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
|
|---|---|---|---|---|---|---|
mul_left_eq_self₀
|
Mathlib/Algebra/GroupWithZero/Basic.lean
|
theorem mul_left_eq_self₀ : a * b = b ↔ a = 1 ∨ b = 0 :=
calc
a * b = b ↔ a * b = 1 * b
|
M₀ : Type u_1
inst✝ : CancelMonoidWithZero M₀
a b : M₀
⊢ a * b = b ↔ a * b = 1 * b
|
rw [one_mul]
|
no goals
|
1cb8410ec0539713
|
not_uncountable_iff
|
Mathlib/Data/Countable/Defs.lean
|
lemma not_uncountable_iff : ¬Uncountable α ↔ Countable α
|
α : Sort u
⊢ ¬Uncountable α ↔ Countable α
|
rw [uncountable_iff_not_countable, not_not]
|
no goals
|
00dd98f7f7556bb5
|
intervalIntegral.continuousOn_primitive_interval_left
|
Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
|
theorem continuousOn_primitive_interval_left (h_int : IntegrableOn f (uIcc a b) μ) :
ContinuousOn (fun x => ∫ t in x..b, f t ∂μ) (uIcc a b)
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NoAtoms μ
h_int : IntegrableOn f [[b, a]] μ
⊢ ContinuousOn (fun x => ∫ (t : ℝ) in x..b, f t ∂μ) [[b, a]]
|
simp only [integral_symm b]
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
μ : Measure ℝ
f : ℝ → E
inst✝ : NoAtoms μ
h_int : IntegrableOn f [[b, a]] μ
⊢ ContinuousOn (fun x => -∫ (t : ℝ) in b..x, f t ∂μ) [[b, a]]
|
f2a5f8820cfc225b
|
NumberField.mixedEmbedding.normAtPlace_real
|
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
|
theorem normAtPlace_real (w : InfinitePlace K) (c : ℝ) :
normAtPlace w ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = |c|
|
K : Type u_1
inst✝ : Field K
w : InfinitePlace K
c : ℝ
⊢ (normAtPlace w) (fun x => c, fun x => ↑c) = |c|
|
rw [show ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = c • 1 by ext <;> simp, normAtPlace_smul,
map_one, mul_one]
|
no goals
|
142af2ef6d8da7ec
|
Irrational.eventually_forall_le_dist_cast_div
|
Mathlib/Topology/Instances/Irrational.lean
|
theorem eventually_forall_le_dist_cast_div (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ m : ℤ, ε ≤ dist x (m / n)
|
case intro.intro
x : ℝ
hx : Irrational x
n : ℕ
A : IsClosed (range fun m => (↑n)⁻¹ * ↑m)
B : x ∉ range fun m => (↑n)⁻¹ * ↑m
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ
δ : ℝ
hδ : δ ≤ ε
m : ℤ
hlt : dist (↑m / ↑n) x < δ
⊢ False
|
refine hε (ball_subset_ball hδ hlt) ⟨m, ?_⟩
|
case intro.intro
x : ℝ
hx : Irrational x
n : ℕ
A : IsClosed (range fun m => (↑n)⁻¹ * ↑m)
B : x ∉ range fun m => (↑n)⁻¹ * ↑m
ε : ℝ
ε0 : ε > 0
hε : ball x ε ⊆ (range fun m => (↑n)⁻¹ * ↑m)ᶜ
δ : ℝ
hδ : δ ≤ ε
m : ℤ
hlt : dist (↑m / ↑n) x < δ
⊢ (fun m => (↑n)⁻¹ * ↑m) m = ↑m / ↑n
|
be2d5b844aa1fb4f
|
SimpleGraph.chromaticNumber_bot
|
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
theorem chromaticNumber_bot [Nonempty V] : (⊥ : SimpleGraph V).chromaticNumber = 1
|
V : Type u
inst✝ : Nonempty V
⊢ ⊥.chromaticNumber = 1
|
have : (⊥ : SimpleGraph V).Colorable 1 := ⟨.mk 0 <| by simp⟩
|
V : Type u
inst✝ : Nonempty V
this : ⊥.Colorable 1
⊢ ⊥.chromaticNumber = 1
|
86642397865942bd
|
IsIntegrallyClosed.eq_map_mul_C_of_dvd
|
Mathlib/RingTheory/Polynomial/GaussLemma.lean
|
theorem IsIntegrallyClosed.eq_map_mul_C_of_dvd [IsIntegrallyClosed R] {f : R[X]} (hf : f.Monic)
{g : K[X]} (hg : g ∣ f.map (algebraMap R K)) :
∃ g' : R[X], g'.map (algebraMap R K) * (C <| leadingCoeff g) = g
|
case h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
f : R[X]
hf : f.Monic
g : K[X]
hg : g ∣ map (algebraMap R K) f
g_ne_0 : g ≠ 0
g' : R[X]
hg' : map (algebraMap R K) g' = g * C g.leadingCoeff⁻¹
⊢ map (algebraMap R K) g' * C g.leadingCoeff = g
|
rw [hg', mul_assoc, ← C_mul, inv_mul_cancel₀ (leadingCoeff_ne_zero.mpr g_ne_0), C_1, mul_one]
|
no goals
|
918c3567824cf666
|
Batteries.UnionFind.lt_rankD_findAux
|
Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean
|
theorem lt_rankD_findAux {self : UnionFind} {x : Fin self.size} :
parentD (findAux self x).s i ≠ i →
self.rank i < self.rank (parentD (findAux self x).s i)
|
case isFalse
i : Nat
self : UnionFind
x : Fin self.size
h'✝ : ¬self.arr[↑x].parent = ↑x
h : ¬i = ↑x
h' : parentD (self.findAux ⟨self.arr[↑x].parent, ⋯⟩).s i ≠ i
this : self.rankMax - self.rank self.arr[↑x].parent < self.rankMax - self.rank ↑x
⊢ self.rank i < self.rank (parentD (self.findAux ⟨self.arr[↑x].parent, ⋯⟩).s i)
|
apply lt_rankD_findAux h'
|
no goals
|
bf3344308c344af2
|
List.dropInfix?_go_eq_some_iff
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem dropInfix?_go_eq_some_iff [BEq α] {i l acc p s : List α} :
dropInfix?.go i l acc = some (p, s) ↔ ∃ p',
p = acc.reverse ++ p' ∧
-- `i` is an infix up to `==`
(∃ i', l = p' ++ i' ++ s ∧ i' == i) ∧
-- and there is no shorter prefix for which that is the case
(∀ p'' i'' s'', l = p'' ++ i'' ++ s'' → i'' == i → p''.length ≥ p'.length)
|
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inr.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
i' : List α
h₂ : (i' == i) = true
p' : List α
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w :
∀ (p'' i'' s'' : List α), a :: (p' ++ i' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ (a :: p').length
p'' i'' s'' : List α
h₃ : p' ++ (i' ++ s) = p'' ++ (i'' ++ s'')
h₄ : (i'' == i) = true
⊢ p'.length ≤ p''.length
|
rw [← append_assoc] at h₃
|
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inr.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
i' : List α
h₂ : (i' == i) = true
p' : List α
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w :
∀ (p'' i'' s'' : List α), a :: (p' ++ i' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ (a :: p').length
p'' i'' s'' : List α
h₃ : p' ++ i' ++ s = p'' ++ (i'' ++ s'')
h₄ : (i'' == i) = true
⊢ p'.length ≤ p''.length
|
2a30116d5cc9db54
|
Finset.mulEnergy_empty_left
|
Mathlib/Combinatorics/Additive/Energy.lean
|
@[to_additive (attr := simp)] lemma mulEnergy_empty_left : Eₘ[∅, t] = 0
|
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Mul α
t : Finset α
⊢ Eₘ[∅, t] = 0
|
simp [mulEnergy]
|
no goals
|
68ed322687dae6da
|
interior_Ici'
|
Mathlib/Topology/Order/DenselyOrdered.lean
|
theorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a
|
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a : α
ha : (Iio a).Nonempty
⊢ interior (Ici a) = Ioi a
|
rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]
|
no goals
|
67c68fb7a49efdf1
|
Unitization.starMap_inr
|
Mathlib/Algebra/Algebra/Unitization.lean
|
@[simp high]
lemma starMap_inr (φ : A →⋆ₙₐ[R] B) (a : A) :
starMap φ (inr a) = inr (φ a)
|
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹² : CommSemiring R
inst✝¹¹ : StarRing R
inst✝¹⁰ : NonUnitalSemiring A
inst✝⁹ : StarRing A
inst✝⁸ : Module R A
inst✝⁷ : SMulCommClass R A A
inst✝⁶ : IsScalarTower R A A
inst✝⁵ : NonUnitalSemiring B
inst✝⁴ : StarRing B
inst✝³ : Module R B
inst✝² : SMulCommClass R B B
inst✝¹ : IsScalarTower R B B
inst✝ : StarModule R B
φ : A →⋆ₙₐ[R] B
a : A
⊢ (starMap φ) ↑a = ↑(φ a)
|
simp
|
no goals
|
7b776f9ea6f62df4
|
CoxeterSystem.prod_alternatingWord_eq_prod_alternatingWord_sub
|
Mathlib/GroupTheory/Coxeter/Basic.lean
|
theorem prod_alternatingWord_eq_prod_alternatingWord_sub (i i' : B) (m : ℕ) (hm : m ≤ M i i' * 2) :
π (alternatingWord i i' m) = π (alternatingWord i' i (M i i' * 2 - m))
|
case intro.inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (cs.simple i' * cs.simple i) ^ (↑(M.M i i') - k)
|
rw [zpow_sub, zpow_natCast, simple_mul_simple_pow' cs i i', ← inv_zpow]
|
case intro.inl
B : Type u_1
W : Type u_3
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
i i' : B
m : ℕ
k : ℤ
⊢ 1 * (cs.simple i * cs.simple i') ^ k = 1 * (1 * (cs.simple i' * cs.simple i)⁻¹ ^ k)
|
8b276f2d43ad82bd
|
ZMod.valMinAbs_mul_two_eq_iff
|
Mathlib/Data/ZMod/ValMinAbs.lean
|
lemma valMinAbs_mul_two_eq_iff (a : ZMod n) : a.valMinAbs * 2 = n ↔ 2 * a.val = n
|
n : ℕ
a : ZMod (n + 1)
h : ¬a.val ≤ n.succ / 2
⊢ ¬a.valMinAbs * 2 = ↑(n + 1)
|
intro he
|
n : ℕ
a : ZMod (n + 1)
h : ¬a.val ≤ n.succ / 2
he : a.valMinAbs * 2 = ↑(n + 1)
⊢ False
|
23849678f97214de
|
geometric_hahn_banach_open
|
Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean
|
theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
(disj : Disjoint s t) : ∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b
|
case inr.intro.inr.intro
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝¹ : 0 ∈ C
this✝ : Convex ℝ C
this : x₀ ∉ C
⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b
|
obtain ⟨f, hf₁, hf₂⟩ := separate_convex_open_set ‹0 ∈ C› ‹_› (hs₂.sub_right.vadd _) ‹x₀ ∉ C›
|
case inr.intro.inr.intro.intro.intro
E : Type u_2
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s t : Set E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousSMul ℝ E
hs₁ : Convex ℝ s
hs₂ : IsOpen s
ht : Convex ℝ t
disj : Disjoint s t
a₀ : E
ha₀ : a₀ ∈ s
b₀ : E
hb₀ : b₀ ∈ t
x₀ : E := b₀ - a₀
C : Set E := x₀ +ᵥ s - t
this✝¹ : 0 ∈ C
this✝ : Convex ℝ C
this : x₀ ∉ C
f : E →L[ℝ] ℝ
hf₁ : f x₀ = 1
hf₂ : ∀ x ∈ C, f x < 1
⊢ ∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b
|
2f61733fca4a9fa5
|
List.prev_getElem
|
Mathlib/Data/List/Cycle.lean
|
theorem prev_getElem (l : List α) (h : Nodup l) (i : Nat) (hi : i < l.length) :
prev l l[i] (get_mem _ _) =
(l[(i + (l.length - 1)) % l.length]'(Nat.mod_lt _ (by omega))) :=
match l with
| [] => by simp at hi
| x::l => by
induction l generalizing i x with
| nil => simp
| cons y l hl =>
rcases i with (_ | _ | i)
· simp [getLast_eq_getElem]
· simp only [mem_cons, nodup_cons] at h
push_neg at h
simp only [zero_add, getElem_cons_succ, getElem_cons_zero,
List.prev_cons_cons_of_ne _ _ _ _ h.left.left.symm, length, add_comm,
Nat.add_sub_cancel_left, Nat.mod_self]
· rw [prev_ne_cons_cons]
· convert hl i.succ y h.of_cons (Nat.le_of_succ_le_succ hi) using 1
have : ∀ k hk, (y :: l)[k] = (x :: y :: l)[k + 1]'(Nat.succ_lt_succ hk)
|
case h.e'_3
α : Type u_1
inst✝ : DecidableEq α
l✝ : List α
y : α
l : List α
hl :
∀ (i : ℕ) (x : α),
(x :: l).Nodup →
∀ (hi : i < (x :: l).length),
(x :: l).prev (x :: l)[i] ⋯ = (x :: l)[(i + ((x :: l).length - 1)) % (x :: l).length]
x : α
h : (x :: y :: l).Nodup
i : ℕ
hi : i + 1 + 1 < (x :: y :: l).length
this : ∀ (k : ℕ) (hk : k < (y :: l).length), (y :: l)[k] = (x :: y :: l)[k + 1]
⊢ (x :: y :: l)[(i + 1 + 1 + ((x :: y :: l).length - 1)) % (x :: y :: l).length] =
(x :: y :: l)[(i.succ + ((y :: l).length - 1)) % (y :: l).length + 1]
|
congr
|
case h.e'_3.e_i
α : Type u_1
inst✝ : DecidableEq α
l✝ : List α
y : α
l : List α
hl :
∀ (i : ℕ) (x : α),
(x :: l).Nodup →
∀ (hi : i < (x :: l).length),
(x :: l).prev (x :: l)[i] ⋯ = (x :: l)[(i + ((x :: l).length - 1)) % (x :: l).length]
x : α
h : (x :: y :: l).Nodup
i : ℕ
hi : i + 1 + 1 < (x :: y :: l).length
this : ∀ (k : ℕ) (hk : k < (y :: l).length), (y :: l)[k] = (x :: y :: l)[k + 1]
⊢ (i + 1 + 1 + ((x :: y :: l).length - 1)) % (x :: y :: l).length =
(i.succ + ((y :: l).length - 1)) % (y :: l).length + 1
|
9ab0f8d6d9da20dc
|
Real.inv_sign
|
Mathlib/Data/Real/Sign.lean
|
theorem inv_sign (r : ℝ) : (sign r)⁻¹ = sign r
|
case inr.inr
r : ℝ
hp : r.sign = 1
⊢ r.sign⁻¹ = r.sign
|
rw [hp]
|
case inr.inr
r : ℝ
hp : r.sign = 1
⊢ 1⁻¹ = 1
|
be6545dd2dc24fd8
|
List.mapFinIdx_eq_mapFinIdx_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MapIdx.lean
|
theorem mapFinIdx_eq_mapFinIdx_iff {l : List α} {f g : (i : Nat) → α → (h : i < l.length) → β} :
l.mapFinIdx f = l.mapFinIdx g ↔ ∀ (i : Nat) (h : i < l.length), f i l[i] h = g i l[i] h
|
α : Type u_1
β : Type u_2
l : List α
f g : (i : Nat) → α → i < l.length → β
⊢ (∃ h, ∀ (i : Nat) (h_1 : i < l.length), (l.mapFinIdx f)[i] = g i l[i] h_1) ↔
∀ (i : Nat) (h : i < l.length), f i l[i] h = g i l[i] h
|
simp [Fin.forall_iff]
|
no goals
|
729dedc47897c4be
|
MeasureTheory.pdf_of_not_aemeasurable
|
Mathlib/Probability/Density.lean
|
theorem pdf_of_not_aemeasurable {_ : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
{X : Ω → E} (hX : ¬AEMeasurable X ℙ) : pdf X ℙ μ =ᵐ[μ] 0
|
Ω : Type u_1
E : Type u_2
inst✝ : MeasurableSpace E
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
μ : Measure E
X : Ω → E
hX : ¬AEMeasurable X ℙ
⊢ rnDeriv 0 μ =ᶠ[ae μ] 0
|
exact rnDeriv_zero μ
|
no goals
|
126a8f10dcaa5b27
|
Finset.small_pos_neg_pos_mul
|
Mathlib/Combinatorics/Additive/SmallTripling.lean
|
@[to_additive]
private lemma small_pos_neg_pos_mul (hA : #(A ^ 3) ≤ K * #A) : #(A * A⁻¹ * A) ≤ K ^ 3 * #A
|
G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
K : ℝ
hA : ↑(#(A ^ 3)) ≤ K * ↑(#A)
hA₀ : A.Nonempty
this : 0 ≤ K
⊢ ↑(#(A * (A * A⁻¹))) * ↑(#(A * A)) = ↑(#(A * A * A⁻¹)) * ↑(#(A ^ 2))
|
simp [pow_succ, mul_assoc]
|
no goals
|
3191000204fbb864
|
HomologicalComplex.cylinder.πCompι₀Homotopy.inrX_nullHomotopy_f
|
Mathlib/Algebra/Homology/HomotopyCofiber.lean
|
lemma inrX_nullHomotopy_f (j : ι) :
inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 _).f j
|
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
⊢ biprod.lift (𝟙 K) (-𝟙 K) ≫ biprod.snd = (biprod.inl - biprod.inr) ≫ biprod.snd
|
simp
|
no goals
|
5ad448f88d86edef
|
WeierstrassCurve.Affine.degree_polynomial
|
Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
|
@[simp]
lemma degree_polynomial [Nontrivial R] : W'.polynomial.degree = 2
|
R : Type r
inst✝¹ : CommRing R
W' : Affine R
inst✝ : Nontrivial R
⊢ { a := 0, b := 1, c := { a := 0, b := 0, c := W'.a₁, d := W'.a₃ }.toPoly,
d := { a := -1, b := -W'.a₂, c := -W'.a₄, d := -W'.a₆ }.toPoly }.toPoly.degree =
2
|
exact Cubic.degree_of_b_ne_zero' one_ne_zero
|
no goals
|
1caef35f1804e142
|
CategoryTheory.Limits.IsZero.hasProjectiveDimensionLT_zero
|
Mathlib/CategoryTheory/Abelian/Projective/Dimension.lean
|
lemma Limits.IsZero.hasProjectiveDimensionLT_zero (hX : IsZero X) :
HasProjectiveDimensionLT X 0
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
hX : IsZero X
this : HasExt C := HasExt.standard C
⊢ HasProjectiveDimensionLT X 0
|
rw [hasProjectiveDimensionLT_iff]
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
hX : IsZero X
this : HasExt C := HasExt.standard C
⊢ ∀ (i : ℕ), 0 ≤ i → ∀ ⦃Y : C⦄ (e : Ext X Y i), e = 0
|
f0254761bc9fcaab
|
continuousOn_update_iff
|
Mathlib/Topology/Separation/Basic.lean
|
theorem continuousOn_update_iff [T1Space X] [DecidableEq X] [TopologicalSpace Y] {f : X → Y}
{s : Set X} {x : X} {y : Y} :
ContinuousOn (Function.update f x y) s ↔
ContinuousOn f (s \ {x}) ∧ (x ∈ s → Tendsto f (𝓝[s \ {x}] x) (𝓝 y))
|
case refine_2
X : Type u_1
Y : Type u_2
inst✝³ : TopologicalSpace X
inst✝² : T1Space X
inst✝¹ : DecidableEq X
inst✝ : TopologicalSpace Y
f : X → Y
s : Set X
x : X
y : Y
H : ContinuousOn f (s \ {x})
z : X
hzx : z ≠ x
hzs : z ∈ s
⊢ (fun a => a ∈ {x} → False) ∈ 𝓝 z
|
exact isOpen_ne.mem_nhds hzx
|
no goals
|
a296cad250456eac
|
MeasureTheory.integral_fin_nat_prod_eq_prod
|
Mathlib/MeasureTheory/Integral/Pi.lean
|
theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) → E i → 𝕜) :
∫ x : (i : Fin n) → E i, ∏ i, f i (x i) = ∏ i, ∫ x, f i x
|
case zero
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Fin 0 → Type u_2
inst✝¹ : (i : Fin 0) → MeasureSpace (E i)
inst✝ : ∀ (i : Fin 0), SigmaFinite volume
f : (i : Fin 0) → E i → 𝕜
⊢ ∫ (x : (i : Fin 0) → E i), ∏ i : Fin 0, f i (x i) = ∏ i : Fin 0, ∫ (x : E i), f i x
|
simp only [volume_pi, Finset.univ_eq_empty, Finset.prod_empty, integral_const,
pi_empty_univ, ENNReal.one_toReal, smul_eq_mul, mul_one, pow_zero, one_smul]
|
no goals
|
657f0fd4de2ed11e
|
Fin.dfoldrM_loop
|
Mathlib/.lake/packages/batteries/Batteries/Data/Fin/Fold.lean
|
theorem dfoldrM_loop [Monad m] [LawfulMonad m] (f : (i : Fin (n+1)) → α i.succ → m (α i.castSucc))
(x) : dfoldrM.loop (n+1) α f (i+1) h x =
dfoldrM.loop n (α ∘ succ) (f ·.succ) i (by omega) x >>= f 0
|
case zero
m : Type u_1 → Type u_2
n : Nat
α : Fin (n + 1 + 1) → Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : (i : Fin (n + 1)) → α i.succ → m (α i.castSucc)
h : 0 + 1 < n + 1 + 1
x : α ⟨0 + 1, h⟩
⊢ f ⟨0, ⋯⟩ x >>= dfoldrM.loop (n + 1) α f 0 ⋯ = f 0 x >>= pure
|
rfl
|
no goals
|
b74a59ce684ddcb3
|
MeasureTheory.Integrable.smul_measure
|
Mathlib/MeasureTheory/Function/L1Space/Integrable.lean
|
theorem Integrable.smul_measure {f : α → β} (h : Integrable f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) :
Integrable f (c • μ)
|
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
h : MemLp f 1 μ
c : ℝ≥0∞
hc : c ≠ ⊤
⊢ MemLp f 1 (c • μ)
|
exact h.smul_measure hc
|
no goals
|
c9a6952834a810b3
|
intervalIntegral.measure_integral_sub_linear_isLittleO_of_tendsto_ae'
|
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae' [IsMeasurablyGenerated l']
[TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ)
(hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l)
(hv : Tendsto v lt l) :
(fun t => (∫ x in u t..v t, f x ∂μ) - ∫ _ in u t..v t, c ∂μ) =o[lt] fun t =>
∫ _ in u t..v t, (1 : ℝ) ∂μ
|
case neg
ι : Type u_1
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
f : ℝ → E
c : E
l l' : Filter ℝ
lt : Filter ι
μ : Measure ℝ
u v : ι → ℝ
inst✝¹ : l'.IsMeasurablyGenerated
inst✝ : TendstoIxxClass Ioc l l'
hfm : StronglyMeasurableAtFilter f l' μ
hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)
hl : μ.FiniteAtFilter l'
hu : Tendsto u lt l
hv : Tendsto v lt l
hE : ¬CompleteSpace E
⊢ (fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ
|
simp [intervalIntegral, integral, hE]
|
no goals
|
890770c8df7add59
|
DihedralGroup.orderOf_sr
|
Mathlib/GroupTheory/SpecificGroups/Dihedral.lean
|
theorem orderOf_sr (i : ZMod n) : orderOf (sr i) = 2
|
case hg1
n : ℕ
i : ZMod n
⊢ sr i ≠ 1
|
simp [← r_zero]
|
no goals
|
782d4d28e2598f6d
|
isUnit_zero_iff
|
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
theorem isUnit_zero_iff : IsUnit (0 : M₀) ↔ (0 : M₀) = 1 :=
⟨fun ⟨⟨_, a, (a0 : 0 * a = 1), _⟩, rfl⟩ => by rwa [zero_mul] at a0, fun h =>
@isUnit_of_subsingleton _ _ (subsingleton_of_zero_eq_one h) 0⟩
|
M₀ : Type u_2
inst✝ : MonoidWithZero M₀
x✝ : IsUnit 0
a : M₀
a0 : 0 * a = 1
inv_val✝ : a * 0 = 1
⊢ 0 = 1
|
rwa [zero_mul] at a0
|
no goals
|
698e6d12376fa5ba
|
List.getLast_filter_of_pos
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem getLast_filter_of_pos {p : α → Bool} {l : List α} (w : l ≠ []) (h : p (getLast l w) = true) :
getLast (filter p l) (ne_nil_of_mem (mem_filter.2 ⟨getLast_mem w, h⟩)) = getLast l w
|
case h
α : Type u_1
p : α → Bool
l : List α
w : l ≠ []
h : p (l.getLast w) = true
⊢ p (l.reverse.head ⋯) = true
|
simp_all
|
no goals
|
fff70265a48e3d7c
|
Composition.sizeUpTo_sizeUpTo_add
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem sizeUpTo_sizeUpTo_add (a : Composition n) (b : Composition a.length) {i j : ℕ}
(hi : i < b.length) (hj : j < blocksFun b ⟨i, hi⟩) :
sizeUpTo a (sizeUpTo b i + j) =
sizeUpTo (a.gather b) i +
sizeUpTo (sigmaCompositionAux a b ⟨i, (length_gather a b).symm ▸ hi⟩) j
|
case succ
n : ℕ
a : Composition n
b : Composition a.length
i : ℕ
hi : i < b.length
j : ℕ
IHj :
j < b.blocksFun ⟨i, hi⟩ →
a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
hj : j + 1 < b.blocksFun ⟨i, hi⟩
A : j < b.blocksFun ⟨i, hi⟩
B : j < (a.sigmaCompositionAux b ⟨i, ⋯⟩).length
C : b.sizeUpTo i + j < b.sizeUpTo (i + 1)
D : b.sizeUpTo i + j < a.length
this : b.sizeUpTo i + j.succ = (b.sizeUpTo i + j).succ
⊢ (a.gather b).sizeUpTo i +
({ blocks := (a.blocks.splitWrtComposition b)[i], blocks_pos := ⋯, blocks_sum := ⋯ }.sizeUpTo j +
a.blocks[b.sizeUpTo i + j]) =
(a.gather b).sizeUpTo i +
({ blocks := (a.blocks.splitWrtComposition b)[i], blocks_pos := ⋯, blocks_sum := ⋯ }.sizeUpTo j +
(a.blocks.splitWrtComposition b)[i][j])
|
rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop, getElem_take' _ _ C]
|
no goals
|
43568ecc50088850
|
isClosed_Ioo_iff
|
Mathlib/Topology/Order/DenselyOrdered.lean
|
theorem isClosed_Ioo_iff {a b : α} : IsClosed (Set.Ioo a b) ↔ b ≤ a
|
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ IsClosed (Ioo a b) ↔ b ≤ a
|
refine ⟨fun h => le_of_not_lt fun hab => ?_, by simp_all⟩
|
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
h : IsClosed (Ioo a b)
hab : a < b
⊢ False
|
85aac5ac9cc53185
|
Orientation.kahler_mul
|
Mathlib/Analysis/InnerProductSpace/TwoDim.lean
|
theorem kahler_mul (a x y : E) : o.kahler x a * o.kahler a y = ‖a‖ ^ 2 * o.kahler x y
|
case a
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)
a x y : E
⊢ ((o.kahler x) a * (o.kahler a) y).re = (↑(‖a‖ ^ 2) * (o.kahler x) y).re
|
simp only [o.kahler_apply_apply, Complex.add_im, Complex.add_re, Complex.I_im, Complex.I_re,
Complex.mul_im, Complex.mul_re, Complex.ofReal_im, Complex.ofReal_re, Complex.real_smul]
|
case a
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)
a x y : E
⊢ (inner x a + ((o.areaForm x) a * 0 - 0 * 1)) * (inner a y + ((o.areaForm a) y * 0 - 0 * 1)) -
(0 + ((o.areaForm x) a * 1 + 0 * 0)) * (0 + ((o.areaForm a) y * 1 + 0 * 0)) =
‖a‖ ^ 2 * (inner x y + ((o.areaForm x) y * 0 - 0 * 1)) - 0 * (0 + ((o.areaForm x) y * 1 + 0 * 0))
|
579eb46c30f78600
|
toIocMod_add_toIocDiv_zsmul
|
Mathlib/Algebra/Order/ToIntervalMod.lean
|
theorem toIocMod_add_toIocDiv_zsmul (a b : α) : toIocMod hp a b + toIocDiv hp a b • p = b
|
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
⊢ toIocMod hp a b + toIocDiv hp a b • p = b
|
rw [toIocMod, sub_add_cancel]
|
no goals
|
5bccd736bcb7957f
|
CategoryTheory.Retract.hasProjectiveDimensionLT
|
Mathlib/CategoryTheory/Abelian/Projective/Dimension.lean
|
lemma Retract.hasProjectiveDimensionLT {X Y : C} (h : Retract X Y) (n : ℕ)
[HasProjectiveDimensionLT Y n] :
HasProjectiveDimensionLT X n
|
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
X Y : C
h : Retract X Y
n : ℕ
inst✝ : HasProjectiveDimensionLT Y n
this : HasExt C := HasExt.standard C
i : ℕ
hi : n ≤ i
T : C
x : Ext X T i
⊢ x = 0
|
rw [← x.mk₀_id_comp, ← h.retract, ← Ext.mk₀_comp_mk₀,
Ext.comp_assoc_of_second_deg_zero,
((Ext.mk₀ h.r).comp x (zero_add i)).eq_zero_of_hasProjectiveDimensionLT n hi,
Ext.comp_zero]
|
no goals
|
985db82b65314f96
|
Orientation.inner_mul_areaForm_sub'
|
Mathlib/Analysis/InnerProductSpace/TwoDim.lean
|
theorem inner_mul_areaForm_sub' (a x : E) : ⟪a, x⟫ • ω a - ω a x • innerₛₗ ℝ a = ‖a‖ ^ 2 • ω x
|
case pos
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)
a x : E
ha : a = 0
⊢ inner a x • o.areaForm a - (o.areaForm a) x • (innerₛₗ ℝ) a = ‖a‖ ^ 2 • o.areaForm x
|
simp [ha]
|
no goals
|
4e6e1b3ea7eb9571
|
Ideal.span_singleton_absNorm
|
Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean
|
theorem span_singleton_absNorm {I : Ideal S} (hI : (Ideal.absNorm I).Prime) :
Ideal.span (singleton (Ideal.absNorm I : ℤ)) = I.comap (algebraMap ℤ S)
|
S : Type u_1
inst✝³ : CommRing S
inst✝² : Nontrivial S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
hI : Nat.Prime (absNorm I)
this : (span {↑(absNorm I)}).IsPrime
⊢ span {↑(absNorm I)} ≠ ⊥
|
rw [Ne, span_singleton_eq_bot]
|
S : Type u_1
inst✝³ : CommRing S
inst✝² : Nontrivial S
inst✝¹ : IsDedekindDomain S
inst✝ : Module.Free ℤ S
I : Ideal S
hI : Nat.Prime (absNorm I)
this : (span {↑(absNorm I)}).IsPrime
⊢ ¬↑(absNorm I) = 0
|
09c2b8399dcaa577
|
bsupr_limsup_dimH
|
Mathlib/Topology/MetricSpace/HausdorffDimension.lean
|
theorem bsupr_limsup_dimH (s : Set X) : ⨆ x ∈ s, limsup dimH (𝓝[s] x).smallSets = dimH s
|
case refine_1
X : Type u_2
inst✝¹ : EMetricSpace X
inst✝ : SecondCountableTopology X
s : Set X
x : X
x✝ : x ∈ s
⊢ ∀ᶠ (n : Set X) in (𝓝[s] x).smallSets, dimH n ≤ dimH s
|
exact eventually_smallSets.2 ⟨s, self_mem_nhdsWithin, fun t => dimH_mono⟩
|
no goals
|
ce953fa85a665a27
|
Complex.one_sub_prime_cpow_ne_zero
|
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
lemma one_sub_prime_cpow_ne_zero {p : ℕ} (hp : p.Prime) {s : ℂ} (hs : 1 < s.re) :
1 - (p : ℂ) ^ (-s) ≠ 0
|
p : ℕ
hp : Nat.Prime p
s : ℂ
hs : 1 < s.re
H : 1 = ↑p ^ (-s)
⊢ False
|
have := norm_prime_cpow_le_one_half ⟨p, hp⟩ hs
|
p : ℕ
hp : Nat.Prime p
s : ℂ
hs : 1 < s.re
H : 1 = ↑p ^ (-s)
this : ‖↑↑⟨p, hp⟩ ^ (-s)‖ ≤ 1 / 2
⊢ False
|
347b3c0a49844399
|
SetTheory.Game.birthday_eq_zero
|
Mathlib/SetTheory/Game/Birthday.lean
|
theorem birthday_eq_zero {x : Game} : birthday x = 0 ↔ x = 0
|
case mp
x : Game
h : x.birthday = 0
⊢ x = 0
|
let ⟨y, hy₁, hy₂⟩ := birthday_eq_pGameBirthday x
|
case mp
x : Game
h : x.birthday = 0
y : PGame
hy₁ : ⟦y⟧ = x
hy₂ : y.birthday = x.birthday
⊢ x = 0
|
adf47eec13e763d8
|
FractionalIdeal.div_spanSingleton
|
Mathlib/RingTheory/FractionalIdeal/Operations.lean
|
theorem div_spanSingleton (J : FractionalIdeal R₁⁰ K) (d : K) :
J / spanSingleton R₁⁰ d = spanSingleton R₁⁰ d⁻¹ * J
|
case neg
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 0
h_spand : spanSingleton R₁⁰ d ≠ 0
⊢ J / spanSingleton R₁⁰ d = 1 / spanSingleton R₁⁰ d * J
|
apply le_antisymm
|
case neg.a
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 0
h_spand : spanSingleton R₁⁰ d ≠ 0
⊢ J / spanSingleton R₁⁰ d ≤ 1 / spanSingleton R₁⁰ d * J
case neg.a
R₁ : Type u_3
inst✝⁴ : CommRing R₁
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra R₁ K
inst✝¹ : IsFractionRing R₁ K
inst✝ : IsDomain R₁
J : FractionalIdeal R₁⁰ K
d : K
hd : ¬d = 0
h_spand : spanSingleton R₁⁰ d ≠ 0
⊢ 1 / spanSingleton R₁⁰ d * J ≤ J / spanSingleton R₁⁰ d
|
3a4f3cf63940a3cf
|
Filter.mem_pi
|
Mathlib/Order/Filter/Pi.lean
|
theorem mem_pi {s : Set (∀ i, α i)} :
s ∈ pi f ↔ ∃ I : Set ι, I.Finite ∧ ∃ t : ∀ i, Set (α i), (∀ i, t i ∈ f i) ∧ I.pi t ⊆ s
|
case mp
ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : Set ((i : ι) → α i)
⊢ s ∈ pi f → ∃ I, I.Finite ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ I.pi t ⊆ s
|
simp only [pi, mem_iInf', mem_comap, pi_def]
|
case mp
ι : Type u_1
α : ι → Type u_2
f : (i : ι) → Filter (α i)
s : Set ((i : ι) → α i)
⊢ (∃ I,
I.Finite ∧
∃ V, (∀ (i : ι), ∃ t ∈ f i, eval i ⁻¹' t ⊆ V i) ∧ (∀ i ∉ I, V i = univ) ∧ s = ⋂ i ∈ I, V i ∧ s = ⋂ i, V i) →
∃ I, I.Finite ∧ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ ⋂ a ∈ I, eval a ⁻¹' t a ⊆ s
|
2c2dd46cf85728eb
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_motive
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean
|
theorem confirmRupHint_preserves_motive {n : Nat} (f : DefaultFormula n) (rupHints : Array Nat)
(idx : Fin rupHints.size) (acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool)
(ih : ConfirmRupHintFoldEntailsMotive f idx.1 acc) :
ConfirmRupHintFoldEntailsMotive f (idx.1 + 1) ((confirmRupHint f.clauses) acc rupHints[idx])
|
case isTrue
n : Nat
f : DefaultFormula n
rupHints : Array Nat
idx : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
h1 : Limplies (PosFin n) f acc.fst
h2 : acc.snd.snd.fst = true → Incompatible (PosFin n) acc.fst f
acc2_eq_false : acc.2.2.snd = false ∧ acc.2.2.fst = false
x✝¹ : Option (Option (DefaultClause n))
c : DefaultClause n
hc : f.clauses[rupHints[↑idx]]? = some (some c)
c_in_f : c ∈ f.toList
x✝ : ReduceResult (PosFin n)
l : PosFin n
b : Bool
heq : c.reduce acc.fst = reducedToUnit (l, b)
h✝ : hasAssignment b acc.fst[l.val]! = true
⊢ (acc.fst, acc.2.fst, false, false).fst.size = n ∧
Limplies (PosFin n) f (acc.fst, acc.2.fst, false, false).fst ∧
((acc.fst, acc.2.fst, false, false).snd.snd.fst = true →
Incompatible (PosFin n) (acc.fst, acc.2.fst, false, false).fst f)
|
simp [h1, hsize]
|
no goals
|
610b60376ef97394
|
Field.primitive_element_inf_aux
|
Mathlib/FieldTheory/PrimitiveElement.lean
|
theorem primitive_element_inf_aux [Algebra.IsSeparable F E] : ∃ γ : E, F⟮α, β⟯ = F⟮γ⟯
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField := algebraMap E (Polynomial.map ιFE g).SplittingField
c : F
hc :
∀ α' ∈ (Polynomial.map (ιEE'.comp ιFE) f).roots,
∀ β' ∈ (Polynomial.map (ιEE'.comp ιFE) g).roots, -(α' - ιEE' α) / (β' - ιEE' β) ≠ (ιEE'.comp ιFE) c
γ : E := α + c • β
β_in_Fγ : β ∈ F⟮γ⟯
⊢ ∃ γ, F⟮α, β⟯ = F⟮γ⟯
|
use γ
|
case h
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Infinite F
E : Type u_2
inst✝² : Field E
α β : E
inst✝¹ : Algebra F E
inst✝ : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField := algebraMap E (Polynomial.map ιFE g).SplittingField
c : F
hc :
∀ α' ∈ (Polynomial.map (ιEE'.comp ιFE) f).roots,
∀ β' ∈ (Polynomial.map (ιEE'.comp ιFE) g).roots, -(α' - ιEE' α) / (β' - ιEE' β) ≠ (ιEE'.comp ιFE) c
γ : E := α + c • β
β_in_Fγ : β ∈ F⟮γ⟯
⊢ F⟮α, β⟯ = F⟮γ⟯
|
c759c025da469f0d
|
IsGalois.of_card_aut_eq_finrank
|
Mathlib/FieldTheory/Galois/Basic.lean
|
theorem of_card_aut_eq_finrank [FiniteDimensional F E]
(h : Fintype.card (E ≃ₐ[F] E) = finrank F E) : IsGalois F E
|
case h
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
h : Fintype.card (E ≃ₐ[F] E) = finrank F E
p : 0 < finrank (↥(fixedField ⊤)) E
⊢ fixedField ⊤ = ⊥
|
classical
rw [← IntermediateField.finrank_eq_one_iff, ← mul_left_inj' (ne_of_lt p).symm,
finrank_mul_finrank, ← h, one_mul, IntermediateField.finrank_fixedField_eq_card]
apply Fintype.card_congr
exact
{ toFun := fun g => ⟨g, Subgroup.mem_top g⟩
invFun := (↑)
left_inv := fun g => rfl
right_inv := fun _ => by ext; rfl }
|
no goals
|
3fb9be27b8da33a5
|
Matroid.fundCocircuit_isCocircuit
|
Mathlib/Data/Matroid/Circuit.lean
|
lemma fundCocircuit_isCocircuit (he : e ∈ B) (hB : M.IsBase B) :
M.IsCocircuit <| M.fundCocircuit e B
|
α : Type u_1
M : Matroid α
e : α
B : Set α
he : e ∈ B
hB : M.IsBase B
⊢ e ∈ M✶.closure (M.E \ B)
|
rw [hB.compl_isBase_dual.closure_eq, dual_ground]
|
α : Type u_1
M : Matroid α
e : α
B : Set α
he : e ∈ B
hB : M.IsBase B
⊢ e ∈ M.E
|
9f804a51a0212ed5
|
Cardinal.mk_biUnion_le_lift
|
Mathlib/SetTheory/Cardinal/Basic.lean
|
theorem mk_biUnion_le_lift {α : Type u} {ι : Type v} (A : ι → Set α) (s : Set ι) :
lift.{v} #(⋃ x ∈ s, A x) ≤ lift.{u} #s * ⨆ x : s, lift.{v} #(A x.1)
|
α : Type u
ι : Type v
A : ι → Set α
s : Set ι
⊢ lift.{v, u} #↑(⋃ x, A ↑x) ≤ lift.{u, v} #↑s * ⨆ x, lift.{v, u} #↑(A ↑x)
|
apply mk_iUnion_le_lift
|
no goals
|
d8c9f880707f7e65
|
Bool.right_le_or
|
Mathlib/Data/Bool/Basic.lean
|
theorem right_le_or : ∀ x y : Bool, y ≤ (x || y)
|
⊢ ∀ (x y : Bool), y ≤ (x || y)
|
decide
|
no goals
|
31daf6d66171fb30
|
BitVec.lt_of_getMsbD
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem lt_of_getMsbD {x : BitVec w} {i : Nat} : getMsbD x i = true → i < w
|
w : Nat
x : BitVec w
i : Nat
h : i < w
⊢ x.getMsbD i = true → i < w
|
simp [h]
|
no goals
|
ca368506eb0e50ad
|
Array.foldl_induction
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean
|
theorem foldl_induction
{as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive 0 init) {f : β → α → β}
(hf : ∀ i : Fin as.size, ∀ b, motive i.1 b → motive (i.1 + 1) (f b as[i])) :
motive as.size (as.foldl f init)
|
α : Type u_1
β : Type u_2
as : Array α
motive : Nat → β → Prop
init : β
h0 : motive 0 init
f : β → α → β
hf : ∀ (i : Fin as.size) (b : β), motive (↑i) b → motive (↑i + 1) (f b as[i])
i j : Nat
b : β
h₁ : j ≤ as.size
h₂ : as.size ≤ i + j
H : motive j b
hj : j < as.size
⊢ motive as.size
(match i with
| 0 => pure b
| i'.succ =>
let_fun this := ⋯;
do
let __do_lift ← f b as[j]
foldlM.loop f as as.size ⋯ i' (j + 1) __do_lift)
|
split
|
case h_1
α : Type u_1
β : Type u_2
as : Array α
motive : Nat → β → Prop
init : β
h0 : motive 0 init
f : β → α → β
hf : ∀ (i : Fin as.size) (b : β), motive (↑i) b → motive (↑i + 1) (f b as[i])
j : Nat
b : β
h₁ : j ≤ as.size
H : motive j b
hj : j < as.size
i✝ : Nat
h₂ : as.size ≤ 0 + j
⊢ motive as.size (pure b)
case h_2
α : Type u_1
β : Type u_2
as : Array α
motive : Nat → β → Prop
init : β
h0 : motive 0 init
f : β → α → β
hf : ∀ (i : Fin as.size) (b : β), motive (↑i) b → motive (↑i + 1) (f b as[i])
j : Nat
b : β
h₁ : j ≤ as.size
H : motive j b
hj : j < as.size
i✝ i'✝ : Nat
h₂ : as.size ≤ i'✝.succ + j
⊢ motive as.size
(let_fun this := ⋯;
do
let __do_lift ← f b as[j]
foldlM.loop f as as.size ⋯ i'✝ (j + 1) __do_lift)
|
e35b353412732eec
|
torusIntegral_dim1
|
Mathlib/MeasureTheory/Integral/TorusIntegral.lean
|
theorem torusIntegral_dim1 (f : ℂ¹ → E) (c : ℂ¹) (R : ℝ¹) :
(∯ x in T(c, R), f x) = ∮ z in C(c 0, R 0), f fun _ => z
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x => 2 * π) = Icc 0 (2 * π)
⊢ torusMap c R = fun θ x => circleMap (c 0) (R 0) (θ 0)
|
ext θ i : 2
|
case h.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : (Fin 1 → ℂ) → E
c : Fin 1 → ℂ
R : Fin 1 → ℝ
H₁ : (⇑(MeasurableEquiv.funUnique (Fin 1) ℝ).symm ⁻¹' Icc 0 fun x => 2 * π) = Icc 0 (2 * π)
θ : Fin 1 → ℝ
i : Fin 1
⊢ torusMap c R θ i = circleMap (c 0) (R 0) (θ 0)
|
34148b6ecc6aa62c
|
Nimber.add_zero
|
Mathlib/SetTheory/Nimber/Basic.lean
|
theorem add_zero (a : Nimber) : a + 0 = a
|
case a.h₁
a a' : Nimber
ha : a' < a
⊢ a' ≠ a
|
exact ha.ne
|
no goals
|
03f116e4360d0b64
|
IsLocalRing.quotient_span_eq_top_iff_span_eq_top
|
Mathlib/RingTheory/LocalRing/Quotient.lean
|
theorem quotient_span_eq_top_iff_span_eq_top (s : Set S) :
span (R ⧸ p) ((Ideal.Quotient.mk (I := pS)) '' s) = ⊤ ↔ span R s = ⊤
|
case mp.hN'
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalRing R
inst✝ : Module.Finite R S
s : Set S
H :
restrictScalars R (span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s)) = map (IsScalarTower.toAlgHom R S (S ⧸ pS)) (span R s)
hs : span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s) = ⊤
⊢ ⊤.FG
|
exact Module.finite_def.mp ‹_›
|
no goals
|
c2f006104663815d
|
Composition.mem_range_embedding
|
Mathlib/Combinatorics/Enumerative/Composition.lean
|
theorem mem_range_embedding (j : Fin n) : j ∈ Set.range (c.embedding (c.index j))
|
n : ℕ
c : Composition n
j : Fin n
⊢ j ∈ Set.range ⇑(c.embedding (c.index j))
|
have : c.embedding (c.index j) (c.invEmbedding j) ∈ Set.range (c.embedding (c.index j)) :=
Set.mem_range_self _
|
n : ℕ
c : Composition n
j : Fin n
this : (c.embedding (c.index j)) (c.invEmbedding j) ∈ Set.range ⇑(c.embedding (c.index j))
⊢ j ∈ Set.range ⇑(c.embedding (c.index j))
|
21f3ddb51c46e27d
|
DirichletCharacter.IsPrimitive.completedLFunction_one_sub
|
Mathlib/NumberTheory/LSeries/DirichletContinuation.lean
|
theorem completedLFunction_one_sub {χ : DirichletCharacter ℂ N} (hχ : IsPrimitive χ) (s : ℂ) :
completedLFunction χ (1 - s) = N ^ (s - 1 / 2) * rootNumber χ * completedLFunction χ⁻¹ s
|
case inr
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j : ZMod N, χ j = 0
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
|
let ε := I ^ (if χ.Even then 0 else 1)
|
case inr
N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ.IsPrimitive
s : ℂ
hN : N ≠ 1
h_sum : ∑ j : ZMod N, χ j = 0
ε : ℂ := I ^ if χ.Even then 0 else 1
⊢ completedLFunction χ (1 - s) = ↑N ^ (s - 1 / 2) * χ.rootNumber * completedLFunction χ⁻¹ s
|
ff38d9c55b1ef998
|
StarAlgebra.elemental.starAlgHomClass_ext
|
Mathlib/Topology/Algebra/StarSubalgebra.lean
|
theorem starAlgHomClass_ext [T2Space B] {F : Type*} {a : A}
[FunLike F (elemental R a) B] [AlgHomClass F R _ B] [StarHomClass F _ B]
{φ ψ : F} (hφ : Continuous φ)
(hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ
|
R : Type u_1
A : Type u_2
B : Type u_3
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : TopologicalSpace A
inst✝¹³ : Semiring A
inst✝¹² : StarRing A
inst✝¹¹ : IsTopologicalSemiring A
inst✝¹⁰ : ContinuousStar A
inst✝⁹ : Algebra R A
inst✝⁸ : StarModule R A
inst✝⁷ : TopologicalSpace B
inst✝⁶ : Semiring B
inst✝⁵ : StarRing B
inst✝⁴ : Algebra R B
inst✝³ : T2Space B
F : Type u_4
a : A
inst✝² : FunLike F (↥(elemental R a)) B
inst✝¹ : AlgHomClass F R (↥(elemental R a)) B
inst✝ : StarHomClass F (↥(elemental R a)) B
φ ψ : F
hφ : Continuous ⇑φ
hψ : Continuous ⇑ψ
h : φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩
x✝ x y : ↥(adjoin R {a})
hx : φ ((inclusion ⋯) x) = ψ ((inclusion ⋯) x)
hy : φ ((inclusion ⋯) y) = ψ ((inclusion ⋯) y)
⊢ φ ((inclusion ⋯) (x + y)) = ψ ((inclusion ⋯) (x + y))
|
simp only [map_add, hx, hy]
|
no goals
|
5c269248ad380788
|
Complex.eq_of_isMaxOn_of_ball_subset
|
Mathlib/Analysis/Complex/AbsMax.lean
|
theorem eq_of_isMaxOn_of_ball_subset {f : E → F} {s : Set E} {z w : E} (hd : DiffContOnCl ℂ f s)
(hz : IsMaxOn (norm ∘ f) s z) (hsub : ball z (dist w z) ⊆ s) : f w = f z :=
have H₁ : ‖f w‖ = ‖f z‖ := norm_eq_norm_of_isMaxOn_of_ball_subset hd hz hsub
have H₂ : ‖f w + f z‖ = ‖f z + f z‖ :=
norm_eq_norm_of_isMaxOn_of_ball_subset (hd.add_const _) hz.norm_add_self hsub
eq_of_norm_eq_of_norm_add_eq H₁ <| by simp only [H₂, SameRay.rfl.norm_add, H₁]
|
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℂ E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : StrictConvexSpace ℝ F
f : E → F
s : Set E
z w : E
hd : DiffContOnCl ℂ f s
hz : IsMaxOn (norm ∘ f) s z
hsub : ball z (dist w z) ⊆ s
H₁ : ‖f w‖ = ‖f z‖
H₂ : ‖f w + f z‖ = ‖f z + f z‖
⊢ ‖f w + f z‖ = ‖f w‖ + ‖f z‖
|
simp only [H₂, SameRay.rfl.norm_add, H₁]
|
no goals
|
9593661ff2f483c5
|
Std.DHashMap.Internal.List.getKey_append_of_containsKey_eq_false
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean
|
theorem getKey_append_of_containsKey_eq_false [BEq α] [PartialEquivBEq α]
{l l' : List ((a : α) × β a)} {a : α} {h} (hl' : containsKey a l' = false) :
getKey a (l ++ l') h =
getKey a l ((containsKey_append_of_not_contains_right hl').symm.trans h)
|
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
l l' : List ((a : α) × β a)
a : α
h : containsKey a (l ++ l') = true
hl' : containsKey a l' = false
⊢ getKey a (l ++ l') h = getKey a l ⋯
|
rw [← Option.some_inj, ← getKey?_eq_some_getKey, ← getKey?_eq_some_getKey,
getKey?_append_of_containsKey_eq_false hl']
|
no goals
|
2f1b4d1b448b422c
|
MeasureTheory.tendsto_of_lintegral_tendsto_of_monotone
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
/-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. -/
lemma tendsto_of_lintegral_tendsto_of_monotone {α : Type*} {mα : MeasurableSpace α}
{f : ℕ → α → ℝ≥0∞} {F : α → ℝ≥0∞} {μ : Measure α}
(hF_meas : AEMeasurable F μ)
(hf_tendsto : Tendsto (fun i ↦ ∫⁻ a, f i a ∂μ) atTop (𝓝 (∫⁻ a, F a ∂μ)))
(hf_mono : ∀ᵐ a ∂μ, Monotone (fun i ↦ f i a))
(h_bound : ∀ᵐ a ∂μ, ∀ i, f i a ≤ F a) (h_int_finite : ∫⁻ a, F a ∂μ ≠ ∞) :
∀ᵐ a ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))
|
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
this : ∀ (n : ℕ), ∃ g, Measurable g ∧ g ≤ f n ∧ ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
|
choose g gmeas gf hg using this
|
α : Type u_5
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))
|
1cdd224ac57038c3
|
ProperSpace.of_locallyCompactSpace
|
Mathlib/Analysis/Normed/Module/FiniteDimension.lean
|
/-- A locally compact normed vector space is proper. -/
lemma ProperSpace.of_locallyCompactSpace (𝕜 : Type*) [NontriviallyNormedField 𝕜]
{E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [LocallyCompactSpace E] :
ProperSpace E
|
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : LocallyCompactSpace E
r : ℝ
rpos : 0 < r
hr : IsCompact (closedBall 0 r)
c : 𝕜
hc : 1 < ‖c‖
n : ℕ
this : c ^ n ≠ 0
⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))
|
simpa [_root_.smul_closedBall' this] using hr.smul (c ^ n)
|
no goals
|
d9806832454f6786
|
CategoryTheory.isPreconnected_induction
|
Mathlib/CategoryTheory/IsConnected.lean
|
theorem isPreconnected_induction [IsPreconnected J] (Z : J → Sort*)
(h₁ : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), Z j₁ → Z j₂) (h₂ : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), Z j₂ → Z j₁)
{j₀ : J} (x : Z j₀) (j : J) : Nonempty (Z j) :=
(induct_on_objects { j | Nonempty (Z j) } ⟨x⟩
(fun f => ⟨by rintro ⟨y⟩; exact ⟨h₁ f y⟩, by rintro ⟨y⟩; exact ⟨h₂ f y⟩⟩)
j :)
|
case intro
J : Type u₁
inst✝¹ : Category.{v₁, u₁} J
inst✝ : IsPreconnected J
Z : J → Sort u_1
h₁ : {j₁ j₂ : J} → (j₁ ⟶ j₂) → Z j₁ → Z j₂
h₂ : {j₁ j₂ : J} → (j₁ ⟶ j₂) → Z j₂ → Z j₁
j₀ : J
x : Z j₀
j j₁✝ j₂✝ : J
f : j₁✝ ⟶ j₂✝
y : Z j₁✝
⊢ j₂✝ ∈ {j | Nonempty (Z j)}
|
exact ⟨h₁ f y⟩
|
no goals
|
8fd9ae1c40916cd1
|
Matrix.eval_matrixOfPolynomials_eq_vandermonde_mul_matrixOfPolynomials
|
Mathlib/LinearAlgebra/Vandermonde.lean
|
theorem eval_matrixOfPolynomials_eq_vandermonde_mul_matrixOfPolynomials {n : ℕ}
(v : Fin n → R) (p : Fin n → R[X]) (h_deg : ∀ i, (p i).natDegree ≤ i) :
Matrix.of (fun i j => ((p j).eval (v i))) =
(Matrix.vandermonde v) * (Matrix.of (fun (i j : Fin n) => (p j).coeff i))
|
case a
R : Type u_1
inst✝ : CommRing R
n : ℕ
v : Fin n → R
p : Fin n → R[X]
h_deg : ∀ (i : Fin n), (p i).natDegree ≤ ↑i
i j : Fin n
this : (p j).support ⊆ range n
⊢ ((p j).sum fun e a => (RingHom.id R) a * v i ^ e) =
∑ x : Fin n, of (fun i j => v i ^ ↑j) i x * of (fun i j => (p j).coeff ↑i) x j
|
rw [sum_eq_of_subset _ (fun j => zero_mul ((v i) ^ j)) this, ← Fin.sum_univ_eq_sum_range]
|
case a
R : Type u_1
inst✝ : CommRing R
n : ℕ
v : Fin n → R
p : Fin n → R[X]
h_deg : ∀ (i : Fin n), (p i).natDegree ≤ ↑i
i j : Fin n
this : (p j).support ⊆ range n
⊢ ∑ i_1 : Fin n, (RingHom.id R) ((p j).coeff ↑i_1) * v i ^ ↑i_1 =
∑ x : Fin n, of (fun i j => v i ^ ↑j) i x * of (fun i j => (p j).coeff ↑i) x j
|
9ae3c6287bae2d19
|
WeakFEPair.Λ_residue_zero
|
Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean
|
theorem Λ_residue_zero :
Tendsto (fun s : ℂ ↦ s • P.Λ s) (𝓝[≠] 0) (𝓝 (-P.f₀))
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s => s • P.Λ₀ s - s • (1 / s) • P.f₀ - s • (P.ε / (↑P.k - s)) • P.g₀) (𝓝[≠] 0)
(𝓝 (0 • P.Λ₀ 0 - P.f₀ - 0))
|
refine ((Tendsto.mono_left ?_ nhdsWithin_le_nhds).sub ?_).sub ?_
|
case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s => s • P.Λ₀ s) (𝓝 0) (𝓝 (0 • P.Λ₀ 0))
case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s => s • (1 / s) • P.f₀) (𝓝[≠] 0) (𝓝 P.f₀)
case refine_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
P : WeakFEPair E
⊢ Tendsto (fun s => s • (P.ε / (↑P.k - s)) • P.g₀) (𝓝[≠] 0) (𝓝 0)
|
ae6236b4fb4a7ef0
|
Sym.coe_equivNatSumOfFintype_symm_apply
|
Mathlib/Data/Finsupp/Multiset.lean
|
@[simp] lemma coe_equivNatSumOfFintype_symm_apply [Fintype α] (P : {P : α → ℕ // ∑ i, P i = n}) :
((equivNatSumOfFintype α n).symm P : Multiset α) = ∑ a, ((P : α → ℕ) a) • {a}
|
case mk
α : Type u_1
inst✝¹ : DecidableEq α
n : ℕ
inst✝ : Fintype α
P : α → ℕ
hP : ∑ i : α, P i = n
⊢ Finsupp.toMultiset (Finsupp.equivFunOnFinite.symm P) = (Multiset.map (fun a => ↑⟨P, hP⟩ a • {a}) univ.val).sum
|
ext a
|
case mk.a
α : Type u_1
inst✝¹ : DecidableEq α
n : ℕ
inst✝ : Fintype α
P : α → ℕ
hP : ∑ i : α, P i = n
a : α
⊢ Multiset.count a (Finsupp.toMultiset (Finsupp.equivFunOnFinite.symm P)) =
Multiset.count a (Multiset.map (fun a => ↑⟨P, hP⟩ a • {a}) univ.val).sum
|
702aa4a986a2f605
|
finprod_cond_eq_prod_of_cond_iff
|
Mathlib/Algebra/BigOperators/Finprod.lean
|
theorem finprod_cond_eq_prod_of_cond_iff (f : α → M) {p : α → Prop} {t : Finset α}
(h : ∀ {x}, f x ≠ 1 → (p x ↔ x ∈ t)) : (∏ᶠ (i) (_ : p i), f i) = ∏ i ∈ t, f i
|
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
p : α → Prop
t : Finset α
h : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)
s : Set α := {x | p x}
this : mulSupport (s.mulIndicator f) ⊆ ↑t
⊢ ∏ i ∈ t, s.mulIndicator f i = ∏ i ∈ t, f i
|
refine Finset.prod_congr rfl fun x hx => mulIndicator_apply_eq_self.2 fun hxs => ?_
|
α : Type u_1
M : Type u_5
inst✝ : CommMonoid M
f : α → M
p : α → Prop
t : Finset α
h : ∀ {x : α}, f x ≠ 1 → (p x ↔ x ∈ t)
s : Set α := {x | p x}
this : mulSupport (s.mulIndicator f) ⊆ ↑t
x : α
hx : x ∈ t
hxs : x ∉ s
⊢ f x = 1
|
d1687f56373c7d13
|
Set.graphOn_univ_inj
|
Mathlib/Data/Set/Function.lean
|
lemma graphOn_univ_inj {g : α → β} : univ.graphOn f = univ.graphOn g ↔ f = g
|
α : Type u_1
β : Type u_2
f g : α → β
⊢ graphOn f univ = graphOn g univ ↔ f = g
|
simp
|
no goals
|
0cad94e96f19a2c5
|
WeierstrassCurve.Projective.negAddY_eq'
|
Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
|
lemma negAddY_eq' {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) :
W'.negAddY P Q * (P z * Q z) ^ 2 =
(P y * Q z - Q y * P z) * ((P y * Q z - Q y * P z) ^ 2 * P z * Q z
+ W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z)
- W'.a₂ * P z * Q z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2
- Q x * P z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2)
+ P y * Q z * (P x * Q z - Q x * P z) ^ 3
|
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ W'.negAddY P Q * (P z * Q z) ^ 2 =
(P y * Q z - Q y * P z) *
((P y * Q z - Q y * P z) ^ 2 * P z * Q z +
W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z) -
W'.a₂ * P z * Q z * (P x * Q z - Q x * P z) ^ 2 -
P x * Q z * (P x * Q z - Q x * P z) ^ 2 -
Q x * P z * (P x * Q z - Q x * P z) ^ 2 -
P x * Q z * (P x * Q z - Q x * P z) ^ 2) +
P y * Q z * (P x * Q z - Q x * P z) ^ 3
|
linear_combination (norm := (rw [negAddY]; ring1))
(2 * Q y * P z * Q z ^ 3 - P y * Q z ^ 4) * (equation_iff _).mp hP
+ (Q y * P z ^ 4 - 2 * P y * P z ^ 3 * Q z) * (equation_iff _).mp hQ
|
no goals
|
1350480939e3501d
|
Set.ncard_singleton
|
Mathlib/Data/Set/Card.lean
|
theorem ncard_singleton (a : α) : ({a} : Set α).ncard = 1
|
α : Type u_1
a : α
⊢ {a}.ncard = 1
|
simp [ncard]
|
no goals
|
4e248f62061f37fe
|
lipschitzGroup.involute_act_ι_mem_range_ι
|
Mathlib/LinearAlgebra/CliffordAlgebra/SpinGroup.lean
|
theorem involute_act_ι_mem_range_ι [Invertible (2 : R)]
{x : (CliffordAlgebra Q)ˣ} (hx : x ∈ lipschitzGroup Q) (b : M) :
involute (Q := Q) ↑x * ι Q b * ↑x⁻¹ ∈ LinearMap.range (ι Q)
|
case one
R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
inst✝ : Invertible 2
x : (CliffordAlgebra Q)ˣ
b : M
⊢ involute ↑1 * (ι Q) b * ↑1⁻¹ ∈ LinearMap.range (ι Q)
|
simp_rw [inv_one, Units.val_one, map_one, one_mul, mul_one, LinearMap.mem_range_self]
|
no goals
|
0035a32f523e5bf4
|
infinite_not_isOfFinOrder
|
Mathlib/GroupTheory/OrderOfElement.lean
|
theorem infinite_not_isOfFinOrder {x : G} (h : ¬IsOfFinOrder x) :
{ y : G | ¬IsOfFinOrder y }.Infinite
|
G : Type u_1
inst✝ : LeftCancelMonoid G
x : G
h : ¬IsOfFinOrder x
s : Set G := (fun n => x ^ n) '' {n | 0 < n}
hs : s ⊆ {y | ¬IsOfFinOrder y}
this : s.Infinite
⊢ {y | ¬IsOfFinOrder y}.Infinite
|
exact this.mono hs
|
no goals
|
56e04058092d4731
|
SimpleGraph.TripartiteFromTriangles.map_toTriangle_disjoint
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean
|
lemma map_toTriangle_disjoint [ExplicitDisjoint t] :
(t.map toTriangle : Set (Finset (α ⊕ β ⊕ γ))).Pairwise
fun x y ↦ (x ∩ y : Set (α ⊕ β ⊕ γ)).Subsingleton
|
case refine_1.intro
α : Type u_1
β : Type u_2
γ : Type u_3
t : Finset (α × β × γ)
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : DecidableEq γ
inst✝ : ExplicitDisjoint t
a : α
b : β
c : γ
habc : (a, b, c) ∈ t
z : γ
hxyz : (a, b, z) ∈ t
h' : ¬toTriangle (a, b, c) = toTriangle (a, b, z)
this : a = a → b = b → ¬c = z
⊢ False
|
exact this rfl rfl (ExplicitDisjoint.inj₂ habc hxyz)
|
no goals
|
baa93415b8cbec8b
|
IsBaseChange.of_lift_unique
|
Mathlib/RingTheory/IsTensorProduct.lean
|
theorem IsBaseChange.of_lift_unique
(h : ∀ (Q : Type max v₁ v₂ v₃) [AddCommMonoid Q],
∀ [Module R Q] [Module S Q], ∀ [IsScalarTower R S Q],
∀ g : M →ₗ[R] Q, ∃! g' : N →ₗ[S] Q, (g'.restrictScalars R).comp f = g) :
IsBaseChange S f
|
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
h :
∀ (Q : Type (max v₁ v₂ v₃)) [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module S Q]
[inst_3 : IsScalarTower R S Q] (g : M →ₗ[R] Q), ∃! g', ↑R g' ∘ₗ f = g
g : N →ₗ[S] ULift.{v₂, max v₁ v₃} (S ⊗[R] M)
hg : ↑R g ∘ₗ f = ↑ULift.moduleEquiv.symm ∘ₗ (mk R S M) 1
f' : S ⊗[R] M →ₗ[R] N := TensorProduct.lift (↑R ((Algebra.ofId S (Module.End S (M →ₗ[R] N))).toLinearMap.flip f))
⊢ S ⊗[R] M →ₗ[S] N
|
refine
{ f' with
map_smul' := fun s x =>
TensorProduct.induction_on x ?_ (fun s' y => smul_assoc s s' _) fun x y hx hy => ?_ }
|
case refine_1
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
h :
∀ (Q : Type (max v₁ v₂ v₃)) [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module S Q]
[inst_3 : IsScalarTower R S Q] (g : M →ₗ[R] Q), ∃! g', ↑R g' ∘ₗ f = g
g : N →ₗ[S] ULift.{v₂, max v₁ v₃} (S ⊗[R] M)
hg : ↑R g ∘ₗ f = ↑ULift.moduleEquiv.symm ∘ₗ (mk R S M) 1
f' : S ⊗[R] M →ₗ[R] N := TensorProduct.lift (↑R ((Algebra.ofId S (Module.End S (M →ₗ[R] N))).toLinearMap.flip f))
s : S
x : S ⊗[R] M
⊢ f'.toFun (s • 0) = (RingHom.id S) s • f'.toFun 0
case refine_2
R : Type u_1
M : Type v₁
N : Type v₂
S : Type v₃
inst✝⁸ : AddCommMonoid M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra R S
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : Module S N
inst✝ : IsScalarTower R S N
f : M →ₗ[R] N
h :
∀ (Q : Type (max v₁ v₂ v₃)) [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module S Q]
[inst_3 : IsScalarTower R S Q] (g : M →ₗ[R] Q), ∃! g', ↑R g' ∘ₗ f = g
g : N →ₗ[S] ULift.{v₂, max v₁ v₃} (S ⊗[R] M)
hg : ↑R g ∘ₗ f = ↑ULift.moduleEquiv.symm ∘ₗ (mk R S M) 1
f' : S ⊗[R] M →ₗ[R] N := TensorProduct.lift (↑R ((Algebra.ofId S (Module.End S (M →ₗ[R] N))).toLinearMap.flip f))
s : S
x✝ x y : S ⊗[R] M
hx : f'.toFun (s • x) = (RingHom.id S) s • f'.toFun x
hy : f'.toFun (s • y) = (RingHom.id S) s • f'.toFun y
⊢ f'.toFun (s • (x + y)) = (RingHom.id S) s • f'.toFun (x + y)
|
f4d4256bf13274f3
|
MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive
|
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
theorem of_inter_eq_of_symmDiff_eq_zero_positive (hu : MeasurableSet u) (hv : MeasurableSet v)
(hw : MeasurableSet w) (hsu : 0 ≤[u] s) (hsv : 0 ≤[v] s) (hs : s (u ∆ v) = 0) :
s (w ∩ u) = s (w ∩ v)
|
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : 0 ≤[u] s
hsv : 0 ≤[v] s
hs : ↑s (u ∆ v) = 0
hwuv : ↑s ((w ∩ u) ∆ (w ∩ v)) = 0
⊢ ↑s (w ∩ u) = ↑s (w ∩ v)
|
obtain ⟨huv, hvu⟩ :=
of_diff_eq_zero_of_symmDiff_eq_zero_positive (hw.inter hu) (hw.inter hv)
(restrict_le_restrict_subset _ _ hu hsu (w.inter_subset_right))
(restrict_le_restrict_subset _ _ hv hsv (w.inter_subset_right)) hwuv
|
case intro
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : 0 ≤[u] s
hsv : 0 ≤[v] s
hs : ↑s (u ∆ v) = 0
hwuv : ↑s ((w ∩ u) ∆ (w ∩ v)) = 0
huv : ↑s ((w ∩ u) \ (w ∩ v)) = 0
hvu : ↑s ((w ∩ v) \ (w ∩ u)) = 0
⊢ ↑s (w ∩ u) = ↑s (w ∩ v)
|
f9268e003122084b
|
ForInStep.bindList_cons'
|
Mathlib/.lake/packages/batteries/Batteries/Control/ForInStep/Lemmas.lean
|
theorem ForInStep.bindList_cons' [Monad m] [LawfulMonad m]
(f : α → β → m (ForInStep β)) (s : ForInStep β) (a l) :
s.bindList f (a::l) = s.bind (f a) >>= (·.bindList f l)
|
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → β → m (ForInStep β)
s : ForInStep β
a : α
l : List α
⊢ bindList f (a :: l) s = do
let x ← s.bind (f a)
bindList f l x
|
simp
|
no goals
|
5cf1ad3c52f00b5d
|
Set.indicator_smul_apply
|
Mathlib/Algebra/Module/Basic.lean
|
lemma indicator_smul_apply (s : Set α) (r : α → R) (f : α → M) (a : α) :
indicator s (fun a ↦ r a • f a) a = r a • indicator s f a
|
α : Type u_1
R : Type u_2
M : Type u_3
inst✝¹ : Zero M
inst✝ : SMulZeroClass R M
s : Set α
r : α → R
f : α → M
a : α
⊢ s.indicator (fun a => r a • f a) a = r a • s.indicator f a
|
dsimp only [indicator]
|
α : Type u_1
R : Type u_2
M : Type u_3
inst✝¹ : Zero M
inst✝ : SMulZeroClass R M
s : Set α
r : α → R
f : α → M
a : α
⊢ (if a ∈ s then r a • f a else 0) = r a • if a ∈ s then f a else 0
|
9045b92b42c07472
|
DividedPowers.dpow_sum'
|
Mathlib/RingTheory/DividedPowers/Basic.lean
|
theorem dpow_sum' {M : Type*} [AddCommMonoid M] {I : AddSubmonoid M} (dpow : ℕ → M → A)
(dpow_zero : ∀ {x} (_ : x ∈ I), dpow 0 x = 1)
(dpow_add : ∀ {n x y} (_ : x ∈ I) (_ : y ∈ I),
dpow n (x + y) = (antidiagonal n).sum fun k ↦ dpow k.1 x * dpow k.2 y)
(dpow_eval_zero : ∀ {n : ℕ} (_ : n ≠ 0), dpow n 0 = 0)
{ι : Type*} [DecidableEq ι] {s : Finset ι} {x : ι → M} (hx : ∀ i ∈ s, x i ∈ I) {n : ℕ} :
dpow n (s.sum x) = (s.sym n).sum fun k ↦ s.prod fun i ↦ dpow (Multiset.count i k) (x i)
|
A : Type u_1
inst✝² : CommSemiring A
M : Type u_2
inst✝¹ : AddCommMonoid M
I : AddSubmonoid M
dpow : ℕ → M → A
dpow_zero : ∀ {x : M}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_3
inst✝ : DecidableEq ι
x : ι → M
dpow_add : ∀ {n : ℕ} {x y : M}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3 x * dpow (n - x_3) y
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, x i ∈ I) → ∀ {n : ℕ}, dpow n (s.sum x) = ∑ k ∈ s.sym n, ∏ i ∈ s, dpow (Multiset.count i ↑k) (x i)
hx : ∀ i ∈ insert a s, x i ∈ I
n : ℕ
hx' : ∀ i ∈ s, x i ∈ I
m : (i : Fin n.succ) × Sym ι (n - ↑i)
hm : m ∈ univ.sigma fun a => s.sym (n - ↑a)
⊢ (fun m x => Sym.fill a m.fst m.snd) m hm ∈ (insert a s).sym n
|
simp only [succ_eq_add_one, mem_sym_iff, mem_insert, Sym.mem_fill_iff]
|
A : Type u_1
inst✝² : CommSemiring A
M : Type u_2
inst✝¹ : AddCommMonoid M
I : AddSubmonoid M
dpow : ℕ → M → A
dpow_zero : ∀ {x : M}, x ∈ I → dpow 0 x = 1
dpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0
ι : Type u_3
inst✝ : DecidableEq ι
x : ι → M
dpow_add : ∀ {n : ℕ} {x y : M}, x ∈ I → y ∈ I → dpow n (x + y) = ∑ x_3 ∈ range n.succ, dpow x_3 x * dpow (n - x_3) y
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, x i ∈ I) → ∀ {n : ℕ}, dpow n (s.sum x) = ∑ k ∈ s.sym n, ∏ i ∈ s, dpow (Multiset.count i ↑k) (x i)
hx : ∀ i ∈ insert a s, x i ∈ I
n : ℕ
hx' : ∀ i ∈ s, x i ∈ I
m : (i : Fin n.succ) × Sym ι (n - ↑i)
hm : m ∈ univ.sigma fun a => s.sym (n - ↑a)
⊢ ∀ (a_1 : ι), ↑m.fst ≠ 0 ∧ a_1 = a ∨ a_1 ∈ m.snd → a_1 = a ∨ a_1 ∈ s
|
991ee9aaa08f10a1
|
IsUnit.div
|
Mathlib/Algebra/Group/Units/Defs.lean
|
@[to_additive] lemma div (ha : IsUnit a) (hb : IsUnit b) : IsUnit (a / b)
|
α : Type u
inst✝ : DivisionMonoid α
a b : α
ha : IsUnit a
hb : IsUnit b
⊢ IsUnit (a / b)
|
rw [div_eq_mul_inv]
|
α : Type u
inst✝ : DivisionMonoid α
a b : α
ha : IsUnit a
hb : IsUnit b
⊢ IsUnit (a * b⁻¹)
|
8356aa3753022b97
|
DifferentiableAt.inversion
|
Mathlib/Geometry/Euclidean/Inversion/Calculus.lean
|
theorem DifferentiableAt.inversion (hc : DifferentiableAt ℝ c a)
(hR : DifferentiableAt ℝ R a) (hx : DifferentiableAt ℝ x a) (hne : x a ≠ c a) :
DifferentiableAt ℝ (fun a ↦ inversion (c a) (R a) (x a)) a
|
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
c x : E → F
R : E → ℝ
a : E
hc : DifferentiableWithinAt ℝ c univ a
hR : DifferentiableWithinAt ℝ R univ a
hx : DifferentiableWithinAt ℝ x univ a
hne : x a ≠ c a
⊢ DifferentiableWithinAt ℝ (fun a => inversion (c a) (R a) (x a)) univ a
|
exact hc.inversion hR hx hne
|
no goals
|
ec4976b9d72d9a2a
|
exists_continuous_add_one_of_isCompact_nnreal
|
Mathlib/MeasureTheory/Integral/RieszMarkovKakutani/Basic.lean
|
lemma exists_continuous_add_one_of_isCompact_nnreal
{s₀ s₁ : Set X} {t : Set X} (s₀_compact : IsCompact s₀) (s₁_compact : IsCompact s₁)
(t_compact : IsCompact t) (disj : Disjoint s₀ s₁) (hst : s₀ ∪ s₁ ⊆ t) :
∃ (f₀ f₁ : C_c(X, ℝ≥0)), EqOn f₀ 1 s₀ ∧ EqOn f₁ 1 s₁ ∧ EqOn (f₀ + f₁) 1 t
|
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j => if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j => if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i : Fin 2, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₁
⊢ { toContinuousMap := f 0, hasCompactSupport' := ⋯ }.nnrealPart x = 1 x
|
simp only [Fin.isValue, nnrealPart_apply,
CompactlySupportedContinuousMap.coe_mk, Pi.one_apply, Real.toNNReal_eq_one]
|
case h.refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : T2Space X
inst✝ : LocallyCompactSpace X
s₀ s₁ t : Set X
s₀_compact : IsCompact s₀
s₁_compact : IsCompact s₁
t_compact : IsCompact t
disj : Disjoint s₀ s₁
hst : s₀ ∪ s₁ ⊆ t
so : Fin 2 → Set X := fun j => if j = 0 then s₀ᶜ else s₁ᶜ
hso : so = fun j => if j = 0 then s₀ᶜ else s₁ᶜ
soopen : ∀ (j : Fin 2), IsOpen (so j)
hsot : t ⊆ ⋃ j, so j
f : Fin 2 → C(X, ℝ)
f_supp_in_so : ∀ (i : Fin 2), tsupport ⇑(f i) ⊆ so i
sum_f_one_on_t : EqOn (∑ i : Fin 2, ⇑(f i)) 1 t
f_in_icc : ∀ (i : Fin 2) (x : X), (f i) x ∈ Icc 0 1
f_hcs : ∀ (i : Fin 2), HasCompactSupport ⇑(f i)
sum_one_x : ∀ x ∈ t, (f 0) x + (f 1) x = 1
x : X
hx : x ∈ s₁
⊢ (f 0) x = 1
|
c2d96a23593bed78
|
CategoryTheory.MorphismProperty.rlp_retracts
|
Mathlib/CategoryTheory/MorphismProperty/LiftingProperty.lean
|
@[simp]
lemma rlp_retracts : T.retracts.rlp = T.rlp
|
case a
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
⊢ T.retracts.rlp ≤ T.rlp
|
exact antitone_rlp T.le_retracts
|
no goals
|
e9cff8f695494cc4
|
AnalyticOnNhd.eqOn_zero_or_eventually_ne_zero_of_preconnected
|
Mathlib/Analysis/Analytic/IsolatedZeros.lean
|
theorem eqOn_zero_or_eventually_ne_zero_of_preconnected (hf : AnalyticOnNhd 𝕜 f U)
(hU : IsPreconnected U) : EqOn f 0 U ∨ ∀ᶠ x in codiscreteWithin U, f x ≠ 0
|
case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : AnalyticOnNhd 𝕜 f U
hU : IsPreconnected U
x : 𝕜
hx : x ∈ U
hx2 : (U \ {x | ¬f x = 0})ᶜ ∉ 𝓝[≠] x
nh : ∀ᶠ (x : 𝕜) in 𝓝[≠] x, ¬(fun z => f z = 0) x
a : 𝕜
ha : ¬f a = 0
⊢ a ∈ (U \ {x | ¬f x = 0})ᶜ
|
simp_all
|
no goals
|
5009e0fa4f7f373d
|
Polynomial.coeff_le_of_roots_le
|
Mathlib/Topology/Algebra/Polynomial.lean
|
theorem coeff_le_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : p.Monic)
(h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * p.natDegree.choose i
|
case inr.inl
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : Splits f p
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : (map f p).natDegree < i
⊢ 0 ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).natDegree.choose i)
|
positivity
|
no goals
|
15177c00698e7d89
|
SetTheory.PGame.mul_moveRight_inr
|
Mathlib/SetTheory/Game/Basic.lean
|
theorem mul_moveRight_inr {x y : PGame} {i j} :
(x * y).moveRight (toRightMovesMul (Sum.inr (i, j))) =
x.moveRight i * y + x * y.moveLeft j - x.moveRight i * y.moveLeft j
|
case mk.mk
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
i : (mk α✝¹ β✝¹ a✝³ a✝²).RightMoves
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
j : (mk α✝ β✝ a✝¹ a✝).LeftMoves
⊢ (mk α✝¹ β✝¹ a✝³ a✝² * mk α✝ β✝ a✝¹ a✝).moveRight (toRightMovesMul (Sum.inr (i, j))) =
(mk α✝¹ β✝¹ a✝³ a✝²).moveRight i * mk α✝ β✝ a✝¹ a✝ + mk α✝¹ β✝¹ a✝³ a✝² * (mk α✝ β✝ a✝¹ a✝).moveLeft j -
(mk α✝¹ β✝¹ a✝³ a✝²).moveRight i * (mk α✝ β✝ a✝¹ a✝).moveLeft j
|
rfl
|
no goals
|
65d0c21dcfd64a62
|
FirstOrder.Language.Hom.range_eq_map
|
Mathlib/ModelTheory/Substructures.lean
|
theorem range_eq_map (f : M →[L] N) : f.range = map f ⊤
|
L : Language
M : Type w
N : Type u_1
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M →[L] N
⊢ f.range = Substructure.map f ⊤
|
ext
|
case h
L : Language
M : Type w
N : Type u_1
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f : M →[L] N
x✝ : N
⊢ x✝ ∈ f.range ↔ x✝ ∈ Substructure.map f ⊤
|
8b0dba7354f1f4ba
|
FormalMultilinearSeries.comp_partialSum
|
Mathlib/Analysis/Analytic/Composition.lean
|
theorem comp_partialSum (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
(M N : ℕ) (z : E) :
q.partialSum M (∑ i ∈ Finset.Ico 1 N, p i fun _j => z) =
∑ i ∈ compPartialSumTarget 0 M N, q.compAlongComposition p i.2 fun _j => z
|
case h.mk
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource 0 M N
⊢ ∀ (i : ℕ) (him : i < ⟨k, blocks_fun⟩.fst) (hin : i < (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.length),
((p (⟨k, blocks_fun⟩.snd ⟨i, him⟩)) fun _j => z) =
p.applyComposition (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd (fun _j => z) ⟨i, hin⟩
|
intros
|
case h.mk
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
q : FormalMultilinearSeries 𝕜 F G
p : FormalMultilinearSeries 𝕜 E F
M N : ℕ
z : E
k : ℕ
blocks_fun : Fin k → ℕ
H : ⟨k, blocks_fun⟩ ∈ compPartialSumSource 0 M N
i✝ : ℕ
him✝ : i✝ < ⟨k, blocks_fun⟩.fst
hin✝ : i✝ < (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd.length
⊢ ((p (⟨k, blocks_fun⟩.snd ⟨i✝, him✝⟩)) fun _j => z) =
p.applyComposition (compChangeOfVariables 0 M N ⟨k, blocks_fun⟩ H).snd (fun _j => z) ⟨i✝, hin✝⟩
|
87e4d5de6857eb93
|
not_mulDissociated
|
Mathlib/Combinatorics/Additive/Dissociation.lean
|
@[to_additive (attr := simp)]
lemma not_mulDissociated :
¬ MulDissociated s ↔
∃ t : Finset α, ↑t ⊆ s ∧ ∃ u : Finset α, ↑u ⊆ s ∧ t ≠ u ∧ ∏ x ∈ t, x = ∏ x ∈ u, x
|
α : Type u_1
inst✝ : CommGroup α
s : Set α
⊢ (∃ x, ↑x ⊆ s ∧ ∃ x_1, ↑x_1 ⊆ s ∧ ∏ x ∈ x, x = ∏ x ∈ x_1, x ∧ ¬x = x_1) ↔
∃ t, ↑t ⊆ s ∧ ∃ u, ↑u ⊆ s ∧ ¬t = u ∧ ∏ x ∈ t, x = ∏ x ∈ u, x
|
aesop
|
no goals
|
6251bda22602c701
|
Lean.Grind.ite_true_false
|
Mathlib/.lake/packages/lean4/src/lean/Init/Grind/Norm.lean
|
theorem ite_true_false {_ : Decidable p} : (ite p True False) = p
|
p : Prop
x✝ : Decidable p
⊢ (if p then True else False) = p
|
by_cases p <;> simp
|
no goals
|
9df15ab4f0db74ad
|
GromovHausdorff.ghDist_le_of_approx_subsets
|
Mathlib/Topology/MetricSpace/GromovHausdorff.lean
|
theorem ghDist_le_of_approx_subsets {s : Set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
(H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) : ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃
|
X : Type u
inst✝⁵ : MetricSpace X
inst✝⁴ : CompactSpace X
inst✝³ : Nonempty X
Y : Type v
inst✝² : MetricSpace Y
inst✝¹ : CompactSpace Y
inst✝ : Nonempty Y
s : Set X
Φ : ↑s → Y
ε₁ ε₂ ε₃ : ℝ
hs : ∀ (x : X), ∃ y ∈ s, dist x y ≤ ε₁
hs' : ∀ (x : Y), ∃ y, dist x (Φ y) ≤ ε₃
H : ∀ (x y : ↑s), |dist x y - dist (Φ x) (Φ y)| ≤ ε₂
δ : ℝ
δ0 : 0 < δ
xX : X
h✝ : xX ∈ univ
xs : X
hxs : xs ∈ s
Dxs : dist xX xs ≤ ε₁
sne : s.Nonempty
this✝⁶ : Nonempty ↑s := Nonempty.to_subtype sne
this✝⁵ : 0 ≤ ε₂
this✝⁴ : ∀ (p q : ↑s), |dist p q - dist (Φ p) (Φ q)| ≤ 2 * (ε₂ / 2 + δ)
this✝³ : MetricSpace (X ⊕ Y) := glueMetricApprox (fun x => ↑x) (fun x => Φ x) (ε₂ / 2 + δ) ⋯ this✝⁴
Fl : X → X ⊕ Y := inl
Fr : Y → X ⊕ Y := inr
Il : Isometry Fl
Ir : Isometry Fr
this✝² : ghDist X Y ≤ hausdorffDist (range Fl) (range Fr)
this✝¹ : hausdorffDist (range Fl) (range Fr) ≤ hausdorffDist (range Fl) (Fl '' s) + hausdorffDist (Fl '' s) (range Fr)
this✝ :
hausdorffDist (Fl '' s) (range Fr) ≤
hausdorffDist (Fl '' s) (Fr '' range Φ) + hausdorffDist (Fr '' range Φ) (range Fr)
this : 0 ≤ ε₁
x : X
x✝ : x ∈ s
⊢ dist x x ≤ ε₁
|
simpa only [dist_self]
|
no goals
|
ca73aa8d17706ff3
|
Algebra.smoothLocus_eq_compl_support_inter
|
Mathlib/RingTheory/Smooth/Locus.lean
|
lemma smoothLocus_eq_compl_support_inter [EssFiniteType R A] :
smoothLocus R A = (Module.support A (H1Cotangent R A))ᶜ ∩ Module.freeLocus A (Ω[A⁄R])
|
R A : Type u
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : EssFiniteType R A
p : PrimeSpectrum A
this✝ : LocalizedModule p.asIdeal.primeCompl (Ω[A⁄R]) ≃ₗ[A] Ω[Localization.AtPrime p.asIdeal⁄R]
this :
LocalizedModule p.asIdeal.primeCompl (Ω[A⁄R]) ≃ₗ[Localization.AtPrime p.asIdeal] Ω[Localization.AtPrime p.asIdeal⁄R]
⊢ Module.Free (Localization.AtPrime p.asIdeal) (Ω[Localization.AtPrime p.asIdeal⁄R]) ↔
Module.Free (Localization.AtPrime p.asIdeal) (LocalizedModule p.asIdeal.primeCompl (Ω[A⁄R]))
|
exact ⟨fun H ↦ H.of_equiv' this.symm, fun H ↦ H.of_equiv' this⟩
|
no goals
|
ff988bcda6ac40cb
|
IsOpen.exists_smooth_support_eq
|
Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
|
theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :
∃ f : E → ℝ, f.support = s ∧ ContDiff ℝ ∞ f ∧ Set.range f ⊆ Set.Icc 0 1
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : IsOpen s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg : T = range g0
g : ℕ → E → ℝ := fun n => ↑(g0 n)
g_s : ∀ (n : ℕ), support (g n) ⊆ s
s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)
g_smooth : ∀ (n : ℕ), ContDiff ℝ ∞ (g n)
g_comp_supp : ∀ (n : ℕ), HasCompactSupport (g n)
g_nonneg : ∀ (n : ℕ) (x : E), 0 ≤ g n x
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
δc : HasSum δ c
c_lt : c < 1
n : ℕ
R : ℕ → ℝ
hR : ∀ (i : ℕ) (x : E), ‖iteratedFDeriv ℝ i (fun x => g n x) x‖ ≤ R i
M : ℝ := (Finset.image R (Finset.range (n + 1))).max' ⋯ ⊔ 1
δnpos : 0 < δ n
IR : ∀ i ≤ n, R i ≤ M
⊢ ∃ r, 0 < r ∧ ∀ i ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ i (r • g n) x‖ ≤ ↑(δ n)
|
refine ⟨M⁻¹ * δ n, by positivity, fun i hi x => ?_⟩
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : IsOpen s
h's : s.Nonempty
ι : Type (max 0 u_1) := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f ∧ range f ⊆ Icc 0 1 }
T : Set ι
T_count : T.Countable
hT : ⋃ f ∈ T, support ↑f = s
g0 : ℕ → ι
hg : T = range g0
g : ℕ → E → ℝ := fun n => ↑(g0 n)
g_s : ∀ (n : ℕ), support (g n) ⊆ s
s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)
g_smooth : ∀ (n : ℕ), ContDiff ℝ ∞ (g n)
g_comp_supp : ∀ (n : ℕ), HasCompactSupport (g n)
g_nonneg : ∀ (n : ℕ) (x : E), 0 ≤ g n x
δ : ℕ → ℝ≥0
δpos : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
δc : HasSum δ c
c_lt : c < 1
n : ℕ
R : ℕ → ℝ
hR : ∀ (i : ℕ) (x : E), ‖iteratedFDeriv ℝ i (fun x => g n x) x‖ ≤ R i
M : ℝ := (Finset.image R (Finset.range (n + 1))).max' ⋯ ⊔ 1
δnpos : 0 < δ n
IR : ∀ i ≤ n, R i ≤ M
i : ℕ
hi : i ≤ n
x : E
⊢ ‖iteratedFDeriv ℝ i ((M⁻¹ * ↑(δ n)) • g n) x‖ ≤ ↑(δ n)
|
415898bf3b6091fb
|
Plausible.TotalFunction.apply_eq_dlookup
|
Mathlib/Testing/Plausible/Functions.lean
|
theorem apply_eq_dlookup (m : List (Σ _ : α, β)) (y : β) (x : α) :
(withDefault m y).apply x = (m.dlookup x).getD y
|
case neg.a
α : Type u
β : Type v
inst✝ : DecidableEq α
y : β
x : α
m : List ((_ : α) × β)
ih : Option.map Sigma.snd (List.find? (fun x_1 => decide (x_1.fst = x)) m) = List.dlookup x m
fst : α
snd : β
heq : ¬fst = x
⊢ ⟨fst, snd⟩.fst ≠ x
|
simp [heq]
|
no goals
|
ff22460d80a20ae8
|
MulChar.apply_mem_algebraAdjoin_of_pow_eq_one
|
Mathlib/NumberTheory/MulChar/Lemmas.lean
|
/-- The values of a multiplicative character `χ` such that `χ^n = 1` are contained in `ℤ[μ]` when
`μ` is a primitive `n`th root of unity. -/
lemma apply_mem_algebraAdjoin_of_pow_eq_one {χ : MulChar F R} {n : ℕ} [NeZero n] (hχ : χ ^ n = 1)
{μ : R} (hμ : IsPrimitiveRoot μ n) (a : F) :
χ a ∈ Algebra.adjoin ℤ {μ}
|
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Finite F
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
χ : MulChar F R
n : ℕ
inst✝ : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
⊢ χ a ∈ Algebra.adjoin ℤ {μ}
|
rcases eq_or_ne a 0 with rfl | h
|
case inl
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Finite F
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
χ : MulChar F R
n : ℕ
inst✝ : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
⊢ χ 0 ∈ Algebra.adjoin ℤ {μ}
case inr
F : Type u_1
inst✝⁴ : Field F
inst✝³ : Finite F
R : Type u_2
inst✝² : CommRing R
inst✝¹ : IsDomain R
χ : MulChar F R
n : ℕ
inst✝ : NeZero n
hχ : χ ^ n = 1
μ : R
hμ : IsPrimitiveRoot μ n
a : F
h : a ≠ 0
⊢ χ a ∈ Algebra.adjoin ℤ {μ}
|
7781d0f060e3ce37
|
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)
|
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
⊢ Injective ⇑(rTensor N M'.subtype)
|
apply (injective_iff_map_eq_zero _).mpr
|
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
⊢ ∀ (a : ↥M' ⊗[R] N), (rTensor N M'.subtype) a = 0 → a = 0
|
3e74d15b86771baa
|
AkraBazziRecurrence.eventually_atTop_sumTransform_ge
|
Mathlib/Computability/AkraBazzi/AkraBazzi.lean
|
lemma eventually_atTop_sumTransform_ge :
∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, c * g n ≤ sumTransform (p a b) g (r i n) n
|
case h.h.hc
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 > p a b + 1
u : ℕ
hu : u ∈ Ico (r i n) n
⊢ 0 < ↑u ^ (p a b + 1)
|
rw [Finset.mem_Ico] at hu
|
case h.h.hc
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
c₃ : ℝ
hc₃_mem : c₃ ∈ Set.Ioo 0 1
hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hc₁_pos : 0 < c₁
hc₃' : 0 < 1 - c₃
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u
hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : 0 > p a b + 1
u : ℕ
hu : r i n ≤ u ∧ u < n
⊢ 0 < ↑u ^ (p a b + 1)
|
aff66f2e59f80837
|
Ordinal.principal_mul_of_le_two
|
Mathlib/SetTheory/Ordinal/Principal.lean
|
theorem principal_mul_of_le_two (ho : o ≤ 2) : Principal (· * ·) o
|
case inl.inl
o : Ordinal.{u}
ho✝¹ : o ≤ 2
ho✝ : o ≤ 1
ho : o < 1
⊢ Principal (fun x1 x2 => x1 * x2) 0
|
exact principal_zero
|
no goals
|
d5ce674088ebbc9e
|
MeasureTheory.SimpleFunc.map_setToSimpleFunc
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem map_setToSimpleFunc (T : Set α → F →L[ℝ] F') (h_add : FinMeasAdditive μ T) {f : α →ₛ G}
(hf : Integrable f μ) {g : G → F} (hg : g 0 = 0) :
(f.map g).setToSimpleFunc T = ∑ x ∈ f.range, T (f ⁻¹' {x}) (g x)
|
case neg.h_disj
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ⊤
a : α
hb : f a ∈ f.range
h0 : ¬g (f a) = 0
h_left_eq : (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = (T (⇑f ⁻¹' ↑(filter (fun b => g b = g (f a)) f.range))) (g (f a))
h_left_eq' :
(T (⇑f ⁻¹' ↑(filter (fun b => g b = g (f a)) f.range))) (g (f a)) =
(T (⋃ y ∈ filter (fun b => g b = g (f a)) f.range, ⇑f ⁻¹' {y})) (g (f a))
i : G
_j : i ∈ filter (fun b => g b = g (f a)) f.range
hi : G
a✝ : hi ∈ filter (fun b => g b = g (f a)) f.range
hij : i ≠ hi
x : α
hx : x ∈ ⇑f ⁻¹' {i} ∩ ⇑f ⁻¹' {hi}
⊢ x ∈ ∅
|
rw [Set.mem_inter_iff, Set.mem_preimage, Set.mem_preimage, Set.mem_singleton_iff,
Set.mem_singleton_iff] at hx
|
case neg.h_disj
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ⊤
a : α
hb : f a ∈ f.range
h0 : ¬g (f a) = 0
h_left_eq : (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = (T (⇑f ⁻¹' ↑(filter (fun b => g b = g (f a)) f.range))) (g (f a))
h_left_eq' :
(T (⇑f ⁻¹' ↑(filter (fun b => g b = g (f a)) f.range))) (g (f a)) =
(T (⋃ y ∈ filter (fun b => g b = g (f a)) f.range, ⇑f ⁻¹' {y})) (g (f a))
i : G
_j : i ∈ filter (fun b => g b = g (f a)) f.range
hi : G
a✝ : hi ∈ filter (fun b => g b = g (f a)) f.range
hij : i ≠ hi
x : α
hx : f x = i ∧ f x = hi
⊢ x ∈ ∅
|
f53dabe35f13a020
|
Int.ediv_left_inj
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean
|
theorem ediv_left_inj {a b d : Int}
(hda : d ∣ a) (hdb : d ∣ b) : a / d = b / d ↔ a = b
|
a b d : Int
hda : d ∣ a
hdb : d ∣ b
h : a / d = b / d
⊢ a = b
|
rw [← Int.mul_ediv_cancel' hda, ← Int.mul_ediv_cancel' hdb, h]
|
no goals
|
68b82413b539d76b
|
MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂
|
Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
|
theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
(f : ℝⁿ⁺¹ → Eⁿ⁺¹)
(f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
(s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
(Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
(Hi : IntegrableOn (∑ i, f' · (e i) i) (Box.Icc I)) :
(∫ x in Box.Icc I, ∑ i, f' x (e i) i) =
∑ i : Fin (n + 1),
((∫ x in Box.Icc (I.face i), f (i.insertNth (I.upper i) x) i) -
∫ x in Box.Icc (I.face i), f (i.insertNth (I.lower i) x) i)
|
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (Box.Icc I) volume
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : ∀ (x : Fin (n + 1)), Tendsto (fun i => (Box.lower ∘ ⇑J) i x) atTop (𝓝 (I.lower x))
hJu : ∀ (x : Fin (n + 1)), Tendsto (fun i => (Box.upper ∘ ⇑J) i x) atTop (𝓝 (I.upper x))
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
∀ (k : ℕ),
∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i : Fin (n + 1), (f' x) (e i) i =
∑ i : Fin (n + 1),
((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
i : Fin (n + 1)
c : ℕ → ℝ
d : ℝ
hc : ∀ (k : ℕ), c k ∈ Set.Icc (I.lower i) (I.upper i)
hcd : Tendsto c atTop (𝓝 d)
hd : d ∈ Set.Icc (I.lower i) (I.upper i)
Hic : ∀ (k : ℕ), IntegrableOn (fun x => f (i.insertNth (c k) x) i) (Box.Icc (I.face i)) volume
Hid : IntegrableOn (fun x => f (i.insertNth d x) i) (Box.Icc (I.face i)) volume
H :
Tendsto (fun k => ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth d x) i) atTop
(𝓝 (∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth d x) i))
ε : ℝ
εpos : 0 < ε
hvol_pos : ∀ (J : Box (Fin n)), 0 < ∏ j : Fin n, (J.upper j - J.lower j)
δ : ℝ
δpos : δ > 0
hδ :
∀ x ∈ Box.Icc I,
∀ y ∈ Box.Icc I, dist x y ≤ δ → dist (f x) (f y) ≤ ε / ∏ j : Fin n, ((I.face i).upper j - (I.face i).lower j)
k : ℕ
hk : dist (c k) d < δ
Hsub : Box.Icc ((J k).face i) ⊆ Box.Icc (I.face i)
⊢ ‖∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth d x) i - f (i.insertNth (c k) x) i‖ ≤
(ε / ∏ j : Fin n, ((I.face i).upper j - (I.face i).lower j)) * (volume (Box.Icc ((J k).face i))).toReal
|
refine norm_setIntegral_le_of_norm_le_const' (((J k).face i).measure_Icc_lt_top _)
((J k).face i).measurableSet_Icc fun x hx => ?_
|
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
n : ℕ
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hc : ContinuousOn f (Box.Icc I)
Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x
Hi : IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (Box.Icc I) volume
J : ℕ →o Box (Fin (n + 1))
hJ_sub : ∀ (n_1 : ℕ), Box.Icc (J n_1) ⊆ Box.Ioo I
hJl : ∀ (x : Fin (n + 1)), Tendsto (fun i => (Box.lower ∘ ⇑J) i x) atTop (𝓝 (I.lower x))
hJu : ∀ (x : Fin (n + 1)), Tendsto (fun i => (Box.upper ∘ ⇑J) i x) atTop (𝓝 (I.upper x))
hJ_sub' : ∀ (k : ℕ), Box.Icc (J k) ⊆ Box.Icc I
hJ_le : ∀ (k : ℕ), J k ≤ I
HcJ : ∀ (k : ℕ), ContinuousOn f (Box.Icc (J k))
HdJ : ∀ (k : ℕ), ∀ x ∈ Box.Icc (J k) \ s, HasFDerivWithinAt f (f' x) (Box.Icc (J k)) x
HiJ : ∀ (k : ℕ), IntegrableOn (fun x => ∑ i : Fin (n + 1), (f' x) (e i) i) (Box.Icc (J k)) volume
HJ_eq :
∀ (k : ℕ),
∫ (x : Fin (n + 1) → ℝ) in Box.Icc (J k), ∑ i : Fin (n + 1), (f' x) (e i) i =
∑ i : Fin (n + 1),
((∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).upper i) x) i) -
∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth ((J k).lower i) x) i)
i : Fin (n + 1)
c : ℕ → ℝ
d : ℝ
hc : ∀ (k : ℕ), c k ∈ Set.Icc (I.lower i) (I.upper i)
hcd : Tendsto c atTop (𝓝 d)
hd : d ∈ Set.Icc (I.lower i) (I.upper i)
Hic : ∀ (k : ℕ), IntegrableOn (fun x => f (i.insertNth (c k) x) i) (Box.Icc (I.face i)) volume
Hid : IntegrableOn (fun x => f (i.insertNth d x) i) (Box.Icc (I.face i)) volume
H :
Tendsto (fun k => ∫ (x : Fin n → ℝ) in Box.Icc ((J k).face i), f (i.insertNth d x) i) atTop
(𝓝 (∫ (x : Fin n → ℝ) in Box.Icc (I.face i), f (i.insertNth d x) i))
ε : ℝ
εpos : 0 < ε
hvol_pos : ∀ (J : Box (Fin n)), 0 < ∏ j : Fin n, (J.upper j - J.lower j)
δ : ℝ
δpos : δ > 0
hδ :
∀ x ∈ Box.Icc I,
∀ y ∈ Box.Icc I, dist x y ≤ δ → dist (f x) (f y) ≤ ε / ∏ j : Fin n, ((I.face i).upper j - (I.face i).lower j)
k : ℕ
hk : dist (c k) d < δ
Hsub : Box.Icc ((J k).face i) ⊆ Box.Icc (I.face i)
x : Fin n → ℝ
hx : x ∈ Box.Icc ((J k).face i)
⊢ ‖f (i.insertNth d x) i - f (i.insertNth (c k) x) i‖ ≤ ε / ∏ j : Fin n, ((I.face i).upper j - (I.face i).lower j)
|
d25a0c4e5a28a1f3
|
max_zero_add_max_neg_zero_eq_abs_self
|
Mathlib/Algebra/Order/Group/Abs.lean
|
theorem max_zero_add_max_neg_zero_eq_abs_self (a : α) : max a 0 + max (-a) 0 = |a|
|
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
a : α
⊢ |a| = a ⊔ 0 + -a ⊔ 0
|
rcases le_total 0 a with (ha | ha) <;> simp [ha]
|
no goals
|
c19353226e58057f
|
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