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... | 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... | 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'... | 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 ((Lin... | 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 ... | 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 ... | 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... | 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... | 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 ... | 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
⊢ ‖... | 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... | 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
... | 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.car... | 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)... | 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 ⟶ ⊤ := h... | 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 ⟶ ⊤ := h... | 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), Si... | 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
s... | 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, ‖... | 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 ... | 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 ... | 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 ... | 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... | 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_integrab... | 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 : In... | 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 : In... | 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 ... | 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 ... | 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 } { ... | 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 } =
utf8G... | 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
in... | 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
... | 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 ⊆ Met... | 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 ... | 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 ... | 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)... | 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 ... | 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),... | 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 : ... | 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_con... | 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 → IsHomogeneousEle... | 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 → IsHomogeneousEle... | 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 ... | 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... | 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.Injecti... | 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, ⊤) := ... | 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, ⊤) := ... | 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✝ ≫... | 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... | 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 s... | 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... | 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... | 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... | 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 : ... | 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 ... | 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 ^ ... | 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_... | 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)) * ... | 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)) * ... | 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 ∧
... | 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✝ :... | 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... | 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... | 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.si... | 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_ad... | 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... | 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... | 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... | 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)... | 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)... | 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 re... | 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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