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
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|---|---|---|---|---|---|---|
Nat.bitwise_lt_two_pow
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Lemmas.lean
|
theorem bitwise_lt_two_pow (left : x < 2^n) (right : y < 2^n) : (Nat.bitwise f x y) < 2^n
|
case zero
f : Bool → Bool → Bool
x y : Nat
left : x < 2 ^ 0
right : y < 2 ^ 0
⊢ bitwise f x y < 2 ^ 0
|
simp only [eq_0_of_lt] at left right
|
case zero
f : Bool → Bool → Bool
x y : Nat
left : x = 0
right : y = 0
⊢ bitwise f x y < 2 ^ 0
|
b3846a598e8ae70f
|
MeasureTheory.mul_upcrossingsBefore_le
|
Mathlib/Probability/Martingale/Upcrossing.lean
|
theorem mul_upcrossingsBefore_le (hf : a ≤ f N ω) (hab : a < b) :
(b - a) * upcrossingsBefore a b f N ω ≤
∑ k ∈ Finset.range N, upcrossingStrat a b f N k ω * (f (k + 1) - f k) ω
|
case neg
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n ∈ Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
x✝ : i ∈ Finset.range N
hi : i ∉ Finset.range (upcrossingsBefore a b f N ω)
hi' : ¬i = upcrossingsBefore a b f N ω
⊢ upcrossingsBefore a b f N ω < i
|
rw [Finset.mem_range, not_lt] at hi
|
case neg
Ω : Type u_1
a b : ℝ
f : ℕ → Ω → ℝ
N : ℕ
ω : Ω
hf : a ≤ f N ω
hab : a < b
hN : ¬N = 0
h₁ :
∀ (k : ℕ),
∑ n ∈ Finset.range N,
(Set.Ico (lowerCrossingTime a b f N k ω) (upperCrossingTime a b f N (k + 1) ω)).indicator
(fun m => f (m + 1) ω - f m ω) n =
stoppedValue f (upperCrossingTime a b f N (k + 1)) ω - stoppedValue f (lowerCrossingTime a b f N k) ω
i : ℕ
x✝ : i ∈ Finset.range N
hi : upcrossingsBefore a b f N ω ≤ i
hi' : ¬i = upcrossingsBefore a b f N ω
⊢ upcrossingsBefore a b f N ω < i
|
fc87e1e3d79bf5b5
|
HomologicalComplex.extend.rightHomologyData.d_comp_desc_eq_zero_iff'
|
Mathlib/Algebra/Homology/Embedding/ExtendHomology.lean
|
lemma d_comp_desc_eq_zero_iff' ⦃W : C⦄ (f' : cocone.pt ⟶ K.X k)
(hf' : cocone.π ≫ f' = K.d j k)
(f'' : cocone.pt ⟶ (K.extend e).X k')
(hf'' : (extendXIso K e hj').hom ≫ cocone.π ≫ f'' = (K.extend e).d j' k')
(φ : W ⟶ cocone.pt) :
φ ≫ f' = 0 ↔ φ ≫ f'' = 0
|
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : ¬c.Rel j k
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0
|
have h₁ : f' = 0 := by
apply Cofork.IsColimit.hom_ext hcocone
simp only [hf', comp_zero, K.shape _ _ hjk]
|
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : ¬c.Rel j k
h₁ : f' = 0
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0
|
0908a43e60e3dcb1
|
CategoryTheory.ShortComplex.ShortExact.hasProjectiveDimensionLT_X₃
|
Mathlib/CategoryTheory/Abelian/Projective/Dimension.lean
|
lemma hasProjectiveDimensionLT_X₃ (h₁ : HasProjectiveDimensionLT S.X₁ n)
(h₂ : HasProjectiveDimensionLT S.X₂ (n + 1)) :
HasProjectiveDimensionLT S.X₃ (n + 1)
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : S.ShortExact
n : ℕ
h₁ : HasProjectiveDimensionLT S.X₁ n
h₂ : HasProjectiveDimensionLT S.X₂ (n + 1)
this : HasExt C := HasExt.standard C
i : ℕ
hi : n + 1 ≤ i + 1
Y : C
x₁ : Ext S.X₁ Y i
⊢ n ≤ i
|
omega
|
no goals
|
d12926ed9ced8939
|
Matrix.PosSemidef.intCast
|
Mathlib/LinearAlgebra/Matrix/PosDef.lean
|
theorem intCast [StarOrderedRing R] [DecidableEq n] (d : ℤ) (hd : 0 ≤ d) :
PosSemidef (d : Matrix n n R) :=
⟨isHermitian_intCast _, fun x => by
simp only [intCast_mulVec, dotProduct_smul]
rw [Int.cast_smul_eq_zsmul]
exact zsmul_nonneg (dotProduct_star_self_nonneg _) hd⟩
|
n : Type u_2
R : Type u_3
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarRing R
inst✝¹ : StarOrderedRing R
inst✝ : DecidableEq n
d : ℤ
hd : 0 ≤ d
x : n → R
⊢ 0 ≤ ↑d • (star x ⬝ᵥ x)
|
rw [Int.cast_smul_eq_zsmul]
|
n : Type u_2
R : Type u_3
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarRing R
inst✝¹ : StarOrderedRing R
inst✝ : DecidableEq n
d : ℤ
hd : 0 ≤ d
x : n → R
⊢ 0 ≤ d • (star x ⬝ᵥ x)
|
ca53ab52753daf24
|
t1Space_TFAE
|
Mathlib/Topology/Separation/Basic.lean
|
theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] :
List.TFAE [T1Space X,
∀ x, IsClosed ({ x } : Set X),
∀ x, IsOpen ({ x }ᶜ : Set X),
Continuous (@CofiniteTopology.of X),
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x,
∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ y ∉ U,
∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
∀ ⦃x y : X⦄, x ⤳ y → x = y]
|
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE
|
tfae_have 2 ↔ 3 := by
simp only [isOpen_compl_iff]
|
X : Type u
inst✝ : TopologicalSpace X
tfae_1_iff_2 : T1Space X ↔ ∀ (x : X), IsClosed {x}
tfae_2_iff_3 : (∀ (x : X), IsClosed {x}) ↔ ∀ (x : X), IsOpen {x}ᶜ
⊢ [T1Space X, ∀ (x : X), IsClosed {x}, ∀ (x : X), IsOpen {x}ᶜ, Continuous ⇑CofiniteTopology.of,
∀ ⦃x y : X⦄, x ≠ y → {y}ᶜ ∈ 𝓝 x, ∀ ⦃x y : X⦄, x ≠ y → ∃ s ∈ 𝓝 x, y ∉ s,
∀ ⦃x y : X⦄, x ≠ y → ∃ U, IsOpen U ∧ x ∈ U ∧ y ∉ U, ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y), ∀ ⦃x y : X⦄, x ⤳ y → x = y].TFAE
|
92ea821073dea3c3
|
FirstOrder.Language.equiv_between_cg
|
Mathlib/ModelTheory/PartialEquiv.lean
|
theorem equiv_between_cg (M_cg : Structure.CG L M) (N_cg : Structure.CG L N)
(g : L.FGEquiv M N)
(ext_dom : L.IsExtensionPair M N)
(ext_cod : L.IsExtensionPair N M) :
∃ f : M ≃[L] N, g ≤ f.toEmbedding.toPartialEquiv
|
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
ext_dom : L.IsExtensionPair M N
ext_cod : L.IsExtensionPair N M
X : Set M
X_count : X.Countable
X_gen : (closure L).toFun X = ⊤
Y : Set N
Y_count : Y.Countable
Y_gen : (closure L).toFun Y = ⊤
x✝⁵ : Countable ↑X
x✝⁴ : Encodable ↑X
x✝³ : Countable ↑Y
x✝² : Encodable ↑Y
D : ↑X ⊕ ↑Y → Order.Cofinal (L.FGEquiv M N) :=
fun p => Sum.recOn p (fun x => ext_dom.definedAtLeft ↑x) fun y => ext_cod.definedAtRight ↑y
S : ℕ →o M ≃ₚ[L] N := { toFun := Subtype.val ∘ Order.sequenceOfCofinals g D, monotone' := ⋯ }
F : M ≃ₚ[L] N := partialEquivLimit S
x✝¹ : X ⊆ ↑F.dom
x✝ : Y ⊆ ↑F.cod
⊢ F.dom = ⊤
|
rwa [← top_le_iff, ← X_gen, Substructure.closure_le]
|
no goals
|
d1b972cd720b6e06
|
Set.Iio_False
|
Mathlib/Order/Interval/Set/Basic.lean
|
@[simp] lemma Iio_False : Iio False = ∅
|
⊢ Iio False = ∅
|
aesop
|
no goals
|
35705b919cf0ed0c
|
Nat.choose_succ_right_eq
|
Mathlib/Data/Nat/Choose/Basic.lean
|
theorem choose_succ_right_eq (n k : ℕ) : choose n (k + 1) * (k + 1) = choose n k * (n - k)
|
n k : ℕ
⊢ (n + 1) * n.choose k = n.choose (k + 1) * (k + 1) + n.choose k * (k + 1)
|
rw [← Nat.add_mul, Nat.add_comm (choose _ _), ← choose_succ_succ, succ_mul_choose_eq]
|
no goals
|
14fca551c1aaeff0
|
SimpleGraph.is3Clique_iff_exists_cycle_length_three
|
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
theorem is3Clique_iff_exists_cycle_length_three :
(∃ s : Finset α, G.IsNClique 3 s) ↔ ∃ (u : α) (w : G.Walk u u), w.IsCycle ∧ w.length = 3
|
α : Type u_1
G : SimpleGraph α
⊢ (∃ s, G.IsNClique 3 s) ↔ ∃ u w, w.IsCycle ∧ w.length = 3
|
classical
simp_rw [is3Clique_iff, isCycle_def]
exact
⟨(fun ⟨_, a, _, _, hab, hac, hbc, _⟩ => ⟨a, cons hab (cons hbc (cons hac.symm nil)), by aesop⟩),
(fun ⟨_, .cons hab (.cons hbc (.cons hca nil)), _, _⟩ => ⟨_, _, _, _, hab, hca.symm, hbc, rfl⟩)⟩
|
no goals
|
f922e3fda39a22ed
|
Mathlib.Tactic.Ring.pow_one_cast
|
Mathlib/Tactic/Ring/Basic.lean
|
theorem pow_one_cast (a : R) : a ^ (nat_lit 1).rawCast = a
|
R : Type u_1
inst✝ : CommSemiring R
a : R
⊢ a ^ Nat.rawCast 1 = a
|
simp
|
no goals
|
bdef837d0eaffab1
|
isCoprime_of_gcd_eq_one_of_FLT
|
Mathlib/NumberTheory/FLT/Basic.lean
|
lemma isCoprime_of_gcd_eq_one_of_FLT {n : ℕ} {a b c : ℤ} (Hgcd : Finset.gcd {a, b, c} id = 1)
(HF : a ^ n + b ^ n + c ^ n = 0) : IsCoprime a b
|
case inr.refine_1
n : ℕ
a b c : ℤ
Hgcd : {a, b, c}.gcd id = 1
HF : a ^ n + b ^ n + c ^ n = 0
hn : n ≠ 0
⊢ ¬(a = 0 ∧ b = 0)
|
rintro ⟨rfl, rfl⟩
|
case inr.refine_1.intro
n : ℕ
c : ℤ
hn : n ≠ 0
Hgcd : {0, 0, c}.gcd id = 1
HF : 0 ^ n + 0 ^ n + c ^ n = 0
⊢ False
|
cb0f80522bad7b59
|
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 = ⊥
|
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 < f i, b = ⊥
|
intro b hb
|
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
⊢ b = ⊥
|
116210e72cdaa30b
|
AbsoluteValue.listSum_le
|
Mathlib/Algebra/Order/AbsoluteValue/Basic.lean
|
/-- The triangle inequality for an `AbsoluteValue` applied to a list. -/
lemma listSum_le (l : List R) : abv l.sum ≤ (l.map abv).sum
|
case nil
R : Type u_5
S : Type u_6
inst✝¹ : Semiring R
inst✝ : OrderedSemiring S
abv : AbsoluteValue R S
⊢ abv [].sum ≤ (List.map ⇑abv []).sum
|
simp
|
no goals
|
228ddf2540343717
|
Ordnode.Valid'.eraseMax_aux
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem Valid'.eraseMax_aux {s l x r o₁ o₂} (H : Valid' o₁ (.node s l x r) o₂) :
Valid' o₁ (@eraseMax α (.node' l x r)) ↑(findMax' x r) ∧
size (.node' l x r) = size (eraseMax (.node' l x r)) + 1
|
α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (Ordnode.node s l x r) o₂
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
|
have := H.2.eq_node'
|
α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (Ordnode.node s l x r) o₂
this : Ordnode.node s l x r = l.node' x r
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
|
df61be37cf649dc4
|
Fin.isAddFreimanIso_Iic
|
Mathlib/Combinatorics/Additive/FreimanHom.lean
|
/-- **No wrap-around principle**.
The first `k + 1` elements of `Fin (n + 1)` are `m`-Freiman isomorphic to the first `k + 1` elements
of `ℕ` assuming there is no wrap-around. -/
lemma isAddFreimanIso_Iic (hm : m ≠ 0) (hkmn : m * k ≤ n) :
IsAddFreimanIso m (Iic (k : Fin (n + 1))) (Iic k) val where
bijOn.left
|
k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
⊢ (map val s).sum = (map val t).sum ↔
(Nat.castRingHom (Fin (n + 1))) (map val s).sum = (Nat.castRingHom (Fin (n + 1))) (map val t).sum
|
have {u : Multiset (Fin (n + 1))} (huk : ∀ x ∈ u, x ≤ k) (hu : card u = m) :
(u.map val).sum < (n + 1) := Nat.lt_succ_iff.2 <| hkmn.trans' <| by
rw [← hu, ← card_map]
refine sum_le_card_nsmul (u.map val) k ?_
simpa [le_iff_val_le_val, -val_fin_le, Nat.mod_eq_of_lt, aux hm hkmn] using huk
|
k m n : ℕ
hm : m ≠ 0
hkmn : m * k ≤ n
s t : Multiset (Fin (n + 1))
hsA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ s → x ∈ Iic ↑k
htA : ∀ ⦃x : Fin (n + 1)⦄, x ∈ t → x ∈ Iic ↑k
hs : s.card = m
ht : t.card = m
this✝ : ∀ (u : Multiset (Fin (n + 1))), (Nat.castRingHom (Fin (n + 1))) (map val u).sum = u.sum
this : ∀ {u : Multiset (Fin (n + 1))}, (∀ x ∈ u, x ≤ ↑k) → u.card = m → (map val u).sum < n + 1
⊢ (map val s).sum = (map val t).sum ↔
(Nat.castRingHom (Fin (n + 1))) (map val s).sum = (Nat.castRingHom (Fin (n + 1))) (map val t).sum
|
d8123ea7c56d3190
|
LieAlgebra.nilpotent_of_nilpotent_quotient
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h₁ : I ≤ center R L)
(h₂ : IsNilpotent (L ⧸ I)) : IsNilpotent L
|
case h
R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
I : LieIdeal R L
h₁ : I ≤ center R L
h₂ : ∃ k, lowerCentralSeries R (L ⧸ I) (L ⧸ I) k = ⊥
k : ℕ
hk : lowerCentralSeries R (L ⧸ I) (L ⧸ I) k = ⊥
⊢ lowerCentralSeries R L (L ⧸ I) k = ⊥
|
simp [← LieSubmodule.toSubmodule_inj, coe_lowerCentralSeries_ideal_quot_eq, hk]
|
no goals
|
93bdd301b98959d0
|
ValuationRing.isFractionRing_iff
|
Mathlib/RingTheory/Valuation/ValuationRing.lean
|
theorem isFractionRing_iff [ValuationRing 𝒪] :
IsFractionRing 𝒪 K ↔
(∀ (x : K), ∃ a : 𝒪, x = algebraMap 𝒪 K a ∨ x⁻¹ = algebraMap 𝒪 K a) ∧
Function.Injective (algebraMap 𝒪 K)
|
case refine_1.inl.intro
𝒪 : Type u
K : Type v
inst✝⁴ : CommRing 𝒪
inst✝³ : IsDomain 𝒪
inst✝² : Field K
inst✝¹ : Algebra 𝒪 K
inst✝ : ValuationRing 𝒪
h : IsFractionRing 𝒪 K
x : K
a : 𝒪
e : (algebraMap 𝒪 K) a = x
⊢ ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
case refine_1.inr.intro
𝒪 : Type u
K : Type v
inst✝⁴ : CommRing 𝒪
inst✝³ : IsDomain 𝒪
inst✝² : Field K
inst✝¹ : Algebra 𝒪 K
inst✝ : ValuationRing 𝒪
h : IsFractionRing 𝒪 K
x : K
a : 𝒪
e : (algebraMap 𝒪 K) a = x⁻¹
⊢ ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a
|
exacts [⟨a, .inl e.symm⟩, ⟨a, .inr e.symm⟩]
|
no goals
|
6f74c68f271272bd
|
MeasureTheory.condExp_nonneg
|
Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
|
lemma condExp_nonneg (hf : 0 ≤ᵐ[μ] f) : 0 ≤ᵐ[μ] μ[f|m]
|
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
hf : 0 ≤ᶠ[ae μ] f
⊢ 0 ≤ᶠ[ae μ] μ[f|m]
|
by_cases hfint : Integrable f μ
|
case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
hf : 0 ≤ᶠ[ae μ] f
hfint : Integrable f μ
⊢ 0 ≤ᶠ[ae μ] μ[f|m]
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝³ : NormedLatticeAddCommGroup E
inst✝² : CompleteSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : OrderedSMul ℝ E
hf : 0 ≤ᶠ[ae μ] f
hfint : ¬Integrable f μ
⊢ 0 ≤ᶠ[ae μ] μ[f|m]
|
513c855a0f76e22e
|
Mathlib.Tactic.LinearCombination.lt_of_eq
|
Mathlib/Tactic/LinearCombination/Lemmas.lean
|
theorem lt_of_eq [OrderedCancelAddCommMonoid α] (p : (a:α) = b) (H : a' + b < b' + a) :
a' < b'
|
α : Type u_1
a a' b b' : α
inst✝ : OrderedCancelAddCommMonoid α
p : a = b
H : a' + b < b' + a
⊢ a' < b'
|
rwa [p, add_lt_add_iff_right] at H
|
no goals
|
90aa093ad49ea228
|
SzemerediRegularity.edgeDensity_star_not_uniform
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
|
theorem edgeDensity_star_not_uniform [Nonempty α]
(hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5)
(hε₁ : ε ≤ 1) {hU : U ∈ P.parts} {hV : V ∈ P.parts} (hUVne : U ≠ V) (hUV : ¬G.IsUniform ε U V) :
↑3 / ↑4 * ε ≤
|(∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), (G.edgeDensity ab.1 ab.2 : ℝ)) /
(#(star hP G ε hU V) * #(star hP G ε hV U)) -
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts,
(G.edgeDensity ab.1 ab.2 : ℝ)) / (16 : ℝ) ^ #P.parts|
|
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
hUVne : U ≠ V
hUV : ¬G.IsUniform ε U V
p : ℝ :=
(∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) /
(↑(#(star hP G ε hU V)) * ↑(#(star hP G ε hV U)))
q : ℝ :=
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) /
(4 ^ #P.parts * 4 ^ #P.parts)
r : ℝ := ↑(G.edgeDensity ((star hP G ε hU V).biUnion id) ((star hP G ε hV U).biUnion id))
s : ℝ := ↑(G.edgeDensity (G.nonuniformWitness ε U V) (G.nonuniformWitness ε V U))
t : ℝ := ↑(G.edgeDensity U V)
hrs : |r - s| ≤ ε / 5
⊢ 3 / 4 * ε ≤ |p - q|
|
have hst : ε ≤ |s - t| := by
unfold s t
exact mod_cast G.nonuniformWitness_spec hUVne hUV
|
α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
U V : Finset α
inst✝ : Nonempty α
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
hε₁ : ε ≤ 1
hU : U ∈ P.parts
hV : V ∈ P.parts
hUVne : U ≠ V
hUV : ¬G.IsUniform ε U V
p : ℝ :=
(∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) /
(↑(#(star hP G ε hU V)) * ↑(#(star hP G ε hV U)))
q : ℝ :=
(∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) /
(4 ^ #P.parts * 4 ^ #P.parts)
r : ℝ := ↑(G.edgeDensity ((star hP G ε hU V).biUnion id) ((star hP G ε hV U).biUnion id))
s : ℝ := ↑(G.edgeDensity (G.nonuniformWitness ε U V) (G.nonuniformWitness ε V U))
t : ℝ := ↑(G.edgeDensity U V)
hrs : |r - s| ≤ ε / 5
hst : ε ≤ |s - t|
⊢ 3 / 4 * ε ≤ |p - q|
|
96200eb905ae7b75
|
Nat.shiftLeft'_false
|
Mathlib/Data/Nat/Bits.lean
|
@[simp]
lemma shiftLeft'_false : ∀ n, shiftLeft' false m n = m <<< n
| 0 => rfl
| n + 1 => by
have : 2 * (m * 2^n) = 2^(n+1)*m
|
m n : ℕ
⊢ m * 2 ^ n.succ = 2 ^ (n + 1) * m
|
simp
|
no goals
|
79783b30dc8fc6e9
|
Cardinal.nat_add_eq
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem nat_add_eq {a : Cardinal} (n : ℕ) (ha : ℵ₀ ≤ a) : n + a = a
|
a : Cardinal.{u_1}
n : ℕ
ha : ℵ₀ ≤ a
⊢ ↑n + a = a
|
rw [add_comm, add_nat_eq n ha]
|
no goals
|
9933e59bb311e99c
|
Submodule.span_insert
|
Mathlib/LinearAlgebra/Span/Defs.lean
|
theorem span_insert (x) (s : Set M) : span R (insert x s) = (R ∙ x) ⊔ span R s
|
R : Type u_1
M : Type u_4
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
s : Set M
⊢ span R (insert x s) = span R {x} ⊔ span R s
|
rw [insert_eq, span_union]
|
no goals
|
b1ddb9e984e88013
|
CategoryTheory.Limits.MonoCoprod.mono_binaryCofanSum_inr'
|
Mathlib/CategoryTheory/Limits/MonoCoprod.lean
|
lemma mono_binaryCofanSum_inr' [MonoCoprod C] (inr : c₂.pt ⟶ c.pt)
(hinr : ∀ (i₂ : I₂), c₂.inj i₂ ≫ inr = c.inj (Sum.inr i₂)) :
Mono inr
|
C : Type u_1
inst✝¹ : Category.{u_4, u_1} C
I₁ : Type u_2
I₂ : Type u_3
X : I₁ ⊕ I₂ → C
c : Cofan X
c₁ : Cofan (X ∘ Sum.inl)
c₂ : Cofan (X ∘ Sum.inr)
hc : IsColimit c
hc₁ : IsColimit c₁
hc₂ : IsColimit c₂
inst✝ : MonoCoprod C
inr : c₂.pt ⟶ c.pt
hinr : ∀ (i₂ : I₂), c₂.inj i₂ ≫ inr = c.inj (Sum.inr i₂)
⊢ inr = (binaryCofanSum c c₁ c₂ hc₁ hc₂).inr
|
exact Cofan.IsColimit.hom_ext hc₂ _ _ (by simpa using hinr)
|
no goals
|
fb7a1916ba471baa
|
Order.pred_eq_iff_covBy
|
Mathlib/Order/SuccPred/Basic.lean
|
theorem pred_eq_iff_covBy : pred b = a ↔ a ⋖ b :=
⟨by
rintro rfl
exact pred_covBy _, CovBy.pred_eq⟩
|
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : PredOrder α
a b : α
inst✝ : NoMinOrder α
⊢ pred b = a → a ⋖ b
|
rintro rfl
|
α : Type u_1
inst✝² : PartialOrder α
inst✝¹ : PredOrder α
b : α
inst✝ : NoMinOrder α
⊢ pred b ⋖ b
|
a4bb58b00414f956
|
Stream'.WSeq.mem_rec_on
|
Mathlib/Data/Seq/WSeq.lean
|
theorem mem_rec_on {C : WSeq α → Prop} {a s} (M : a ∈ s) (h1 : ∀ b s', a = b ∨ C s' → C (cons b s'))
(h2 : ∀ s, C s → C (think s)) : C s
|
case none
α : Type u
C : WSeq α → Prop
a : α
s : WSeq α
M : a ∈ s
h1 : ∀ (b : α) (s' : WSeq α), a = b ∨ C s' → C (cons b s')
h2 : ∀ (s : WSeq α), C s → C s.think
s' : Seq (Option α)
h : some a = none ∨ C s'
⊢ C (Seq.cons none s')
|
apply h2
|
case none.a
α : Type u
C : WSeq α → Prop
a : α
s : WSeq α
M : a ∈ s
h1 : ∀ (b : α) (s' : WSeq α), a = b ∨ C s' → C (cons b s')
h2 : ∀ (s : WSeq α), C s → C s.think
s' : Seq (Option α)
h : some a = none ∨ C s'
⊢ C s'
|
33df928e7244e2c7
|
Mathlib.Tactic.Monoidal.evalHorizontalCompAux'_whisker
|
Mathlib/Tactic/CategoryTheory/Monoidal/Normalize.lean
|
theorem evalHorizontalCompAux'_whisker {f f' g g' h : C} {η : g ⟶ h} {θ : f' ⟶ g'}
{ηθ : g ⊗ f' ⟶ h ⊗ g'} {η₁ : f ⊗ (g ⊗ f') ⟶ f ⊗ (h ⊗ g')}
{η₂ : f ⊗ (g ⊗ f') ⟶ (f ⊗ h) ⊗ g'} {η₃ : (f ⊗ g) ⊗ f' ⟶ (f ⊗ h) ⊗ g'}
(e_ηθ : η ⊗ θ = ηθ) (e_η₁ : f ◁ ηθ = η₁)
(e_η₂ : η₁ ≫ (α_ _ _ _).inv = η₂) (e_η₃ : (α_ _ _ _).hom ≫ η₂ = η₃) :
(f ◁ η) ⊗ θ = η₃
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : MonoidalCategory C
f f' g g' h : C
η : g ⟶ h
θ : f' ⟶ g'
ηθ : g ⊗ f' ⟶ h ⊗ g'
η₁ : f ⊗ g ⊗ f' ⟶ f ⊗ h ⊗ g'
η₂ : f ⊗ g ⊗ f' ⟶ (f ⊗ h) ⊗ g'
η₃ : (f ⊗ g) ⊗ f' ⟶ (f ⊗ h) ⊗ g'
e_ηθ : η ⊗ θ = ηθ
e_η₁ : f ◁ ηθ = η₁
e_η₂ : η₁ ≫ (α_ f h g').inv = η₂
e_η₃ : (α_ f g f').hom ≫ η₂ = η₃
⊢ f ◁ η ⊗ θ = (α_ f g f').hom ≫ f ◁ (η ⊗ θ) ≫ (α_ f h g').inv
|
simp [MonoidalCategory.tensorHom_def]
|
no goals
|
ce213c10e1423cfd
|
t2Space_iff_disjoint_nhds
|
Mathlib/Topology/Separation/Hausdorff.lean
|
theorem t2Space_iff_disjoint_nhds : T2Space X ↔ Pairwise fun x y : X => Disjoint (𝓝 x) (𝓝 y)
|
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
x✝ : x ≠ y
⊢ (fun x y => ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Disjoint u v) x y ↔ (fun x y => Disjoint (𝓝 x) (𝓝 y)) x y
|
simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens y), exists_prop, ← exists_and_left,
and_assoc, and_comm, and_left_comm]
|
no goals
|
0f8dfb3d26d0b4f3
|
CategoryTheory.Pretriangulated.exists_iso_binaryBiproduct_of_distTriang
|
Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
|
lemma exists_iso_binaryBiproduct_of_distTriang (T : Triangle C) (hT : T ∈ distTriang C)
(zero : T.mor₃ = 0) :
∃ (e : T.obj₂ ≅ T.obj₁ ⊞ T.obj₃), T.mor₁ ≫ e.hom = biprod.inl ∧
T.mor₂ = e.hom ≫ biprod.snd
|
case intro
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ := binaryBiproductData T hT zero (section_ T.mor₂) ⋯ fst ⋯
⊢ T.mor₁ ≫ (biprod.uniqueUpToIso T.obj₁ T.obj₃ d.isBilimit).hom = biprod.inl
|
ext
|
case intro.h₀
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ := binaryBiproductData T hT zero (section_ T.mor₂) ⋯ fst ⋯
⊢ (T.mor₁ ≫ (biprod.uniqueUpToIso T.obj₁ T.obj₃ d.isBilimit).hom) ≫ biprod.fst = biprod.inl ≫ biprod.fst
case intro.h₁
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
zero : T.mor₃ = 0
this✝ : Epi T.mor₂
this : IsSplitEpi T.mor₂
fst : T.obj₂ ⟶ T.obj₁
hfst : 𝟙 T.obj₂ - T.mor₂ ≫ section_ T.mor₂ = fst ≫ T.mor₁
d : BinaryBiproductData T.obj₁ T.obj₃ := binaryBiproductData T hT zero (section_ T.mor₂) ⋯ fst ⋯
⊢ (T.mor₁ ≫ (biprod.uniqueUpToIso T.obj₁ T.obj₃ d.isBilimit).hom) ≫ biprod.snd = biprod.inl ≫ biprod.snd
|
fc47a2ec2f16a2ec
|
NNRat.cast_divNat
|
Mathlib/Data/Rat/Cast/CharZero.lean
|
@[simp]
lemma cast_divNat (a b : ℕ) : (divNat a b : α) = a / b
|
case e_a.a
α : Type u_3
inst✝¹ : DivisionSemiring α
inst✝ : CharZero α
a b : ℕ
⊢ ↑(divNat a b) = ↑(↑a / ↑b)
|
apply Rat.mkRat_eq_div
|
no goals
|
3b04fb0e64c29267
|
ContMDiffWithinAt.mfderivWithin
|
Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
|
theorem ContMDiffWithinAt.mfderivWithin {x₀ : N} {f : N → M → M'} {g : N → M}
{t : Set N} {u : Set M}
(hf : ContMDiffWithinAt (J.prod I) I' n (Function.uncurry f) (t ×ˢ u) (x₀, g x₀))
(hg : ContMDiffWithinAt J I m g t x₀) (hx₀ : x₀ ∈ t)
(hu : MapsTo g t u) (hmn : m + 1 ≤ n) (h'u : UniqueMDiffOn I u) :
haveI : IsManifold I 1 M := .of_le (le_trans le_add_self hmn)
haveI : IsManifold I' 1 M' := .of_le (le_trans le_add_self hmn)
ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
(inTangentCoordinates I I' g (fun x => f x (g x))
(fun x => mfderivWithin I I' (f x) u (g x)) x₀) t x₀
|
case a.a
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J n N
Is : IsManifold I n M
I's : IsManifold I' n M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiffOn I u
this✝⁴ : IsManifold I 1 M
this✝³ : IsManifold I' 1 M'
this✝² : IsManifold J 1 N
this✝¹ : IsManifold J m N
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
hx₀gx₀ : (x₀, g x₀) ∈ t ×ˢ u
h4f✝ : ContinuousWithinAt (fun x => f x (g x)) t x₀
h4f : (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝[t] x₀
h3f : ∀ᶠ (x' : N × M) in 𝓝[t ×ˢ u] (x₀, g x₀), ContMDiffWithinAt (J.prod I) I' 1 (uncurry f) (t ×ˢ u) x'
h2f : ∀ᶠ (x₂ : N) in 𝓝[t] x₀, ContMDiffWithinAt I I' 1 (f x₂) u (g x₂)
h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝[t] x₀
this✝ :
ContDiffWithinAt 𝕜 m
(fun x =>
fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm)
((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u)
(↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x))))
(↑(extChartAt J x₀).symm ⁻¹' t' ∩ range ↑J) (↑(extChartAt J x₀) x₀)
this :
ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
(fun x =>
fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm)
((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u) (↑(extChartAt I (g x₀)) (g x)))
t' x₀
x : N
hx : ContMDiffWithinAt I I' 1 (f x) u (g x)
h'x : x ∈ (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source
h2 : x ∈ g ⁻¹' (extChartAt I (g x₀)).source
hxt : x ∈ t
h1 : g x ∈ u
⊢ ↑(extChartAt I (g x₀)) (g x) ∈ (extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u
|
refine ⟨PartialEquiv.map_source _ h2, ?_⟩
|
case a.a
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
m n : WithTop ℕ∞
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
F : Type u_8
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
G : Type u_9
inst✝² : TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_10
inst✝¹ : TopologicalSpace N
inst✝ : ChartedSpace G N
Js : IsManifold J n N
Is : IsManifold I n M
I's : IsManifold I' n M'
x₀ : N
f : N → M → M'
g : N → M
t : Set N
u : Set M
hf : ContMDiffWithinAt (J.prod I) I' n (uncurry f) (t ×ˢ u) (x₀, g x₀)
hg : ContMDiffWithinAt J I m g t x₀
hx₀ : x₀ ∈ t
hu : MapsTo g t u
hmn : m + 1 ≤ n
h'u : UniqueMDiffOn I u
this✝⁴ : IsManifold I 1 M
this✝³ : IsManifold I' 1 M'
this✝² : IsManifold J 1 N
this✝¹ : IsManifold J m N
t' : Set N := t ∩ g ⁻¹' (extChartAt I (g x₀)).source
ht't : t' ⊆ t
hx₀gx₀ : (x₀, g x₀) ∈ t ×ˢ u
h4f✝ : ContinuousWithinAt (fun x => f x (g x)) t x₀
h4f : (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source ∈ 𝓝[t] x₀
h3f : ∀ᶠ (x' : N × M) in 𝓝[t ×ˢ u] (x₀, g x₀), ContMDiffWithinAt (J.prod I) I' 1 (uncurry f) (t ×ˢ u) x'
h2f : ∀ᶠ (x₂ : N) in 𝓝[t] x₀, ContMDiffWithinAt I I' 1 (f x₂) u (g x₂)
h2g : g ⁻¹' (extChartAt I (g x₀)).source ∈ 𝓝[t] x₀
this✝ :
ContDiffWithinAt 𝕜 m
(fun x =>
fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f (↑(extChartAt J x₀).symm x) ∘ ↑(extChartAt I (g x₀)).symm)
((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u)
(↑(extChartAt I (g x₀)) (g (↑(extChartAt J x₀).symm x))))
(↑(extChartAt J x₀).symm ⁻¹' t' ∩ range ↑J) (↑(extChartAt J x₀) x₀)
this :
ContMDiffWithinAt J 𝓘(𝕜, E →L[𝕜] E') m
(fun x =>
fderivWithin 𝕜 (↑(extChartAt I' (f x₀ (g x₀))) ∘ f x ∘ ↑(extChartAt I (g x₀)).symm)
((extChartAt I (g x₀)).target ∩ ↑(extChartAt I (g x₀)).symm ⁻¹' u) (↑(extChartAt I (g x₀)) (g x)))
t' x₀
x : N
hx : ContMDiffWithinAt I I' 1 (f x) u (g x)
h'x : x ∈ (fun x => f x (g x)) ⁻¹' (extChartAt I' (f x₀ (g x₀))).source
h2 : x ∈ g ⁻¹' (extChartAt I (g x₀)).source
hxt : x ∈ t
h1 : g x ∈ u
⊢ ↑(extChartAt I (g x₀)) (g x) ∈ ↑(extChartAt I (g x₀)).symm ⁻¹' u
|
b011e1f26afdb375
|
Finmap.ext_lookup
|
Mathlib/Data/Finmap.lean
|
theorem ext_lookup {s₁ s₂ : Finmap β} : (∀ x, s₁.lookup x = s₂.lookup x) → s₁ = s₂ :=
induction_on₂ s₁ s₂ fun s₁ s₂ h => by
simp only [AList.lookup, lookup_toFinmap] at h
rw [AList.toFinmap_eq]
apply lookup_ext s₁.nodupKeys s₂.nodupKeys
intro x y
rw [h]
|
α : Type u
β : α → Type v
inst✝ : DecidableEq α
s₁✝ s₂✝ : Finmap β
s₁ s₂ : AList β
h : ∀ (x : α), lookup x ⟦s₁⟧ = lookup x ⟦s₂⟧
⊢ ⟦s₁⟧ = ⟦s₂⟧
|
simp only [AList.lookup, lookup_toFinmap] at h
|
α : Type u
β : α → Type v
inst✝ : DecidableEq α
s₁✝ s₂✝ : Finmap β
s₁ s₂ : AList β
h : ∀ (x : α), dlookup x s₁.entries = dlookup x s₂.entries
⊢ ⟦s₁⟧ = ⟦s₂⟧
|
85e1fdcd0e824010
|
Vector.foldrM_filterMap
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Monadic.lean
|
theorem foldrM_filterMap [Monad m] [LawfulMonad m] (f : α → Option β) (g : β → γ → m γ) (l : Vector α n) (init : γ) :
(l.filterMap f).foldrM g init =
l.foldrM (fun x y => match f x with | some b => g b y | none => pure y) init
|
case mk
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_4
γ : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → Option β
g : β → γ → m γ
init : γ
l : Array α
⊢ Array.foldrM
(fun x y =>
match f x with
| some b => g b y
| none => pure y)
init l =
Array.foldrM
(fun x y =>
match f x with
| some b => g b y
| none => pure y)
init l
|
rfl
|
no goals
|
8712f7f8cb1de212
|
AlgebraicGeometry.sourceAffineLocally_isLocal
|
Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
|
theorem sourceAffineLocally_isLocal (h₁ : RingHom.RespectsIso P)
(h₂ : RingHom.LocalizationAwayPreserves P) (h₃ : RingHom.OfLocalizationSpan P) :
(sourceAffineLocally P).IsLocal
|
case of_basicOpenCover
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] => P
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
s : Finset ↑Γ(Y, ⊤)
hs : Ideal.span ↑s = ⊤
hs' : ∀ (r : { x // x ∈ s }), sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] => P) (f ∣_ Y.basicOpen ↑r)
U : ↑X.affineOpens
r : ↑↑s
⊢ P (Localization.awayMap (CommRingCat.Hom.hom (Scheme.Hom.appLE f ⊤ ↑U ⋯)) ↑r)
|
simp_rw [sourceAffineLocally_morphismRestrict] at hs'
|
case of_basicOpenCover
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] => P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] => P
X Y : Scheme
inst✝ : IsAffine Y
f : X ⟶ Y
s : Finset ↑Γ(Y, ⊤)
hs : Ideal.span ↑s = ⊤
U : ↑X.affineOpens
r : ↑↑s
hs' :
∀ (r : { x // x ∈ s }) (V : ↑X.affineOpens) (e : ↑V ≤ f ⁻¹ᵁ Y.basicOpen ↑r),
P (CommRingCat.Hom.hom (Scheme.Hom.appLE f (Y.basicOpen ↑r) (↑V) e))
⊢ P (Localization.awayMap (CommRingCat.Hom.hom (Scheme.Hom.appLE f ⊤ ↑U ⋯)) ↑r)
|
222ac1262a013627
|
Polynomial.coeff_divByMonic_X_sub_C
|
Mathlib/Algebra/Polynomial/Div.lean
|
theorem coeff_divByMonic_X_sub_C (p : R[X]) (a : R) (n : ℕ) :
(p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i
|
case inr.refine_1.e_a.refine_2
R : Type u
inst✝ : Ring R
p : R[X]
a : R
n✝ : ℕ
this :
∀ (n : ℕ), p.natDegree ≤ n → (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i
h : ¬p.natDegree ≤ n✝
n : ℕ
hn : n < p.natDegree
x✝ : n✝ ≤ n
ih : (p /ₘ (X - C a)).coeff (n + 1) = ∑ i ∈ Icc (n + 1 + 1) p.natDegree, a ^ (i - (n + 1 + 1)) * p.coeff i
i : ℕ
hi : i ∈ Icc (n + 1 + 1) p.natDegree
⊢ 1 ≤ i - (n + 1)
|
apply Nat.le_sub_of_add_le
|
case inr.refine_1.e_a.refine_2.h
R : Type u
inst✝ : Ring R
p : R[X]
a : R
n✝ : ℕ
this :
∀ (n : ℕ), p.natDegree ≤ n → (p /ₘ (X - C a)).coeff n = ∑ i ∈ Icc (n + 1) p.natDegree, a ^ (i - (n + 1)) * p.coeff i
h : ¬p.natDegree ≤ n✝
n : ℕ
hn : n < p.natDegree
x✝ : n✝ ≤ n
ih : (p /ₘ (X - C a)).coeff (n + 1) = ∑ i ∈ Icc (n + 1 + 1) p.natDegree, a ^ (i - (n + 1 + 1)) * p.coeff i
i : ℕ
hi : i ∈ Icc (n + 1 + 1) p.natDegree
⊢ 1 + (n + 1) ≤ i
|
b5156308e39e2482
|
Fin.contractNth_apply_of_eq
|
Mathlib/Data/Fin/Tuple/Basic.lean
|
theorem contractNth_apply_of_eq (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n)
(h : (k : ℕ) = j) : contractNth j op g k = op (g (Fin.castSucc k)) (g k.succ)
|
n : ℕ
α : Sort u_1
j : Fin (n + 1)
op : α → α → α
g : Fin (n + 1) → α
k : Fin n
h : ↑k = ↑j
⊢ j.contractNth op g k = op (g k.castSucc) (g k.succ)
|
have : ¬(k : ℕ) < j := not_lt.2 (le_of_eq h.symm)
|
n : ℕ
α : Sort u_1
j : Fin (n + 1)
op : α → α → α
g : Fin (n + 1) → α
k : Fin n
h : ↑k = ↑j
this : ¬↑k < ↑j
⊢ j.contractNth op g k = op (g k.castSucc) (g k.succ)
|
32cad5442fb3a7d6
|
IsClosed.isClopenable
|
Mathlib/Topology/MetricSpace/Polish.lean
|
theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s : Set α}
(hs : IsClosed s) : IsClopenable s
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PolishSpace α
s : Set α
hs : IsClosed s
this : PolishSpace ↑s
⊢ IsClopenable s
|
let t : Set α := sᶜ
|
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PolishSpace α
s : Set α
hs : IsClosed s
this : PolishSpace ↑s
t : Set α := sᶜ
⊢ IsClopenable s
|
86268e182c814247
|
HahnSeries.order_mul
|
Mathlib/RingTheory/HahnSeries/Multiplication.lean
|
theorem order_mul {Γ} [LinearOrderedCancelAddCommMonoid Γ] [NonUnitalNonAssocSemiring R]
[NoZeroDivisors R] {x y : HahnSeries Γ R} (hx : x ≠ 0) (hy : y ≠ 0) :
(x * y).order = x.order + y.order
|
case a.h
R : Type u_3
Γ : Type u_6
inst✝² : LinearOrderedCancelAddCommMonoid Γ
inst✝¹ : NonUnitalNonAssocSemiring R
inst✝ : NoZeroDivisors R
x y : HahnSeries Γ R
hx : x ≠ 0
hy : y ≠ 0
⊢ x.leadingCoeff * y.leadingCoeff ≠ 0
|
exact mul_ne_zero (leadingCoeff_ne_iff.mpr hx) (leadingCoeff_ne_iff.mpr hy)
|
no goals
|
e6cf92b660be132a
|
Matrix.blockDiagonal_tsum
|
Mathlib/Topology/Instances/Matrix.lean
|
theorem Matrix.blockDiagonal_tsum [DecidableEq p] [T2Space R] {f : X → p → Matrix m n R} :
blockDiagonal (∑' x, f x) = ∑' x, blockDiagonal (f x)
|
X : Type u_1
m : Type u_4
n : Type u_5
p : Type u_6
R : Type u_8
inst✝³ : AddCommMonoid R
inst✝² : TopologicalSpace R
inst✝¹ : DecidableEq p
inst✝ : T2Space R
f : X → p → Matrix m n R
⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)
|
by_cases hf : Summable f
|
case pos
X : Type u_1
m : Type u_4
n : Type u_5
p : Type u_6
R : Type u_8
inst✝³ : AddCommMonoid R
inst✝² : TopologicalSpace R
inst✝¹ : DecidableEq p
inst✝ : T2Space R
f : X → p → Matrix m n R
hf : Summable f
⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)
case neg
X : Type u_1
m : Type u_4
n : Type u_5
p : Type u_6
R : Type u_8
inst✝³ : AddCommMonoid R
inst✝² : TopologicalSpace R
inst✝¹ : DecidableEq p
inst✝ : T2Space R
f : X → p → Matrix m n R
hf : ¬Summable f
⊢ blockDiagonal (∑' (x : X), f x) = ∑' (x : X), blockDiagonal (f x)
|
baf1709573783a76
|
Multiset.le_bind
|
Mathlib/Data/Multiset/Bind.lean
|
theorem le_bind {α β : Type*} {f : α → Multiset β} (S : Multiset α) {x : α} (hx : x ∈ S) :
f x ≤ S.bind f
|
case intro
α : Type u_4
β : Type u_5
f : α → Multiset β
S : Multiset α
x : α
hx : x ∈ S
a : β
m' : Multiset ℕ
hm' : map (fun b => count a (f b)) S = (fun b => count a (f b)) x ::ₘ m'
⊢ count a (f x) ≤ (fun b => count a (f b)) x + m'.sum
|
exact Nat.le_add_right _ _
|
no goals
|
2e61a8e5a411e4b9
|
OreLocalization.eq_of_num_factor_eq
|
Mathlib/GroupTheory/OreLocalization/Basic.lean
|
theorem eq_of_num_factor_eq {r r' r₁ r₂ : R} {s t : S} (h : t * r = t * r') :
r₁ * r * r₂ /ₒ s = r₁ * r' * r₂ /ₒ s
|
R : Type u_1
inst✝¹ : Monoid R
S : Submonoid R
inst✝ : OreSet S
r r' r₁ r₂ : R
s t : ↥S
h : ↑t * r = ↑t * r'
r₁' : R
t' : ↥S
hr₁ : ↑t' * r₁ = r₁' * ↑t
⊢ r₁' * (↑t * r') * r₂ = r₁' * ↑t * r' * r₂
|
simp [← mul_assoc]
|
no goals
|
20ac2fecf822d2fd
|
LieSubmodule.lcs_add_le_iff
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
theorem lcs_add_le_iff (l k : ℕ) : N₁.lcs (l + k) ≤ N₂ ↔ N₁.lcs l ≤ N₂.ucs k
|
case zero
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
N₁ N₂ : LieSubmodule R L M
inst✝ : LieModule R L M
l : ℕ
⊢ lcs (l + 0) N₁ ≤ N₂ ↔ lcs l N₁ ≤ ucs 0 N₂
|
simp
|
no goals
|
38260d5d543126fc
|
BitVec.toNat_ushiftRight_lt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem toNat_ushiftRight_lt (x : BitVec w) (n : Nat) (hn : n ≤ w) :
(x >>> n).toNat < 2 ^ (w - n)
|
case h
w : Nat
x : BitVec w
n : Nat
hn : n ≤ w
⊢ n ≤ w
|
apply hn
|
no goals
|
e2b3a134eb083317
|
PrimeSpectrum.isClosed_image_of_stableUnderSpecialization
|
Mathlib/RingTheory/Spectrum/Prime/Topology.lean
|
@[stacks 05JL]
lemma isClosed_image_of_stableUnderSpecialization
(Z : Set (PrimeSpectrum S)) (hZ : IsClosed Z)
(hf : StableUnderSpecialization (comap f '' Z)) :
IsClosed (comap f '' Z)
|
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
Z : Set (PrimeSpectrum S)
hZ : IsClosed Z
hf : StableUnderSpecialization (⇑(comap f) '' Z)
⊢ IsClosed (⇑(comap f) '' Z)
|
obtain ⟨I, rfl⟩ := (PrimeSpectrum.isClosed_iff_zeroLocus_ideal Z).mp hZ
|
case intro
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
I : Ideal S
hZ : IsClosed (zeroLocus ↑I)
hf : StableUnderSpecialization (⇑(comap f) '' zeroLocus ↑I)
⊢ IsClosed (⇑(comap f) '' zeroLocus ↑I)
|
26a6df79a3998cc2
|
SetTheory.PGame.birthday_add
|
Mathlib/SetTheory/Game/Birthday.lean
|
theorem birthday_add : ∀ x y : PGame.{u}, (x + y).birthday = x.birthday ♯ y.birthday
| ⟨xl, xr, xL, xR⟩, ⟨yl, yr, yL, yR⟩ => by
rw [birthday_def, nadd, lsub_sum, lsub_sum]
simp only [mk_add_moveLeft_inl, mk_add_moveLeft_inr, mk_add_moveRight_inl, mk_add_moveRight_inr,
moveLeft_mk, moveRight_mk]
conv_lhs => left; left; right; intro a; rw [birthday_add (xL a) ⟨yl, yr, yL, yR⟩]
conv_lhs => left; right; right; intro b; rw [birthday_add ⟨xl, xr, xL, xR⟩ (yL b)]
conv_lhs => right; left; right; intro a; rw [birthday_add (xR a) ⟨yl, yr, yL, yR⟩]
conv_lhs => right; right; right; intro b; rw [birthday_add ⟨xl, xr, xL, xR⟩ (yR b)]
rw [max_max_max_comm]
congr <;> apply le_antisymm
any_goals
refine max_le_iff.2 ⟨?_, ?_⟩
all_goals
refine lsub_le_iff.2 fun i ↦ ?_
rw [← Order.succ_le_iff]
refine Ordinal.le_iSup (fun _ : Set.Iio _ ↦ _) ⟨_, ?_⟩
apply_rules [birthday_moveLeft_lt, birthday_moveRight_lt]
all_goals
rw [Ordinal.iSup_le_iff]
rintro ⟨i, hi⟩
obtain ⟨j, hj⟩ | ⟨j, hj⟩ := lt_birthday_iff.1 hi <;> rw [Order.succ_le_iff]
· exact lt_max_of_lt_left ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _))
· exact lt_max_of_lt_right ((nadd_le_nadd_right hj _).trans_lt (lt_lsub _ _))
· exact lt_max_of_lt_left ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _))
· exact lt_max_of_lt_right ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _))
termination_by a b => (a, b)
|
case e_a.a.refine_2
xl xr : Type u
xL : xl → PGame
xR : xr → PGame
yl yr : Type u
yL : yl → PGame
yR : yr → PGame
i : yr
⊢ (yR i).birthday ∈ Set.Iio (mk yl yr yL yR).birthday
|
apply_rules [birthday_moveLeft_lt, birthday_moveRight_lt]
|
no goals
|
33d4fe8b94c4ad64
|
MeasureTheory.Measure.sum_comm
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
theorem sum_comm {ι' : Type*} (μ : ι → ι' → Measure α) :
(sum fun n => sum (μ n)) = sum fun m => sum fun n => μ n m
|
case h
α : Type u_1
ι : Type u_5
m0 : MeasurableSpace α
ι' : Type u_8
μ : ι → ι' → Measure α
s : Set α
hs : MeasurableSet s
⊢ ∑' (i : ι) (i_1 : ι'), (μ i i_1) s = ∑' (i : ι') (i_1 : ι), (μ i_1 i) s
|
rw [ENNReal.tsum_comm]
|
no goals
|
0c3b2d99d7a8c1f4
|
Finpartition.IsEquipartition.card_interedges_sparsePairs_le'
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
lemma IsEquipartition.card_interedges_sparsePairs_le' (hP : P.IsEquipartition)
(hε : 0 ≤ ε) :
#((P.sparsePairs G ε).biUnion fun (U, V) ↦ G.interedges U V) ≤ ε * (#A + #P.parts) ^ 2
|
case calc_2.calc_1
α : Type u_1
𝕜 : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
hP : P.IsEquipartition
hε : 0 ≤ ε
⊢ ∀ (a b : Finset α), a ∈ P.parts → b ∈ P.parts → a ≠ b → #a * #b ≤ (#A / #P.parts + 1) * (#A / #P.parts + 1)
|
rintro U V hU hV -
|
case calc_2.calc_1
α : Type u_1
𝕜 : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
hP : P.IsEquipartition
hε : 0 ≤ ε
U V : Finset α
hU : U ∈ P.parts
hV : V ∈ P.parts
⊢ #U * #V ≤ (#A / #P.parts + 1) * (#A / #P.parts + 1)
|
2382db98fe2b8214
|
comap_map_eq_map_adjoin_of_coprime_conductor
|
Mathlib/NumberTheory/KummerDedekind.lean
|
theorem comap_map_eq_map_adjoin_of_coprime_conductor
(hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤)
(h_alg : Function.Injective (algebraMap R<x> S)) :
(I.map (algebraMap R S)).comap (algebraMap R<x> S) = I.map (algebraMap R R<x>)
|
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S)
z : S
hz : z ∈ adjoin R {x}
hy : ⟨z, hz⟩ ∈ comap (algebraMap (↥(adjoin R {x})) S) (Ideal.map (algebraMap R S) I)
p : R
hp : p ∈ comap (algebraMap R S) (conductor R x)
q : R
hq : q ∈ I
hpq : p + q = 1
temp : (algebraMap R S) p * z + (algebraMap R S) q * z = z
this :
z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I) ↔
⟨z, hz⟩ ∈ Ideal.map (algebraMap R ↥(adjoin R {x})) I
⊢ (algebraMap R S) p * z + (algebraMap R S) q * z ∈
⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
|
obtain ⟨a, ha⟩ := (Set.mem_image _ _ _).mp (prod_mem_ideal_map_of_mem_conductor hp
(show z ∈ I.map (algebraMap R S) by rwa [Ideal.mem_comap] at hy))
|
case intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
x : S
I : Ideal R
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
h_alg : Function.Injective ⇑(algebraMap (↥(adjoin R {x})) S)
z : S
hz : z ∈ adjoin R {x}
hy : ⟨z, hz⟩ ∈ comap (algebraMap (↥(adjoin R {x})) S) (Ideal.map (algebraMap R S) I)
p : R
hp : p ∈ comap (algebraMap R S) (conductor R x)
q : R
hq : q ∈ I
hpq : p + q = 1
temp : (algebraMap R S) p * z + (algebraMap R S) q * z = z
this :
z ∈ ⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I) ↔
⟨z, hz⟩ ∈ Ideal.map (algebraMap R ↥(adjoin R {x})) I
a : ↥(adjoin R {x})
ha : a ∈ ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I) ∧ (algebraMap (↥(adjoin R {x})) S) a = (algebraMap R S) p * z
⊢ (algebraMap R S) p * z + (algebraMap R S) q * z ∈
⇑(algebraMap (↥(adjoin R {x})) S) '' ↑(Ideal.map (algebraMap R ↥(adjoin R {x})) I)
|
1d9a3ec969c1bfed
|
Matroid.coindep_iff_compl_spanning
|
Mathlib/Data/Matroid/Closure.lean
|
lemma coindep_iff_compl_spanning (hI : I ⊆ M.E
|
α : Type u_2
M : Matroid α
I : Set α
hI : autoParam (I ⊆ M.E) _auto✝
⊢ M.Coindep I ↔ M.Spanning (M.E \ I)
|
rw [coindep_iff_exists, spanning_iff_exists_isBase_subset]
|
no goals
|
c54a3a9bf19be3c5
|
Int.preimage_Ioi
|
Mathlib/Algebra/Order/Floor.lean
|
theorem preimage_Ioi : ((↑) : ℤ → α) ⁻¹' Set.Ioi a = Set.Ioi ⌊a⌋
|
case h
α : Type u_2
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
a : α
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Ioi a ↔ x✝ ∈ Ioi ⌊a⌋
|
simp [floor_lt]
|
no goals
|
abd1e7e7a65c9c57
|
fourierCoeff_bernoulli_eq
|
Mathlib/NumberTheory/ZetaValues.lean
|
theorem fourierCoeff_bernoulli_eq {k : ℕ} (hk : k ≠ 0) (n : ℤ) :
fourierCoeff ((↑) ∘ periodizedBernoulli k : 𝕌 → ℂ) n = -k ! / (2 * π * I * n) ^ k
|
k : ℕ
hk : k ≠ 0
n : ℤ
this : ofReal ∘ periodizedBernoulli k = AddCircle.liftIco 1 0 (ofReal ∘ bernoulliFun k)
⊢ fourierCoeffOn ⋯ (ofReal ∘ bernoulliFun k) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k
|
simpa only [zero_add] using bernoulliFourierCoeff_eq hk n
|
no goals
|
b829b68637ea7172
|
Finset.singleton_product
|
Mathlib/Data/Finset/Prod.lean
|
theorem singleton_product {a : α} :
({a} : Finset α) ×ˢ t = t.map ⟨Prod.mk a, Prod.mk.inj_left _⟩
|
case h.mk
α : Type u_1
β : Type u_2
t : Finset β
a x : α
y : β
⊢ (x, y) ∈ {a} ×ˢ t ↔ (x, y) ∈ map { toFun := Prod.mk a, inj' := ⋯ } t
|
simp [and_left_comm, eq_comm]
|
no goals
|
5a2a921959a1b1aa
|
FormalMultilinearSeries.changeOriginSeries_sum_eq_partialSum_of_finite
|
Mathlib/Analysis/Analytic/CPolynomialDef.lean
|
lemma changeOriginSeries_sum_eq_partialSum_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
(hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ) :
(p.changeOriginSeries k).sum = (p.changeOriginSeries k).partialSum (n - k)
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
⊢ (p.changeOriginSeries k).sum = (p.changeOriginSeries k).partialSum (n - k)
|
ext x
|
case h.H
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
n : ℕ
hn : ∀ (m : ℕ), n ≤ m → p m = 0
k : ℕ
x : E
x✝ : Fin k → E
⊢ ((p.changeOriginSeries k).sum x) x✝ = ((p.changeOriginSeries k).partialSum (n - k) x) x✝
|
916b6ae2b7c15f41
|
AddCircle.homeomorphCircle_apply
|
Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean
|
theorem homeomorphCircle_apply (hT : T ≠ 0) (x : AddCircle T) :
homeomorphCircle hT x = toCircle x
|
T : ℝ
hT : T ≠ 0
x : AddCircle T
⊢ (homeomorphCircle hT) x = x.toCircle
|
induction' x using QuotientAddGroup.induction_on with x
|
case H
T : ℝ
hT : T ≠ 0
x : ℝ
⊢ (homeomorphCircle hT) ↑x = toCircle ↑x
|
f21d65266bed3e5f
|
Finset.sum_sym2_filter_not_isDiag
|
Mathlib/Algebra/BigOperators/Sym.lean
|
theorem Finset.sum_sym2_filter_not_isDiag {ι α} [LinearOrder ι] [AddCommMonoid α]
(s : Finset ι) (p : Sym2 ι → α) :
∑ i ∈ s.sym2 with ¬ i.IsDiag, p i = ∑ i ∈ s.offDiag with i.1 < i.2, p s(i.1, i.2)
|
case refine_1.mk.mk
ι : Type u_1
α : Type u_2
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid α
s : Finset ι
p : Sym2 ι → α
i₁ j₁ : ι
hij₁ : (i₁, j₁).1 ≤ (i₁, j₁).2
⊢ ⟨(i₁, j₁), hij₁⟩ ∈ Finset.subtype (fun i => i.1 ≤ i.2) s.offDiag ↔
Sym2.sortEquiv.symm ⟨(i₁, j₁), hij₁⟩ ∈ filter (fun i => ¬i.IsDiag) s.sym2
|
simp [and_assoc]
|
no goals
|
ec5378fc923a84da
|
FreeGroup.Red.inv_of_red_of_ne
|
Mathlib/GroupTheory/FreeGroup/Basic.lean
|
theorem inv_of_red_of_ne {x1 b1 x2 b2} (H1 : (x1, b1) ≠ (x2, b2))
(H2 : Red ((x1, b1) :: L₁) ((x2, b2) :: L₂)) : Red L₁ ((x1, not b1) :: (x2, b2) :: L₂)
|
case intro.cons.intro.intro.intro.refl
α : Type u
L₂ : List (α × Bool)
x1 : α
b1 : Bool
x2 : α
b2 : Bool
H1 : (x1, b1) ≠ (x2, b2)
L₃ L₄ : List (α × Bool)
h₂ : Red L₄ L₂
h₁ : Red ((x1, b1) :: L₃) [(x2, b2)]
H2 : Red ((x1, b1) :: L₃.append L₄) ((x2, b2) :: L₂)
this : Red ((x1, b1) :: L₃.append L₄) ([(x2, b2)] ++ L₂)
⊢ Red (L₃ ++ L₄) ([(x1, !b1), (x2, b2)] ++ L₂)
|
apply append_append _ h₂
|
α : Type u
L₂ : List (α × Bool)
x1 : α
b1 : Bool
x2 : α
b2 : Bool
H1 : (x1, b1) ≠ (x2, b2)
L₃ L₄ : List (α × Bool)
h₂ : Red L₄ L₂
h₁ : Red ((x1, b1) :: L₃) [(x2, b2)]
H2 : Red ((x1, b1) :: L₃.append L₄) ((x2, b2) :: L₂)
this : Red ((x1, b1) :: L₃.append L₄) ([(x2, b2)] ++ L₂)
⊢ Red L₃ [(x1, !b1), (x2, b2)]
|
fbf48e09e0ebe5b0
|
MonomialOrder.degree_prod
|
Mathlib/RingTheory/MvPolynomial/MonomialOrder.lean
|
theorem degree_prod [IsDomain R] {ι : Type*} {P : ι → MvPolynomial σ R} {s : Finset ι}
(H : ∀ i ∈ s, P i ≠ 0) :
m.degree (∏ i ∈ s, P i) = ∑ i ∈ s, m.degree (P i)
|
case H.a0
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
ι : Type u_3
P : ι → MvPolynomial σ R
s : Finset ι
H : ∀ i ∈ s, P i ≠ 0
i : ι
hi : i ∈ s
⊢ m.leadingCoeff (P i) ≠ 0
|
rw [leadingCoeff_ne_zero_iff]
|
case H.a0
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : IsDomain R
ι : Type u_3
P : ι → MvPolynomial σ R
s : Finset ι
H : ∀ i ∈ s, P i ≠ 0
i : ι
hi : i ∈ s
⊢ P i ≠ 0
|
3cfd73bd8df87df1
|
ProbabilityTheory.integrable_preCDF
|
Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
|
lemma integrable_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℚ) :
Integrable (fun a ↦ (preCDF ρ x a).toReal) ρ.fst
|
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x : ℚ
⊢ Integrable (fun a => (preCDF ρ x a).toReal) ρ.fst
|
refine integrable_of_forall_fin_meas_le _ (measure_lt_top ρ.fst univ) ?_ fun t _ _ ↦ ?_
|
case refine_1
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x : ℚ
⊢ AEStronglyMeasurable (fun a => (preCDF ρ x a).toReal) ρ.fst
case refine_2
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
x : ℚ
t : Set α
x✝¹ : MeasurableSet t
x✝ : ρ.fst t ≠ ⊤
⊢ ∫⁻ (x_1 : α) in t, ‖(preCDF ρ x x_1).toReal‖ₑ ∂ρ.fst ≤ ρ.fst univ
|
4c5dd33b70a92091
|
Path.range_reparam
|
Mathlib/Topology/Path.lean
|
theorem range_reparam (γ : Path x y) {f : I → I} (hfcont : Continuous f) (hf₀ : f 0 = 0)
(hf₁ : f 1 = 1) : range (γ.reparam f hfcont hf₀ hf₁) = range γ
|
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Surjective f
|
intro t
|
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
γ : Path x y
f : ↑I → ↑I
hfcont : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
t : ↑I
⊢ ∃ a, f a = t
|
ce2c5842516e1041
|
NormedAddGroupHom.SurjectiveOnWith.mono
|
Mathlib/Analysis/Normed/Group/Hom.lean
|
theorem SurjectiveOnWith.mono {f : NormedAddGroupHom V₁ V₂} {K : AddSubgroup V₂} {C C' : ℝ}
(h : f.SurjectiveOnWith K C) (H : C ≤ C') : f.SurjectiveOnWith K C'
|
V₁ : Type u_2
V₂ : Type u_3
inst✝¹ : SeminormedAddCommGroup V₁
inst✝ : SeminormedAddCommGroup V₂
f : NormedAddGroupHom V₁ V₂
K : AddSubgroup V₂
C C' : ℝ
h : f.SurjectiveOnWith K C
H : C ≤ C'
g : V₁
k_in : f g ∈ K
hg : ‖g‖ ≤ C * ‖f g‖
Hg : ¬‖f g‖ = 0
⊢ C * ‖f g‖ ≤ C' * ‖f g‖
|
gcongr
|
no goals
|
0ec223564f13d644
|
IsCompact.exists_isOpen_closure_subset
|
Mathlib/Topology/Separation/Regular.lean
|
theorem IsCompact.exists_isOpen_closure_subset {K U : Set X} (hK : IsCompact K) (hU : U ∈ 𝓝ˢ K) :
∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U
|
case intro.intro.intro.intro.intro.intro
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : RegularSpace X
K U : Set X
hK : IsCompact K
hU : U ∈ 𝓝ˢ K
hd : Disjoint (𝓝ˢ K) (𝓝ˢ Uᶜ)
V : Set X
hVo : IsOpen V
hKV : K ⊆ V
W : Set X
hVW : Disjoint V W
hW : IsOpen W
hUW : Uᶜ ⊆ W
⊢ closure V ⊆ Wᶜ
|
exact closure_minimal hVW.subset_compl_right hW.isClosed_compl
|
no goals
|
6fe64539d616c336
|
Filter.countable_biInf_eq_iInf_seq'
|
Mathlib/Order/Filter/CountablyGenerated.lean
|
theorem countable_biInf_eq_iInf_seq' [CompleteLattice α] {B : Set ι} (Bcbl : B.Countable)
(f : ι → α) {i₀ : ι} (h : f i₀ = ⊤) : ∃ x : ℕ → ι, ⨅ t ∈ B, f t = ⨅ i, f (x i)
|
α : Type u_1
ι : Type u_4
inst✝ : CompleteLattice α
B : Set ι
Bcbl : B.Countable
f : ι → α
i₀ : ι
h : f i₀ = ⊤
⊢ ∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i)
|
rcases B.eq_empty_or_nonempty with hB | Bnonempty
|
case inl
α : Type u_1
ι : Type u_4
inst✝ : CompleteLattice α
B : Set ι
Bcbl : B.Countable
f : ι → α
i₀ : ι
h : f i₀ = ⊤
hB : B = ∅
⊢ ∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i)
case inr
α : Type u_1
ι : Type u_4
inst✝ : CompleteLattice α
B : Set ι
Bcbl : B.Countable
f : ι → α
i₀ : ι
h : f i₀ = ⊤
Bnonempty : B.Nonempty
⊢ ∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i)
|
1109f96365705c26
|
Nat.lt_size_self
|
Mathlib/Data/Nat/Size.lean
|
theorem lt_size_self (n : ℕ) : n < 2 ^ size n
|
n : ℕ
this : ∀ {n : ℕ}, n = 0 → n < 1 <<< n.size
⊢ n < 1 <<< n.size
|
refine binaryRec ?_ ?_ n
|
case refine_1
n : ℕ
this : ∀ {n : ℕ}, n = 0 → n < 1 <<< n.size
⊢ 0 < 1 <<< size 0
case refine_2
n : ℕ
this : ∀ {n : ℕ}, n = 0 → n < 1 <<< n.size
⊢ ∀ (b : Bool) (n : ℕ), n < 1 <<< n.size → bit b n < 1 <<< (bit b n).size
|
cc1b3f57abedd490
|
LinearMap.map_eq_top_iff
|
Mathlib/LinearAlgebra/Span/Basic.lean
|
theorem map_eq_top_iff {f : F} (hf : range f = ⊤) {p : Submodule R M} :
p.map f = ⊤ ↔ p ⊔ LinearMap.ker f = ⊤
|
R : Type u_1
R₂ : Type u_2
M : Type u_4
M₂ : Type u_5
inst✝⁸ : Semiring R
inst✝⁷ : Semiring R₂
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
τ₁₂ : R →+* R₂
inst✝² : RingHomSurjective τ₁₂
F : Type u_8
inst✝¹ : FunLike F M M₂
inst✝ : SemilinearMapClass F τ₁₂ M M₂
f : F
hf : range f = ⊤
p : Submodule R M
⊢ map f p = ⊤ ↔ p ⊔ ker f = ⊤
|
simp_rw [← top_le_iff, ← hf, range_eq_map, LinearMap.map_le_map_iff]
|
no goals
|
45dc5dbaa96645dc
|
Set.iInter_setOf
|
Mathlib/Data/Set/Lattice.lean
|
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x }
|
case h
α : Type u_1
ι : Sort u_5
P : ι → α → Prop
x✝ : α
⊢ x✝ ∈ ⋂ i, {x | P i x} ↔ x✝ ∈ {x | ∀ (i : ι), P i x}
|
exact mem_iInter
|
no goals
|
27beddd378adba5f
|
List.sublist_replicate_iff
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem sublist_replicate_iff : l <+ replicate m a ↔ ∃ n, n ≤ m ∧ l = replicate n a
|
α✝ : Type u_1
a : α✝
m n : Nat
le : n ≤ m
ih : ∀ {m : Nat}, replicate n a <+ replicate m a ↔ ∃ n_1, n_1 ≤ m ∧ replicate n a = replicate n_1 a
w : replicate n a <+ replicate m a
⊢ a :: replicate n a = replicate (n + 1) a
|
simp [replicate_succ]
|
no goals
|
b36e6729c3d1c9f6
|
AlgebraicGeometry.Scheme.Hom.range_subset_ker_support
|
Mathlib/AlgebraicGeometry/IdealSheaf.lean
|
lemma Hom.range_subset_ker_support (f : X.Hom Y) :
Set.range f.base ⊆ f.ker.support
|
X Y : Scheme
f : X.Hom Y
⊢ Set.range ⇑(ConcreteCategory.hom f.base) ⊆ f.ker.support
|
rintro _ ⟨x, rfl⟩
|
case intro
X Y : Scheme
f : X.Hom Y
x : ↑↑X.toPresheafedSpace
⊢ (ConcreteCategory.hom f.base) x ∈ f.ker.support
|
9d460a55329e679b
|
continuous_parametric_integral_of_continuous
|
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
theorem continuous_parametric_integral_of_continuous
[FirstCountableTopology X] [LocallyCompactSpace X]
[SecondCountableTopologyEither Y E] [IsLocallyFiniteMeasure μ]
{f : X → Y → E} (hf : Continuous f.uncurry) {s : Set Y} (hs : IsCompact s) :
Continuous (∫ y in s, f · y ∂μ)
|
Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FirstCountableTopology X
inst✝² : LocallyCompactSpace X
inst✝¹ : SecondCountableTopologyEither Y E
inst✝ : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous (uncurry f)
s : Set Y
hs : IsCompact s
⊢ ∀ (x : X), ContinuousAt (fun x => ∫ (y : Y) in s, f x y ∂μ) x
|
intro x₀
|
Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FirstCountableTopology X
inst✝² : LocallyCompactSpace X
inst✝¹ : SecondCountableTopologyEither Y E
inst✝ : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
⊢ ContinuousAt (fun x => ∫ (y : Y) in s, f x y ∂μ) x₀
|
77c93b2e7b9a838f
|
Equiv.Perm.cycle_is_cycleOf
|
Mathlib/GroupTheory/Perm/Cycle/Factors.lean
|
theorem cycle_is_cycleOf {f c : Equiv.Perm α} {a : α} (ha : a ∈ c.support)
(hc : c ∈ f.cycleFactorsFinset) : c = f.cycleOf a
|
α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f c : Perm α
a : α
ha : a ∈ c.support
hc : c ∈ f.cycleFactorsFinset
hfc : c.Disjoint (f * c⁻¹) := Disjoint.symm (disjoint_mul_inv_of_mem_cycleFactorsFinset hc)
hfc2 : Commute c (f * c⁻¹) := Disjoint.commute hfc
⊢ f.cycleOf a = (c * (f * c⁻¹)).cycleOf a
|
simp only [hfc2.eq, inv_mul_cancel_right]
|
no goals
|
51f66f0a14646a04
|
Matrix.derivative_det_one_add_X_smul_aux
|
Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
|
lemma derivative_det_one_add_X_smul_aux {n} (M : Matrix (Fin n) (Fin n) R) :
(derivative <| det (1 + (X : R[X]) • M.map C)).eval 0 = trace M
|
case succ
R : Type u
inst✝ : CommRing R
n : ℕ
IH : ∀ (M : Matrix (Fin n) (Fin n) R), eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace
M : Matrix (Fin (n + 1)) (Fin (n + 1)) R
⊢ eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace
|
rw [det_succ_row_zero, map_sum, eval_finset_sum]
|
case succ
R : Type u
inst✝ : CommRing R
n : ℕ
IH : ∀ (M : Matrix (Fin n) (Fin n) R), eval 0 (derivative (1 + X • M.map ⇑C).det) = M.trace
M : Matrix (Fin (n + 1)) (Fin (n + 1)) R
⊢ ∑ i : Fin n.succ,
eval 0
(derivative ((-1) ^ ↑i * (1 + X • M.map ⇑C) 0 i * ((1 + X • M.map ⇑C).submatrix Fin.succ i.succAbove).det)) =
M.trace
|
a88f93f677573a35
|
NumberField.mixedEmbedding.logMap_zero
|
Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean
|
theorem logMap_zero : logMap (0 : mixedSpace K) = 0
|
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ logMap 0 = 0
|
ext
|
case h
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x✝ : { w // w ≠ w₀ }
⊢ logMap 0 x✝ = 0 x✝
|
84b5e372e1e35332
|
MonoidAlgebra.mapDomainAlgHom_comp
|
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
@[simp]
lemma mapDomainAlgHom_comp (k A) {G₁ G₂ G₃} [CommSemiring k] [Semiring A] [Algebra k A]
[Monoid G₁] [Monoid G₂] [Monoid G₃] (f : G₁ →* G₂) (g : G₂ →* G₃) :
mapDomainAlgHom k A (g.comp f) = (mapDomainAlgHom k A g).comp (mapDomainAlgHom k A f)
|
k : Type u_4
A : Type u_5
G₁ : Type u_6
G₂ : Type u_7
G₃ : Type u_8
inst✝⁵ : CommSemiring k
inst✝⁴ : Semiring A
inst✝³ : Algebra k A
inst✝² : Monoid G₁
inst✝¹ : Monoid G₂
inst✝ : Monoid G₃
f : G₁ →* G₂
g : G₂ →* G₃
⊢ mapDomainAlgHom k A (g.comp f) = (mapDomainAlgHom k A g).comp (mapDomainAlgHom k A f)
|
ext
|
case H.H
k : Type u_4
A : Type u_5
G₁ : Type u_6
G₂ : Type u_7
G₃ : Type u_8
inst✝⁵ : CommSemiring k
inst✝⁴ : Semiring A
inst✝³ : Algebra k A
inst✝² : Monoid G₁
inst✝¹ : Monoid G₂
inst✝ : Monoid G₃
f : G₁ →* G₂
g : G₂ →* G₃
x✝¹ : MonoidAlgebra A G₁
x✝ : G₃
⊢ ((mapDomainAlgHom k A (g.comp f)) x✝¹) x✝ = (((mapDomainAlgHom k A g).comp (mapDomainAlgHom k A f)) x✝¹) x✝
|
d66390a8e2edadda
|
Multiset.measurable_prod
|
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
theorem Multiset.measurable_prod (s : Multiset (α → M)) (hs : ∀ f ∈ s, Measurable f) :
Measurable fun x => (s.map fun f : α → M => f x).prod
|
M : Type u_2
α : Type u_4
inst✝² : CommMonoid M
inst✝¹ : MeasurableSpace M
inst✝ : MeasurableMul₂ M
m : MeasurableSpace α
s : Multiset (α → M)
hs : ∀ f ∈ s, Measurable f
⊢ Measurable fun x => (map (fun f => f x) s).prod
|
simpa only [← Pi.multiset_prod_apply] using s.measurable_prod' hs
|
no goals
|
f42bc86598a100e1
|
YoungDiagram.exists_not_mem_row
|
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
theorem exists_not_mem_row (μ : YoungDiagram) (i : ℕ) : ∃ j, (i, j) ∉ μ
|
μ : YoungDiagram
i : ℕ
⊢ ∃ j, (i, j) ∉ μ
|
obtain ⟨j, hj⟩ :=
Infinite.exists_not_mem_finset
(μ.cells.preimage (Prod.mk i) fun _ _ _ _ h => by
cases h
rfl)
|
case intro
μ : YoungDiagram
i j : ℕ
hj : j ∉ μ.cells.preimage (Prod.mk i) ⋯
⊢ ∃ j, (i, j) ∉ μ
|
0d89f12ff14aa88b
|
EuclideanGeometry.angle_eq_angle_of_dist_eq
|
Mathlib/Geometry/Euclidean/Triangle.lean
|
theorem angle_eq_angle_of_dist_eq {p₁ p₂ p₃ : P} (h : dist p₁ p₂ = dist p₁ p₃) :
∠ p₁ p₂ p₃ = ∠ p₁ p₃ p₂
|
case h.e'_2.h.e'_5
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ : P
h : ‖p₁ -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖
⊢ p₃ -ᵥ p₂ = p₁ -ᵥ p₂ - (p₁ -ᵥ p₃)
|
exact (vsub_sub_vsub_cancel_left p₃ p₂ p₁).symm
|
no goals
|
ef7ff6838bfcad30
|
Valued.closure_coe_completion_v_lt
|
Mathlib/Topology/Algebra/Valued/ValuedField.lean
|
theorem closure_coe_completion_v_lt {γ : Γ₀ˣ} :
closure ((↑) '' { x : K | v x < (γ : Γ₀) }) =
{ x : hat K | extensionValuation x < (γ : Γ₀) }
|
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
γ : Γ₀ˣ
⊢ closure (Completion.coe' '' {x | v x < ↑γ}) = {x | extensionValuation x < ↑γ}
|
ext x
|
case h
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
γ : Γ₀ˣ
x : hat K
⊢ x ∈ closure (Completion.coe' '' {x | v x < ↑γ}) ↔ x ∈ {x | extensionValuation x < ↑γ}
|
543e7e76040e703b
|
List.Sublist.of_sublist_append_left
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean
|
theorem Sublist.of_sublist_append_left (w : ∀ a, a ∈ l → a ∉ l₂) (h : l <+ l₁ ++ l₂) : l <+ l₁
|
α✝ : Type u_1
l l₂ l₁ : List α✝
w : ∀ (a : α✝), a ∈ l → ¬a ∈ l₂
h : l <+ l₁ ++ l₂
⊢ l <+ l₁
|
rw [sublist_append_iff] at h
|
α✝ : Type u_1
l l₂ l₁ : List α✝
w : ∀ (a : α✝), a ∈ l → ¬a ∈ l₂
h : ∃ l₁_1 l₂_1, l = l₁_1 ++ l₂_1 ∧ l₁_1 <+ l₁ ∧ l₂_1 <+ l₂
⊢ l <+ l₁
|
6d7cd8226795f0b9
|
List.Vector.mapAccumr₂_unused_input_right
|
Mathlib/Data/Vector/MapLemmas.lean
|
theorem mapAccumr₂_unused_input_right [Inhabited β] (f : α → β → σ → σ × γ)
(h : ∀ a b s, f a default s = f a b s) :
mapAccumr₂ f xs ys s = mapAccumr (fun a s => f a default s) xs s
|
case snoc
α : Type u_1
β : Type u_2
γ : Type u_3
σ : Type u_5
n : ℕ
inst✝ : Inhabited β
f : α → β → σ → σ × γ
h : ∀ (a : α) (b : β) (s : σ), f a default s = f a b s
n✝ : ℕ
xs : Vector α n✝
ys : Vector β n✝
x : α
y : β
ih : ∀ {s : σ}, mapAccumr₂ f xs ys s = mapAccumr (fun a s => f a default s) xs s
s : σ
⊢ mapAccumr₂ f (xs.snoc x) (ys.snoc y) s = mapAccumr (fun a s => f a default s) (xs.snoc x) s
|
simp [h x y s, ih]
|
no goals
|
67a351372db5a81c
|
CochainComplex.HomComplex.δ_comp_zero_cocycle
|
Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
|
@[simp]
lemma δ_comp_zero_cocycle {n : ℤ} (z₁ : Cochain F G n) (z₂ : Cocycle G K 0) (m : ℤ) :
δ n m (z₁.comp z₂.1 (add_zero n)) =
(δ n m z₁).comp z₂.1 (add_zero m)
|
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n : ℤ
z₁ : Cochain F G n
z₂ : Cocycle G K 0
m : ℤ
⊢ δ n m (z₁.comp ↑z₂ ⋯) = (δ n m z₁).comp ↑z₂ ⋯
|
by_cases hnm : n + 1 = m
|
case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n : ℤ
z₁ : Cochain F G n
z₂ : Cocycle G K 0
m : ℤ
hnm : n + 1 = m
⊢ δ n m (z₁.comp ↑z₂ ⋯) = (δ n m z₁).comp ↑z₂ ⋯
case neg
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n : ℤ
z₁ : Cochain F G n
z₂ : Cocycle G K 0
m : ℤ
hnm : ¬n + 1 = m
⊢ δ n m (z₁.comp ↑z₂ ⋯) = (δ n m z₁).comp ↑z₂ ⋯
|
51f9ff0fa213941d
|
MeasureTheory.Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite
|
Mathlib/MeasureTheory/Measure/SeparableMeasure.lean
|
theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite (h𝒜 : IsSetAlgebra 𝒜)
(S : μ.FiniteSpanningSetsIn 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) :
μ.MeasureDense 𝒜 where
measurable s hs := hgen ▸ measurableSet_generateFrom hs
approx s ms hμs ε ε_pos
|
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ⊤
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := Accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ⊤
T_spanning : ⋃ n, T n = univ
⊢ ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε
|
have mono : Monotone (fun n ↦ (T n) ∩ s) := fun m n hmn ↦ inter_subset_inter_left s
(biUnion_subset_biUnion_left fun k hkm ↦ Nat.le_trans hkm hmn)
|
X : Type u_1
m : MeasurableSpace X
μ : Measure X
𝒜 : Set (Set X)
h𝒜 : IsSetAlgebra 𝒜
S : μ.FiniteSpanningSetsIn 𝒜
hgen : m = MeasurableSpace.generateFrom 𝒜
s : Set X
ms : MeasurableSet s
hμs : μ s ≠ ⊤
ε : ℝ
ε_pos : 0 < ε
T : ℕ → Set X := Accumulate S.set
T_mem : ∀ (n : ℕ), T n ∈ 𝒜
T_finite : ∀ (n : ℕ), μ (T n) < ⊤
T_spanning : ⋃ n, T n = univ
mono : Monotone fun n => T n ∩ s
⊢ ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε
|
9296ab2335ceb925
|
AlgebraicGeometry.Spec.basicOpen_hom_ext
|
Mathlib/AlgebraicGeometry/Spec.lean
|
theorem Spec.basicOpen_hom_ext {X : RingedSpace.{u}} {R : CommRingCat.{u}}
{α β : X ⟶ Spec.sheafedSpaceObj R} (w : α.base = β.base)
(h : ∀ r : R,
let U := PrimeSpectrum.basicOpen r
(toOpen R U ≫ α.c.app (op U)) ≫ X.presheaf.map (eqToHom (by rw [w])) =
toOpen R U ≫ β.c.app (op U)) :
α = β
|
case h
X : RingedSpace
R : CommRingCat
α β : X ⟶ sheafedSpaceObj R
w : α.base = β.base
h :
∀ (r : ↑R),
let U := PrimeSpectrum.basicOpen r;
(toOpen (↑R) U ≫ α.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.c.app (op U)
r : ↑R
⊢ (α.c ≫ whiskerRight (eqToHom ⋯) X.presheaf).app (op (PrimeSpectrum.basicOpen r)) =
β.c.app (op (PrimeSpectrum.basicOpen r))
|
apply (StructureSheaf.to_basicOpen_epi R r).1
|
case h.a
X : RingedSpace
R : CommRingCat
α β : X ⟶ sheafedSpaceObj R
w : α.base = β.base
h :
∀ (r : ↑R),
let U := PrimeSpectrum.basicOpen r;
(toOpen (↑R) U ≫ α.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.c.app (op U)
r : ↑R
⊢ toOpen (↑R) (PrimeSpectrum.basicOpen r) ≫
(α.c ≫ whiskerRight (eqToHom ⋯) X.presheaf).app (op (PrimeSpectrum.basicOpen r)) =
toOpen (↑R) (PrimeSpectrum.basicOpen r) ≫ β.c.app (op (PrimeSpectrum.basicOpen r))
|
2f8e4e9bbb1515d3
|
List.dropSlice_eq_dropSliceTR
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Basic.lean
|
theorem dropSlice_eq_dropSliceTR : @dropSlice = @dropSliceTR
|
case h.h.h.h.h_1
α : Type u_1
n : Nat
l : List α
m✝ : Nat
⊢ dropSlice n 0 l = l
case h.h.h.h.h_2
α : Type u_1
n : Nat
l : List α
m✝¹ m✝ : Nat
⊢ dropSlice n m✝.succ l = dropSliceTR.go l m✝ l n #[]
|
{ rw [dropSlice_zero₂] }
|
case h.h.h.h.h_2
α : Type u_1
n : Nat
l : List α
m✝¹ m✝ : Nat
⊢ dropSlice n m✝.succ l = dropSliceTR.go l m✝ l n #[]
|
f858ea09a79c5819
|
lowerSemicontinuousOn_iff_le_liminf
|
Mathlib/Topology/Semicontinuous.lean
|
theorem lowerSemicontinuousOn_iff_le_liminf {f : α → γ} :
LowerSemicontinuousOn f s ↔ ∀ x ∈ s, f x ≤ liminf f (𝓝[s] x)
|
α : Type u_1
inst✝² : TopologicalSpace α
s : Set α
γ : Type u_3
inst✝¹ : CompleteLinearOrder γ
inst✝ : DenselyOrdered γ
f : α → γ
⊢ LowerSemicontinuousOn f s ↔ ∀ x ∈ s, f x ≤ liminf f (𝓝[s] x)
|
simp only [← lowerSemicontinuousWithinAt_iff_le_liminf, LowerSemicontinuousOn]
|
no goals
|
edac2a2a6372dc13
|
AffineSubspace.wOppSide_iff_exists_right
|
Mathlib/Analysis/Convex/Side.lean
|
theorem wOppSide_iff_exists_right {s : AffineSubspace R P} {x y p₂ : P} (h : p₂ ∈ s) :
s.WOppSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
|
case mpr.inr.intro.intro
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₂ : P
h : p₂ ∈ s
p : P
hp : p ∈ s
hr : SameRay R (x -ᵥ p) (p₂ -ᵥ y)
⊢ SameRay R (y -ᵥ p₂) (p -ᵥ x)
|
rwa [SameRay.sameRay_comm, ← sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]
|
no goals
|
0339b77a1fbe19db
|
hasSum_mellin_pi_mul
|
Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
|
/-- Shortcut version for the commonly arising special case when `p i = π * q i` for some other
sequence `q`. -/
lemma hasSum_mellin_pi_mul {a : ι → ℂ} {q : ι → ℝ} {F : ℝ → ℂ} {s : ℂ}
(hq : ∀ i, a i = 0 ∨ 0 < q i) (hs : 0 < s.re)
(hF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * rexp (-π * q i * t)) (F t))
(h_sum : Summable fun i ↦ ‖a i‖ / (q i) ^ s.re) :
HasSum (fun i ↦ π ^ (-s) * Gamma s * a i / q i ^ s) (mellin F s)
|
case inr
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
q : ι → ℝ
F : ℝ → ℂ
s : ℂ
hq : ∀ (i : ι), a i = 0 ∨ 0 < q i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a i * ↑(rexp (-π * q i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / q i ^ s.re
hp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i
i : ι
h : 0 < q i
⊢ ‖a i‖ / (π * q i) ^ s.re = π ^ (-s.re) * ‖a i‖ / q i ^ s.re
|
rw [mul_rpow pi_pos.le h.le, ← div_div, rpow_neg pi_pos.le, ← div_eq_inv_mul]
|
no goals
|
29b92a7c6424e7e6
|
Cardinal.mul_eq_left_iff
|
Mathlib/SetTheory/Cardinal/Arithmetic.lean
|
theorem mul_eq_left_iff {a b : Cardinal} : a * b = a ↔ max ℵ₀ b ≤ a ∧ b ≠ 0 ∨ b = 1 ∨ a = 0
|
a b : Cardinal.{u_1}
⊢ a * b = a ↔ ℵ₀ ⊔ b ≤ a ∧ b ≠ 0 ∨ b = 1 ∨ a = 0
|
rw [max_le_iff]
|
a b : Cardinal.{u_1}
⊢ a * b = a ↔ (ℵ₀ ≤ a ∧ b ≤ a) ∧ b ≠ 0 ∨ b = 1 ∨ a = 0
|
ce26cd2fc398ff3e
|
SimpleGraph.good_vertices_triangle_card
|
Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean
|
private lemma good_vertices_triangle_card [DecidableEq α] (dst : 2 * ε ≤ G.edgeDensity s t)
(dsu : 2 * ε ≤ G.edgeDensity s u) (dtu : 2 * ε ≤ G.edgeDensity t u) (utu : G.IsUniform ε t u)
(x : α) (hx : x ∈ s \ (badVertices G ε s t ∪ badVertices G ε s u)) :
ε ^ 3 * #t * #u ≤ #((({y ∈ t | G.Adj x y} ×ˢ {y ∈ u | G.Adj x y}).filter
fun (y, z) ↦ G.Adj y z).image (x, ·))
|
α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝ : DecidableEq α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
dtu : 2 * ε ≤ ↑(G.edgeDensity t u)
utu : G.IsUniform ε t u
x : α
hx :
x ∈ s ∧
↑(#t) * (↑(G.edgeDensity s t) - ε) ≤ ↑(#(filter (fun y => G.Adj x y) t)) ∧
↑(#u) * (↑(G.edgeDensity s u) - ε) ≤ ↑(#(filter (fun y => G.Adj x y) u))
⊢ ε ^ 3 * ↑(#t) * ↑(#u) ≤
↑(#(image (fun x_1 => (x, x_1))
(filter
(fun x =>
match x with
| (y, z) => G.Adj y z)
(filter (fun y => G.Adj x y) t ×ˢ filter (fun y => G.Adj x y) u))))
|
obtain ⟨-, hxY, hsu⟩ := hx
|
case intro.intro
α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
s t u : Finset α
inst✝ : DecidableEq α
dst : 2 * ε ≤ ↑(G.edgeDensity s t)
dsu : 2 * ε ≤ ↑(G.edgeDensity s u)
dtu : 2 * ε ≤ ↑(G.edgeDensity t u)
utu : G.IsUniform ε t u
x : α
hxY : ↑(#t) * (↑(G.edgeDensity s t) - ε) ≤ ↑(#(filter (fun y => G.Adj x y) t))
hsu : ↑(#u) * (↑(G.edgeDensity s u) - ε) ≤ ↑(#(filter (fun y => G.Adj x y) u))
⊢ ε ^ 3 * ↑(#t) * ↑(#u) ≤
↑(#(image (fun x_1 => (x, x_1))
(filter
(fun x =>
match x with
| (y, z) => G.Adj y z)
(filter (fun y => G.Adj x y) t ×ˢ filter (fun y => G.Adj x y) u))))
|
abd3ebe3cd7fa6c4
|
IsFractionRing.num_zero
|
Mathlib/RingTheory/Localization/NumDen.lean
|
@[simp]
lemma num_zero : IsFractionRing.num A (0 : K) = 0
|
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ num A 0 = 0
|
have := mk'_num_den' A (0 : K)
|
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
this : (algebraMap A K) (num A 0) / (algebraMap A K) ↑(den A 0) = 0
⊢ num A 0 = 0
|
7d57af980224f592
|
nnnorm_cfcₙ_nnreal_le
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Isometric.lean
|
lemma nnnorm_cfcₙ_nnreal_le {f : ℝ≥0 → ℝ≥0} {a : A} {c : ℝ≥0} (h : ∀ x ∈ σₙ ℝ≥0 a, f x ≤ c) :
‖cfcₙ f a‖₊ ≤ c
|
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x ≤ c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
⊢ (fun x => ‖x‖₊ ≤ c) ((cfcₙHom ha) { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯, map_zero' := hf0 })
|
simp only [← cfcₙ_apply f a, isLUB_le_iff (IsGreatest.nnnorm_cfcₙ_nnreal f a hf hf0 ha |>.isLUB)]
|
A : Type u_1
inst✝⁸ : NonUnitalNormedRing A
inst✝⁷ : StarRing A
inst✝⁶ : NormedSpace ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f : ℝ≥0 → ℝ≥0
a : A
c : ℝ≥0
h : ∀ x ∈ σₙ ℝ≥0 a, f x ≤ c
hf : ContinuousOn f (σₙ ℝ≥0 a)
hf0 : { toFun := (σₙ ℝ≥0 a).restrict f, continuous_toFun := ⋯ } 0 = 0
ha : 0 ≤ a
⊢ c ∈ upperBounds (f '' σₙ ℝ≥0 a)
|
3c2311247b002212
|
Nat.toDigitsCore_lens_eq
|
Mathlib/Data/Nat/Digits.lean
|
lemma toDigitsCore_lens_eq (b f : Nat) : ∀ (n : Nat) (c : Char) (tl : List Char),
(Nat.toDigitsCore b f n (c :: tl)).length = (Nat.toDigitsCore b f n tl).length + 1
|
b f n : ℕ
c : Char
tl : List Char
hnb : ¬n / b = 0
x : Char
hx : (n % b).digitChar = x
ih : (b.toDigitsCore f (n / b) (c :: x :: tl)).length = (b.toDigitsCore f (n / b) (x :: tl)).length + 1
lens_eq : (x :: c :: tl).length = (c :: x :: tl).length
⊢ (b.toDigitsCore f (n / b) (x :: c :: tl)).length = (b.toDigitsCore f (n / b) (c :: x :: tl)).length
|
apply toDigitsCore_lens_eq_aux
|
case a
b f n : ℕ
c : Char
tl : List Char
hnb : ¬n / b = 0
x : Char
hx : (n % b).digitChar = x
ih : (b.toDigitsCore f (n / b) (c :: x :: tl)).length = (b.toDigitsCore f (n / b) (x :: tl)).length + 1
lens_eq : (x :: c :: tl).length = (c :: x :: tl).length
⊢ (x :: c :: tl).length = (c :: x :: tl).length
|
4f5c924780b41cd4
|
splits_X_pow_sub_one_of_X_pow_sub_C
|
Mathlib/FieldTheory/AbelRuffini.lean
|
theorem splits_X_pow_sub_one_of_X_pow_sub_C {F : Type*} [Field F] {E : Type*} [Field E]
(i : F →+* E) (n : ℕ) {a : F} (ha : a ≠ 0) (h : (X ^ n - C a).Splits i) :
(X ^ n - 1 : F[X]).Splits i
|
case neg.intro
F : Type u_3
inst✝¹ : Field F
E : Type u_4
inst✝ : Field E
i : F →+* E
n : ℕ
a : F
ha : a ≠ 0
h : Splits i (X ^ n - C a)
ha' : i a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : (X ^ n - C a).degree ≠ 0
b : E
hb : b ^ n = i a
hb' : b ≠ 0
s : Multiset E := (Polynomial.map i (X ^ n - C a)).roots
hs : Polynomial.map i (X ^ n - C a) = (Multiset.map (fun a => X - C a) s).prod
hs' : s.card = n
⊢ Polynomial.map i (X ^ n - 1) =
C (i (X ^ n - 1).leadingCoeff) * (Multiset.map (fun a => X - C a) (Multiset.map (fun c => c / b) s)).prod
|
rw [leadingCoeff_X_pow_sub_one hn', RingHom.map_one, C_1, one_mul, Multiset.map_map]
|
case neg.intro
F : Type u_3
inst✝¹ : Field F
E : Type u_4
inst✝ : Field E
i : F →+* E
n : ℕ
a : F
ha : a ≠ 0
h : Splits i (X ^ n - C a)
ha' : i a ≠ 0
hn : ¬n = 0
hn' : 0 < n
hn'' : (X ^ n - C a).degree ≠ 0
b : E
hb : b ^ n = i a
hb' : b ≠ 0
s : Multiset E := (Polynomial.map i (X ^ n - C a)).roots
hs : Polynomial.map i (X ^ n - C a) = (Multiset.map (fun a => X - C a) s).prod
hs' : s.card = n
⊢ Polynomial.map i (X ^ n - 1) = (Multiset.map ((fun a => X - C a) ∘ fun c => c / b) s).prod
|
3246ca7f177f02a2
|
AffineSubspace.wOppSide_smul_vsub_vadd_left
|
Mathlib/Analysis/Convex/Side.lean
|
theorem wOppSide_smul_vsub_vadd_left {s : AffineSubspace R P} {p₁ p₂ : P} (x : P) (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.WOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x
|
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : StrictOrderedCommRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
p₁ p₂ x : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
t : R
ht : t ≤ 0
⊢ s.WOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x
|
refine ⟨p₂, hp₂, p₁, hp₁, ?_⟩
|
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : StrictOrderedCommRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
p₁ p₂ x : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
t : R
ht : t ≤ 0
⊢ SameRay R ((t • (x -ᵥ p₁) +ᵥ p₂) -ᵥ p₂) (p₁ -ᵥ x)
|
b0da55667c351b77
|
MeasureTheory.SimpleFunc.setToSimpleFunc_sub
|
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
theorem setToSimpleFunc_sub (T : Set α → E →L[ℝ] F) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
(hf : Integrable f μ) (hg : Integrable g μ) :
setToSimpleFunc T (f - g) = setToSimpleFunc T f - setToSimpleFunc T g
|
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : ∀ (y : E), y ≠ 0 → μ (⇑g ⁻¹' {y}) < ⊤
x : E
hx_ne : x ≠ 0
⊢ μ (⇑g ⁻¹' {-x}) < ⊤
|
refine hg (-x) ?_
|
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
f g : α →ₛ E
hf : Integrable (⇑f) μ
hg : ∀ (y : E), y ≠ 0 → μ (⇑g ⁻¹' {y}) < ⊤
x : E
hx_ne : x ≠ 0
⊢ -x ≠ 0
|
640edecad994037c
|
LinearMap.rank_diagonal
|
Mathlib/LinearAlgebra/Matrix/Diagonal.lean
|
theorem rank_diagonal [DecidableEq m] [DecidableEq K] (w : m → K) :
LinearMap.rank (toLin' (diagonal w)) = Fintype.card { i // w i ≠ 0 }
|
m : Type u_1
inst✝³ : Fintype m
K : Type u
inst✝² : Field K
inst✝¹ : DecidableEq m
inst✝ : DecidableEq K
w : m → K
hu : univ ⊆ {i | w i = 0}ᶜ ∪ {i | w i = 0}
hd : Disjoint {i | w i ≠ 0} {i | w i = 0}
⊢ (toLin' (Matrix.diagonal w)).rank = ↑(Fintype.card { i // w i ≠ 0 })
|
have B₁ := iSup_range_single_eq_iInf_ker_proj K (fun _ : m => K) hd hu (Set.toFinite _)
|
m : Type u_1
inst✝³ : Fintype m
K : Type u
inst✝² : Field K
inst✝¹ : DecidableEq m
inst✝ : DecidableEq K
w : m → K
hu : univ ⊆ {i | w i = 0}ᶜ ∪ {i | w i = 0}
hd : Disjoint {i | w i ≠ 0} {i | w i = 0}
B₁ : ⨆ i ∈ {i | w i ≠ 0}, range (single K (fun x => K) i) = ⨅ i ∈ {i | w i = 0}, ker (proj i)
⊢ (toLin' (Matrix.diagonal w)).rank = ↑(Fintype.card { i // w i ≠ 0 })
|
7528f6e5f9343b3c
|
Finpartition.IsEquipartition.exists_partsEquiv
|
Mathlib/Order/Partition/Equipartition.lean
|
theorem IsEquipartition.exists_partsEquiv (hP : P.IsEquipartition) :
∃ f : P.parts ≃ Fin #P.parts, ∀ t, #t.1 = #s / #P.parts + 1 ↔ f t < #s % #P.parts
|
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
el : { x // x ∈ P.parts ∧ #x = #s / #P.parts + 1 } ≃ Fin (#s % #P.parts)
es : { x // x ∈ P.parts ∧ #x = #s / #P.parts } ≃ Fin (#P.parts - #s % #P.parts)
x✝ : Finset α
ha : x✝ ∈ P.parts
⊢ ¬#x✝ = #s / #P.parts + 1 ↔ #x✝ = #s / #P.parts
|
rw [hP.card_part_eq_average_iff ha, ne_eq]
|
no goals
|
21d39a55d89e227d
|
CategoryTheory.Iso.inv_eq_inv
|
Mathlib/CategoryTheory/Iso.lean
|
theorem inv_eq_inv (f g : X ≅ Y) : f.inv = g.inv ↔ f.hom = g.hom :=
have : ∀ {X Y : C} (f g : X ≅ Y), f.hom = g.hom → f.inv = g.inv := fun f g h => by rw [ext h]
⟨this f.symm g.symm, this f g⟩
|
C : Type u
inst✝ : Category.{v, u} C
X Y : C
f✝ g✝ : X ≅ Y
X✝ Y✝ : C
f g : X✝ ≅ Y✝
h : f.hom = g.hom
⊢ f.inv = g.inv
|
rw [ext h]
|
no goals
|
e2234ec76460ed0e
|
MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos'
|
Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
|
theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
(hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0
|
case h.e'_4.h.e'_7.h
α : Type u_1
inst✝ : MeasurableSpace α
s : SignedMeasure α
i : Set α
hi₁ : MeasurableSet i
hi₂ : ↑s i < 0
h✝ : ¬s ≤[i] 0
hn : ∃ n, s ≤[i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
k : ℕ := Nat.find hn
hk₂ : s ≤[i \ ⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l] 0
hmeas : MeasurableSet (⋃ l, ⋃ (_ : l < k), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ∀ l < k, 0 ≤ ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
l : ℕ
h : ¬l < k
x✝ : α
⊢ l < k → x✝ ∉ MeasureTheory.SignedMeasure.restrictNonposSeq s i l
|
exact fun h' => False.elim (h h')
|
no goals
|
37ca223bdee11c33
|
CategoryTheory.Functor.relativelyRepresentable.lift_snd
|
Mathlib/CategoryTheory/MorphismProperty/Representable.lean
|
@[reassoc (attr := simp)]
lemma lift_snd [Full F] [Faithful F] : hf.lift i h hi ≫ hf.snd g = h :=
F.map_injective <| by simpa [lift] using PullbackCone.IsLimit.lift_snd _ _ _ _
|
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
F : C ⥤ D
X Y : D
f : X ⟶ Y
hf : F.relativelyRepresentable f
a : C
g : F.obj a ⟶ Y
c : C
i : F.obj c ⟶ X
h : c ⟶ a
hi : i ≫ f = F.map h ≫ g
inst✝¹ : F.Full
inst✝ : F.Faithful
⊢ F.map (hf.lift i h hi ≫ hf.snd g) = F.map h
|
simpa [lift] using PullbackCone.IsLimit.lift_snd _ _ _ _
|
no goals
|
00155b860d7f41af
|
HomologicalComplex₂.ι_totalShift₁Iso_hom_f
|
Mathlib/Algebra/Homology/TotalComplexShift.lean
|
@[reassoc]
lemma ι_totalShift₁Iso_hom_f (a b n : ℤ) (h : a + b = n) (a' : ℤ) (ha' : a' = a + x)
(n' : ℤ) (hn' : n' = n + x) :
((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) a b n h ≫ (K.totalShift₁Iso x).hom.f n =
(K.shiftFunctor₁XXIso a x a' ha' b).hom ≫ K.ιTotal (up ℤ) a' b n' (by dsimp; omega) ≫
(CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) x n n' hn').inv
|
C : Type u_1
inst✝² : Category.{?u.83805, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
a b n : ℤ
h : a + b = n
a' : ℤ
ha' : a' = a + x
n' : ℤ
hn' : n' = n + x
⊢ (up ℤ).π (up ℤ) (up ℤ) (a', b) = n'
|
dsimp
|
C : Type u_1
inst✝² : Category.{?u.83805, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
a b n : ℤ
h : a + b = n
a' : ℤ
ha' : a' = a + x
n' : ℤ
hn' : n' = n + x
⊢ a' + b = n'
|
94ce43fcd0071317
|
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