℕ → Set α :=
fun p =>
(μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1
e : ℕ ≃ ℕ × ... | 858f4f82daf7e411 |
Set.pairwise_disjoint_Ioo_add_intCast | Mathlib/Algebra/Order/Interval/Set/Group.lean | theorem pairwise_disjoint_Ioo_add_intCast :
Pairwise (Disjoint on fun n : ℤ => Ioo (a + n) (a + n + 1)) | α : Type u_1
inst✝ : OrderedRing α
a : α
⊢ Pairwise (Disjoint on fun n => Ioo (a + ↑n) (a + ↑n + 1)) | simpa only [zsmul_one, Int.cast_add, Int.cast_one, ← add_assoc] using
pairwise_disjoint_Ioo_add_zsmul a (1 : α) | no goals | 6061a4166d66960b |
MeasureTheory.Measure.measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | /-- Two left invariant measures give the same mass to level sets of continuous compactly supported
functions, up to the scalar `haarScalarFactor μ' μ`.
Auxiliary lemma in the proof of the more general
`measure_isMulInvariant_eq_smul_of_isCompact_closure`, which works for any set with
compact closure. -/
@[to_additive ... | G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
⊢ μ' (f ⁻¹' {1}) = ... | obtain ⟨u, -, u_mem, u_lim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Ioo 0 1)
∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto' (zero_lt_one : (0 : ℝ) < 1) | case intro.intro.intro
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSuppor... | 16895233429fcfcd |
Orientation.areaForm_swap | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | theorem areaForm_swap (x y : E) : ω x y = -ω y x | case h.e'_2.h.e'_6.h
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
i : Fin (Nat.succ 1)
⊢ ![x, y] i = (![y, x] ∘ ⇑(Equiv.swap 0 1)) i | fin_cases i <;> rfl | no goals | 722c1eb48ae967e0 |
AddCircle.ergodic_zsmul_add | Mathlib/Dynamics/Ergodic/AddCircle.lean | theorem ergodic_zsmul_add (x : AddCircle T) {n : ℤ} (h : 1 < |n|) : Ergodic fun y => n • y + x | T : ℝ
hT : Fact (0 < T)
x : AddCircle T
n : ℤ
h : |1| < |n|
f : AddCircle T → AddCircle T := fun y => n • y + x
e : AddCircle T ≃ᵐ AddCircle T := MeasurableEquiv.addLeft (DivisibleBy.div x (n - 1))
he : MeasurePreserving (⇑e) volume volume
⊢ n ≠ 1 | exact ne_of_apply_ne _ (ne_of_gt h) | no goals | 05c20060c1761793 |
dotProduct_sub | Mathlib/Data/Matrix/Mul.lean | theorem dotProduct_sub : u ⬝ᵥ (v - w) = u ⬝ᵥ v - u ⬝ᵥ w | m : Type u_2
α : Type v
inst✝¹ : Fintype m
inst✝ : NonUnitalNonAssocRing α
u v w : m → α
⊢ u ⬝ᵥ (v - w) = u ⬝ᵥ v - u ⬝ᵥ w | simp [sub_eq_add_neg] | no goals | 838081043b2f8019 |
MeasureTheory.Measure.HasTemperateGrowth.exists_eLpNorm_lt_top | Mathlib/Analysis/Distribution/SchwartzSpace.lean | theorem _root_.MeasureTheory.Measure.HasTemperateGrowth.exists_eLpNorm_lt_top (p : ℝ≥0∞)
{μ : Measure D} (hμ : μ.HasTemperateGrowth) :
∃ k : ℕ, eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-k : ℝ)) p μ < ⊤ | case h
D : Type u_3
inst✝¹ : NormedAddCommGroup D
inst✝ : MeasurableSpace D
μ : Measure D
hμ : μ.HasTemperateGrowth
p : ℝ≥0
hp : ↑p ≠ 0
h_one_add : ∀ (x : D), 0 < 1 + ‖x‖
hp_pos : 0 < ↑p
l : ℕ
hl : Integrable (fun x => (1 + ‖x‖) ^ (-↑l)) μ
k : ℕ := ⌈↑l / ↑p⌉₊
hlk : ↑l ≤ ↑k * ↑p
x : D
⊢ ‖(1 + ‖x‖) ^ (-(↑k * ↑p))‖ ≤ (1 +... | rw [Real.norm_of_nonneg (Real.rpow_nonneg (h_one_add x).le _)] | case h
D : Type u_3
inst✝¹ : NormedAddCommGroup D
inst✝ : MeasurableSpace D
μ : Measure D
hμ : μ.HasTemperateGrowth
p : ℝ≥0
hp : ↑p ≠ 0
h_one_add : ∀ (x : D), 0 < 1 + ‖x‖
hp_pos : 0 < ↑p
l : ℕ
hl : Integrable (fun x => (1 + ‖x‖) ^ (-↑l)) μ
k : ℕ := ⌈↑l / ↑p⌉₊
hlk : ↑l ≤ ↑k * ↑p
x : D
⊢ (1 + ‖x‖) ^ (-(↑k * ↑p)) ≤ (1 + ‖... | c7102e41e06a4fde |
IsPreconnected.preimage_of_isClosedMap | Mathlib/Topology/Connected/Basic.lean | theorem IsPreconnected.preimage_of_isClosedMap [TopologicalSpace β] {s : Set β}
(hs : IsPreconnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f)
(hsf : s ⊆ range f) : IsPreconnected (f ⁻¹' s) :=
isPreconnected_closed_iff.2 fun u v hu hv hsuv hsu hsv => by
replace hsf : f '' (f ⁻¹' s)... | case refine_3
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
s : Set β
hs : IsPreconnected s
f : α → β
hinj : Injective f
hf : IsClosedMap f
u v : Set α
hu : IsClosed u
hv : IsClosed v
hsuv : f ⁻¹' s ⊆ u ∪ v
hsu : (f ⁻¹' s ∩ u).Nonempty
hsv : (f ⁻¹' s ∩ v).Nonempty
hsf : f '' (f ⁻¹' s) = s... | simpa only [image_preimage_inter] using hsv.image f | no goals | cf3a0afc7154a636 |
Metric.cthickening_eq_iInter_cthickening' | Mathlib/Topology/MetricSpace/Thickening.lean | theorem cthickening_eq_iInter_cthickening' {δ : ℝ} (s : Set ℝ) (hsδ : s ⊆ Ioi δ)
(hs : ∀ ε, δ < ε → (s ∩ Ioc δ ε).Nonempty) (E : Set α) :
cthickening δ E = ⋂ ε ∈ s, cthickening ε E | α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set ℝ
hsδ : s ⊆ Ioi δ
hs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty
E : Set α
⊢ cthickening δ E = ⋂ ε ∈ s, cthickening ε E | apply Subset.antisymm | case h₁
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set ℝ
hsδ : s ⊆ Ioi δ
hs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty
E : Set α
⊢ cthickening δ E ⊆ ⋂ ε ∈ s, cthickening ε E
case h₂
α : Type u
inst✝ : PseudoEMetricSpace α
δ : ℝ
s : Set ℝ
hsδ : s ⊆ Ioi δ
hs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty
E : Set α
... | 6f877c5032a5d223 |
CategoryTheory.tensorRightHomEquiv_symm_naturality | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | theorem tensorRightHomEquiv_symm_naturality {X X' Y Y' Z : C} [ExactPairing Y Y'] (f : X ⟶ X')
(g : X' ⟶ Z ⊗ Y') :
(tensorRightHomEquiv X Y Y' Z).symm (f ≫ g) =
f ▷ Y ≫ (tensorRightHomEquiv X' Y Y' Z).symm g | C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X X' Y Y' Z : C
inst✝ : ExactPairing Y Y'
f : X ⟶ X'
g : X' ⟶ Z ⊗ Y'
⊢ (tensorRightHomEquiv X Y Y' Z).symm (f ≫ g) = f ▷ Y ≫ (tensorRightHomEquiv X' Y Y' Z).symm g | simp [tensorRightHomEquiv] | no goals | 9c613ce563937b47 |
Pi.isAtom_iff | Mathlib/Order/Atoms.lean | theorem isAtom_iff {f : ∀ i, π i} [∀ i, PartialOrder (π i)] [∀ i, OrderBot (π i)] :
IsAtom f ↔ ∃ i, IsAtom (f i) ∧ ∀ j, j ≠ i → f j = ⊥ | case pos
ι : Type u_4
π : ι → Type u
f : (i : ι) → π i
inst✝¹ : (i : ι) → PartialOrder (π i)
inst✝ : (i : ι) → OrderBot (π i)
hbot✝ : f ≠ ⊥
h✝ : ∀ b < f, b = ⊥
i : ι
hbot : f i ≠ ⊥
b : π i
hb : b < f i
this :
(∀ (i_1 : ι), Function.update ⊥ i b i_1 ≤ f i_1) →
∀ (x : ι), Function.update ⊥ i b x < f x → ∀ (i_1 : ι)... | { subst h; simpa using le_of_lt hb } | case neg
ι : Type u_4
π : ι → Type u
f : (i : ι) → π i
inst✝¹ : (i : ι) → PartialOrder (π i)
inst✝ : (i : ι) → OrderBot (π i)
hbot✝ : f ≠ ⊥
h✝ : ∀ b < f, b = ⊥
i : ι
hbot : f i ≠ ⊥
b : π i
hb : b < f i
this :
(∀ (i_1 : ι), Function.update ⊥ i b i_1 ≤ f i_1) →
∀ (x : ι), Function.update ⊥ i b x < f x → ∀ (i_1 : ι)... | 116210e72cdaa30b |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRupUnits_of_assignmentsInvariant | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem assignmentsInvariant_insertRupUnits_of_assignmentsInvariant {n : Nat} (f : DefaultFormula n) (f_readyForRupAdd : ReadyForRupAdd f)
(units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRupUnits f units).1 | case neg.h.left
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsiz... | rw [← h1] | case neg.h.left
n : Nat
f : DefaultFormula n
f_readyForRupAdd : f.ReadyForRupAdd
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRupUnits units).fst.assignments;
let_fun hsize := ⋯;
let rupUnits := (f.insertRupUnits units).fst.rupUnits;
InsertUnitInvariant f.assignments ⋯ rupUnits assignments hsiz... | 7537b9cac0fe9952 |
Subring.comap_map_eq | Mathlib/Algebra/Ring/Subring/Basic.lean | theorem comap_map_eq (f : R →+* S) (s : Subring R) :
(s.map f).comap f = s ⊔ closure (f ⁻¹' {0}) | case a.intro.intro
R : Type u
S : Type v
inst✝¹ : Ring R
inst✝ : Ring S
f : R →+* S
s : Subring R
x y : R
hy : y ∈ s
hxy : x - y ∈ ⇑f ⁻¹' {0}
⊢ y + (x - y) ∈ closure (↑s ∪ ⇑f ⁻¹' {0}) | exact Subring.add_mem _ (subset_closure <| Or.inl hy) (subset_closure <| Or.inr hxy) | no goals | 1888ddc34813b7fe |
MeasureTheory.VectorMeasure.MutuallySingular.add_left | Mathlib/MeasureTheory/VectorMeasure/Basic.lean | theorem add_left [T2Space N] [ContinuousAdd M] (h₁ : v₁ ⟂ᵥ w) (h₂ : v₂ ⟂ᵥ w) : v₁ + v₂ ⟂ᵥ w | case intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
m : MeasurableSpace α
M : Type u_4
N : Type u_5
inst✝⁵ : AddCommMonoid M
inst✝⁴ : TopologicalSpace M
inst✝³ : AddCommMonoid N
inst✝² : TopologicalSpace N
v₁ v₂ : VectorMeasure α M
w : VectorMeasure α N
inst✝¹ : T2Space N
inst✝ : ContinuousAdd M
u : Set α
hm... | rw [add_apply, hu₁ _ (Set.subset_inter_iff.1 ht).1, hv₁ _ (Set.subset_inter_iff.1 ht).2,
zero_add] | no goals | a173a954f6dae6d1 |
Turing.PartrecToTM2.trNormal_supports | Mathlib/Computability/TMToPartrec.lean | theorem trNormal_supports {S c k} (Hk : codeSupp c k ⊆ S) : (trNormal c k).Supports S | case tail
S : Finset Λ'
k : Cont'
Hk : codeSupp Code.tail k ⊆ S
⊢ contSupp k ⊆ S | exact Finset.union_subset_right Hk | no goals | 279b0f47b2c1d269 |
Finset.smul_neg | Mathlib/Algebra/GroupWithZero/Pointwise/Finset.lean | @[simp]
protected lemma smul_neg (s : Finset α) (t : Finset β) : s • -t = -(s • t) | α : Type u_1
β : Type u_2
inst✝³ : DecidableEq β
inst✝² : Monoid α
inst✝¹ : AddGroup β
inst✝ : DistribMulAction α β
s : Finset α
t : Finset β
⊢ s • image (fun x => -x) t = image (fun x => -x) (s • t) | exact image_image₂_right_comm smul_neg | no goals | 22510c172674c599 |
Besicovitch.TauPackage.color_lt | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N | case inr
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 ... | simp only [G] | case inr
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 ... | 70d4d5d8aa8a7de2 |
MeasureTheory.lintegral_fintype | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_fintype [MeasurableSingletonClass α] [Fintype α] (f : α → ℝ≥0∞) :
∫⁻ x, f x ∂μ = ∑ x, f x * μ {x} | α : Type u_1
m : MeasurableSpace α
μ : Measure α
inst✝¹ : MeasurableSingletonClass α
inst✝ : Fintype α
f : α → ℝ≥0∞
⊢ ∫⁻ (x : α), f x ∂μ = ∑ x : α, f x * μ {x} | rw [← lintegral_finset, Finset.coe_univ, Measure.restrict_univ] | no goals | 5faedcd9b5fba8e2 |
Polynomial.reflect_map | Mathlib/Algebra/Polynomial/Reverse.lean | lemma reflect_map {S : Type*} [Semiring S] (f : R →+* S) (p : R[X]) (n : ℕ) :
(p.map f).reflect n = (p.reflect n).map f | case a
R : Type u_1
inst✝¹ : Semiring R
S : Type u_2
inst✝ : Semiring S
f : R →+* S
p : R[X]
n n✝ : ℕ
⊢ (reflect n (map f p)).coeff n✝ = (map f (reflect n p)).coeff n✝ | simp | no goals | 325b496f582a8fe9 |
MeasureTheory.addHaar_image_eq_zero_of_det_fderivWithin_eq_zero | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem addHaar_image_eq_zero_of_det_fderivWithin_eq_zero
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (h'f' : ∀ x ∈ s, (f' x).det = 0) :
μ (f '' s) = 0 | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
h'f' : ∀ x ∈ s, (f' x).det = 0
H : ∀ (R : ℝ), ... | simp only [H, tsum_zero, nonpos_iff_eq_zero] | no goals | 3a2e04836bc0a78b |
MultilinearMap.map_coord_zero | Mathlib/LinearAlgebra/Multilinear/Basic.lean | theorem map_coord_zero {m : ∀ i, M₁ i} (i : ι) (h : m i = 0) : f m = 0 | R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁴ : Semiring R
inst✝³ : (i : ι) → AddCommMonoid (M₁ i)
inst✝² : AddCommMonoid M₂
inst✝¹ : (i : ι) → Module R (M₁ i)
inst✝ : Module R M₂
f : MultilinearMap R M₁ M₂
m : (i : ι) → M₁ i
i : ι
h : m i = 0
this : 0 • 0 = 0
⊢ f m = 0 | rw [← update_eq_self i m, h, ← this, f.map_update_smul, zero_smul R (M := M₂)] | no goals | 9b8ad45246a78c98 |
Metric.thickening_cthickening_subset | Mathlib/Topology/MetricSpace/Thickening.lean | theorem thickening_cthickening_subset (ε : ℝ) (hδ : 0 ≤ δ) (s : Set α) :
thickening ε (cthickening δ s) ⊆ thickening (ε + δ) s | case inr.intro.intro
α : Type u
inst✝ : PseudoEMetricSpace α
δ ε : ℝ
hδ : 0 ≤ δ
s : Set α
hε : 0 ≤ ε
x y : α
hy : infEdist y s ≤ ENNReal.ofReal δ
hxy : edist x y < ENNReal.ofReal ε
⊢ infEdist x s < ENNReal.ofReal ε + ENNReal.ofReal δ | exact infEdist_le_edist_add_infEdist.trans_lt
(ENNReal.add_lt_add_of_lt_of_le (hy.trans_lt ENNReal.ofReal_lt_top).ne hxy hy) | no goals | 9806ac5af9699dfe |
MeasureTheory.LevyProkhorov.continuous_equiv_symm_probabilityMeasure | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | lemma LevyProkhorov.continuous_equiv_symm_probabilityMeasure :
Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)).symm | case intro.intro.intro.intro.intro.intro
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), Measur... | filter_upwards [(Finset.iInter_mem_sets Js_finite.toFinset).mpr <|
fun J _ ↦ mem_nhds_P _ (Gs_open J)] with Q hQ | case h
Ω : Type u_1
inst✝³ : PseudoMetricSpace Ω
inst✝² : MeasurableSpace Ω
inst✝¹ : OpensMeasurableSpace Ω
inst✝ : SeparableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ),... | 1196bd5a9632bf03 |
GromovHausdorff.hausdorffDist_optimal_le_HD | Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f | case refine_1.intro.intro
X : Type u
Y : Type v
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
f : GromovHausdorff.Cb X Y
h : f ∈ GromovHausdorff.candidatesB X Y
r : ℝ
hr : HD (GromovHausdorff.optimalGHDist X Y) < r
A : ∀ x ∈ range (o... | exact le_trans dist_nonneg hy | no goals | 805af102df252446 |
NumberField.InfinitePlace.mk_eq_iff | Mathlib/NumberTheory/NumberField/Embeddings.lean | theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ | case mp
K : Type u_2
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ → φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ | intro h₀ | case mp
K : Type u_2
inst✝ : Field K
φ ψ : K →+* ℂ
h₀ : mk φ = mk ψ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ | 1d2ef4e3545a7979 |
Rat.AbsoluteValue.eq_one_of_not_dvd | Mathlib/NumberTheory/Ostrowski.lean | /-- A natural number not divible by `p` has absolute value 1. -/
lemma eq_one_of_not_dvd {m : ℕ} (hpm : ¬ p ∣ m) : f m = 1 | f : AbsoluteValue ℚ ℝ
bdd : ∀ (n : ℕ), f ↑n ≤ 1
p : ℕ
hp0 : 0 < f ↑p
hp1 : f ↑p < 1
hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m
m : ℕ
hpm : ¬p ∣ m
hm : f ↑m < 1
M : ℝ := f ↑p ⊔ f ↑m
hM : M = f ↑p ⊔ f ↑m
k : ℕ := ⌈logb M (1 / 2)⌉₊ + 1
hk : k = ⌈logb M (1 / 2)⌉₊ + 1
⊢ False | obtain ⟨a, b, bezout⟩ : IsCoprime (p ^ k : ℤ) (m ^ k) :=
is_prime_of_minimal_nat_zero_lt_and_lt_one hp0 hp1 hmin
|>.coprime_iff_not_dvd |>.mpr hpm |>.isCoprime |>.pow | case intro.intro
f : AbsoluteValue ℚ ℝ
bdd : ∀ (n : ℕ), f ↑n ≤ 1
p : ℕ
hp0 : 0 < f ↑p
hp1 : f ↑p < 1
hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m
m : ℕ
hpm : ¬p ∣ m
hm : f ↑m < 1
M : ℝ := f ↑p ⊔ f ↑m
hM : M = f ↑p ⊔ f ↑m
k : ℕ := ⌈logb M (1 / 2)⌉₊ + 1
hk : k = ⌈logb M (1 / 2)⌉₊ + 1
a b : ℤ
bezout : a * ↑p ^ k + b * ↑m... | 581c58a2894f4e47 |
IntermediateField.LinearDisjoint.linearIndependent_left | Mathlib/FieldTheory/LinearDisjoint.lean | theorem linearIndependent_left (H : A.LinearDisjoint L)
{ι : Type*} {a : ι → A} (ha : LinearIndependent F a) : LinearIndependent L (A.val ∘ a) :=
(Subalgebra.LinearDisjoint.linearIndependent_left_of_flat H ha).map_of_injective_injective
(AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom F L E)) (AddMonoidHom.... | F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
ι : Type u_1
a : ι → ↥A
ha : LinearIndependent F a
x✝¹ : L
x✝ : E
⊢ (AddMonoidHom.id E) ((AlgEqu... | simp_rw [Algebra.smul_def] | F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
ι : Type u_1
a : ι → ↥A
ha : LinearIndependent F a
x✝¹ : L
x✝ : E
⊢ (AddMonoidHom.id E)
((... | 276f60b2b182ddad |
preimage_connectedComponent_connected | Mathlib/Topology/Connected/Clopen.lean | theorem preimage_connectedComponent_connected
(connected_fibers : ∀ t : β, IsConnected (f ⁻¹' {t}))
(hcl : ∀ T : Set β, IsClosed T ↔ IsClosed (f ⁻¹' T)) (t : β) :
IsConnected (f ⁻¹' connectedComponent t) | α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
⊢ IsPreconnected (f ⁻¹' connectedComponent t) | have hT : IsClosed (f ⁻¹' connectedComponent t) :=
(hcl (connectedComponent t)).1 isClosed_connectedComponent | α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
f : α → β
connected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})
hcl : ∀ (T : Set β), IsClosed T ↔ IsClosed (f ⁻¹' T)
t : β
hf : Surjective f
hT : IsClosed (f ⁻¹' connectedComponent t)
⊢ IsPreconnected (f ⁻¹' connectedComponent t) | 4b0d93be45591aa1 |
PrimeSpectrum.zeroLocus_empty_of_one_mem | Mathlib/RingTheory/Spectrum/Prime/Basic.lean | theorem zeroLocus_empty_of_one_mem {s : Set R} (h : (1 : R) ∈ s) : zeroLocus s = ∅ | R : Type u
inst✝ : CommSemiring R
s : Set R
h : 1 ∈ s
x : PrimeSpectrum R
hx : s ⊆ ↑x.asIdeal
x_prime : x.asIdeal.IsPrime
⊢ False | have eq_top : x.asIdeal = ⊤ := by
rw [Ideal.eq_top_iff_one]
exact hx h | R : Type u
inst✝ : CommSemiring R
s : Set R
h : 1 ∈ s
x : PrimeSpectrum R
hx : s ⊆ ↑x.asIdeal
x_prime : x.asIdeal.IsPrime
eq_top : x.asIdeal = ⊤
⊢ False | 5067e5311eeaeb23 |
Function.periodicOrbit_chain | Mathlib/Dynamics/PeriodicPts/Defs.lean | theorem periodicOrbit_chain (r : α → α → Prop) {f : α → α} {x : α} :
(periodicOrbit f x).Chain r ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x) | case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x) | have hM := Nat.sub_add_cancel (succ_le_iff.2 hx') | case pos
α : Type u_1
r : α → α → Prop
f : α → α
x : α
hx : x ∈ periodicPts f
hx' : 0 < minimalPeriod f x
hM : minimalPeriod f x - succ 0 + succ 0 = minimalPeriod f x
⊢ Cycle.Chain r (periodicOrbit f x) ↔ ∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x) | 13df9f8318a3fe41 |
SimpleGraph.isBridge_iff_adj_and_forall_cycle_not_mem | Mathlib/Combinatorics/SimpleGraph/Path.lean | theorem isBridge_iff_adj_and_forall_cycle_not_mem {v w : V} : G.IsBridge s(v, w) ↔
G.Adj v w ∧ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges | V : Type u
G : SimpleGraph V
v w : V
⊢ G.Adj v w → (¬(G \ fromEdgeSet {s(v, w)}).Reachable v w ↔ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges) | intro h | V : Type u
G : SimpleGraph V
v w : V
h : G.Adj v w
⊢ ¬(G \ fromEdgeSet {s(v, w)}).Reachable v w ↔ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges | 63b9c269e10026c2 |
monotoneOn_of_pred_le | Mathlib/Order/SuccPred/Archimedean.lean | lemma monotoneOn_of_pred_le (hs : s.OrdConnected)
(hf : ∀ a, ¬ IsMin a → a ∈ s → pred a ∈ s → f (pred a) ≤ f a) : MonotoneOn f s | case pos
α : Type u_3
β : Type u_4
inst✝³ : PartialOrder α
inst✝² : Preorder β
inst✝¹ : PredOrder α
inst✝ : IsPredArchimedean α
s : Set α
f : α → β
hs : s.OrdConnected
hf : ∀ (a : α), ¬IsMin a → a ∈ s → pred a ∈ s → f (pred a) ≤ f a
b : α
hb : b ∈ s
n : ℕ
hn : pred^[n] b ∈ s → f (pred^[n] b) ≤ f b
ha : pred (pred^[n] b... | rw [pred_eq_iff_isMin.2 ha'] | case pos
α : Type u_3
β : Type u_4
inst✝³ : PartialOrder α
inst✝² : Preorder β
inst✝¹ : PredOrder α
inst✝ : IsPredArchimedean α
s : Set α
f : α → β
hs : s.OrdConnected
hf : ∀ (a : α), ¬IsMin a → a ∈ s → pred a ∈ s → f (pred a) ≤ f a
b : α
hb : b ∈ s
n : ℕ
hn : pred^[n] b ∈ s → f (pred^[n] b) ≤ f b
ha : pred (pred^[n] b... | ac8d154143bd872a |
Pi.ωScottContinuous_curry | Mathlib/Control/LawfulFix.lean | theorem ωScottContinuous_curry :
ωScottContinuous (monotoneCurry α β γ) :=
ωScottContinuous.of_map_ωSup_of_orderHom fun c ↦ by
ext x y
dsimp [curry, ωSup]
rw [map_comp, map_comp]
rfl
| case h.h
α : Type u_1
β : α → Type u_2
γ : (a : α) → β a → Type u_3
inst✝ : (x : α) → (y : β x) → OmegaCompletePartialOrder (γ x y)
c : Chain ((x : (a : α) × β a) → γ x.fst x.snd)
x : α
y : β x
⊢ ωSup (c.map (evalOrderHom ⟨x, y⟩)) =
ωSup (c.map (((evalOrderHom y).comp (evalOrderHom x)).comp (monotoneCurry α β γ))) | rfl | no goals | ae498409bc676625 |
MeasureTheory.uniformIntegrable_of' | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem uniformIntegrable_of' [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ i, StronglyMeasurable (f i))
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
UniformIntegrable f p μ | α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofRe... | have : ∀ᵐ x ∂μ, ‖{ x : α | ‖f i x‖₊ < C }.indicator (f i) x‖₊ ≤ C := by
filter_upwards
simp_rw [nnnorm_indicator_eq_indicator_nnnorm]
exact Set.indicator_le fun x (hx : _ < _) => hx.le | α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofRe... | 539fc25109976788 |
Algebra.FinitePresentation.quotient | Mathlib/RingTheory/FinitePresentation.lean | theorem quotient {I : Ideal A} (h : I.FG) [FinitePresentation R A] :
FinitePresentation R (A ⧸ I) where
out | case intro.intro.refine_2
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
I : Ideal A
h : I.FG
inst✝ : FinitePresentation R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
⊢ (RingHom.ker ((Ideal.Quotient.mkₐ R I).comp f).toRingHom).FG | refine Ideal.fg_ker_comp _ _ hf.2 ?_ hf.1 | case intro.intro.refine_2
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
I : Ideal A
h : I.FG
inst✝ : FinitePresentation R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG
⊢ (RingHom.ker (Ideal.Quotient.mkₐ R I).toRingHom).FG | c0eaad945fef2e4b |
integral_withDensity_eq_integral_smul₀ | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem integral_withDensity_eq_integral_smul₀ {f : X → ℝ≥0} (hf : AEMeasurable f μ) (g : X → E) :
∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ | X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
⊢ (∫ (x : X), g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • g x ∂μ | let f' := hf.mk _ | X : Type u_1
E : Type u_3
inst✝² : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → ℝ≥0
hf : AEMeasurable f μ
g : X → E
f' : X → ℝ≥0 := AEMeasurable.mk f hf
⊢ (∫ (x : X), g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • g x ∂μ | 71e9ad39a9402ab5 |
memPartitionSet_eq_iff | Mathlib/Data/Set/MemPartition.lean | lemma memPartitionSet_eq_iff {f : ℕ → Set α} {n : ℕ} (a : α) {s : Set α}
(hs : s ∈ memPartition f n) :
memPartitionSet f n a = s ↔ a ∈ s | α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n
h : a ∈ s
h_ne : ¬memPartitionSet f n a = s
⊢ False | have h_disj : Disjoint s (memPartitionSet f n a) :=
disjoint_memPartition f n hs (memPartitionSet_mem f n a) (Ne.symm h_ne) | α : Type u_1
f : ℕ → Set α
n : ℕ
a : α
s : Set α
hs : s ∈ memPartition f n
h : a ∈ s
h_ne : ¬memPartitionSet f n a = s
h_disj : Disjoint s (memPartitionSet f n a)
⊢ False | b8e5acdb14ce0b5e |
Matroid.map_loopyOn | Mathlib/Data/Matroid/Map.lean | @[simp] lemma map_loopyOn (f : α → β) (hf) : (loopyOn E).map f hf = loopyOn (f '' E) | α : Type u_1
β : Type u_2
E : Set α
f : α → β
hf : InjOn f (loopyOn E).E
⊢ (loopyOn E).map f hf = loopyOn (f '' E) | simp [eq_loopyOn_iff] | no goals | 28ead1e85c03fe4c |
geom_series_mul_neg | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem geom_series_mul_neg (x : R) (h : ‖x‖ < 1) : (∑' i : ℕ, x ^ i) * (1 - x) = 1 | case h.e'_3.h
R : Type u_2
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
this✝ : HasSum (fun i => x ^ i * (1 - x)) ((∑' (b : ℕ), x ^ b) * (1 - x))
this : Tendsto (fun n => 1 - x ^ n) atTop (𝓝 1)
x✝ : ℕ
⊢ 1 - x ^ x✝ = ∑ i ∈ Finset.range x✝, x ^ i * (1 - x) | rw [← geom_sum_mul_neg, Finset.sum_mul] | no goals | acaf2eb12ca8d928 |
SimpleGraph.nonempty_hom_of_forall_finite_subgraph_hom | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | theorem nonempty_hom_of_forall_finite_subgraph_hom [Finite W]
(h : ∀ G' : G.Subgraph, G'.verts.Finite → G'.coe →g F) : Nonempty (G →g F) | case intro.intro.refine_2
V : Type u
W : Type v
G : SimpleGraph V
F : SimpleGraph W
inst✝ : Finite W
h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F
val✝ : Fintype W
this✝ : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')
this : (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F)... | refine Hom.map_adj (u (Opposite.op (finsubgraphOfAdj e))) ?_ | case intro.intro.refine_2
V : Type u
W : Type v
G : SimpleGraph V
F : SimpleGraph W
inst✝ : Finite W
h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F
val✝ : Fintype W
this✝ : ∀ (G' : G.Finsubgraphᵒᵖ), Nonempty ((G.finsubgraphHomFunctor F).obj G')
this : (G' : G.Finsubgraphᵒᵖ) → Fintype ((G.finsubgraphHomFunctor F)... | 136c3c8727823ef9 |
Polynomial.Monic.eq_X_add_C | Mathlib/Algebra/Polynomial/Degree/SmallDegree.lean | theorem Monic.eq_X_add_C (hm : p.Monic) (hnd : p.natDegree = 1) : p = X + C (p.coeff 0) | R : Type u
inst✝ : Semiring R
p : R[X]
hm : p.Monic
hnd : p.natDegree = 1
⊢ p = X + C (p.coeff 0) | rw [← one_mul X, ← C_1, ← hm.coeff_natDegree, hnd, ← eq_X_add_C_of_natDegree_le_one hnd.le] | no goals | d72858360a765589 |
eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open | Mathlib/Topology/Compactness/Compact.lean | lemma eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U) :
∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i | case intro.intro.intro.intro.refine_2.h.intro.intro
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
Y : Type u
f' : Y → ι
e : U = ⋃ i, (b ∘ f') i
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Finset Y
ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i... | rw [e] | case intro.intro.intro.intro.refine_2.h.intro.intro
X : Type u
ι : Type u_1
inst✝ : TopologicalSpace X
b : ι → Set X
hb : IsTopologicalBasis (range b)
U : Set X
hUc : IsCompact U
hUo : IsOpen U
Y : Type u
f' : Y → ι
e : U = ⋃ i, (b ∘ f') i
hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i
t : Finset Y
ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i... | 0a3b3cc3a1e26713 |
String.Iterator.ValidFor.out' | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem out' : ∀ {it}, ValidFor l r it → it = ⟨⟨l.reverse ++ r⟩, ⟨utf8Len l.reverse⟩⟩
| _, ⟨⟩ => by simp [List.reverseAux_eq]
| l r : List Char
⊢ { s := { data := l.reverseAux r }, i := { byteIdx := utf8Len l } } =
{ s := { data := l.reverse ++ r }, i := { byteIdx := utf8Len l.reverse } } | simp [List.reverseAux_eq] | no goals | 017de1ead61f4cbb |
Finset.sum_const_nat | Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean | theorem sum_const_nat {m : ℕ} {f : α → ℕ} (h₁ : ∀ x ∈ s, f x = m) : ∑ x ∈ s, f x = #s * m | α : Type u_3
s : Finset α
m : ℕ
f : α → ℕ
h₁ : ∀ x ∈ s, f x = m
⊢ ∑ x ∈ s, f x = #s * m | rw [← Nat.nsmul_eq_mul, ← sum_const] | α : Type u_3
s : Finset α
m : ℕ
f : α → ℕ
h₁ : ∀ x ∈ s, f x = m
⊢ ∑ x ∈ s, f x = ∑ _x ∈ s, m | 2c4e67c2e38a06a2 |
Bornology.isBounded_empty | Mathlib/Topology/Bornology/Basic.lean | theorem isBounded_empty : IsBounded (∅ : Set α) | α : Type u_2
x✝ : Bornology α
⊢ univ ∈ cobounded α | exact univ_mem | no goals | f66fdda22727a5d7 |
pos_nilpotencyClass_iff | Mathlib/RingTheory/Nilpotent/Defs.lean | @[simp] lemma pos_nilpotencyClass_iff [Nontrivial R] :
0 < nilpotencyClass x ↔ IsNilpotent x | R : Type u_1
x : R
inst✝¹ : MonoidWithZero R
inst✝ : Nontrivial R
hx' : nilpotencyClass x = 0
hx : x ^ nilpotencyClass x = 0
⊢ False | rw [hx', pow_zero] at hx | R : Type u_1
x : R
inst✝¹ : MonoidWithZero R
inst✝ : Nontrivial R
hx' : nilpotencyClass x = 0
hx : 1 = 0
⊢ False | 70ecea08e0cd3db0 |
DiffContOnCl.ball_subset_image_closedBall | Mathlib/Analysis/Complex/OpenMapping.lean | theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
(hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r | case intro.intro
f : ℂ → ℂ
z₀ : ℂ
ε r : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : Analyti... | have h10 : f z = f z₀ := (h9 (mem_ball_self hr)).symm | case intro.intro
f : ℂ → ℂ
z₀ : ℂ
ε r : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : Analyti... | 2479e7c39939bc28 |
integrable_one_add_norm | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | theorem integrable_one_add_norm {r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) :
Integrable (fun x ↦ (1 + ‖x‖) ^ (-r)) μ | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
x✝ : E
⊢ 0 ≤ 1 + ‖x✝‖ | positivity | no goals | 5716a99ff42f9848 |
Turing.TM2to1.tr_respects | Mathlib/Computability/TuringMachine.lean | theorem tr_respects : Respects (TM2.step M) (TM1.step (tr M)) TrCfg | case H₃
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
l : Λ
p : σ → Bool
q₁ q₂ : TM2.Stmt Γ Λ σ
IH₁ :
∀ {v : σ} {S : (k : K) → List (Γ k)} (L : ListBlank ((k : K) → Option (Γ k))),
(∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reve... | cases p v <;> [exact IH₂ _ hT; exact IH₁ _ hT] | no goals | d13c9b9233bfce82 |
Ideal.IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count | Mathlib/NumberTheory/RamificationInertia/Basic.lean | theorem ramificationIdx_eq_normalizedFactors_count [DecidableEq (Ideal S)]
(hp0 : map f p ≠ ⊥) (hP : P.IsPrime)
(hP0 : P ≠ ⊥) : ramificationIdx f p P = (normalizedFactors (map f p)).count P | R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
inst✝¹ : IsDedekindDomain S
inst✝ : DecidableEq (Ideal S)
hp0 : map f p ≠ ⊥
hP : P.IsPrime
hP0 : P ≠ ⊥
⊢ ramificationIdx f p P = Multiset.count P (normalizedFactors (map f p)) | have hPirr := (Ideal.prime_of_isPrime hP0 hP).irreducible | R : Type u
inst✝³ : CommRing R
S : Type v
inst✝² : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
inst✝¹ : IsDedekindDomain S
inst✝ : DecidableEq (Ideal S)
hp0 : map f p ≠ ⊥
hP : P.IsPrime
hP0 : P ≠ ⊥
hPirr : Irreducible P
⊢ ramificationIdx f p P = Multiset.count P (normalizedFactors (map f p)) | 0d0eb4eb10dd10be |
isPurelyInseparable_iff_fd_isPurelyInseparable | Mathlib/FieldTheory/PurelyInseparable/Basic.lean | theorem isPurelyInseparable_iff_fd_isPurelyInseparable [Algebra.IsAlgebraic F E] :
IsPurelyInseparable F E ↔
∀ L : IntermediateField F E, FiniteDimensional F L → IsPurelyInseparable F L | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
⊢ IsPurelyInseparable F E ↔ ∀ (L : IntermediateField F E), FiniteDimensional F ↥L → IsPurelyInseparable F ↥L | refine ⟨fun _ _ _ ↦ IsPurelyInseparable.tower_bot F _ E,
fun h ↦ isPurelyInseparable_iff.2 fun x ↦ ?_⟩ | F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsAlgebraic F E
h : ∀ (L : IntermediateField F E), FiniteDimensional F ↥L → IsPurelyInseparable F ↥L
x : E
⊢ IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range) | c61123c2356ba443 |
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