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
|
|---|---|---|---|---|---|---|
Set.star_mem_centralizer'
|
Mathlib/Algebra/Star/Center.lean
|
theorem Set.star_mem_centralizer' (h : ∀ a : R, a ∈ s → star a ∈ s) (ha : a ∈ Set.centralizer s) :
star a ∈ Set.centralizer s := fun y hy => by simpa using congr_arg star (ha _ (h _ hy)).symm
|
R : Type u_1
inst✝¹ : Mul R
inst✝ : StarMul R
a : R
s : Set R
h : ∀ a ∈ s, star a ∈ s
ha : a ∈ s.centralizer
y : R
hy : y ∈ s
⊢ y * star a = star a * y
|
simpa using congr_arg star (ha _ (h _ hy)).symm
|
no goals
|
8edae8b21c24f8fa
|
Std.DHashMap.Raw.get!_insert
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean
|
theorem get!_insert [LawfulBEq α] (h : m.WF) {k a : α} [Inhabited (β a)] {v : β k} :
(m.insert k v).get! a = if h : k == a then cast (congrArg β (eq_of_beq h)) v else m.get! a
|
α : Type u
β : α → Type v
m : Raw α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : LawfulBEq α
h : m.WF
k a : α
inst✝ : Inhabited (β a)
v : β k
⊢ (m.insert k v).get! a = if h : (k == a) = true then cast ⋯ v else m.get! a
|
simp_to_raw using Raw₀.get!_insert
|
no goals
|
3598cf39ad211877
|
List.getElem_succ_scanl
|
Mathlib/Data/List/Scan.lean
|
theorem getElem_succ_scanl {i : ℕ} (h : i + 1 < (scanl f b l).length) :
(scanl f b l)[i + 1] =
f ((scanl f b l)[i]'(Nat.lt_of_succ_lt h))
(l[i]'(Nat.lt_of_succ_lt_succ (h.trans_eq (length_scanl b l))))
|
case succ.nil
α : Type u_1
β : Type u_2
f : β → α → β
i : ℕ
hi : ∀ {b : β} {l : List α} (h : i + 1 < (scanl f b l).length), (scanl f b l)[i + 1] = f (scanl f b l)[i] l[i]
b : β
h : i + 1 + 1 < (scanl f b []).length
⊢ (scanl f b [])[i + 1 + 1] = f (scanl f b [])[i + 1] [][i + 1]
|
simp only [scanl, length] at h
|
case succ.nil
α : Type u_1
β : Type u_2
f : β → α → β
i : ℕ
hi : ∀ {b : β} {l : List α} (h : i + 1 < (scanl f b l).length), (scanl f b l)[i + 1] = f (scanl f b l)[i] l[i]
b : β
h : i + 1 + 1 < 0 + 1
⊢ (scanl f b [])[i + 1 + 1] = f (scanl f b [])[i + 1] [][i + 1]
|
376c53f19ff35d3f
|
SimpleGraph.Colorable.chromaticNumber_le
|
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
theorem Colorable.chromaticNumber_le {n : ℕ} (hc : G.Colorable n) : G.chromaticNumber ≤ n
|
V : Type u
G : SimpleGraph V
n : ℕ
hc : G.Colorable n
⊢ n ∈ {n | G.Colorable n}
|
exact hc
|
no goals
|
041236dd6b2afd6d
|
iter_deriv_zpow'
|
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
theorem iter_deriv_zpow' (m : ℤ) (k : ℕ) :
(deriv^[k] fun x : 𝕜 => x ^ m) =
fun x => (∏ i ∈ Finset.range k, ((m : 𝕜) - i)) * x ^ (m - k)
|
case zero
𝕜 : Type u
inst✝ : NontriviallyNormedField 𝕜
m : ℤ
⊢ (deriv^[0] fun x => x ^ m) = fun x => (∏ i ∈ Finset.range 0, (↑m - ↑i)) * x ^ (m - ↑0)
|
simp only [one_mul, Int.ofNat_zero, id, sub_zero, Finset.prod_range_zero,
Function.iterate_zero]
|
no goals
|
95c061e275d189c2
|
Monotone.countable_not_continuousWithinAt_Ioi
|
Mathlib/Topology/Order/LeftRightLim.lean
|
theorem countable_not_continuousWithinAt_Ioi [SecondCountableTopology β] :
Set.Countable { x | ¬ContinuousWithinAt f (Ioi x) x }
|
case refine_2.intro.intro
α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
f : α → β
hf : Monotone f
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology β
x : α
hx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z
u : β
hu : u > f x
v : α
xv : x < v
fvu : f v < u
⊢ ∀ᶠ (b : α) in 𝓝[>] x, f b < u
|
have : Ioo x v ∈ 𝓝[>] x := Ioo_mem_nhdsGT xv
|
case refine_2.intro.intro
α : Type u_1
β : Type u_2
inst✝⁶ : LinearOrder α
inst✝⁵ : ConditionallyCompleteLinearOrder β
inst✝⁴ : TopologicalSpace β
inst✝³ : OrderTopology β
f : α → β
hf : Monotone f
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology β
x : α
hx : ∀ (z : β), f x < z → ∃ y, x < y ∧ f y < z
u : β
hu : u > f x
v : α
xv : x < v
fvu : f v < u
this : Ioo x v ∈ 𝓝[>] x
⊢ ∀ᶠ (b : α) in 𝓝[>] x, f b < u
|
2ac3359eebee2a2f
|
Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero
|
Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
|
theorem oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
(θ : Real.Angle) : o.oangle x y = θ ↔ y = (‖y‖ / ‖x‖) • o.rotation θ x
|
case mpr
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x ≠ 0
hy : y ≠ 0
θ : Real.Angle
hp : 0 < ‖y‖ / ‖x‖
hye : y = (‖y‖ / ‖x‖) • (o.rotation θ) x
⊢ o.oangle x y = θ
|
rw [hye, o.oangle_smul_right_of_pos _ _ hp, o.oangle_rotation_self_right hx]
|
no goals
|
005d5b02112f49d4
|
Submodule.map_mul
|
Mathlib/Algebra/Algebra/Operations.lean
|
theorem map_mul {A'} [Semiring A'] [Algebra R A'] (f : A →ₐ[R] A') :
map f.toLinearMap (M * N) = map f.toLinearMap M * map f.toLinearMap N :=
calc
map f.toLinearMap (M * N) = ⨆ i : M, (N.map (LinearMap.mul R A i)).map f.toLinearMap
|
case h
R : Type u
inst✝⁴ : CommSemiring R
A : Type v
inst✝³ : Semiring A
inst✝² : Algebra R A
M N : Submodule R A
A' : Type u_1
inst✝¹ : Semiring A'
inst✝ : Algebra R A'
f : A →ₐ[R] A'
S : Submodule R A'
y : ↥M
hy : (fun i => map f.toLinearMap (map ((LinearMap.mul R A) ↑i) N)) y = S
⊢ (fun s => map ((LinearMap.mul R A') ↑s) (map f.toLinearMap N)) ⟨f ↑y, ⋯⟩ =
(fun i => map f.toLinearMap (map ((LinearMap.mul R A) ↑i) N)) y
|
ext
|
case h.h
R : Type u
inst✝⁴ : CommSemiring R
A : Type v
inst✝³ : Semiring A
inst✝² : Algebra R A
M N : Submodule R A
A' : Type u_1
inst✝¹ : Semiring A'
inst✝ : Algebra R A'
f : A →ₐ[R] A'
S : Submodule R A'
y : ↥M
hy : (fun i => map f.toLinearMap (map ((LinearMap.mul R A) ↑i) N)) y = S
x✝ : A'
⊢ x✝ ∈ (fun s => map ((LinearMap.mul R A') ↑s) (map f.toLinearMap N)) ⟨f ↑y, ⋯⟩ ↔
x✝ ∈ (fun i => map f.toLinearMap (map ((LinearMap.mul R A) ↑i) N)) y
|
ba1dc7e92341582b
|
Polynomial.div_C_mul
|
Mathlib/Algebra/Polynomial/FieldDivision.lean
|
theorem div_C_mul : p / (C a * q) = C a⁻¹ * (p / q)
|
case neg
R : Type u
a : R
inst✝ : Field R
p q : R[X]
ha : ¬a = 0
⊢ C a⁻¹ * (C q.leadingCoeff⁻¹ * (p /ₘ (C a * (q * (C a⁻¹ * C q.leadingCoeff⁻¹))))) =
C a⁻¹ * (C q.leadingCoeff⁻¹ * (p /ₘ (q * C q.leadingCoeff⁻¹)))
|
congr 3
|
case neg.e_a.e_a.e_q
R : Type u
a : R
inst✝ : Field R
p q : R[X]
ha : ¬a = 0
⊢ C a * (q * (C a⁻¹ * C q.leadingCoeff⁻¹)) = q * C q.leadingCoeff⁻¹
|
7bbaa64313a1e147
|
Continuous.strictMonoOn_of_inj_rigidity
|
Mathlib/Topology/Order/IntermediateValue.lean
|
theorem Continuous.strictMonoOn_of_inj_rigidity {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) {a b : α} (hab : a < b)
(hf_mono : StrictMonoOn f (Icc a b)) : StrictMono f
|
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
t : α := b ⊔ y
⊢ f x < f y
|
have hsa : s ≤ a := min_le_left a x
|
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
f : α → δ
hf_c : Continuous f
hf_i : Injective f
a b : α
hab : a < b
hf_mono : StrictMonoOn f (Icc a b)
x y : α
hxy : x < y
s : α := a ⊓ x
t : α := b ⊔ y
hsa : s ≤ a
⊢ f x < f y
|
3b8e874111e057b3
|
left_mem_openSegment_iff
|
Mathlib/Analysis/Convex/Segment.lean
|
theorem left_mem_openSegment_iff [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] :
x ∈ openSegment 𝕜 x y ↔ x = y
|
case mp
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : LinearOrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
x y : E
inst✝¹ : DenselyOrdered 𝕜
inst✝ : NoZeroSMulDivisors 𝕜 E
⊢ x ∈ openSegment 𝕜 x y → x = y
|
rintro ⟨a, b, _, hb, hab, hx⟩
|
case mp.intro.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : LinearOrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
x y : E
inst✝¹ : DenselyOrdered 𝕜
inst✝ : NoZeroSMulDivisors 𝕜 E
a b : 𝕜
left✝ : 0 < a
hb : 0 < b
hab : a + b = 1
hx : a • x + b • y = x
⊢ x = y
|
f5008e74233a27c3
|
Ordinal.blsub_le_of_brange_subset
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
theorem blsub_le_of_brange_subset {o o'} {f : ∀ a < o, Ordinal} {g : ∀ a < o', Ordinal}
(h : brange o f ⊆ brange o' g) : blsub.{u, max v w} o f ≤ blsub.{v, max u w} o' g :=
bsup_le_of_brange_subset.{u, v, w} fun a ⟨b, hb, hb'⟩ => by
obtain ⟨c, hc, hc'⟩ := h ⟨b, hb, rfl⟩
simp_rw [← hc'] at hb'
exact ⟨c, hc, hb'⟩
|
case intro.intro
o : Ordinal.{u}
o' : Ordinal.{v}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max (max u v) w}
g : (a : Ordinal.{v}) → a < o' → Ordinal.{max (max u v) w}
h : o.brange f ⊆ o'.brange g
a : Ordinal.{max (max u v) w}
x✝ : a ∈ o.brange fun a ha => succ (f a ha)
b : Ordinal.{u}
hb : b < o
hb' : (fun a ha => succ (f a ha)) b hb = a
c : Ordinal.{v}
hc : c < o'
hc' : g c hc = f b hb
⊢ a ∈ o'.brange fun a ha => succ (g a ha)
|
simp_rw [← hc'] at hb'
|
case intro.intro
o : Ordinal.{u}
o' : Ordinal.{v}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max (max u v) w}
g : (a : Ordinal.{v}) → a < o' → Ordinal.{max (max u v) w}
h : o.brange f ⊆ o'.brange g
a : Ordinal.{max (max u v) w}
x✝ : a ∈ o.brange fun a ha => succ (f a ha)
b : Ordinal.{u}
hb : b < o
c : Ordinal.{v}
hc : c < o'
hc' : g c hc = f b hb
hb' : succ (g c hc) = a
⊢ a ∈ o'.brange fun a ha => succ (g a ha)
|
4b7287eb9fb78e0a
|
BitVec.extractLsb_ofNat
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean
|
theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
extractLsb hi lo (BitVec.ofNat n x) = .ofNat (hi - lo + 1) ((x % 2^n) >>> lo)
|
x n hi lo : Nat
⊢ extractLsb hi lo (BitVec.ofNat n x) = BitVec.ofNat (hi - lo + 1) ((x % 2 ^ n) >>> lo)
|
ext i
|
case pred
x n hi lo i : Nat
a✝ : i < hi - lo + 1
⊢ (extractLsb hi lo (BitVec.ofNat n x)).getLsbD i = (BitVec.ofNat (hi - lo + 1) ((x % 2 ^ n) >>> lo)).getLsbD i
|
5af7ac0f3c16929a
|
FirstOrder.Language.distinctConstantsTheory_eq_iUnion
|
Mathlib/ModelTheory/Syntax.lean
|
theorem distinctConstantsTheory_eq_iUnion (s : Set α) :
L.distinctConstantsTheory s =
⋃ t : Finset s,
L.distinctConstantsTheory (t.map (Function.Embedding.subtype fun x => x ∈ s))
|
case h.mk.refine_3
L : Language
α : Type u'
s : Set α
i j : α
⊢ (∃ i_1, (∃ (x : i ∈ s), ⟨i, ⋯⟩ ∈ i_1) ∧ ∃ (x : j ∈ s), ⟨j, ⋯⟩ ∈ i_1) → i ∈ s ∧ j ∈ s
|
rintro ⟨t, ⟨is, _⟩, ⟨js, _⟩⟩
|
case h.mk.refine_3.intro.intro.intro.intro
L : Language
α : Type u'
s : Set α
i j : α
t : Finset ↑s
is : i ∈ s
h✝¹ : ⟨i, ⋯⟩ ∈ t
js : j ∈ s
h✝ : ⟨j, ⋯⟩ ∈ t
⊢ i ∈ s ∧ j ∈ s
|
15983f098b5f3c8e
|
tendsto_tsum_div_pow_atTop_integral
|
Mathlib/Analysis/BoxIntegral/UnitPartition.lean
|
theorem _root_.tendsto_tsum_div_pow_atTop_integral (hF : Continuous F) (hs₁ : IsBounded s)
(hs₂ : MeasurableSet s) (hs₃ : volume (frontier s) = 0) :
Tendsto (fun n : ℕ ↦ (∑' x : ↑(s ∩ (n : ℝ)⁻¹ • L), F x) / n ^ card ι)
atTop (nhds (∫ x in s, F x))
|
case pos
ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ≤ ↑B
ε : ℝ
hε : ε > 0
C₀ : ℝ
h₀ : ∀ x ∈ Box.Icc B, ‖F x‖ ≤ C₀
x : ι → ℝ
hx : x ∈ Box.Icc B
hs : x ∈ s
⊢ ‖F x‖ ≤ 0 ⊔ C₀
|
exact le_max_of_le_right (h₀ x hx)
|
no goals
|
e99fafdae745475e
|
NNRat.addSubmonoid_closure_range_pow
|
Mathlib/Data/Rat/Star.lean
|
@[simp] lemma addSubmonoid_closure_range_pow {n : ℕ} (hn₀ : n ≠ 0) :
closure (range fun x : ℚ≥0 ↦ x ^ n) = ⊤
|
n : ℕ
hn₀ : n ≠ 0
x : ℚ≥0
this : x = (x.num * x.den ^ (n - 1)) • (↑x.den)⁻¹ ^ n
⊢ x ∈ closure (range fun x => x ^ n)
|
rw [this]
|
n : ℕ
hn₀ : n ≠ 0
x : ℚ≥0
this : x = (x.num * x.den ^ (n - 1)) • (↑x.den)⁻¹ ^ n
⊢ (x.num * x.den ^ (n - 1)) • (↑x.den)⁻¹ ^ n ∈ closure (range fun x => x ^ n)
|
7fc9ce8a3377d839
|
Nat.prime_def
|
Mathlib/Data/Nat/Prime/Defs.lean
|
theorem prime_def {p : ℕ} : Prime p ↔ 2 ≤ p ∧ ∀ m, m ∣ p → m = 1 ∨ m = p
|
p : ℕ
h : 2 ≤ p ∧ ∀ (m : ℕ), m ∣ p → m = 1 ∨ m = p
h1 : 1 < p
a b : ℕ
hab : p = a * b
⊢ a = p → b = 1
|
rintro rfl
|
a b : ℕ
h : 2 ≤ a ∧ ∀ (m : ℕ), m ∣ a → m = 1 ∨ m = a
h1 : 1 < a
hab : a = a * b
⊢ b = 1
|
2723fa131234f025
|
jacobiTheta₂_add_left'
|
Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
|
/-- The two-variable Jacobi theta function is quasi-periodic in `z` with period `τ`. -/
lemma jacobiTheta₂_add_left' (z τ : ℂ) :
jacobiTheta₂ (z + τ) τ = cexp (-π * I * (τ + 2 * z)) * jacobiTheta₂ z τ
|
z τ : ℂ
⊢ jacobiTheta₂ (z + τ) τ = ∑' (c : ℤ), cexp (-↑π * I * (τ + 2 * z)) * jacobiTheta₂_term ((Equiv.addRight 1) c) z τ
|
refine tsum_congr (fun n ↦ ?_)
|
z τ : ℂ
n : ℤ
⊢ jacobiTheta₂_term n (z + τ) τ = cexp (-↑π * I * (τ + 2 * z)) * jacobiTheta₂_term ((Equiv.addRight 1) n) z τ
|
87d6327b328d6fd4
|
CategoryTheory.Arrow.finite_iff
|
Mathlib/CategoryTheory/Comma/CardinalArrow.lean
|
lemma Arrow.finite_iff (C : Type u) [SmallCategory C] :
Finite (Arrow C) ↔ Nonempty (FinCategory C)
|
case mpr
C : Type u
inst✝ : SmallCategory C
⊢ Nonempty (FinCategory C) → Finite (Arrow C)
|
rintro ⟨_⟩
|
case mpr.intro
C : Type u
inst✝ : SmallCategory C
val✝ : FinCategory C
⊢ Finite (Arrow C)
|
98b324129af2b9de
|
DoubleQuot.quotQuotEquivQuotOfLE_comp_quotQuotMk
|
Mathlib/RingTheory/Ideal/Quotient/Operations.lean
|
theorem quotQuotEquivQuotOfLE_comp_quotQuotMk (h : I ≤ J) :
RingHom.comp (↑(quotQuotEquivQuotOfLE h)) (quotQuotMk I J) = (Ideal.Quotient.mk J)
|
case a
R : Type u
inst✝ : CommRing R
I J : Ideal R
h : I ≤ J
x✝ : R
⊢ ((↑(quotQuotEquivQuotOfLE h)).comp (quotQuotMk I J)) x✝ = (Ideal.Quotient.mk J) x✝
|
rfl
|
no goals
|
d5928e7fe0ad4fb1
|
IsPrimePow.deficient
|
Mathlib/NumberTheory/FactorisationProperties.lean
|
theorem _root_.IsPrimePow.deficient (h : IsPrimePow n) : Deficient n
|
n : ℕ
h : IsPrimePow n
⊢ n.Deficient
|
obtain ⟨p, k, hp, -, rfl⟩ := h
|
case intro.intro.intro.intro
p k : ℕ
hp : _root_.Prime p
⊢ (p ^ k).Deficient
|
1836780b272ad835
|
MeasureTheory.lmarginal_union
|
Mathlib/MeasureTheory/Integral/Marginal.lean
|
theorem lmarginal_union (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f)
(hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_s, ∫⋯∫⁻_t, f ∂μ ∂μ
|
case hf.h
δ : Type u_1
π : δ → Type u_3
inst✝² : (x : δ) → MeasurableSpace (π x)
μ : (i : δ) → Measure (π i)
inst✝¹ : DecidableEq δ
s t : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → π i) → ℝ≥0∞
hf : Measurable f
hst : Disjoint s t
x : (i : δ) → π i
e : ((i : { x // x ∈ s }) → π ↑i) × ((i : { x // x ∈ t }) → π ↑i) ≃ᵐ ((i : { x // x ∈ s ∪ t }) → π ↑i) :=
MeasurableEquiv.piFinsetUnion π hst
⊢ Measurable fun z => f (updateFinset x (s ∪ t) ((MeasurableEquiv.piFinsetUnion π hst) z))
|
exact hf.comp <| measurable_updateFinset.comp e.measurable
|
no goals
|
10d2bec50feaf499
|
HallMarriageTheorem.hall_hard_inductive
|
Mathlib/Combinatorics/Hall/Finite.lean
|
theorem hall_hard_inductive (ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t)) :
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x
|
case intro.ind
α : Type v
inst✝¹ : DecidableEq α
n : ℕ
ih :
∀ m < n,
∀ {ι : Type u} {t : ι → Finset α} [inst : Finite ι],
(∀ (s : Finset ι), #s ≤ #(s.biUnion t)) →
∀ (val : Fintype ι), Fintype.card ι = m → ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
ι : Type u
t : ι → Finset α
inst✝ : Finite ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
val✝ : Fintype ι
hn : Fintype.card ι = n
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
|
rcases n with (_ | n)
|
case intro.ind.zero
α : Type v
inst✝¹ : DecidableEq α
ι : Type u
t : ι → Finset α
inst✝ : Finite ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
val✝ : Fintype ι
ih :
∀ m < 0,
∀ {ι : Type u} {t : ι → Finset α} [inst : Finite ι],
(∀ (s : Finset ι), #s ≤ #(s.biUnion t)) →
∀ (val : Fintype ι), Fintype.card ι = m → ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
hn : Fintype.card ι = 0
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
case intro.ind.succ
α : Type v
inst✝¹ : DecidableEq α
ι : Type u
t : ι → Finset α
inst✝ : Finite ι
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
val✝ : Fintype ι
n : ℕ
ih :
∀ m < n + 1,
∀ {ι : Type u} {t : ι → Finset α} [inst : Finite ι],
(∀ (s : Finset ι), #s ≤ #(s.biUnion t)) →
∀ (val : Fintype ι), Fintype.card ι = m → ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
hn : Fintype.card ι = n + 1
⊢ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x
|
3e81b0be12d2d845
|
ProbabilityTheory.sum_meas_smul_cond_fiber
|
Mathlib/Probability/ConditionalProbability.lean
|
/-- The **law of total probability** for a random variable taking finitely many values: a measure
`μ` can be expressed as a linear combination of its conditional measures `μ[|X ← x]` on fibers of a
random variable `X` valued in a fintype. -/
lemma sum_meas_smul_cond_fiber {X : Ω → α} (hX : Measurable X) (μ : Measure Ω) [IsFiniteMeasure μ] :
∑ x, μ (X ⁻¹' {x}) • μ[|X ← x] = μ
|
Ω : Type u_1
α : Type u_3
m : MeasurableSpace Ω
inst✝³ : Fintype α
inst✝² : MeasurableSpace α
inst✝¹ : DiscreteMeasurableSpace α
X : Ω → α
hX : Measurable X
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
E : Set Ω
hE : MeasurableSet E
⊢ ⋃ x ∈ Finset.univ, X ⁻¹' {x} ∩ E = E
|
ext
|
case h
Ω : Type u_1
α : Type u_3
m : MeasurableSpace Ω
inst✝³ : Fintype α
inst✝² : MeasurableSpace α
inst✝¹ : DiscreteMeasurableSpace α
X : Ω → α
hX : Measurable X
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
E : Set Ω
hE : MeasurableSet E
x✝ : Ω
⊢ x✝ ∈ ⋃ x ∈ Finset.univ, X ⁻¹' {x} ∩ E ↔ x✝ ∈ E
|
184ad315f536fdf8
|
CategoryTheory.GrothendieckTopology.plusMap_toPlus
|
Mathlib/CategoryTheory/Sites/Plus.lean
|
theorem plusMap_toPlus : J.plusMap (J.toPlus P) = J.toPlus (J.plusObj P)
|
case w.h.e_a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Cᵒᵖ ⥤ D
inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : unop S ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =
colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op
|
let ee : (J.pullback (I.map e).f).obj S.unop ⟶ ⊤ := homOfLE (OrderTop.le_top _)
|
case w.h.e_a
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝² : Category.{max v u, w} D
inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
P : Cᵒᵖ ⥤ D
inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D
X : Cᵒᵖ
S : (J.Cover (unop X))ᵒᵖ
e : unop S ⟶ ⊤ := homOfLE ⋯
I : (unop S).shape.L
ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯
⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =
colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op
|
d8ce6a7c6fc86c9e
|
Matroid.sigma_isBasis_iff
|
Mathlib/Data/Matroid/Sum.lean
|
@[simp] lemma sigma_isBasis_iff {I X} :
(Matroid.sigma M).IsBasis I X ↔ ∀ i, (M i).IsBasis (Sigma.mk i ⁻¹' I) (Sigma.mk i ⁻¹' X)
|
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
⊢ (Matroid.sigma M).IsBasis I X ↔ ∀ (i : ι), (M i).IsBasis (Sigma.mk i ⁻¹' I) (Sigma.mk i ⁻¹' X)
|
simp only [IsBasis, sigma_indep_iff, maximal_subset_iff, and_imp, and_assoc, sigma_ground_eq,
forall_and, and_congr_right_iff]
|
ι : Type u_1
α : ι → Type u_2
M : (i : ι) → Matroid (α i)
I X : Set ((i : ι) × α i)
⊢ (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)) →
((I ⊆ X ∧
(∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧
X ⊆ univ.sigma fun i => (M i).E) ↔
(∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧
(∀ (x : ι) ⦃t : Set (α x)⦄,
(M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻¹' I = t) ∧
∀ (x : ι), Sigma.mk x ⁻¹' X ⊆ (M x).E)
|
4e0fd1a0ca8bdf65
|
Trivialization.nhds_eq_inf_comap
|
Mathlib/Topology/FiberBundle/Trivialization.lean
|
theorem nhds_eq_inf_comap {z : Z} (hz : z ∈ e.source) :
𝓝 z = comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ e) (𝓝 (e z).2)
|
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
z : Z
hz : z ∈ e.source
⊢ 𝓝 z = comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ ↑e) (𝓝 (↑e z).2)
|
refine eq_of_forall_le_iff fun l ↦ ?_
|
B : Type u_1
F : Type u_2
Z : Type u_4
inst✝² : TopologicalSpace B
inst✝¹ : TopologicalSpace F
proj : Z → B
inst✝ : TopologicalSpace Z
e : Trivialization F proj
z : Z
hz : z ∈ e.source
l : Filter Z
⊢ l ≤ 𝓝 z ↔ l ≤ comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ ↑e) (𝓝 (↑e z).2)
|
51e3c253850563a3
|
ONote.repr_mul
|
Mathlib/SetTheory/Ordinal/Notation.lean
|
theorem repr_mul : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ * o₂) = repr o₁ * repr o₂
| 0, o, _, h₂ => by cases o <;> exact (zero_mul _).symm
| oadd _ _ _, 0, _, _ => (mul_zero _).symm
| oadd e₁ n₁ a₁, oadd e₂ n₂ a₂, h₁, h₂ => by
have IH : repr (mul _ _) = _ := @repr_mul _ _ h₁ h₂.snd
conv =>
lhs
simp [(· * ·)]
have ao : repr a₁ + ω ^ repr e₁ * (n₁ : ℕ) = ω ^ repr e₁ * (n₁ : ℕ)
|
e₁ : ONote
n₁ : ℕ+
a₁ e₂ : ONote
n₂ : ℕ+
a₂ : ONote
h₁ : (e₁.oadd n₁ a₁).NF
h₂ : (e₂.oadd n₂ a₂).NF
IH : ((e₁.oadd n₁ a₁).mul a₂).repr = (e₁.oadd n₁ a₁).repr * a₂.repr
⊢ ω ^ e₁.repr ≤ ω ^ e₁.repr * ↑↑n₁
|
simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos _ omega0_pos).2 (Nat.cast_le.2 n₁.2)
|
no goals
|
44048fbeadfd4e40
|
MvPolynomial.msymm_one
|
Mathlib/RingTheory/MvPolynomial/Symmetric/Defs.lean
|
theorem msymm_one : msymm σ R (.indiscrete 1) = ∑ i, X i
|
σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : DecidableEq σ
⊢ (fun x => x ∈ Set.univ) = fun x => Nat.Partition.ofSym x = Nat.Partition.indiscrete 1
|
simp_rw [Set.mem_univ, Nat.Partition.ofSym_one]
|
no goals
|
d6c945688db88dc3
|
EMetric.diam_le_iff
|
Mathlib/Topology/EMetricSpace/Diam.lean
|
theorem diam_le_iff {d : ℝ≥0∞} : diam s ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist x y ≤ d
|
α : Type u_1
s : Set α
inst✝ : PseudoEMetricSpace α
d : ℝ≥0∞
⊢ diam s ≤ d ↔ ∀ x ∈ s, ∀ y ∈ s, edist x y ≤ d
|
simp only [diam, iSup_le_iff]
|
no goals
|
00d705ef4bb2faab
|
LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₃
|
Mathlib/NumberTheory/LSeries/SumCoeff.lean
|
theorem LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₃
(hlim : Tendsto (fun n : ℕ ↦ (∑ k ∈ Icc 1 n, f k) / n) atTop (𝓝 l))
(hfS : ∀ s : ℝ, 1 < s → LSeriesSummable f s) {ε : ℝ} (hε : ε > 0) :
∃ C ≥ 0, (fun s : ℝ ↦ ‖(s - 1) * LSeries f s - s * l‖) ≤ᶠ[𝓝[>] 1]
fun s ↦ (s - 1) * s * C + s * ε
|
case e_a.e_a
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
ε : ℝ
hε : ε > 0
T : ℝ
hT₁ : T ≥ 1
hT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y
S : ℝ → ℂ := fun t => ∑ k ∈ Icc 1 ⌊t⌋₊, f k
C : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-1 - 1)
hC : 0 ≤ C
s : ℝ
hs : 1 < s
hs' : 0 ≤ (s - 1) * s
h₀ : LocallyIntegrableOn (fun t => S t - l * ↑t) (Set.Ici 1) volume
h₁ : IntegrableOn (fun t => ‖S t - l * ↑t‖ * t ^ (-s - 1)) (Set.Ici 1) volume
h₂ : IntegrableOn (fun t => ‖S t - l * ↑t‖ * t ^ (-1 - 1)) (Set.Ioc 1 T) volume
h₃ : (↑s - 1) * ∫ (t : ℝ) in Set.Ioi 1, ↑t ^ (-↑s) = 1
Cs : ℝ := ∫ (t : ℝ) in Set.Ioc 1 T, ‖S t - l * ↑t‖ * t ^ (-s - 1)
h₄ : Cs ≤ C
t : ℝ
ht : ↑t ≠ 0
⊢ S t * ↑t ^ (-↑s - 1) - l * ↑t ^ (-↑s) = (S t - l * ↑t) * ↑t ^ (-↑s - 1)
|
rw [sub_mul, cpow_sub _ _ ht, cpow_one, mul_assoc, mul_div_cancel₀ _ ht]
|
no goals
|
bea02eb3aab4bafa
|
List.sorted_merge
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sort/Lemmas.lean
|
theorem sorted_merge
(trans : ∀ (a b c : α), le a b → le b c → le a c)
(total : ∀ (a b : α), le a b || le b a)
(l₁ l₂ : List α) (h₁ : l₁.Pairwise le) (h₂ : l₂.Pairwise le) : (merge l₁ l₂ le).Pairwise le
|
case cons.cons.isTrue.a
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
x : α
l₁ : List α
ih₁ :
∀ (l₂ : List α),
Pairwise (fun a b => le a b = true) l₁ →
Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) (l₁.merge l₂ le)
h₁ : Pairwise (fun a b => le a b = true) (x :: l₁)
y : α
l₂ : List α
ih₂ : Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) ((x :: l₁).merge l₂ le)
h₂ : Pairwise (fun a b => le a b = true) (y :: l₂)
h : le x y = true
z : α
m : z ∈ l₁.merge (y :: l₂) le
⊢ le x z = true
|
rw [mem_merge, mem_cons] at m
|
case cons.cons.isTrue.a
α : Type u_1
le : α → α → Bool
trans : ∀ (a b c : α), le a b = true → le b c = true → le a c = true
total : ∀ (a b : α), (le a b || le b a) = true
x : α
l₁ : List α
ih₁ :
∀ (l₂ : List α),
Pairwise (fun a b => le a b = true) l₁ →
Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) (l₁.merge l₂ le)
h₁ : Pairwise (fun a b => le a b = true) (x :: l₁)
y : α
l₂ : List α
ih₂ : Pairwise (fun a b => le a b = true) l₂ → Pairwise (fun a b => le a b = true) ((x :: l₁).merge l₂ le)
h₂ : Pairwise (fun a b => le a b = true) (y :: l₂)
h : le x y = true
z : α
m : z ∈ l₁ ∨ z = y ∨ z ∈ l₂
⊢ le x z = true
|
8efa7192a6a8e376
|
MeasureTheory.OuterMeasure.mkMetric_mono_smul
|
Mathlib/MeasureTheory/Measure/Hausdorff.lean
|
theorem mkMetric_mono_smul {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ∞) (h0 : c ≠ 0)
(hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂) : (mkMetric m₁ : OuterMeasure X) ≤ c • mkMetric m₂
|
case intro.intro
X : Type u_2
inst✝ : EMetricSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ⊤
h0 : c ≠ 0
hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂
r : ℝ≥0∞
hr0 : r ∈ Ioi 0
hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x}
s : Set X
r' : ℝ≥0∞
hr' : r' ∈ Ioo 0 r
⊢ r' ∈
{x |
(fun x =>
(fun r => (mkMetric'.pre (fun s => m₁ (diam s)) r) s) x ≤
(fun b => c * (mkMetric'.pre (fun s => m₂ (diam s)) b) s) x)
x}
|
simp only [mem_setOf_eq, mkMetric'.pre, RingHom.id_apply]
|
case intro.intro
X : Type u_2
inst✝ : EMetricSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ⊤
h0 : c ≠ 0
hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂
r : ℝ≥0∞
hr0 : r ∈ Ioi 0
hr : Ico 0 r ⊆ {x | (fun x => m₁ x ≤ (c • m₂) x) x}
s : Set X
r' : ℝ≥0∞
hr' : r' ∈ Ioo 0 r
⊢ (boundedBy (extend fun s x => m₁ (diam s))) s ≤ c * (boundedBy (extend fun s x => m₂ (diam s))) s
|
ad17ed8b50787b10
|
hasDerivAt_integral_of_dominated_loc_of_lip
|
Mathlib/Analysis/Calculus/ParametricIntegral.lean
|
theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
(hF'_meas : AEStronglyMeasurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
(bound_integrable : Integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' a) x₀) :
Integrable F' μ ∧ HasDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀
|
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
ε : ℝ
F : 𝕜 → α → E
x₀ : 𝕜
F' : α → E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀
⊢ Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
|
have hm : AEStronglyMeasurable (L ∘ F') μ := L.continuous.comp_aestronglyMeasurable hF'_meas
|
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
ε : ℝ
F : 𝕜 → α → E
x₀ : 𝕜
F' : α → E
ε_pos : 0 < ε
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
⊢ Integrable F' μ ∧ HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀
|
40a70440e6bc4172
|
exteriorPower.pairingDual_apply_apply_eq_one_zero
|
Mathlib/LinearAlgebra/ExteriorPower/Pairing.lean
|
lemma pairingDual_apply_apply_eq_one_zero (a b : Fin n ↪o ι) (h : a ≠ b) :
pairingDual R M n (ιMulti _ _ (f ∘ a)) (ιMulti _ _ (x ∘ b)) = 0
|
R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a b : Fin n ↪o ι
h : a ≠ b
σ : Equiv.Perm (Fin n)
x✝ : σ ∈ Finset.univ
h' : ¬∏ x_1 : Fin n, (f (a x_1)) (x (b (σ x_1))) = 0
this : ⇑a = ⇑b ∘ ⇑σ
⊢ a = b
|
have hσ : Monotone σ := fun i j hij ↦ by
have h'' := congr_fun this
dsimp at h''
rw [← a.map_rel_iff] at hij
simpa only [← b.map_rel_iff, ← h'']
|
R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a b : Fin n ↪o ι
h : a ≠ b
σ : Equiv.Perm (Fin n)
x✝ : σ ∈ Finset.univ
h' : ¬∏ x_1 : Fin n, (f (a x_1)) (x (b (σ x_1))) = 0
this : ⇑a = ⇑b ∘ ⇑σ
hσ : Monotone ⇑σ
⊢ a = b
|
fd3ea1bd40177425
|
Subspace.dualAnnihilator_inf_eq
|
Mathlib/LinearAlgebra/Dual.lean
|
theorem dualAnnihilator_inf_eq (W W' : Subspace K V₁) :
(W ⊓ W').dualAnnihilator = W.dualAnnihilator ⊔ W'.dualAnnihilator
|
K : Type uK
inst✝² : Field K
V₁ : Type uV₁
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W W' : Subspace K V₁
⊢ dualAnnihilator (W ⊓ W') = dualAnnihilator W ⊔ dualAnnihilator W'
|
refine le_antisymm ?_ (sup_dualAnnihilator_le_inf W W')
|
K : Type uK
inst✝² : Field K
V₁ : Type uV₁
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W W' : Subspace K V₁
⊢ dualAnnihilator (W ⊓ W') ≤ dualAnnihilator W ⊔ dualAnnihilator W'
|
3e94dfba4a8edec6
|
String.utf8GetAux_add_right_cancel
|
Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean
|
theorem utf8GetAux_add_right_cancel (s : List Char) (i p n : Nat) :
utf8GetAux s ⟨i + n⟩ ⟨p + n⟩ = utf8GetAux s ⟨i⟩ ⟨p⟩
|
case ind
s : List Char
i✝ p n : Nat
c : Char
cs : List Char
i : Nat
ih :
utf8GetAux cs { byteIdx := { byteIdx := i }.byteIdx + c.utf8Size + n } { byteIdx := p + n } =
utf8GetAux cs ({ byteIdx := i } + c) { byteIdx := p }
h : ¬i = p
⊢ (if i + n = p + n then c else utf8GetAux cs { byteIdx := i + n + c.utf8Size } { byteIdx := p + n }) =
if i = p then c else utf8GetAux cs { byteIdx := i + c.utf8Size } { byteIdx := p }
|
simp only [Nat.add_right_cancel_iff, h, ↓reduceIte]
|
case ind
s : List Char
i✝ p n : Nat
c : Char
cs : List Char
i : Nat
ih :
utf8GetAux cs { byteIdx := { byteIdx := i }.byteIdx + c.utf8Size + n } { byteIdx := p + n } =
utf8GetAux cs ({ byteIdx := i } + c) { byteIdx := p }
h : ¬i = p
⊢ utf8GetAux cs { byteIdx := i + n + c.utf8Size } { byteIdx := p + n } =
utf8GetAux cs { byteIdx := i + c.utf8Size } { byteIdx := p }
|
b5412c14c2cf0dbc
|
MixedCharZero.reduce_to_maximal_ideal
|
Mathlib/Algebra/CharP/MixedCharZero.lean
|
theorem reduce_to_maximal_ideal {p : ℕ} (hp : Nat.Prime p) :
(∃ I : Ideal R, I ≠ ⊤ ∧ CharP (R ⧸ I) p) ↔ ∃ I : Ideal R, I.IsMaximal ∧ CharP (R ⧸ I) p
|
R : Type u_1
inst✝ : CommRing R
p : ℕ
hp : Nat.Prime p
I : Ideal R
hI_not_top : I ≠ ⊤
right✝ : CharP (R ⧸ I) p
M : Ideal R
hM_max : M.IsMaximal
hM_ge : I ≤ M
r : ℕ
hr : CharP (R ⧸ M) r
⊢ ↑p = 0
|
convert congr_arg (Ideal.Quotient.factor hM_ge) (CharP.cast_eq_zero (R ⧸ I) p)
|
no goals
|
69a224ad99562067
|
IsCyclotomicExtension.finite
|
Mathlib/NumberTheory/Cyclotomic/Basic.lean
|
theorem finite [IsDomain B] [h₁ : Finite S] [h₂ : IsCyclotomicExtension S A B] :
Module.Finite A B
|
S✝ : Set ℕ+
n : ℕ+
S : Set ℕ+
a✝ : n ∉ S
hs✝ : S.Finite
H :
∀ (A : Type u) (B : Type v) [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] [inst_3 : IsDomain B]
[h₂ : IsCyclotomicExtension S A B], Module.Finite A B
A : Type u
B : Type v
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsDomain B
h₂ : IsCyclotomicExtension (S ∪ {n}) A B
this✝ : IsCyclotomicExtension S A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1})
this : Module.Finite A ↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1})
x✝ : IsCyclotomicExtension {n} (↥(adjoin A {b | ∃ a ∈ S, b ^ ↑a = 1})) B := union_right S {n} A B
⊢ Module.Finite (↥(adjoin A {b | ∃ n ∈ S, b ^ ↑n = 1})) B
|
exact finite_of_singleton n _ _
|
no goals
|
0050e7a1aa9fbd3d
|
Fin.pred_succ
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean
|
theorem pred_succ (i : Fin n) {h : i.succ ≠ 0} : i.succ.pred h = i
|
n : Nat
i : Fin n
h : i.succ ≠ 0
⊢ i.succ.pred h = i
|
cases i
|
case mk
n val✝ : Nat
isLt✝ : val✝ < n
h : ⟨val✝, isLt✝⟩.succ ≠ 0
⊢ ⟨val✝, isLt✝⟩.succ.pred h = ⟨val✝, isLt✝⟩
|
8cf79eefd36c5991
|
Polynomial.hasseDeriv_comp
|
Mathlib/Algebra/Polynomial/HasseDeriv.lean
|
theorem hasseDeriv_comp (k l : ℕ) :
(@hasseDeriv R _ k).comp (hasseDeriv l) = (k + l).choose k • hasseDeriv (k + l)
|
case neg.a
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑((i - l).choose k * i.choose l) = ↑((k + l).choose k * i.choose (k + l))
|
push_cast
|
case neg.a
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑((i - l).choose k) * ↑(i.choose l) = ↑((k + l).choose k) * ↑(i.choose (k + l))
|
bf976ccfa5367d28
|
intervalIntegral.integral_comp_sub_mul
|
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
theorem integral_comp_sub_mul (hc : c ≠ 0) (d) :
(∫ x in a..b, f (d - c * x)) = c⁻¹ • ∫ x in d - c * b..d - c * a, f x
|
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
hc : c ≠ 0
d : ℝ
⊢ (-c)⁻¹ • -∫ (x : ℝ) in d + -c * b..d + -c * a, f x = c⁻¹ • ∫ (x : ℝ) in d + -c * b..d + -c * a, f x
|
simp only [inv_neg, smul_neg, neg_neg, neg_smul]
|
no goals
|
29471941aea59eaa
|
Multiset.toFinset_eq_singleton_iff
|
Mathlib/Algebra/Order/Group/Finset.lean
|
lemma toFinset_eq_singleton_iff (s : Multiset α) (a : α) :
s.toFinset = {a} ↔ card s ≠ 0 ∧ s = card s • {a}
|
α : Type u_1
inst✝ : DecidableEq α
s : Multiset α
a : α
⊢ s.toFinset = {a} ↔ s.card ≠ 0 ∧ s = s.card • {a}
|
refine ⟨fun H ↦ ⟨fun h ↦ ?_, ext' fun x ↦ ?_⟩, fun H ↦ ?_⟩
|
case refine_1
α : Type u_1
inst✝ : DecidableEq α
s : Multiset α
a : α
H : s.toFinset = {a}
h : s.card = 0
⊢ False
case refine_2
α : Type u_1
inst✝ : DecidableEq α
s : Multiset α
a : α
H : s.toFinset = {a}
x : α
⊢ count x s = count x (s.card • {a})
case refine_3
α : Type u_1
inst✝ : DecidableEq α
s : Multiset α
a : α
H : s.card ≠ 0 ∧ s = s.card • {a}
⊢ s.toFinset = {a}
|
22486ce822cb5377
|
PrincipalIdealRing.factors_spec
|
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
theorem factors_spec (a : R) (h : a ≠ 0) :
(∀ b ∈ factors a, Irreducible b) ∧ Associated (factors a).prod a
|
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
a : R
h : a ≠ 0
⊢ (∀ b ∈ factors a, Irreducible b) ∧ Associated (factors a).prod a
|
unfold factors
|
R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : IsPrincipalIdealRing R
a : R
h : a ≠ 0
⊢ (∀ b ∈ if h : a = 0 then ∅ else Classical.choose ⋯, Irreducible b) ∧
Associated (if h : a = 0 then ∅ else Classical.choose ⋯).prod a
|
6d62ceca800ddf89
|
Derivation.leibniz_of_mul_eq_one
|
Mathlib/RingTheory/Derivation/Basic.lean
|
theorem leibniz_of_mul_eq_one {a b : A} (h : a * b = 1) : D a = -a ^ 2 • D b
|
R : Type u_1
inst✝⁵ : CommRing R
A : Type u_2
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Module A M
inst✝ : Module R M
D : Derivation R A M
a b : A
h : a * b = 1
⊢ D a + a ^ 2 • D b = a • b • D a + a • a • D b
|
simp only [smul_smul, h, one_smul, sq]
|
no goals
|
43a9916ed8bb80f0
|
Stream'.get_even
|
Mathlib/Data/Stream/Init.lean
|
theorem get_even : ∀ (n : ℕ) (s : Stream' α), get (even s) n = get s (2 * n)
| 0, _ => rfl
| succ n, s => by
change get (even s) (succ n) = get s (succ (succ (2 * n)))
rw [get_succ, get_succ, tail_even, get_even n]; rfl
|
α : Type u
n : ℕ
s : Stream' α
⊢ s.even.get n.succ = s.get (2 * n.succ)
|
change get (even s) (succ n) = get s (succ (succ (2 * n)))
|
α : Type u
n : ℕ
s : Stream' α
⊢ s.even.get n.succ = s.get (2 * n).succ.succ
|
fa65c1f32553eb1f
|
BoxIntegral.HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO
|
Mathlib/Analysis/BoxIntegral/Basic.lean
|
theorem HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO (hl : l.bRiemann = false)
(B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : Set ℝⁿ) (hs : s.Countable)
(hlH : s.Nonempty → l.bHenstock = true)
(H₁ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > (0 : ℝ),
∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → x ∈ Box.Icc J →
(l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε)
(H₂ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I \ s, ∀ ε > (0 : ℝ),
∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Metric.closedBall x δ → (l.bHenstock → x ∈ Box.Icc J) →
(l.bDistortion → J.distortion ≤ c) → dist (vol J (f x)) (g J) ≤ ε * B J) :
HasIntegral I l f vol (g I)
|
case intro.intro.intro.intro
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : s.Countable
hlH : s.Nonempty → l.bHenstock = true
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I ∩ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₁ c x ε) →
x ∈ Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I \ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₂ c x ε) →
(l.bHenstock = true → x ∈ Box.Icc J) →
(l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ i ∈ t, 2 ^ Fintype.card ι * εs i ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
δ : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => if x ∈ s then δ₁ c x (εs x) else δ₂ c x ε'
⊢ ∀ x ∈ {π | ∃ c, l.MemBaseSet I c (δ c) π ∧ π.IsPartition}, integralSum f vol x ∈ Metric.closedBall (g I) ε
|
simp only [Set.mem_iUnion, mem_inter_iff, mem_setOf_eq]
|
case intro.intro.intro.intro
ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s : Set (ι → ℝ)
hs : s.Countable
hlH : s.Nonempty → l.bHenstock = true
ε : ℝ
ε0 : 0 < ε
δ₁ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₁ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I ∩ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₁ c x ε) →
x ∈ Box.Icc J → (l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε
δ₂ : ℝ≥0 → (ι → ℝ) → ℝ → { a // 0 < a }
Hδ₂ :
∀ (c : ℝ≥0),
∀ x ∈ Box.Icc I \ s,
∀ (ε : ℝ),
0 < ε →
∀ J ≤ I,
Box.Icc J ⊆ Metric.closedBall x ↑(δ₂ c x ε) →
(l.bHenstock = true → x ∈ Box.Icc J) →
(l.bDistortion = true → J.distortion ≤ c) → dist ((vol J) (f x)) (g J) ≤ ε * B J
ε0' : 0 < ε / 2
H0 : 0 < 2 ^ Fintype.card ι
εs : (ι → ℝ) → ℝ
hεs0 : ∀ (i : ι → ℝ), 0 < εs i
hεs : ∀ (t : Finset (ι → ℝ)), ↑t ⊆ s → ∑ i ∈ t, 2 ^ Fintype.card ι * εs i ≤ ε / 2
ε' : ℝ
ε'0 : 0 < ε'
hεI : B I * ε' < ε / 2
δ : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => if x ∈ s then δ₁ c x (εs x) else δ₂ c x ε'
⊢ ∀ (x : TaggedPrepartition I),
(∃ c, l.MemBaseSet I c (δ c) x ∧ x.IsPartition) → integralSum f vol x ∈ Metric.closedBall (g I) ε
|
f7fbe41cacd6cc55
|
Batteries.RBNode.lowerBound?_le'
|
Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean
|
theorem lowerBound?_le' {t : RBNode α} (H : ∀ {x}, x ∈ lb → cut x ≠ .lt) :
t.lowerBound? cut lb = some x → cut x ≠ .lt
|
α : Type u_1
lb : Option α
cut : α → Ordering
x : α
t : RBNode α
H : ∀ {x : α}, x ∈ lb → cut x ≠ Ordering.lt
⊢ upperBound? (fun x => (cut x).swap) t.reverse lb = some x → ¬(cut x).swap = Ordering.lt.swap
|
exact upperBound?_ge' fun h => by specialize H h; rwa [Ne, ← Ordering.swap_inj] at H
|
no goals
|
2ad89092ca809906
|
GenContFract.of_s_succ
|
Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
|
theorem of_s_succ (n : ℕ) : (of v).s.get? (n + 1) = (of (fract v)⁻¹).s.get? n
|
case inr
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
h : fract v ≠ 0
⊢ (of v).s.get? (n + 1) = (of (fract v)⁻¹).s.get? n
|
rcases eq_or_ne ((of (fract v)⁻¹).s.get? n) none with h₁ | h₁
|
case inr.inl
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
h : fract v ≠ 0
h₁ : (of (fract v)⁻¹).s.get? n = none
⊢ (of v).s.get? (n + 1) = (of (fract v)⁻¹).s.get? n
case inr.inr
K : Type u_1
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
v : K
n : ℕ
h : fract v ≠ 0
h₁ : (of (fract v)⁻¹).s.get? n ≠ none
⊢ (of v).s.get? (n + 1) = (of (fract v)⁻¹).s.get? n
|
3296185bcb61fc29
|
hasFDerivAt_integral_of_dominated_loc_of_lip_interval
|
Mathlib/Analysis/Calculus/ParametricIntegral.lean
|
theorem hasFDerivAt_integral_of_dominated_loc_of_lip_interval [NormedSpace ℝ H] {μ : Measure ℝ}
{F : H → ℝ → E} {F' : ℝ → H →L[ℝ] E} {a b : ℝ} {bound : ℝ → ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) <| μ.restrict (Ι a b))
(hF_int : IntervalIntegrable (F x₀) μ a b)
(hF'_meas : AEStronglyMeasurable F' <| μ.restrict (Ι a b))
(h_lip : ∀ᵐ t ∂μ.restrict (Ι a b),
LipschitzOnWith (Real.nnabs <| bound t) (F · t) (ball x₀ ε))
(bound_integrable : IntervalIntegrable bound μ a b)
(h_diff : ∀ᵐ t ∂μ.restrict (Ι a b), HasFDerivAt (F · t) (F' t) x₀) :
IntervalIntegrable F' μ a b ∧
HasFDerivAt (fun x ↦ ∫ t in a..b, F x t ∂μ) (∫ t in a..b, F' t ∂μ) x₀
|
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
x₀ : H
ε : ℝ
inst✝ : NormedSpace ℝ H
μ : Measure ℝ
F : H → ℝ → E
F' : ℝ → H →L[ℝ] E
a b : ℝ
bound : ℝ → ℝ
ε_pos : 0 < ε
hF_int : IntervalIntegrable (F x₀) μ a b
h_lip :
(∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)) ∧
∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)
bound_integrable : IntervalIntegrable bound μ a b
h_diff :
(∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc a b), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀) ∧
∀ᵐ (x : ℝ) ∂μ.restrict (Set.Ioc b a), HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀
hF'_meas : AEStronglyMeasurable F' (μ.restrict (Set.Ioc a b)) ∧ AEStronglyMeasurable F' (μ.restrict (Set.Ioc b a))
hF_meas :
(∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Set.Ioc a b))) ∧
∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Set.Ioc b a))
H₁ :
Integrable F' (μ.restrict (Set.Ioc a b)) ∧
HasFDerivAt (fun x => ∫ (a : ℝ) in Set.Ioc a b, F x a ∂μ) (∫ (a : ℝ) in Set.Ioc a b, F' a ∂μ) x₀
H₂ :
Integrable F' (μ.restrict (Set.Ioc b a)) ∧
HasFDerivAt (fun x => ∫ (a : ℝ) in Set.Ioc b a, F x a ∂μ) (∫ (a : ℝ) in Set.Ioc b a, F' a ∂μ) x₀
⊢ IntervalIntegrable F' μ a b ∧ HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' t ∂μ) x₀
|
exact ⟨⟨H₁.1, H₂.1⟩, H₁.2.sub H₂.2⟩
|
no goals
|
e223456fe9a772e9
|
zorn_le_nonempty₀
|
Mathlib/Order/Zorn.lean
|
theorem zorn_le_nonempty₀ (s : Set α)
(ih : ∀ c ⊆ s, IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) (x : α) (hxs : x ∈ s) :
∃ m, x ≤ m ∧ Maximal (· ∈ s) m
|
α : Type u_1
inst✝ : Preorder α
s : Set α
ih : ∀ c ⊆ s, IsChain (fun x1 x2 => x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub
x : α
hxs : x ∈ s
⊢ ∃ m, x ≤ m ∧ Maximal (fun x => x ∈ s) m
|
have H := zorn_le₀ ({ y ∈ s | x ≤ y }) fun c hcs hc => ?_
|
case refine_2
α : Type u_1
inst✝ : Preorder α
s : Set α
ih : ∀ c ⊆ s, IsChain (fun x1 x2 => x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub
x : α
hxs : x ∈ s
H : ∃ m, Maximal (fun x_1 => x_1 ∈ {y | y ∈ s ∧ x ≤ y}) m
⊢ ∃ m, x ≤ m ∧ Maximal (fun x => x ∈ s) m
case refine_1
α : Type u_1
inst✝ : Preorder α
s : Set α
ih : ∀ c ⊆ s, IsChain (fun x1 x2 => x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub
x : α
hxs : x ∈ s
c : Set α
hcs : c ⊆ {y | y ∈ s ∧ x ≤ y}
hc : IsChain (fun x1 x2 => x1 ≤ x2) c
⊢ ∃ ub ∈ {y | y ∈ s ∧ x ≤ y}, ∀ z ∈ c, z ≤ ub
|
fad181aa3c90f03a
|
Descriptive.tree_eq_bot
|
Mathlib/SetTheory/Descriptive/Tree.lean
|
@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where
mp
|
A : Type u_1
T : ↥(tree A)
h : [] ∉ T
⊢ T = ⊥
|
ext x
|
case h
A : Type u_1
T : ↥(tree A)
h : [] ∉ T
x : List A
⊢ x ∈ T ↔ x ∈ ⊥
|
bebd3d9a3bdbec7f
|
Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem
|
Mathlib/RingTheory/GradedAlgebra/Radical.lean
|
theorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous 𝒜)
(I_ne_top : I ≠ ⊤)
(homogeneous_mem_or_mem :
∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :
Ideal.IsPrime I :=
⟨I_ne_top, by
intro x y hxy
by_contra! rid
obtain ⟨rid₁, rid₂⟩ := rid
classical
/-
The idea of the proof is the following :
since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.
Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}`;
let `max₁, max₂` be the maximum of the two sets, then `proj (max₁ + max₂) (x * y) ∈ I`.
Then, `proj max₁ x ∉ I` and `proj max₂ j ∉ I`
but `proj i x ∈ I` for all `max₁ < i` and `proj j y ∈ I` for all `max₂ < j`.
` proj (max₁ + max₂) (x * y)`
`= ∑ {(i, j) ∈ supports | i + j = max₁ + max₂}, xᵢ * yⱼ`
`= proj max₁ x * proj max₂ y`
` + ∑ {(i, j) ∈ supports \ {(max₁, max₂)} | i + j = max₁ + max₂}, xᵢ * yⱼ`.
This is a contradiction, because both `proj (max₁ + max₂) (x * y) ∈ I` and the sum on the
right hand side is in `I` however `proj max₁ x * proj max₂ y` is not in `I`.
-/
set set₁ := {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I} with set₁_eq
set set₂ := {i ∈ (decompose 𝒜 y).support | proj 𝒜 i y ∉ I} with set₂_eq
have nonempty :
∀ x : A, x ∉ I → {i ∈ (decompose 𝒜 x).support | proj 𝒜 i x ∉ I}.Nonempty
|
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₁_eq : set₁ = filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₂ : Finset ι := filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
set₂_eq : set₂ = filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
nonempty : ∀ x ∉ I, (filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))).Nonempty
max₁ : ι := set₁.max' ⋯
max₂ : ι := set₂.max' ⋯
mem_max₁ : max₁ ∈ set₁
mem_max₂ : max₂ ∈ set₂
hxy : (proj 𝒜 (max₁ + max₂)) (x * y) ∈ I
mem_I : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
⊢ (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∉ I
|
have neither_mem : proj 𝒜 max₁ x ∉ I ∧ proj 𝒜 max₂ y ∉ I := by
rw [mem_filter] at mem_max₁ mem_max₂
exact ⟨mem_max₁.2, mem_max₂.2⟩
|
ι : Type u_1
σ : Type u_2
A : Type u_3
inst✝⁴ : CommRing A
inst✝³ : LinearOrderedCancelAddCommMonoid ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
I : Ideal A
hI : IsHomogeneous 𝒜 I
I_ne_top : I ≠ ⊤
homogeneous_mem_or_mem : ∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I
x y : A
rid₁ : x ∉ I
rid₂ : y ∉ I
set₁ : Finset ι := filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₁_eq : set₁ = filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))
set₂ : Finset ι := filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
set₂_eq : set₂ = filter (fun i => (proj 𝒜 i) y ∉ I) (DFinsupp.support ((decompose 𝒜) y))
nonempty : ∀ x ∉ I, (filter (fun i => (proj 𝒜 i) x ∉ I) (DFinsupp.support ((decompose 𝒜) x))).Nonempty
max₁ : ι := set₁.max' ⋯
max₂ : ι := set₂.max' ⋯
mem_max₁ : max₁ ∈ set₁
mem_max₂ : max₂ ∈ set₂
hxy : (proj 𝒜 (max₁ + max₂)) (x * y) ∈ I
mem_I : (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∈ I
neither_mem : (proj 𝒜 max₁) x ∉ I ∧ (proj 𝒜 max₂) y ∉ I
⊢ (proj 𝒜 max₁) x * (proj 𝒜 max₂) y ∉ I
|
7b2fd2fea167fa2f
|
Basis.ext_multilinear
|
Mathlib/LinearAlgebra/Multilinear/Basis.lean
|
theorem Basis.ext_multilinear [Finite ι] {f g : MultilinearMap R (fun _ : ι => M₂) M₃} {ι₁ : Type*}
(e : Basis ι₁ R M₂) (h : ∀ v : ι → ι₁, (f fun i => e (v i)) = g fun i => e (v i)) : f = g
|
R : Type u_1
ι : Type u_2
M₂ : Type u_4
M₃ : Type u_5
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M₂
inst✝³ : AddCommMonoid M₃
inst✝² : Module R M₂
inst✝¹ : Module R M₃
inst✝ : Finite ι
f g : MultilinearMap R (fun x => M₂) M₃
ι₁ : Type u_6
e : Basis ι₁ R M₂
h : ∀ (v : ι → ι₁), (f fun i => e (v i)) = g fun i => e (v i)
⊢ f = g
|
cases nonempty_fintype ι
|
case intro
R : Type u_1
ι : Type u_2
M₂ : Type u_4
M₃ : Type u_5
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M₂
inst✝³ : AddCommMonoid M₃
inst✝² : Module R M₂
inst✝¹ : Module R M₃
inst✝ : Finite ι
f g : MultilinearMap R (fun x => M₂) M₃
ι₁ : Type u_6
e : Basis ι₁ R M₂
h : ∀ (v : ι → ι₁), (f fun i => e (v i)) = g fun i => e (v i)
val✝ : Fintype ι
⊢ f = g
|
e8fb20dfd8be9f92
|
AlgebraicGeometry.stalkMap_injective_of_isOpenMap_of_injective
|
Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean
|
/-- If `f : X ⟶ Y` is open, injective, `X` is quasi-compact and `Y` is affine, then `f` is stalkwise
injective if it is injective on global sections. -/
lemma stalkMap_injective_of_isOpenMap_of_injective [CompactSpace X]
(hfopen : IsOpenMap f.base) (hfinj₁ : Function.Injective f.base)
(hfinj₂ : Function.Injective (f.appTop)) (x : X) :
Function.Injective (f.stalkMap x)
|
X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↑↑X.toPresheafedSpace
hfopen : IsOpenMap ⇑(ConcreteCategory.hom f.base)
hfinj₁ : Function.Injective ⇑(ConcreteCategory.hom f.base)
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↑↑X.toPresheafedSpace
φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := Scheme.Hom.appTop f
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
this : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
res : (i : 𝒰.J) → Γ(X, ⊤) ⟶ Γ(𝒰.obj i, ⊤) := fun i => Scheme.Hom.appTop (𝒰.map i)
g : ↑Γ(Y, ⊤)
h :
(ConcreteCategory.hom (Scheme.Hom.stalkMap f x))
((ConcreteCategory.hom (Y.presheaf.Γgerm ((ConcreteCategory.hom f.base) x))) g) =
0
⊢ (ConcreteCategory.hom (Y.presheaf.Γgerm ((ConcreteCategory.hom f.base) x))) g = 0
|
rw [TopCat.Presheaf.Γgerm, Scheme.stalkMap_germ_apply] at h
|
X Y : Scheme
inst✝¹ : IsAffine Y
f : X ⟶ Y
inst✝ : CompactSpace ↑↑X.toPresheafedSpace
hfopen : IsOpenMap ⇑(ConcreteCategory.hom f.base)
hfinj₁ : Function.Injective ⇑(ConcreteCategory.hom f.base)
hfinj₂ : Function.Injective ⇑(ConcreteCategory.hom (Scheme.Hom.appTop f))
x : ↑↑X.toPresheafedSpace
φ : Γ(Y, ⊤) ⟶ Γ(X, ⊤) := Scheme.Hom.appTop f
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
this : ∀ (i : 𝒰.J), IsAffine (𝒰.obj i)
res : (i : 𝒰.J) → Γ(X, ⊤) ⟶ Γ(𝒰.obj i, ⊤) := fun i => Scheme.Hom.appTop (𝒰.map i)
g : ↑Γ(Y, ⊤)
h : (ConcreteCategory.hom (X.presheaf.germ (f ⁻¹ᵁ ⊤) x True.intro)) ((ConcreteCategory.hom (Scheme.Hom.app f ⊤)) g) = 0
⊢ (ConcreteCategory.hom (Y.presheaf.Γgerm ((ConcreteCategory.hom f.base) x))) g = 0
|
b31e93c621949e44
|
TopCat.pullback_fst_image_snd_preimage
|
Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
|
theorem pullback_fst_image_snd_preimage (f : X ⟶ Z) (g : Y ⟶ Z) (U : Set Y) :
(pullback.fst f g) '' ((pullback.snd f g) ⁻¹' U) =
f ⁻¹' (g '' U)
|
case h.mp
X Y Z : TopCat
f : X ⟶ Z
g : Y ⟶ Z
U : Set ↑Y
x : ↑X
⊢ x ∈ ⇑(ConcreteCategory.hom (pullback.fst f g)) '' (⇑(ConcreteCategory.hom (pullback.snd f g)) ⁻¹' U) →
x ∈ ⇑(ConcreteCategory.hom f) ⁻¹' (⇑(ConcreteCategory.hom g) '' U)
|
rintro ⟨y, hy, rfl⟩
|
case h.mp.intro.intro
X Y Z : TopCat
f : X ⟶ Z
g : Y ⟶ Z
U : Set ↑Y
y : ↑(pullback f g)
hy : y ∈ ⇑(ConcreteCategory.hom (pullback.snd f g)) ⁻¹' U
⊢ (ConcreteCategory.hom (pullback.fst f g)) y ∈ ⇑(ConcreteCategory.hom f) ⁻¹' (⇑(ConcreteCategory.hom g) '' U)
|
17d4c30cd8a6d32e
|
CategoryTheory.rightDistributor_inv
|
Mathlib/CategoryTheory/Monoidal/Preadditive.lean
|
theorem rightDistributor_inv {J : Type} [Fintype J] (f : J → C) (X : C) :
(rightDistributor f X).inv = ∑ j : J, biproduct.π _ j ≫ (biproduct.ι f j ▷ X)
|
case w
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
f : J → C
X : C
j✝ : J
⊢ biproduct.ι (fun j => f j ⊗ X) j✝ ≫ (rightDistributor f X).inv =
biproduct.ι (fun j => f j ⊗ X) j✝ ≫ ∑ j : J, biproduct.π (fun j => f j ⊗ X) j ≫ biproduct.ι f j ▷ X
|
dsimp [rightDistributor, Functor.mapBiproduct, Functor.mapBicone]
|
case w
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasFiniteBiproducts C
J : Type
inst✝ : Fintype J
f : J → C
X : C
j✝ : J
⊢ (biproduct.ι (fun j => f j ⊗ X) j✝ ≫ biproduct.desc fun j => biproduct.ι f j ▷ X) =
biproduct.ι (fun j => f j ⊗ X) j✝ ≫ ∑ j : J, biproduct.π (fun j => f j ⊗ X) j ≫ biproduct.ι f j ▷ X
|
787fce203b0e0f35
|
SchwartzMap.integrable_of_le_of_pow_mul_le
|
Mathlib/Analysis/Distribution/SchwartzSpace.lean
|
/-- Given a function such that `f` and `x ^ (k + l) * f` are bounded for a suitable `l`, then
`x ^ k * f` is integrable. The bounds are not relevant for the integrability conclusion, but they
are relevant for bounding the integral in `integral_pow_mul_le_of_le_of_pow_mul_le`. We formulate
the two lemmas with the same set of assumptions for ease of applications. -/
-- We redeclare `E` here to avoid the `NormedSpace ℝ E` typeclass available throughout this file.
lemma integrable_of_le_of_pow_mul_le
{E : Type*} [NormedAddCommGroup E]
{μ : Measure D} [μ.HasTemperateGrowth] {f : D → E} {C₁ C₂ : ℝ} {k : ℕ}
(hf : ∀ x, ‖f x‖ ≤ C₁) (h'f : ∀ x, ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂)
(h''f : AEStronglyMeasurable f μ) :
Integrable (fun x ↦ ‖x‖ ^ k * ‖f x‖) μ
|
case h.h
D : Type u_3
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : MeasurableSpace D
inst✝³ : BorelSpace D
inst✝² : SecondCountableTopology D
E : Type u_8
inst✝¹ : NormedAddCommGroup E
μ : Measure D
inst✝ : μ.HasTemperateGrowth
f : D → E
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : D), ‖f x‖ ≤ C₁
h'f : ∀ (x : D), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
h''f : AEStronglyMeasurable f μ
v : D
⊢ ‖‖v‖ ^ k * ‖f v‖‖ ≤ 2 ^ μ.integrablePower * (C₁ + C₂) * (1 + ‖v‖) ^ (-↑μ.integrablePower)
|
simp only [norm_mul, norm_pow, norm_norm]
|
case h.h
D : Type u_3
inst✝⁵ : NormedAddCommGroup D
inst✝⁴ : MeasurableSpace D
inst✝³ : BorelSpace D
inst✝² : SecondCountableTopology D
E : Type u_8
inst✝¹ : NormedAddCommGroup E
μ : Measure D
inst✝ : μ.HasTemperateGrowth
f : D → E
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : D), ‖f x‖ ≤ C₁
h'f : ∀ (x : D), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
h''f : AEStronglyMeasurable f μ
v : D
⊢ ‖v‖ ^ k * ‖f v‖ ≤ 2 ^ μ.integrablePower * (C₁ + C₂) * (1 + ‖v‖) ^ (-↑μ.integrablePower)
|
30d7f9baab6716a7
|
Submodule.image2_subset_map₂
|
Mathlib/Algebra/Module/Submodule/Bilinear.lean
|
theorem image2_subset_map₂ (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N) :
Set.image2 (fun m n => f m n) (↑p : Set M) (↑q : Set N) ⊆ (↑(map₂ f p q) : Set P)
|
R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N →ₗ[R] P
p : Submodule R M
q : Submodule R N
⊢ image2 (fun m n => (f m) n) ↑p ↑q ⊆ ↑(map₂ f p q)
|
rintro _ ⟨i, hi, j, hj, rfl⟩
|
case intro.intro.intro.intro
R : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid P
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R P
f : M →ₗ[R] N →ₗ[R] P
p : Submodule R M
q : Submodule R N
i : M
hi : i ∈ ↑p
j : N
hj : j ∈ ↑q
⊢ (fun m n => (f m) n) i j ∈ ↑(map₂ f p q)
|
f6c0a5d024c88229
|
TwoSidedIdeal.mem_sup
|
Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean
|
lemma mem_sup {I J : TwoSidedIdeal R} {x : R} :
x ∈ I ⊔ J ↔ ∃ y ∈ I, ∃ z ∈ J, y + z = x
|
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
I J : TwoSidedIdeal R
x : R
⊢ ∀ {x : R}, x ∈ {x | ∃ y ∈ I, ∃ z ∈ J, y + z = x} → -x ∈ {x | ∃ y ∈ I, ∃ z ∈ J, y + z = x}
|
rintro _ ⟨x, ⟨hx, ⟨y, ⟨hy, rfl⟩⟩⟩⟩
|
case intro.intro.intro.intro
R : Type u_1
inst✝ : NonUnitalNonAssocRing R
I J : TwoSidedIdeal R
x✝ x : R
hx : x ∈ I
y : R
hy : y ∈ J
⊢ -(x + y) ∈ {x | ∃ y ∈ I, ∃ z ∈ J, y + z = x}
|
cb59196376d3b2dd
|
measurableSet_bddAbove_range
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
|
lemma measurableSet_bddAbove_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
MeasurableSet {b | BddAbove (range (fun i ↦ f i b))}
|
α : Type u_1
δ : Type u_4
inst✝⁵ : TopologicalSpace α
mα : MeasurableSpace α
inst✝⁴ : BorelSpace α
mδ : MeasurableSpace δ
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : SecondCountableTopology α
ι : Sort u_5
inst✝ : Countable ι
f : ι → δ → α
hf : ∀ (i : ι), Measurable (f i)
hα : Nonempty α
A : ∀ (i : ι) (c : α), MeasurableSet {x | f i x ≤ c}
c : α
⊢ MeasurableSet (⋂ i, {x | f i x ≤ c})
|
exact MeasurableSet.iInter (fun i ↦ A i c)
|
no goals
|
2ea9408e1624118d
|
maximalIdeal_isPrincipal_of_isDedekindDomain
|
Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean
|
theorem maximalIdeal_isPrincipal_of_isDedekindDomain [IsLocalRing R] [IsDomain R]
[IsDedekindDomain R] : (maximalIdeal R).IsPrincipal
|
case neg.intro.intro.zero
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝ : (Ideal.span {a}).radical = maximalIdeal R
this : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
hn : Nat.find this = 0
⊢ Submodule.IsPrincipal (maximalIdeal R)
|
have := Nat.find_spec this
|
case neg.intro.intro.zero
R : Type u_1
inst✝³ : CommRing R
inst✝² : IsLocalRing R
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ne_bot : ¬maximalIdeal R = ⊥
a : R
ha₁ : a ∈ maximalIdeal R
ha₂ : a ≠ 0
hle : Ideal.span {a} ≤ maximalIdeal R
this✝¹ : (Ideal.span {a}).radical = maximalIdeal R
this✝ : ∃ n, maximalIdeal R ^ n ≤ Ideal.span {a}
hn : Nat.find this✝ = 0
this : maximalIdeal R ^ Nat.find this✝ ≤ Ideal.span {a}
⊢ Submodule.IsPrincipal (maximalIdeal R)
|
e801db6f819d1797
|
edist_le_of_edist_le_geometric_of_tendsto
|
Mathlib/Analysis/SpecificLimits/Basic.lean
|
theorem edist_le_of_edist_le_geometric_of_tendsto {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
edist (f n) a ≤ C * r ^ n / (1 - r)
|
α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
a : α
ha : Tendsto f atTop (𝓝 a)
n : ℕ
⊢ edist (f n) a ≤ C * r ^ n / (1 - r)
|
convert edist_le_tsum_of_edist_le_of_tendsto _ hu ha _
|
case h.e'_4
α : Type u_1
inst✝ : PseudoEMetricSpace α
r C : ℝ≥0∞
f : ℕ → α
hu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n
a : α
ha : Tendsto f atTop (𝓝 a)
n : ℕ
⊢ C * r ^ n / (1 - r) = ∑' (m : ℕ), C * r ^ (n + m)
|
3461a6a04afc537b
|
IsLocalization.scaleRoots_commonDenom_mem_lifts
|
Mathlib/RingTheory/Localization/Integral.lean
|
theorem IsLocalization.scaleRoots_commonDenom_mem_lifts (p : Rₘ[X])
(hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range) :
p.scaleRoots (algebraMap R Rₘ <| IsLocalization.commonDenom M p.support p.coeff) ∈
Polynomial.lifts (algebraMap R Rₘ)
|
R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
Rₘ : Type u_3
inst✝² : CommRing Rₘ
inst✝¹ : Algebra R Rₘ
inst✝ : IsLocalization M Rₘ
p : Rₘ[X]
hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range
⊢ p.scaleRoots ((algebraMap R Rₘ) ↑(commonDenom M p.support p.coeff)) ∈ lifts (algebraMap R Rₘ)
|
rw [Polynomial.lifts_iff_coeff_lifts]
|
R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
Rₘ : Type u_3
inst✝² : CommRing Rₘ
inst✝¹ : Algebra R Rₘ
inst✝ : IsLocalization M Rₘ
p : Rₘ[X]
hp : p.leadingCoeff ∈ (algebraMap R Rₘ).range
⊢ ∀ (n : ℕ),
(p.scaleRoots ((algebraMap R Rₘ) ↑(commonDenom M p.support p.coeff))).coeff n ∈ Set.range ⇑(algebraMap R Rₘ)
|
2bf54bf7debabe79
|
Filter.HasBasis.to_hasBasis'
|
Mathlib/Order/Filter/Bases.lean
|
theorem HasBasis.to_hasBasis' (hl : l.HasBasis p s) (h : ∀ i, p i → ∃ i', p' i' ∧ s' i' ⊆ s i)
(h' : ∀ i', p' i' → s' i' ∈ l) : l.HasBasis p' s'
|
α : Type u_1
ι : Sort u_4
ι' : Sort u_5
l : Filter α
p : ι → Prop
s : ι → Set α
p' : ι' → Prop
s' : ι' → Set α
hl : l.HasBasis p s
h : ∀ (i : ι), p i → ∃ i', p' i' ∧ s' i' ⊆ s i
h' : ∀ (i' : ι'), p' i' → s' i' ∈ l
⊢ l.HasBasis p' s'
|
refine ⟨fun t => ⟨fun ht => ?_, fun ⟨i', hi', ht⟩ => mem_of_superset (h' i' hi') ht⟩⟩
|
α : Type u_1
ι : Sort u_4
ι' : Sort u_5
l : Filter α
p : ι → Prop
s : ι → Set α
p' : ι' → Prop
s' : ι' → Set α
hl : l.HasBasis p s
h : ∀ (i : ι), p i → ∃ i', p' i' ∧ s' i' ⊆ s i
h' : ∀ (i' : ι'), p' i' → s' i' ∈ l
t : Set α
ht : t ∈ l
⊢ ∃ i, p' i ∧ s' i ⊆ t
|
a95362fabd2747b3
|
Real.Angle.sin_zero
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
theorem sin_zero : sin (0 : Angle) = 0
|
⊢ sin 0 = 0
|
rw [← coe_zero, sin_coe, Real.sin_zero]
|
no goals
|
3183bb9a63a0554b
|
MeasureTheory.addContent_le_sum_of_subset_sUnion
|
Mathlib/MeasureTheory/Measure/AddContent.lean
|
lemma addContent_le_sum_of_subset_sUnion {m : AddContent C} (hC : IsSetSemiring C)
{J : Finset (Set α)} (h_ss : ↑J ⊆ C) (ht : t ∈ C) (htJ : t ⊆ ⋃₀ ↑J) :
m t ≤ ∑ u ∈ J, m u
|
α : Type u_1
C : Set (Set α)
t : Set α
m : AddContent C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : ↑J ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ ↑J
⊢ m t ≤ ∑ u ∈ J, m u
|
let Jt := J.image (fun u ↦ t ∩ u)
|
α : Type u_1
C : Set (Set α)
t : Set α
m : AddContent C
hC : IsSetSemiring C
J : Finset (Set α)
h_ss : ↑J ⊆ C
ht : t ∈ C
htJ : t ⊆ ⋃₀ ↑J
Jt : Finset (Set α) := Finset.image (fun u => t ∩ u) J
⊢ m t ≤ ∑ u ∈ J, m u
|
81e047e530496612
|
EReal.continuous_toENNReal
|
Mathlib/Topology/Instances/EReal/Lemmas.lean
|
lemma continuous_toENNReal : Continuous EReal.toENNReal
|
⊢ Continuous toENNReal
|
refine continuous_iff_continuousAt.mpr fun x ↦ ?_
|
x : EReal
⊢ ContinuousAt toENNReal x
|
e6b0a6cf0669f603
|
Finset.noncommProd_union_of_disjoint
|
Mathlib/Data/Finset/NoncommProd.lean
|
theorem noncommProd_union_of_disjoint [DecidableEq α] {s t : Finset α} (h : Disjoint s t)
(f : α → β) (comm : { x | x ∈ s ∪ t }.Pairwise (Commute on f)) :
noncommProd (s ∪ t) f comm =
noncommProd s f (comm.mono <| coe_subset.2 subset_union_left) *
noncommProd t f (comm.mono <| coe_subset.2 subset_union_right)
|
case h
α : Type u_3
β : Type u_4
inst✝¹ : Monoid β
inst✝ : DecidableEq α
f : α → β
sl : List α
sl' : sl.Nodup
tl : List α
tl' : tl.Nodup
h : sl.Disjoint tl
comm : {x | x ∈ sl.toFinset ∪ tl.toFinset}.Pairwise (Commute on f)
a✝ : α
⊢ a✝ ∈ sl.toFinset ∪ tl.toFinset ↔ a✝ ∈ { val := ↑(sl ++ tl), nodup := ⋯ }
|
simp
|
no goals
|
b1392de476fe3f45
|
HurwitzZeta.expZeta_one_sub
|
Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
|
/-- Functional equation for the exponential zeta function. -/
lemma expZeta_one_sub (a : UnitAddCircle) {s : ℂ} (hs : ∀ (n : ℕ), s ≠ 1 - n) :
expZeta a (1 - s) = (2 * π) ^ (-s) * Gamma s *
(exp (π * I * s / 2) * hurwitzZeta a s + exp (-π * I * s / 2) * hurwitzZeta (-a) s)
|
a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ 1 - ↑n
hs' : ∀ (n : ℕ), s ≠ -↑n
⊢ 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * ((cexp (↑π * s / 2 * I) + cexp (-(↑π * s / 2) * I)) / 2) *
hurwitzZetaEven a s +
I *
(2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * ((cexp (-(↑π * s / 2) * I) - cexp (↑π * s / 2 * I)) * I / 2) *
hurwitzZetaOdd a s) =
(2 * ↑π) ^ (-s) * Complex.Gamma s *
(cexp (↑π * I * s / 2) * (hurwitzZetaEven a s + hurwitzZetaOdd a s) +
cexp (-↑π * I * s / 2) * (hurwitzZetaEven a s + -hurwitzZetaOdd a s))
|
rw [show ↑π * I * s / 2 = ↑π * s / 2 * I by ring,
show -↑π * I * s / 2 = -(↑π * s / 2) * I by ring]
|
a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ 1 - ↑n
hs' : ∀ (n : ℕ), s ≠ -↑n
⊢ 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * ((cexp (↑π * s / 2 * I) + cexp (-(↑π * s / 2) * I)) / 2) *
hurwitzZetaEven a s +
I *
(2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * ((cexp (-(↑π * s / 2) * I) - cexp (↑π * s / 2 * I)) * I / 2) *
hurwitzZetaOdd a s) =
(2 * ↑π) ^ (-s) * Complex.Gamma s *
(cexp (↑π * s / 2 * I) * (hurwitzZetaEven a s + hurwitzZetaOdd a s) +
cexp (-(↑π * s / 2) * I) * (hurwitzZetaEven a s + -hurwitzZetaOdd a s))
|
e843e482da66a44f
|
List.erase_eq_eraseP'
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean
|
theorem erase_eq_eraseP' (a : α) (l : List α) : l.erase a = l.eraseP (· == a)
|
α : Type u_1
inst✝ : BEq α
a b : α
t : List α
ih : t.erase a = eraseP (fun x => x == a) t
⊢ (if (b == a) = true then t else b :: eraseP (fun x => x == a) t) =
bif b == a then t else b :: eraseP (fun x => x == a) t
|
if h : b == a then simp [h] else simp [h]
|
no goals
|
a7d8cd443cc25e52
|
ArithmeticFunction.zeta_mul_pow_eq_sigma
|
Mathlib/NumberTheory/ArithmeticFunction.lean
|
theorem zeta_mul_pow_eq_sigma {k : ℕ} : ζ * pow k = σ k
|
case h
k x✝ x : ℕ
hx : x ∈ x✝.divisors
⊢ (pow k) x = x ^ k
|
rw [pow_apply, if_neg (not_and_of_not_right _ _)]
|
k x✝ x : ℕ
hx : x ∈ x✝.divisors
⊢ ¬x = 0
|
f10614b722390e1d
|
List.Chain'.two_mul_count_bool_le_length_add_one
|
Mathlib/Data/Bool/Count.lean
|
theorem two_mul_count_bool_le_length_add_one (hl : Chain' (· ≠ ·) l) (b : Bool) :
2 * count b l ≤ length l + 1
|
l : List Bool
hl : Chain' (fun x1 x2 => x1 ≠ x2) l
b : Bool
⊢ 2 * count b l ≤ l.length + 1
|
rw [hl.two_mul_count_bool_eq_ite]
|
l : List Bool
hl : Chain' (fun x1 x2 => x1 ≠ x2) l
b : Bool
⊢ (if Even l.length then l.length else if (some b == l.head?) = true then l.length + 1 else l.length - 1) ≤ l.length + 1
|
1f625c7dba8a04ca
|
Polynomial.natDegree_mul_X
|
Mathlib/Algebra/Polynomial/Degree/Operations.lean
|
@[simp] lemma natDegree_mul_X (hp : p ≠ 0) : natDegree (p * X) = natDegree p + 1
|
R : Type u
inst✝¹ : Semiring R
inst✝ : Nontrivial R
p : R[X]
hp : p ≠ 0
⊢ p.leadingCoeff * X.leadingCoeff ≠ 0
|
simpa
|
no goals
|
bc5b6256cfdb6c9b
|
MeasureTheory.exists_continuous_eLpNorm_sub_le_of_closed
|
Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
|
theorem exists_continuous_eLpNorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ ∞) {s u : Set α}
(s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ∞) (c : E) {ε : ℝ≥0∞}
(hε : ε ≠ 0) :
∃ f : α → E,
Continuous f ∧
eLpNorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ ε ∧
(∀ x, ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ MemLp f p μ
|
case pos.inr
α : Type u_1
inst✝⁶ : TopologicalSpace α
inst✝⁵ : NormalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
E : Type u_2
inst✝² : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝¹ : NormedSpace ℝ E
inst✝ : μ.OuterRegular
hp : p ≠ ⊤
s u : Set α
s_closed : IsClosed s
u_open : IsOpen u
hsu : s ⊆ u
hs✝ : μ s ≠ ⊤
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
η : ℝ≥0
η_pos : 0 < η
hη : ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun _x => c) p μ ≤ ε
ηpos : 0 < ↑η
V : Set α
sV : V ⊇ s
V_open : IsOpen V
h'V : μ V < ⊤
hV : μ (V \ s) < ↑η
v : Set α := u ∩ V
hsv : s ⊆ v
hμv : μ v < ⊤
g : C(α, ℝ)
hgv : EqOn (⇑g) 0 (u ∩ V)ᶜ
hgs : EqOn (⇑g) 1 s
hg_range : ∀ (x : α), g x ∈ Icc 0 1
g_norm : ∀ (x : α), ‖g x‖ = g x
gc_bd0 : ∀ (x : α), ‖g x • c‖ ≤ ‖c‖
x : α
hs : x ∈ s
⊢ ‖g x • c - s.indicator (fun _x => c) x‖ ≤ ‖(v \ s).indicator (fun _x => c) x‖
|
simp [hgs hs, hs]
|
no goals
|
4686c200fbebc327
|
MvPolynomial.eval₂_cast_comp
|
Mathlib/Algebra/MvPolynomial/Rename.lean
|
theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p)
|
σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
c : ℤ →+* R
g : τ → R
p✝ p q : MvPolynomial σ ℤ
hp : eval₂ c (g ∘ f) p = eval₂ c g ((rename f) p)
hq : eval₂ c (g ∘ f) q = eval₂ c g ((rename f) q)
⊢ eval₂ c (g ∘ f) (p + q) = eval₂ c g ((rename f) (p + q))
|
simp only [hp, hq, rename, eval₂_add, map_add]
|
no goals
|
89626df0693e355f
|
StieltjesFunction.measure_Iic
|
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
theorem measure_Iic {l : ℝ} (hf : Tendsto f atBot (𝓝 l)) (x : ℝ) :
f.measure (Iic x) = ofReal (f x - l)
|
f : StieltjesFunction
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : ℝ
⊢ Tendsto (fun x_1 => f.measure (Ioc x_1 x)) atBot (𝓝 (ofReal (↑f x - l)))
|
simp_rw [measure_Ioc]
|
f : StieltjesFunction
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : ℝ
⊢ Tendsto (fun x_1 => ofReal (↑f x - ↑f x_1)) atBot (𝓝 (ofReal (↑f x - l)))
|
029bdab6bc2d0628
|
IsGalois.card_aut_eq_finrank
|
Mathlib/FieldTheory/Galois/Basic.lean
|
theorem card_aut_eq_finrank [FiniteDimensional F E] [IsGalois F E] :
Fintype.card (E ≃ₐ[F] E) = finrank F E
|
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
|
obtain ⟨α, hα⟩ := Field.exists_primitive_element F E
|
case intro
F : Type u_1
inst✝⁴ : Field F
E : Type u_2
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : FiniteDimensional F E
inst✝ : IsGalois F E
α : E
hα : F⟮α⟯ = ⊤
⊢ Fintype.card (E ≃ₐ[F] E) = finrank F E
|
9073114732e73abf
|
MeasureTheory.Measure.addHaar_submodule
|
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
theorem addHaar_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E]
[BorelSpace E] [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] (s : Submodule ℝ E)
(hs : s ≠ ⊤) : μ s = 0
|
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A✝ : Bornology.IsBounded (range fun n => c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
hyn : -(c ^ n • x) + y ∈ s
A : (c ^ n - c ^ m) • x ∈ s
⊢ c ^ n - c ^ m ≠ 0
|
simpa only [sub_eq_zero, Ne] using (pow_right_strictAnti₀ cpos cone).injective.ne hmn.symm
|
no goals
|
0c3f2ca004ff3b96
|
Std.Tactic.BVDecide.LRAT.Internal.limplies_unsat
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Entails.lean
|
theorem limplies_unsat {α : Type u} {σ1 : Type v} {σ2 : Type w} [Entails α σ1] [Entails α σ2]
(f1 : σ1) (f2 : σ2) (h : Limplies α f2 f1) :
Unsatisfiable α f1 → Unsatisfiable α f2
|
α : Type u
σ1 : Type v
σ2 : Type w
inst✝¹ : Entails α σ1
inst✝ : Entails α σ2
f1 : σ1
f2 : σ2
h : Limplies α f2 f1
f1_unsat : Unsatisfiable α f1
a : α → Bool
a_entails_f2 : a ⊨ f2
⊢ False
|
exact f1_unsat a <| h a a_entails_f2
|
no goals
|
7bb96243a6e45cc8
|
LaurentPolynomial.leftInverse_trunc_toLaurent
|
Mathlib/Algebra/Polynomial/Laurent.lean
|
theorem leftInverse_trunc_toLaurent :
Function.LeftInverse (trunc : R[T;T⁻¹] → R[X]) Polynomial.toLaurent
|
case refine_2
R : Type u_1
inst✝ : Semiring R
f : R[X]
⊢ ∀ (n : ℕ) (a : R), trunc (toLaurent ((monomial n) a)) = (monomial n) a
|
intro n r
|
case refine_2
R : Type u_1
inst✝ : Semiring R
f : R[X]
n : ℕ
r : R
⊢ trunc (toLaurent ((monomial n) r)) = (monomial n) r
|
b85722c5169ce13e
|
DedekindDomain.ProdAdicCompletions.IsFiniteAdele.zero
|
Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean
|
theorem zero : (0 : K_hat R K).IsFiniteAdele
|
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
⊢ {v | 0 ∉ adicCompletionIntegers K v} = ∅
|
ext v
|
case h
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
⊢ v ∈ {v | 0 ∉ adicCompletionIntegers K v} ↔ v ∈ ∅
|
dfa7b46906b35ef2
|
ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_le_one
|
Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean
|
lemma IsMeasurableRatCDF.stieltjesFunction_le_one (a : α) (x : ℝ) :
hf.stieltjesFunction a x ≤ 1
|
case intro
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
⊢ ↑(hf.stieltjesFunction a) x ≤ 1
|
rw [← StieltjesFunction.iInf_rat_gt_eq]
|
case intro
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
⊢ ⨅ r, ↑(hf.stieltjesFunction a) ↑↑r ≤ 1
|
969ac33d4dcd9898
|
erase_eq_iff
|
Mathlib/Combinatorics/SetFamily/FourFunctions.lean
|
private lemma erase_eq_iff (hs : a ∉ s) : t.erase a = s ↔ t = s ∨ t = insert a s
|
case neg
α : Type u_1
inst✝ : DecidableEq α
a : α
s t : Finset α
hs : a ∉ s
ht : a ∉ t
⊢ t.erase a = s ↔ t = s ∨ t = insert a s
|
simp [ne_of_mem_of_not_mem', erase_eq_iff_eq_insert, *]
|
case neg
α : Type u_1
inst✝ : DecidableEq α
a : α
s t : Finset α
hs : a ∉ s
ht : a ∉ t
⊢ t = insert a s → t = s
|
a6760e58bbc205dd
|
HahnSeries.coeff_orderTop_ne
|
Mathlib/RingTheory/HahnSeries/Basic.lean
|
theorem coeff_orderTop_ne {x : HahnSeries Γ R} {g : Γ} (hg : x.orderTop = g) :
x.coeff g ≠ 0
|
Γ : Type u_1
R : Type u_3
inst✝¹ : PartialOrder Γ
inst✝ : Zero R
x : HahnSeries Γ R
g : Γ
hg : x.orderTop = ↑g
⊢ x.orderTop ≠ ⊤
|
simp_all only [ne_eq, WithTop.coe_ne_top, not_false_eq_true]
|
no goals
|
bb1d2cdc417587d8
|
IsPreconnected.union'
|
Mathlib/Topology/Connected/Basic.lean
|
theorem IsPreconnected.union' {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ∪ t)
|
case intro.intro
α : Type u
inst✝ : TopologicalSpace α
s t : Set α
hs : IsPreconnected s
ht : IsPreconnected t
x : α
hxs : x ∈ s
hxt : x ∈ t
⊢ IsPreconnected (s ∪ t)
|
exact hs.union x hxs hxt ht
|
no goals
|
a2c85e15ef4c078b
|
Ordnode.Valid'.node4L
|
Mathlib/Data/Ordmap/Ordset.lean
|
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
(hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
(H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
0 < size l ∧
ratio * size r ≤ size m ∧
delta * size l ≤ size m + size r ∧
3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
Valid' o₁ (@node4L α l x m y r) o₂
|
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < (Ordnode.node s ml z mr).size
l0 : 0 < l.size
mr₁ : ratio * r.size ≤ ml.size + mr.size + 1
lr₁ : delta * l.size ≤ ml.size + mr.size + 1 + r.size
lr₂ : 3 * (ml.size + mr.size + 1 + r.size) ≤ 16 * l.size + 9
mr₂ : ml.size + mr.size + 1 ≤ delta * r.size
r0 : r.size > 0
mm : ¬ml.size + mr.size ≤ 1
mm₁ : ml.size ≤ delta * mr.size
mm₂ : mr.size ≤ delta * ml.size
ml0 : ml.size > 0
this✝ : delta * (ratio * l.size) ≤ ratio * (ml.size + mr.size + 1) + ratio * r.size
this : delta * (ratio * l.size) ≤ ratio.succ * (ml.size + mr.size + 1)
⊢ 2 * l.size ≤ ml.size + mr.size + 1
|
exact (mul_le_mul_left (by decide)).1 this
|
no goals
|
4d0cc9fa23e9caa2
|
MonoidHom.exponent_dvd
|
Mathlib/GroupTheory/Exponent.lean
|
theorem MonoidHom.exponent_dvd {F M₁ M₂ : Type*} [Monoid M₁] [Monoid M₂]
[FunLike F M₁ M₂] [MonoidHomClass F M₁ M₂]
{f : F} (hf : Function.Surjective f) : exponent M₂ ∣ exponent M₁
|
case intro
F : Type u_1
M₁ : Type u_2
M₂ : Type u_3
inst✝³ : Monoid M₁
inst✝² : Monoid M₂
inst✝¹ : FunLike F M₁ M₂
inst✝ : MonoidHomClass F M₁ M₂
f : F
hf : Function.Surjective ⇑f
m₁ : M₁
⊢ f m₁ ^ exponent M₁ = 1
|
rw [← map_pow, pow_exponent_eq_one, map_one]
|
no goals
|
1422a75d8bf15951
|
String.ltb_cons_addChar
|
Mathlib/Data/String/Basic.lean
|
theorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos) :
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ = ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩
|
c : Char
cs₁ cs₂ : List Char
i₁ i₂ : Pos
⊢ ltb { s := { data := c :: cs₁ }, i := i₁ + c } { s := { data := c :: cs₂ }, i := i₂ + c } =
ltb { s := { data := cs₁ }, i := i₁ } { s := { data := cs₂ }, i := i₂ }
|
apply ltb.inductionOn ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ (motive := fun ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ ↦
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ =
ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩) <;> simp only <;>
intro ⟨cs₁⟩ ⟨cs₂⟩ i₁ i₂ <;>
intros <;>
(conv => lhs; unfold ltb) <;> (conv => rhs; unfold ltb) <;>
simp only [Iterator.hasNext_cons_addChar, ite_false, ite_true, *, reduceCtorEq]
|
case ind
c : Char
cs₁✝ cs₂✝ : List Char
i₁✝ i₂✝ : Pos
cs₁ cs₂ : List Char
i₁ i₂ : Pos
a✝³ : { s := { data := cs₂ }, i := i₂ }.hasNext = true
a✝² : { s := { data := cs₁ }, i := i₁ }.hasNext = true
a✝¹ : { data := cs₁ }.get i₁ = { data := cs₂ }.get i₂
a✝ :
ltb
{ s := { data := c :: { s := { data := cs₁ }, i := i₁ }.next.1.data },
i := { s := { data := cs₁ }, i := i₁ }.next.i + c }
{ s := { data := c :: { s := { data := cs₂ }, i := i₂ }.next.1.data },
i := { s := { data := cs₂ }, i := i₂ }.next.i + c } =
ltb
{ s := { data := { s := { data := cs₁ }, i := i₁ }.next.1.data }, i := { s := { data := cs₁ }, i := i₁ }.next.i }
{ s := { data := { s := { data := cs₂ }, i := i₂ }.next.1.data }, i := { s := { data := cs₂ }, i := i₂ }.next.i }
⊢ (if { s := { data := c :: cs₁ }, i := i₁ + c }.curr = { s := { data := c :: cs₂ }, i := i₂ + c }.curr then
ltb { s := { data := c :: cs₁ }, i := i₁ + c }.next { s := { data := c :: cs₂ }, i := i₂ + c }.next
else decide ({ s := { data := c :: cs₁ }, i := i₁ + c }.curr < { s := { data := c :: cs₂ }, i := i₂ + c }.curr)) =
if { s := { data := cs₁ }, i := i₁ }.curr = { s := { data := cs₂ }, i := i₂ }.curr then
ltb { s := { data := cs₁ }, i := i₁ }.next { s := { data := cs₂ }, i := i₂ }.next
else decide ({ s := { data := cs₁ }, i := i₁ }.curr < { s := { data := cs₂ }, i := i₂ }.curr)
case eq
c : Char
cs₁✝ cs₂✝ : List Char
i₁✝ i₂✝ : Pos
cs₁ cs₂ : List Char
i₁ i₂ : Pos
a✝² : { s := { data := cs₂ }, i := i₂ }.hasNext = true
a✝¹ : { s := { data := cs₁ }, i := i₁ }.hasNext = true
a✝ : ¬{ data := cs₁ }.get i₁ = { data := cs₂ }.get i₂
⊢ (if { s := { data := c :: cs₁ }, i := i₁ + c }.curr = { s := { data := c :: cs₂ }, i := i₂ + c }.curr then
ltb { s := { data := c :: cs₁ }, i := i₁ + c }.next { s := { data := c :: cs₂ }, i := i₂ + c }.next
else decide ({ s := { data := c :: cs₁ }, i := i₁ + c }.curr < { s := { data := c :: cs₂ }, i := i₂ + c }.curr)) =
if { s := { data := cs₁ }, i := i₁ }.curr = { s := { data := cs₂ }, i := i₂ }.curr then
ltb { s := { data := cs₁ }, i := i₁ }.next { s := { data := cs₂ }, i := i₂ }.next
else decide ({ s := { data := cs₁ }, i := i₁ }.curr < { s := { data := cs₂ }, i := i₂ }.curr)
|
335ab19a00638e55
|
top_le_span_of_aux
|
Mathlib/LinearAlgebra/Basis/Exact.lean
|
private lemma top_le_span_of_aux (v : κ ⊕ σ → M)
(hg : Function.Surjective g) (hslzero : ∀ i, s (v (.inl i)) = 0)
(hli : LinearIndependent R (s ∘ v ∘ .inr)) (hsp : ⊤ ≤ Submodule.span R (Set.range v)) :
⊤ ≤ Submodule.span R (Set.range <| g ∘ v ∘ .inl)
|
case intro.inr
R : Type u_1
M : Type u_2
K : Type u_3
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup K
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R K
inst✝ : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
κ : Type u_6
σ : Type u_7
v : κ ⊕ σ → M
hg : Function.Surjective ⇑g
hslzero : ∀ (i : κ), s (v (Sum.inl i)) = 0
hli : LinearIndependent R (⇑s ∘ v ∘ Sum.inr)
hsp : ⊤ ≤ Submodule.span R (Set.range v)
m : M
this :
∀ {R : Type u_1} {M : Type u_2} {K : Type u_3} {P : Type u_4} [inst : Ring R] [inst_1 : AddCommGroup M]
[inst_2 : AddCommGroup K] [inst_3 : AddCommGroup P] [inst_4 : Module R M] [inst_5 : Module R K]
[inst_6 : Module R P] {f : K →ₗ[R] M} {g : M →ₗ[R] P} {s : M →ₗ[R] K},
s ∘ₗ f = LinearMap.id →
Function.Exact ⇑f ⇑g →
∀ {κ : Type u_6} {σ : Type u_7} (v : κ ⊕ σ → M),
Function.Surjective ⇑g →
(∀ (i : κ), s (v (Sum.inl i)) = 0) →
LinearIndependent R (⇑s ∘ v ∘ Sum.inr) →
⊤ ≤ Submodule.span R (Set.range v) →
∀ m ∈ LinearMap.ker s, g m ∈ Submodule.span R (Set.range (⇑g ∘ v ∘ Sum.inl))
h : m ∉ LinearMap.ker s
x : M := f (s m)
⊢ g (m - f (s m)) ∈ Submodule.span R (Set.range (⇑g ∘ v ∘ Sum.inl))
|
apply this hs hfg v hg hslzero hli hsp
|
case intro.inr.h
R : Type u_1
M : Type u_2
K : Type u_3
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup K
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R K
inst✝ : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
κ : Type u_6
σ : Type u_7
v : κ ⊕ σ → M
hg : Function.Surjective ⇑g
hslzero : ∀ (i : κ), s (v (Sum.inl i)) = 0
hli : LinearIndependent R (⇑s ∘ v ∘ Sum.inr)
hsp : ⊤ ≤ Submodule.span R (Set.range v)
m : M
this :
∀ {R : Type u_1} {M : Type u_2} {K : Type u_3} {P : Type u_4} [inst : Ring R] [inst_1 : AddCommGroup M]
[inst_2 : AddCommGroup K] [inst_3 : AddCommGroup P] [inst_4 : Module R M] [inst_5 : Module R K]
[inst_6 : Module R P] {f : K →ₗ[R] M} {g : M →ₗ[R] P} {s : M →ₗ[R] K},
s ∘ₗ f = LinearMap.id →
Function.Exact ⇑f ⇑g →
∀ {κ : Type u_6} {σ : Type u_7} (v : κ ⊕ σ → M),
Function.Surjective ⇑g →
(∀ (i : κ), s (v (Sum.inl i)) = 0) →
LinearIndependent R (⇑s ∘ v ∘ Sum.inr) →
⊤ ≤ Submodule.span R (Set.range v) →
∀ m ∈ LinearMap.ker s, g m ∈ Submodule.span R (Set.range (⇑g ∘ v ∘ Sum.inl))
h : m ∉ LinearMap.ker s
x : M := f (s m)
⊢ m - f (s m) ∈ LinearMap.ker s
|
d124f5729aeaf148
|
WeierstrassCurve.Φ_four
|
Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean
|
@[simp]
lemma Φ_four : W.Φ 4 = X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)
|
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ W.Φ 4 = X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)
|
rw [show 4 = ((3 : ℕ) + 1 : ℤ) by rfl, Φ_ofNat, preΨ'_four, if_neg <| by decide,
show 3 + 2 = 2 * 2 + 1 by rfl, preΨ'_odd, preΨ'_four, preΨ'_two, if_pos Even.zero, preΨ'_one,
preΨ'_three, if_pos Even.zero, if_neg <| by decide]
|
R : Type r
inst✝ : CommRing R
W : WeierstrassCurve R
⊢ X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - (W.preΨ₄ * 1 ^ 3 * W.Ψ₂Sq ^ 2 - 1 * W.Ψ₃ ^ 3 * 1) * W.Ψ₃ * 1 =
X * W.preΨ₄ ^ 2 * W.Ψ₂Sq - W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)
|
3f9ac080aad09b8e
|
ComplexShape.boundaryLE_embeddingUpIntLE_iff
|
Mathlib/Algebra/Homology/Embedding/Boundary.lean
|
lemma boundaryLE_embeddingUpIntLE_iff (p : ℤ) (n : ℕ) :
(embeddingUpIntGE p).BoundaryGE n ↔ n = 0
|
case mpr.right
p : ℤ
i : ℕ
hi : (up ℤ).Rel ((embeddingUpIntGE p).f i) ((embeddingUpIntGE p).f 0)
⊢ False
|
dsimp at hi
|
case mpr.right
p : ℤ
i : ℕ
hi : p + ↑i + 1 = p + 0
⊢ False
|
14489894ffc1fd9c
|
VitaliFamily.exists_measurable_supersets_limRatio
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
{x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I✝ : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L : Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x))
I : ∀ᶠ (b : Set α) in v.filterAt x, ρ b / μ b < ↑p
a : Set α
ha : ρ a / μ a < ↑p
⊢ ρ a ≤ (p • μ) a
|
rw [coe_nnreal_smul_apply]
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I✝ : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L : Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x))
I : ∀ᶠ (b : Set α) in v.filterAt x, ρ b / μ b < ↑p
a : Set α
ha : ρ a / μ a < ↑p
⊢ ρ a ≤ ↑p * μ a
|
f2acb3dc8c3a0a55
|
Array.findIdx?_mkArray
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean
|
theorem findIdx?_mkArray :
(mkArray n a).findIdx? p = if 0 < n ∧ p a then some 0 else none
|
n : Nat
α✝ : Type u_1
a : α✝
p : α✝ → Bool
⊢ List.findIdx? p (List.replicate n a) = if 0 < n ∧ p a = true then some 0 else none
|
simp
|
no goals
|
751156ffdaeb598a
|
Convex.mem_smul_of_zero_mem
|
Mathlib/Analysis/Convex/Basic.lean
|
theorem Convex.mem_smul_of_zero_mem (h : Convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s) (hx : x ∈ s)
{t : 𝕜} (ht : 1 ≤ t) : x ∈ t • s
|
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
h : Convex 𝕜 s
x : E
zero_mem : 0 ∈ s
hx : x ∈ s
t : 𝕜
ht : 1 ≤ t
⊢ x ∈ t • s
|
rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne']
|
𝕜 : Type u_1
E : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s : Set E
h : Convex 𝕜 s
x : E
zero_mem : 0 ∈ s
hx : x ∈ s
t : 𝕜
ht : 1 ≤ t
⊢ t⁻¹ • x ∈ s
|
e77250cfc151c423
|
ProbabilityTheory.lintegral_exponentialPDF_eq_antiDeriv
|
Mathlib/Probability/Distributions/Exponential.lean
|
lemma lintegral_exponentialPDF_eq_antiDeriv {r : ℝ} (hr : 0 < r) (x : ℝ) :
∫⁻ y in Iic x, exponentialPDF r y
= ENNReal.ofReal (if 0 ≤ x then 1 - exp (-(r * x)) else 0)
|
r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
this : ∫ (a : ℝ) in Ι 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0 ..x, r * rexp (-(r * a))
⊢ ∫ (a : ℝ) in Icc 0 x, r * rexp (-(r * a)) = (ENNReal.ofReal (1 - rexp (-(r * x)))).toReal
|
rw [integral_Icc_eq_integral_Ioc, ← uIoc_of_le h, this]
|
r : ℝ
hr : 0 < r
x : ℝ
h : 0 ≤ x
this : ∫ (a : ℝ) in Ι 0 x, r * rexp (-(r * a)) = ∫ (a : ℝ) in 0 ..x, r * rexp (-(r * a))
⊢ ∫ (a : ℝ) in 0 ..x, r * rexp (-(r * a)) = (ENNReal.ofReal (1 - rexp (-(r * x)))).toReal
|
8690d2dfbd86b4ca
|
CauSeq.mul_not_equiv_zero
|
Mathlib/Algebra/Order/CauSeq/Basic.lean
|
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 :=
fun (this : LimZero (f * g - 0)) => by
have hlz : LimZero (f * g)
|
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : CauSeq β abv
hf : ¬f ≈ 0
hg : ¬g ≈ 0
this : (f * g - 0).LimZero
hlz : (f * g).LimZero
⊢ ¬f.LimZero
|
simpa using show ¬LimZero (f - 0) from hf
|
no goals
|
506124b1239a5e26
|
FirstOrder.Field.realize_genericMonicPolyHasRoot
|
Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean
|
theorem realize_genericMonicPolyHasRoot [Field K] [CompatibleRing K] (n : ℕ) :
K ⊨ genericMonicPolyHasRoot n ↔
∀ p : { p : K[X] // p.Monic ∧ p.natDegree = n }, ∃ x, p.1.eval x = 0
|
K : Type u_1
inst✝¹ : Field K
inst✝ : CompatibleRing K
n : ℕ
x✝ : DecidableEq K := Classical.decEq K
⊢ K ⊨ genericMonicPolyHasRoot n ↔ ∀ (p : { p // p.Monic ∧ p.natDegree = n }), ∃ x, eval x ↑p = 0
|
rw [Equiv.forall_congr_left ((monicEquivDegreeLT n).trans (degreeLTEquiv K n).toEquiv)]
|
K : Type u_1
inst✝¹ : Field K
inst✝ : CompatibleRing K
n : ℕ
x✝ : DecidableEq K := Classical.decEq K
⊢ K ⊨ genericMonicPolyHasRoot n ↔
∀ (b : Fin n → K), ∃ x, eval x ↑(((monicEquivDegreeLT n).trans (degreeLTEquiv K n).toEquiv).symm b) = 0
|
321775160522cd62
|
Std.Tactic.BVDecide.LRAT.Internal.lratCheckerSound
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/LRATCheckerSound.lean
|
theorem lratCheckerSound [DecidableEq α] [Clause α β] [Entails α σ] [Formula α β σ] (f : σ)
(f_readyForRupAdd : ReadyForRupAdd f) (f_readyForRatAdd : ReadyForRatAdd f)
(prf : List (Action β α)) (prfWellFormed : ∀ a : Action β α, a ∈ prf → WellFormedAction a) :
lratChecker f prf = success → Unsatisfiable α f
|
α : Type u_1
β : Type u_2
σ : Type u_3
inst✝³ : DecidableEq α
inst✝² : Clause α β
inst✝¹ : Entails α σ
inst✝ : Formula α β σ
action : Action β α
restPrf : List (Action β α)
ih :
∀ (f : σ),
ReadyForRupAdd f →
ReadyForRatAdd f →
(∀ (a : Action β α), a ∈ restPrf → WellFormedAction a) → lratChecker f restPrf = success → Unsatisfiable α f
f : σ
f_readyForRupAdd : ReadyForRupAdd f
f_readyForRatAdd : ReadyForRatAdd f
actionWellFormed : WellFormedAction action
restPrfWellFormed : ∀ (a : Action β α), a ∈ restPrf → WellFormedAction a
prf✝ : List (Action β α)
id : Nat
rupHints : Array Nat
restPrf' : List (Action β α)
heq✝ : action :: restPrf = Action.addEmpty id rupHints :: restPrf'
rupAddSuccess : (performRupAdd f empty rupHints).snd = true
⊢ Unsatisfiable α f
|
exact addEmptyCaseSound f f_readyForRupAdd rupHints rupAddSuccess
|
no goals
|
daa30b736e5664b3
|
InnerProductGeometry.sin_angle_sub_add_angle_sub_rev_eq_sin_angle
|
Mathlib/Geometry/Euclidean/Triangle.lean
|
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)
|
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
Real.sin (angle x y)
|
have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
|
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
Real.sin (angle x y)
|
cff8220b46e80476
|
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