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 |
|---|---|---|---|---|---|---|
Complex.GammaSeq_eq_betaIntegral_of_re_pos | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | theorem GammaSeq_eq_betaIntegral_of_re_pos {s : ℂ} (hs : 0 < re s) (n : ℕ) :
GammaSeq s n = (n : ℂ) ^ s * betaIntegral s (n + 1) | s : ℂ
hs : 0 < s.re
n : ℕ
⊢ s.GammaSeq n = ↑n ^ s * s.betaIntegral (↑n + 1) | rw [GammaSeq, betaIntegral_eval_nat_add_one_right hs n, ← mul_div_assoc] | no goals | 533b7648112a9dc9 |
AkraBazziRecurrence.GrowsPolynomially.add_isLittleO | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | lemma GrowsPolynomially.add_isLittleO {f g : ℝ → ℝ} (hf : GrowsPolynomially f)
(hfg : g =o[atTop] f) : GrowsPolynomially fun x => f x + g x | f g : ℝ → ℝ
hf✝ : GrowsPolynomially f
b : ℝ
hb : b ∈ Set.Ioo 0 1
hb_ub : b < 1
hf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x
c₁ : ℝ
hc₁_mem : 0 < c₁
c₂ : ℝ
hc₂_mem : 0 < c₂
hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)
hfg : ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ 1 / 2 * ‖f x‖
⊢ c₁ / 3 > 0 | positivity | no goals | dd533a6f58fe3819 |
eq_of_prime_pow_eq | Mathlib/Algebra/GroupWithZero/Associated.lean | theorem eq_of_prime_pow_eq (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₁)
(h : p₁ ^ k₁ = p₂ ^ k₂) : p₁ = p₂ | R : Type u_2
inst✝¹ : CancelCommMonoidWithZero R
inst✝ : Subsingleton Rˣ
p₁ p₂ : R
k₁ k₂ : ℕ
hp₁ : Prime p₁
hp₂ : Prime p₂
hk₁ : 0 < k₁
h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂
⊢ p₁ ~ᵤ p₂ | apply h.of_pow_associated_of_prime hp₁ hp₂ hk₁ | no goals | 4a0e2fc7b4439504 |
FinVec.exists_iff | Mathlib/Data/Fin/Tuple/Reflection.lean | theorem exists_iff : ∀ {m} (P : (Fin m → α) → Prop), Exists P ↔ ∃ x, P x
| 0, P => by
simp only [Exists, Fin.exists_fin_zero_pi, Matrix.vecEmpty]
rfl
| .succ n, P => by simp only [Exists, exists_iff, Fin.exists_fin_succ_pi, Matrix.vecCons]
| α : Type u_1
P : (Fin 0 → α) → Prop
⊢ Exists P ↔ ∃ x, P x | simp only [Exists, Fin.exists_fin_zero_pi, Matrix.vecEmpty] | α : Type u_1
P : (Fin 0 → α) → Prop
⊢ P Fin.elim0 ↔ P finZeroElim | e6560d8534b0b2be |
PartitionOfUnity.exists_finset_nhd' | Mathlib/Topology/PartitionOfUnity.lean | theorem exists_finset_nhd' {s : Set X} (ρ : PartitionOfUnity ι X s) (x₀ : X) :
∃ I : Finset ι, (∀ᶠ x in 𝓝[s] x₀, ∑ i ∈ I, ρ i x = 1) ∧
∀ᶠ x in 𝓝 x₀, support (ρ · x) ⊆ I | case intro
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
I : Finset ι
hI : ∀ᶠ (x : X) in 𝓝 x₀, (support fun i => (ρ i) x) ⊆ ↑I
⊢ ∃ I, (∀ᶠ (x : X) in 𝓝[s] x₀, ∑ i ∈ I, (ρ i) x = 1) ∧ ∀ᶠ (x : X) in 𝓝 x₀, (support fun x_1 => (ρ x_1) x) ⊆ ↑I | refine ⟨I, eventually_nhdsWithin_iff.mpr (hI.mono fun x hx x_in ↦ ?_), hI⟩ | case intro
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
I : Finset ι
hI : ∀ᶠ (x : X) in 𝓝 x₀, (support fun i => (ρ i) x) ⊆ ↑I
x : X
hx : (support fun i => (ρ i) x) ⊆ ↑I
x_in : x ∈ s
⊢ ∑ i ∈ I, (ρ i) x = 1 | cafa9cd274ca9787 |
ENNReal.le_tsum_schlomilch | Mathlib/Analysis/PSeries.lean | theorem le_tsum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
(hu : StrictMono u) :
∑' k , f k ≤ ∑ k ∈ range (u 0), f k + ∑' k : ℕ, (u (k + 1) - u k) * f (u k) | u : ℕ → ℕ
f : ℕ → ℝ≥0∞
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu : StrictMono u
n : ℕ
this : ∀ (k : ℕ), ↑(u (k + 1)) - ↑(u k) = ↑(u (k + 1) - u k)
⊢ ∑ x ∈ range n, ↑(u (x + 1) - u x) * f (u x) ≤ ∑' (k : ℕ), ↑(u (k + 1) - u k) * f (u k) | apply ENNReal.sum_le_tsum | no goals | 9e8ec193db9e8e7f |
SimpleGraph.Walk.IsCycle.getVert_endpoint_iff | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma IsCycle.getVert_endpoint_iff {i : ℕ} {p : G.Walk u u} (hpc : p.IsCycle) (hl : i ≤ p.length) :
p.getVert i = u ↔ i = 0 ∨ i = p.length | V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
hl : i ≤ p.length
⊢ p.getVert i = u → i = 0 ∨ i = p.length | rw [or_iff_not_imp_left] | V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
hl : i ≤ p.length
⊢ p.getVert i = u → ¬i = 0 → i = p.length | ea4c612415e552b0 |
MeasureTheory.OuterMeasure.coe_mkMetric | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem OuterMeasure.coe_mkMetric [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞) :
⇑(OuterMeasure.mkMetric m : OuterMeasure X) = Measure.mkMetric m | X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m : ℝ≥0∞ → ℝ≥0∞
⊢ ⇑(mkMetric m) = ⇑(Measure.mkMetric m) | rw [← Measure.mkMetric_toOuterMeasure, Measure.coe_toOuterMeasure] | no goals | 52020685fb951743 |
List.maximum_cons | Mathlib/Data/List/MinMax.lean | theorem maximum_cons (a : α) (l : List α) : maximum (a :: l) = max ↑a (maximum l) :=
List.reverseRecOn l (by simp [@max_eq_left (WithBot α) _ _ _ bot_le]) fun tl hd ih => by
rw [← cons_append, maximum_concat, ih, maximum_concat, max_assoc]
| α : Type u_1
inst✝ : LinearOrder α
a : α
l tl : List α
hd : α
ih : (a :: tl).maximum = ↑a ⊔ tl.maximum
⊢ (a :: (tl ++ [hd])).maximum = ↑a ⊔ (tl ++ [hd]).maximum | rw [← cons_append, maximum_concat, ih, maximum_concat, max_assoc] | no goals | f1ff425d8cb4f71a |
CategoryTheory.Pretriangulated.Triangle.isZero₁_of_isZero₂₃ | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | lemma isZero₁_of_isZero₂₃ (h₂ : IsZero T.obj₂) (h₃ : IsZero T.obj₃) : IsZero T.obj₁ | C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₂ : IsZero T.obj₂
h₃ : IsZero T.obj₃
⊢ T.mor₁ = 0 ∧ T.mor₃ = 0 | exact ⟨h₂.eq_of_tgt _ _, h₃.eq_of_src _ _⟩ | no goals | a88099a26e4ec974 |
DirichletCharacter.LSeries_eulerProduct_exp_log | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | theorem DirichletCharacter.LSeries_eulerProduct_exp_log {N : ℕ} (χ : DirichletCharacter ℂ N)
{s : ℂ} (hs : 1 < s.re) :
exp (∑' p : Nat.Primes, -log (1 - χ p * p ^ (-s))) = L ↗χ s | N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
⊢ cexp (∑' (p : Primes), -Complex.log (1 - χ ↑↑p * ↑↑p ^ (-s))) = L (fun n => χ ↑n) s | let f := dirichletSummandHom χ <| ne_zero_of_one_lt_re hs | N : ℕ
χ : DirichletCharacter ℂ N
s : ℂ
hs : 1 < s.re
f : ℕ →*₀ ℂ := dirichletSummandHom χ ⋯
⊢ cexp (∑' (p : Primes), -Complex.log (1 - χ ↑↑p * ↑↑p ^ (-s))) = L (fun n => χ ↑n) s | 4a8a3baf0a888228 |
Nat.diag_induction | Mathlib/Data/Nat/Init.lean | theorem diag_induction (P : ℕ → ℕ → Prop) (ha : ∀ a, P (a + 1) (a + 1)) (hb : ∀ b, P 0 (b + 1))
(hd : ∀ a b, a < b → P (a + 1) b → P a (b + 1) → P (a + 1) (b + 1)) : ∀ a b, a < b → P a b
| 0, _ + 1, _ => hb _
| a + 1, b + 1, h => by
apply hd _ _ (Nat.add_lt_add_iff_right.1 h)
· have this : a + 1 = b ∨ a... | case a
P : ℕ → ℕ → Prop
ha : ∀ (a : ℕ), P (a + 1) (a + 1)
hb : ∀ (b : ℕ), P 0 (b + 1)
hd : ∀ (a b : ℕ), a < b → P (a + 1) b → P a (b + 1) → P (a + 1) (b + 1)
a b : ℕ
h : a + 1 < b + 1
⊢ P a (b + 1) | have _ : a + (b + 1) < (a + 1) + (b + 1) := by simp | case a
P : ℕ → ℕ → Prop
ha : ∀ (a : ℕ), P (a + 1) (a + 1)
hb : ∀ (b : ℕ), P 0 (b + 1)
hd : ∀ (a b : ℕ), a < b → P (a + 1) b → P a (b + 1) → P (a + 1) (b + 1)
a b : ℕ
h : a + 1 < b + 1
x✝ : a + (b + 1) < a + 1 + (b + 1)
⊢ P a (b + 1) | 42f141b2edc38622 |
exists_spanning_measurableSet_le | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | theorem exists_spanning_measurableSet_le {f : α → ℝ≥0} (hf : Measurable f) (μ : Measure α)
[SigmaFinite μ] :
∃ s : ℕ → Set α,
(∀ n, MeasurableSet (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, f x ≤ n) ∧
⋃ i, s i = Set.univ | α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
⊢ ∃ s, (∀ (n : ℕ), MeasurableSet (s n) ∧ μ (s n) < ⊤ ∧ ∀ x ∈ s n, f x ≤ ↑n) ∧ ⋃ i, s i = univ | let sigma_finite_sets := spanningSets μ | α : Type u_1
mα : MeasurableSpace α
f : α → ℝ≥0
hf : Measurable f
μ : Measure α
inst✝ : SigmaFinite μ
sigma_finite_sets : ℕ → Set α := spanningSets μ
⊢ ∃ s, (∀ (n : ℕ), MeasurableSet (s n) ∧ μ (s n) < ⊤ ∧ ∀ x ∈ s n, f x ≤ ↑n) ∧ ⋃ i, s i = univ | 447eb51f3f11a510 |
Real.summable_nat_rpow_inv | Mathlib/Analysis/PSeries.lean | theorem summable_nat_rpow_inv {p : ℝ} :
Summable (fun n => ((n : ℝ) ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p | case inl
p : ℝ
hp : 0 ≤ p
⊢ 2 = 0 ∧ 1 - p ≠ 0 ∨ 1 < 2 ∧ 1 - p < 0 ∨ 2 < 1 ∧ 0 < 1 - p ↔ 1 < p | norm_num | no goals | 3c89ea3843e56182 |
IsMin.withTop | Mathlib/Order/WithBot.lean | theorem _root_.IsMin.withTop (h : IsMin a) : IsMin (a : WithTop α) :=
fun x ↦ by cases x <;> simp; simpa using @h _
| α : Type u_1
a : α
inst✝ : LE α
h : IsMin a
x : WithTop α
⊢ x ≤ ↑a → ↑a ≤ x | cases x <;> simp | case coe
α : Type u_1
a : α
inst✝ : LE α
h : IsMin a
a✝ : α
⊢ a✝ ≤ a → a ≤ a✝ | 105ac1a88c1255bc |
Batteries.UnionFind.parentD_findAux_rootD | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | theorem parentD_findAux_rootD {self : UnionFind} {x : Fin self.size} :
parentD (findAux self x).s (self.rootD x) = self.rootD x | case isFalse
self : UnionFind
x : Fin self.size
h : ¬self.arr[↑x].parent = ↑x
this : self.rankMax - self.rank self.arr[↑x].parent < self.rankMax - self.rank ↑x
⊢ parentD (self.findAux ⟨self.arr[↑x].parent, ⋯⟩).s (self.rootD self.arr[↑x].parent) = self.rootD self.arr[↑x].parent | exact parentD_findAux_rootD (x := ⟨_, self.parent'_lt _ x.2⟩) | no goals | b39ae1971cababd7 |
WeierstrassCurve.Jacobian.baseChange_dblZ | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma baseChange_dblZ (P : Fin 3 → A) : (W'.baseChange B).toJacobian.dblZ (f ∘ P) =
f ((W'.baseChange A).toJacobian.dblZ P) | R : Type r
inst✝¹⁰ : CommRing R
W' : Jacobian R
S : Type s
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
A : Type u
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra S A
inst✝⁴ : IsScalarTower R S A
B : Type v
inst✝³ : CommRing B
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P :... | rw [← RingHom.coe_coe, ← map_dblZ, map_baseChange] | no goals | b3f06f97bb4cfef4 |
HahnSeries.SummableFamily.support_hsum_subset | Mathlib/RingTheory/HahnSeries/Summable.lean | theorem support_hsum_subset {s : SummableFamily Γ R α} : s.hsum.support ⊆ ⋃ a : α, (s a).support :=
fun g hg => by
rw [mem_support, coeff_hsum, finsum_eq_sum _ (s.finite_co_support _)] at hg
obtain ⟨a, _, h2⟩ := exists_ne_zero_of_sum_ne_zero hg
rw [Set.mem_iUnion]
exact ⟨a, h2⟩
| case intro.intro
Γ : Type u_1
R : Type u_3
α : Type u_5
inst✝¹ : PartialOrder Γ
inst✝ : AddCommMonoid R
s : SummableFamily Γ R α
g : Γ
hg : ∑ i ∈ ⋯.toFinset, (s i).coeff g ≠ 0
a : α
left✝ : a ∈ ⋯.toFinset
h2 : (s a).coeff g ≠ 0
⊢ g ∈ ⋃ a, (s a).support | rw [Set.mem_iUnion] | case intro.intro
Γ : Type u_1
R : Type u_3
α : Type u_5
inst✝¹ : PartialOrder Γ
inst✝ : AddCommMonoid R
s : SummableFamily Γ R α
g : Γ
hg : ∑ i ∈ ⋯.toFinset, (s i).coeff g ≠ 0
a : α
left✝ : a ∈ ⋯.toFinset
h2 : (s a).coeff g ≠ 0
⊢ ∃ i, g ∈ (s i).support | b452909f8507b00d |
continuousAt_clog | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | theorem continuousAt_clog {x : ℂ} (h : x ∈ slitPlane) : ContinuousAt log x | case refine_2
x : ℂ
h : x ∈ slitPlane
⊢ ContinuousAt (fun x => ↑x.arg * I) x | have h_cont_mul : Continuous fun x : ℂ => x * I := continuous_id'.mul continuous_const | case refine_2
x : ℂ
h : x ∈ slitPlane
h_cont_mul : Continuous fun x => x * I
⊢ ContinuousAt (fun x => ↑x.arg * I) x | 3806bdd6f92461d6 |
FirstOrder.Language.isExtensionPair_iff_exists_embedding_closure_singleton_sup | Mathlib/ModelTheory/PartialEquiv.lean | theorem isExtensionPair_iff_exists_embedding_closure_singleton_sup :
L.IsExtensionPair M N ↔
∀ (S : L.Substructure M) (_ : S.FG) (f : S ↪[L] N) (m : M),
∃ g : (closure L {m} ⊔ S : L.Substructure M) ↪[L] N, f =
g.comp (Substructure.inclusion le_sup_right) | case refine_1
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
h : L.IsExtensionPair M N
S : L.Substructure M
S_FG : S.FG
f : ↥S ↪[L] N
m : M
⊢ ∃ g, f = g.comp (Substructure.inclusion ⋯) | obtain ⟨⟨f', hf'⟩, mf', ff'1, ff'2⟩ := h ⟨⟨S, _, f.equivRange⟩, S_FG⟩ m | case refine_1.intro.mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
h : L.IsExtensionPair M N
S : L.Substructure M
S_FG : S.FG
f : ↥S ↪[L] N
m : M
f' : M ≃ₚ[L] N
hf' : f'.dom.FG
mf' : m ∈ (↑⟨f', hf'⟩).dom
ff'1 : (↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange... | 5f72d9d9514858fa |
Nat.probablePrime_iff_modEq | Mathlib/NumberTheory/FermatPsp.lean | theorem probablePrime_iff_modEq (n : ℕ) {b : ℕ} (h : 1 ≤ b) :
ProbablePrime n b ↔ b ^ (n - 1) ≡ 1 [MOD n] | case mp
n b : ℕ
h : 1 ≤ b
this : 1 ≤ b ^ (n - 1)
h₁ : n.ProbablePrime b
⊢ 1 ≡ b ^ (n - 1) [MOD n] | apply Nat.modEq_of_dvd | case mp.a
n b : ℕ
h : 1 ≤ b
this : 1 ≤ b ^ (n - 1)
h₁ : n.ProbablePrime b
⊢ ↑n ∣ ↑(b ^ (n - 1)) - ↑1 | 7df803935218b1eb |
Array.countP_eq_size_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Count.lean | theorem countP_eq_size_filter (l) : countP p l = (filter p l).size | α : Type u_1
p : α → Bool
l : Array α
⊢ countP p l = (filter p l).size | cases l | case mk
α : Type u_1
p : α → Bool
toList✝ : List α
⊢ countP p { toList := toList✝ } = (filter p { toList := toList✝ }).size | 0a64719b1132430a |
exists_subset_nhds_of_isCompact' | Mathlib/Topology/Compactness/Compact.lean | theorem exists_subset_nhds_of_isCompact' [Nonempty ι] {V : ι → Set X}
(hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) (hV_closed : ∀ i, IsClosed (V i))
{U : Set X} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U | case intro.intro.intro
X : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Nonempty ι
V : ι → Set X
hV : Directed (fun x1 x2 => x1 ⊇ x2) V
hV_cpct : ∀ (i : ι), IsCompact (V i)
hV_closed : ∀ (i : ι), IsClosed (V i)
U : Set X
hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x
W : Set X
hsubW : ⋂ i, V i ⊆ W
W_op : IsOpen W
hWU : W ⊆ U... | have : (⋂ i, V i ∩ Wᶜ).Nonempty := by
refine
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed _ (fun i j => ?_) H
(fun i => (hV_cpct i).inter_right W_op.isClosed_compl) fun i =>
(hV_closed i).inter W_op.isClosed_compl
rcases hV i j with ⟨k, hki, hkj⟩
refine ⟨k, ⟨fun x => ?_, fun x... | case intro.intro.intro
X : Type u
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : Nonempty ι
V : ι → Set X
hV : Directed (fun x1 x2 => x1 ⊇ x2) V
hV_cpct : ∀ (i : ι), IsCompact (V i)
hV_closed : ∀ (i : ι), IsClosed (V i)
U : Set X
hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x
W : Set X
hsubW : ⋂ i, V i ⊆ W
W_op : IsOpen W
hWU : W ⊆ U... | 50965283514ddd14 |
ExpGrowth.expGrowthSup_add | Mathlib/Analysis/Asymptotics/ExpGrowth.lean | lemma expGrowthSup_add {u v : ℕ → ℝ≥0∞} :
expGrowthSup (u + v) = expGrowthSup u ⊔ expGrowthSup v | case a
u v : ℕ → ℝ≥0∞
⊢ expGrowthSup (u + v) ≤ expGrowthSup (u ⊔ v) | refine expGrowthSup_le_of_eventually_le (C := 2) ofNat_ne_top (Eventually.of_forall fun n ↦ ?_) | case a
u v : ℕ → ℝ≥0∞
n : ℕ
⊢ (u + v) n ≤ 2 * (u ⊔ v) n | 28c8d74ad6220fd8 |
Finset.one_mem_div_iff | Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean | theorem one_mem_div_iff : (1 : α) ∈ s / t ↔ ¬Disjoint s t | α : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Group α
s t : Finset α
⊢ 1 ∈ s / t ↔ ¬Disjoint s t | rw [← mem_coe, ← disjoint_coe, coe_div, Set.one_mem_div_iff] | no goals | e7fcc965cfae3496 |
AlgebraicGeometry.HasAffineProperty.affineAnd_isStableUnderComposition | Mathlib/AlgebraicGeometry/Morphisms/AffineAnd.lean | /-- If `P` is a morphism property affine locally defined by `affineAnd Q`, `P` is stable under
composition if `Q` is. -/
lemma HasAffineProperty.affineAnd_isStableUnderComposition {P : MorphismProperty Scheme.{u}}
(hA : HasAffineProperty P (affineAnd Q)) (hQ : RingHom.StableUnderComposition Q) :
P.IsStableUnder... | case inr
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
P : MorphismProperty Scheme
hA : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] => Q)
hQ : RingHom.StableUnderComposition fun {R S} [CommRing R] [CommRing S] => Q
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
hf : P f
... | exact this hA hQ _ _ (IsLocalAtTarget.restrict hf _) (IsLocalAtTarget.restrict hg _) hA U.2 | no goals | de1b90be70353a0a |
IsDiscreteValuationRing.HasUnitMulPowIrreducibleFactorization.toUniqueFactorizationMonoid | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem toUniqueFactorizationMonoid (hR : HasUnitMulPowIrreducibleFactorization R) :
UniqueFactorizationMonoid R :=
let p := Classical.choose hR
let spec := Classical.choose_spec hR
UniqueFactorizationMonoid.of_exists_prime_factors fun x hx => by
use Multiset.replicate (Classical.choose (spec.2 hx)) p
... | case h.left
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
Classical.choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.repli... | intro a b h | case h.left
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
hR : HasUnitMulPowIrreducibleFactorization R
p : R := Classical.choose hR
spec : Irreducible (Classical.choose hR) ∧ ∀ {x : R}, x ≠ 0 → ∃ n, Associated (Classical.choose hR ^ n) x :=
Classical.choose_spec hR
x : R
hx : x ≠ 0
q : R
hq : q ∈ Multiset.repli... | c98f86af3b032a22 |
Pretrivialization.proj_symm_apply | Mathlib/Topology/FiberBundle/Trivialization.lean | theorem proj_symm_apply {x : B × F} (hx : x ∈ e.target) : proj (e.toPartialEquiv.symm x) = x.1 | B : Type u_1
F : Type u_2
Z : Type u_4
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
proj : Z → B
e : Pretrivialization F proj
x : B × F
hx : x ∈ e.target
this : proj (↑e.symm x) = (↑e (↑e.symm x)).1
⊢ proj (↑e.symm x) = x.1 | rwa [← e.coe_coe, e.right_inv hx] at this | no goals | a4606a4a8285f860 |
Finset.truncatedInf_union_left | Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | lemma truncatedInf_union_left (hs : a ∈ upperClosure s) (ht : a ∉ upperClosure t) :
truncatedInf (s ∪ t) a = truncatedInf s a | α : Type u_1
inst✝³ : SemilatticeInf α
s t : Finset α
a : α
inst✝² : DecidableRel fun x1 x2 => x1 ≤ x2
inst✝¹ : BoundedOrder α
inst✝ : DecidableEq α
hs : a ∈ upperClosure ↑s
ht : ∀ x ∈ t, ¬x ≤ a
⊢ (s ∪ t).truncatedInf a = s.truncatedInf a | simp only [truncatedInf_of_mem, hs, filter_union, filter_false_of_mem ht, union_empty,
upper_aux.2 (Or.inl hs), ht] | no goals | 5469642ad7a21d66 |
algebraMap_apply | Mathlib/Topology/ContinuousMap/Algebra.lean | theorem algebraMap_apply (k : R) (a : α) : algebraMap R C(α, A) k a = k • (1 : A) | α : Type u_1
inst✝⁵ : TopologicalSpace α
R : Type u_2
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra R A
inst✝ : IsTopologicalSemiring A
k : R
a : α
⊢ ((algebraMap R C(α, A)) k) a = k • 1 | rw [Algebra.algebraMap_eq_smul_one] | α : Type u_1
inst✝⁵ : TopologicalSpace α
R : Type u_2
inst✝⁴ : CommSemiring R
A : Type u_3
inst✝³ : TopologicalSpace A
inst✝² : Semiring A
inst✝¹ : Algebra R A
inst✝ : IsTopologicalSemiring A
k : R
a : α
⊢ (k • 1) a = k • 1 | 82f9ad05f21f7859 |
ContinuousMultilinearMap.integral_apply | Mathlib/MeasureTheory/Integral/SetIntegral.lean | theorem _root_.ContinuousMultilinearMap.integral_apply {ι : Type*} [Fintype ι] {M : ι → Type*}
[∀ i, NormedAddCommGroup (M i)] [∀ i, NormedSpace 𝕜 (M i)]
{φ : X → ContinuousMultilinearMap 𝕜 M E} (φ_int : Integrable φ μ) (m : ∀ i, M i) :
(∫ x, φ x ∂μ) m = ∫ x, φ x m ∂μ | X : Type u_1
E : Type u_3
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace ℝ E
ι : Type u_6
inst✝² : Fintype ι
M : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace 𝕜 (M i)
φ : X →... | by_cases hE : CompleteSpace E | case pos
X : Type u_1
E : Type u_3
inst✝⁷ : MeasurableSpace X
μ : Measure X
𝕜 : Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace ℝ E
ι : Type u_6
inst✝² : Fintype ι
M : ι → Type u_7
inst✝¹ : (i : ι) → NormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace 𝕜 (M i... | 136928cd4a541ef3 |
Lean.Order.List.monotone_forIn | Mathlib/.lake/packages/lean4/src/lean/Init/Internal/Order/Lemmas.lean | theorem monotone_forIn {α : Type uu}
(as : List α) (init : β) (f : γ → (a : α) → β → m (ForInStep β)) (hmono : monotone f) :
monotone (fun x => forIn as init (f x)) | case h
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
β : Type u
γ : Type w
inst✝ : PartialOrder γ
α : Type uu
as : List α
init : β
f : γ → α → β → m (ForInStep β)
hmono : monotone f
y : α
⊢ monotone fun x x_1 => f x y | apply monotone_of_monotone_apply | case h.h
m : Type u → Type v
inst✝³ : Monad m
inst✝² : (α : Type u) → PartialOrder (m α)
inst✝¹ : MonoBind m
β : Type u
γ : Type w
inst✝ : PartialOrder γ
α : Type uu
as : List α
init : β
f : γ → α → β → m (ForInStep β)
hmono : monotone f
y : α
⊢ y ∈ as → monotone fun x => f x y | d3d5da3772679065 |
Lagrange.nodalWeight_ne_zero | Mathlib/LinearAlgebra/Lagrange.lean | theorem nodalWeight_ne_zero (hvs : Set.InjOn v s) (hi : i ∈ s) : nodalWeight s v i ≠ 0 | F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v : ι → F
i : ι
hvs : Set.InjOn v ↑s
hi : i ∈ s
j : ι
hj : j ∈ s.erase i
⊢ (v i - v j)⁻¹ ≠ 0 | rcases mem_erase.mp hj with ⟨hij, hj⟩ | case intro
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v : ι → F
i : ι
hvs : Set.InjOn v ↑s
hi : i ∈ s
j : ι
hj✝ : j ∈ s.erase i
hij : j ≠ i
hj : j ∈ s
⊢ (v i - v j)⁻¹ ≠ 0 | c09a62be794c1ff8 |
Num.ofNat'_succ | Mathlib/Data/Num/Lemmas.lean | theorem ofNat'_succ : ∀ {n}, ofNat' (n + 1) = ofNat' n + 1 :=
@(Nat.binaryRec (by simp [zero_add]) fun b n ih => by
cases b
· erw [ofNat'_bit true n, ofNat'_bit]
simp only [← bit1_of_bit1, ← bit0_of_bit0, cond]
· rw [show n.bit true + 1 = (n + 1).bit false by simp [Nat.bit, mul_add],
ofNat'_... | ⊢ ofNat' (0 + 1) = ofNat' 0 + 1 | simp [zero_add] | no goals | 8fe2f9f90ef395b9 |
Vector.eq_empty_of_size_eq_zero | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem eq_empty_of_size_eq_zero (xs : Vector α n) (h : n = 0) : xs = #v[].cast h.symm | case mk
α : Type u_1
xs : Array α
h✝ : xs.size = 0
h : xs = #[]
⊢ { toArray := xs, size_toArray := ⋯ }.toArray = (Vector.cast ⋯ { toArray := #[], size_toArray := ⋯ }).toArray | simp [h] | no goals | 1cd9b011a5d13816 |
Nat.ordProj_pos | Mathlib/Data/Nat/Factorization/Basic.lean | theorem ordProj_pos (n p : ℕ) : 0 < ordProj[p] n | n p : ℕ
pp : Prime p
⊢ 0 < p ^ n.factorization p | simp [pow_pos pp.pos] | no goals | 1096efa6604cd9d5 |
ZMod.auxDFT_auxDFT | Mathlib/Analysis/Fourier/ZMod.lean | /-- Fourier inversion formula, discrete case. -/
private lemma auxDFT_auxDFT (Φ : ZMod N → E) : auxDFT (auxDFT Φ) = fun j ↦ (N : ℂ) • Φ (-j) | case h
N : ℕ
inst✝² : NeZero N
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
Φ : ZMod N → E
j : ZMod N
⊢ ∑ y : ZMod N, ∑ x : ZMod N, stdAddChar (-((j + y) * x)) • Φ y = ↑N • Φ (-j) | simp only [← sum_smul, ← neg_mul] | case h
N : ℕ
inst✝² : NeZero N
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
Φ : ZMod N → E
j : ZMod N
⊢ ∑ x : ZMod N, (∑ i : ZMod N, stdAddChar (-(j + x) * i)) • Φ x = ↑N • Φ (-j) | 5bf146112cf481d7 |
isUnit_iff_not_dvd_char_of_ringChar_ne_zero | Mathlib/Algebra/CharP/CharAndCard.lean | theorem isUnit_iff_not_dvd_char_of_ringChar_ne_zero (R : Type*) [CommRing R] (p : ℕ) [Fact p.Prime]
(hR : ringChar R ≠ 0) : IsUnit (p : R) ↔ ¬p ∣ ringChar R | R : Type u_1
inst✝¹ : CommRing R
p : ℕ
inst✝ : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
⊢ IsUnit ↑p ↔ ¬p ∣ ringChar R | have hch := CharP.cast_eq_zero R (ringChar R) | R : Type u_1
inst✝¹ : CommRing R
p : ℕ
inst✝ : Fact (Nat.Prime p)
hR : ringChar R ≠ 0
hch : ↑(ringChar R) = 0
⊢ IsUnit ↑p ↔ ¬p ∣ ringChar R | 6d0053e98206702c |
MeasureTheory.stoppedProcess_eq_of_mem_finset | Mathlib/Probability/Process/Stopping.lean | theorem stoppedProcess_eq_of_mem_finset [LinearOrder ι] [AddCommMonoid E] {s : Finset ι} (n : ι)
(hbdd : ∀ ω, τ ω < n → τ ω ∈ s) : stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) +
∑ i ∈ s.filter (· < n), Set.indicator {ω | τ ω = i} (u i) | case h.inr.h₀
Ω : Type u_1
ι : Type u_3
τ : Ω → ι
E : Type u_4
u : ι → Ω → E
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid E
s : Finset ι
n : ι
hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s
ω : Ω
h : τ ω < n
⊢ ∀ b ∈ Finset.filter (fun x => x < n) s, b ≠ τ ω → {ω | τ ω = b}.indicator (u b) ω = 0 | intro b _ hneq | case h.inr.h₀
Ω : Type u_1
ι : Type u_3
τ : Ω → ι
E : Type u_4
u : ι → Ω → E
inst✝¹ : LinearOrder ι
inst✝ : AddCommMonoid E
s : Finset ι
n : ι
hbdd : ∀ (ω : Ω), τ ω < n → τ ω ∈ s
ω : Ω
h : τ ω < n
b : ι
a✝ : b ∈ Finset.filter (fun x => x < n) s
hneq : b ≠ τ ω
⊢ {ω | τ ω = b}.indicator (u b) ω = 0 | c67823aca57b6859 |
Iio_mem_nhdsSet_Ico | Mathlib/Topology/Order/NhdsSet.lean | theorem Iio_mem_nhdsSet_Ico (h : b ≤ c) : Iio c ∈ 𝓝ˢ (Ico a b) :=
nhdsSet_mono Ico_subset_Iio_self <| by simpa
| α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderClosedTopology α
a b c : α
h : b ≤ c
⊢ Iio c ∈ 𝓝ˢ (Iio b) | simpa | no goals | f7678cfe00e4171f |
nilpotent_iff_lowerCentralSeries | Mathlib/GroupTheory/Nilpotent.lean | theorem nilpotent_iff_lowerCentralSeries : IsNilpotent G ↔ ∃ n, lowerCentralSeries G n = ⊥ | case h
G : Type u_1
inst✝ : Group G
n : ℕ
H : ℕ → Subgroup G
hn : H n = ⊥
h0 : H 0 = ⊤
hs : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), x * g * x⁻¹ * g⁻¹ ∈ H (n + 1)
⊢ lowerCentralSeries G n ≤ H n | exact descending_central_series_ge_lower H ⟨h0, hs⟩ n | no goals | 58985aaf2708f4c4 |
MeasureTheory.Measure.prodAssoc_prod | Mathlib/MeasureTheory/Measure/Prod.lean | theorem prodAssoc_prod [SFinite τ] :
map MeasurableEquiv.prodAssoc ((μ.prod ν).prod τ) = μ.prod (ν.prod τ) | case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
τ : Measure γ
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
s : Set (α × β × γ)
hs : MeasurableSet s
⊢ (sum fun p => (sfiniteSeq μ p.1).prod ((sfiniteSeq ν... | rw [sum_apply _ hs, sum_apply _ hs, ← (Equiv.prodAssoc _ _ _).tsum_eq] | case h
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : MeasurableSpace α
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace γ
μ : Measure α
ν : Measure β
τ : Measure γ
inst✝² : SFinite ν
inst✝¹ : SFinite μ
inst✝ : SFinite τ
s : Set (α × β × γ)
hs : MeasurableSet s
⊢ ∑' (c : (ℕ × ℕ) × ℕ),
((sfiniteSeq μ ((Equiv.p... | 1acdd665c9d8230d |
ProbabilityTheory.Kernel.iIndepSets.indepSets | Mathlib/Probability/Independence/Kernel.lean | theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} (h_indep : iIndepSets s κ μ) {i j : ι} (hij : i ≠ j) :
IndepSets (s i) (s j) κ μ | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
⊢ ∀ x ∈ {i, j}, (if x = i then t₁ else t₂) ∈ s x | intro x hx | α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
s : ι → Set (Set Ω)
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α
h_indep : iIndepSets s κ μ
i j : ι
hij : i ≠ j
t₁ t₂ : Set Ω
ht₁ : t₁ ∈ s i
ht₂ : t₂ ∈ s j
x : ι
hx : x ∈ {i, j}
⊢ (if x = i then t₁ else t₂) ∈ s x | f62ea0e10e361ba7 |
MeasureTheory.IsSetSemiring.subset_disjointOfDiff | Mathlib/MeasureTheory/SetSemiring.lean | lemma subset_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
↑(hC.disjointOfDiff hs ht) ⊆ C | α : Type u_1
C : Set (Set α)
s t : Set α
hC : IsSetSemiring C
hs : s ∈ C
ht : t ∈ C
⊢ ↑⋯.choose ⊆ insert ∅ C | exact (hC.diff_eq_sUnion' s hs t ht).choose_spec.1.trans (Set.subset_insert _ _) | no goals | 93450462dd47925e |
minpoly.neg | Mathlib/FieldTheory/Minpoly/Field.lean | theorem neg {B : Type*} [CommRing B] [Algebra A B] (x : B) :
minpoly A (- x) = (-1) ^ (natDegree (minpoly A x)) * (minpoly A x).comp (- X) | case pos.refine_2
A : Type u_1
inst✝² : Field A
B : Type u_3
inst✝¹ : CommRing B
inst✝ : Algebra A B
x : B
hx : IsIntegral A x
q : A[X]
qmo : q.Monic
hq : (Polynomial.aeval (-x)) q = 0
this : (Polynomial.aeval x) ((-1) ^ q.natDegree * q.comp (-X)) = 0
H : (minpoly A x).degree ≤ ((-1) ^ q.natDegree * q.comp (-X)).degree... | have n1 := ((minpoly.monic hx).neg_one_pow_natDegree_mul_comp_neg_X).ne_zero | case pos.refine_2
A : Type u_1
inst✝² : Field A
B : Type u_3
inst✝¹ : CommRing B
inst✝ : Algebra A B
x : B
hx : IsIntegral A x
q : A[X]
qmo : q.Monic
hq : (Polynomial.aeval (-x)) q = 0
this : (Polynomial.aeval x) ((-1) ^ q.natDegree * q.comp (-X)) = 0
H : (minpoly A x).degree ≤ ((-1) ^ q.natDegree * q.comp (-X)).degree... | 66280e634dfbed23 |
SimpleGraph.Subgraph.IsMatching.toEdge_eq_toEdge_of_adj | Mathlib/Combinatorics/SimpleGraph/Matching.lean | theorem IsMatching.toEdge_eq_toEdge_of_adj (h : M.IsMatching)
(hv : v ∈ M.verts) (hw : w ∈ M.verts) (ha : M.Adj v w) :
h.toEdge ⟨v, hv⟩ = h.toEdge ⟨w, hw⟩ | V : Type u_1
G : SimpleGraph V
M : G.Subgraph
v w : V
h : M.IsMatching
hv : v ∈ M.verts
hw : w ∈ M.verts
ha : M.Adj v w
⊢ h.toEdge ⟨v, hv⟩ = h.toEdge ⟨w, hw⟩ | rw [h.toEdge_eq_of_adj hv ha, h.toEdge_eq_of_adj hw (M.symm ha), Subtype.mk_eq_mk, Sym2.eq_swap] | no goals | 51fc56d44c6c9558 |
hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall | Mathlib/Analysis/Analytic/Within.lean | /-- `f` has power series `p` at `x` iff some local extension of `f` has that series -/
lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpace F] {f : E → F}
{p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ℝ≥0∞} :
HasFPowerSeriesWithinOnBall f p s x r ↔
∃ g, EqOn f ... | case mp.refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWi... | simp only [add_sub_cancel_left] | case mp.refine_2
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWi... | 978cd62b7e47ef4b |
isClosed_innerDualCone | Mathlib/Analysis/Convex/Cone/InnerDual.lean | theorem isClosed_innerDualCone : IsClosed (s.innerDualCone : Set H) | H : Type u_1
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
s : Set H
⊢ IsClosed (⋂ i, ↑{↑i}.innerDualCone) | apply isClosed_iInter | case h
H : Type u_1
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
s : Set H
⊢ ∀ (i : ↑s), IsClosed ↑{↑i}.innerDualCone | 7e2be3df05d0cca5 |
HasStrictFDerivAt.hasStrictFDerivAt_norm_smul | Mathlib/Analysis/Calculus/FDeriv/Norm.lean | theorem HasStrictFDerivAt.hasStrictFDerivAt_norm_smul
(ht : t ≠ 0) (h : HasStrictFDerivAt (‖·‖) f x) :
HasStrictFDerivAt (‖·‖) ((SignType.sign t : ℝ) • f) (t • x) | case h.e'_12
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E →L[ℝ] ℝ
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x => ‖x‖) f x
h1 : HasStrictFDerivAt (fun y => t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y => |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
⊢ ↑(SignT... | ext y | case h.e'_12.h
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E →L[ℝ] ℝ
x : E
t : ℝ
ht : t ≠ 0
h : HasStrictFDerivAt (fun x => ‖x‖) f x
h1 : HasStrictFDerivAt (fun y => t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)
h2 : HasStrictFDerivAt (fun y => |t| * ‖y‖) (|t| • f) (t⁻¹ • t • x)
y : E
⊢... | 2e9a1e0a1454442d |
Nat.add_pred_div_lt | Mathlib/Data/Nat/Log.lean | theorem add_pred_div_lt {b n : ℕ} (hb : 1 < b) (hn : 2 ≤ n) : (n + b - 1) / b < n | b n : ℕ
hb : 1 < b
hn : 2 ≤ n
⊢ 0 < n + b | omega | no goals | db2a8bd8e7a61779 |
vadd_vsub_assoc | Mathlib/Algebra/AddTorsor/Defs.lean | theorem vadd_vsub_assoc (g : G) (p₁ p₂ : P) : (g +ᵥ p₁) -ᵥ p₂ = g + (p₁ -ᵥ p₂) | G : Type u_1
P : Type u_2
inst✝ : AddGroup G
T : AddTorsor G P
g : G
p₁ p₂ : P
⊢ ((g +ᵥ p₁) -ᵥ p₂) +ᵥ p₂ = (g + (p₁ -ᵥ p₂)) +ᵥ p₂ | rw [vsub_vadd, add_vadd, vsub_vadd] | no goals | a569223f29550e20 |
SemiconjBy.tprod | Mathlib/RingTheory/PiTensorProduct.lean | theorem _root_.SemiconjBy.tprod {a₁ a₂ a₃ : Π i, A i}
(ha : SemiconjBy a₁ a₂ a₃) :
SemiconjBy (tprod R a₁) (tprod R a₂) (tprod R a₃) | ι : Type u_1
R : Type u_3
A : ι → Type u_4
inst✝⁴ : CommSemiring R
inst✝³ : (i : ι) → NonUnitalNonAssocSemiring (A i)
inst✝² : (i : ι) → Module R (A i)
inst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)
inst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)
a₁ a₂ a₃ : (i : ι) → A i
ha : SemiconjBy a₁ a₂ a₃
⊢ SemiconjBy ((PiTensor... | rw [SemiconjBy, tprod_mul_tprod, tprod_mul_tprod, ha] | no goals | 3ef893cdfa4214c5 |
CategoryTheory.Presheaf.isLocallySurjective_iff_whisker_forget | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | theorem isLocallySurjective_iff_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ IsLocallySurjective J (whiskerRight f (forget A)) | C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
⊢ IsLocallySurjective J f ↔ IsLocallySurjective J (whiskerRight f (fo... | simp only [isLocallySurjective_iff_range_sheafify_eq_top] | C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝ : ConcreteCategory A FA
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
⊢ Subpresheaf.sheafify J (Subpresheaf.range (whiskerRight f (forget A... | d0eb4e3fc5235358 |
Nat.Prime.emultiplicity_factorial_mul_succ | Mathlib/Data/Nat/Multiplicity.lean | theorem emultiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
emultiplicity p (p * (n + 1))! = emultiplicity p (p * n)! + emultiplicity p (n + 1) + 1 | n p : ℕ
hp : Prime p
hp' : _root_.Prime p
h0 : 2 ≤ p
h1 : 1 ≤ p * n + 1
h2 : p * n + 1 ≤ p * (n + 1)
h3 : p * n + 1 ≤ p * (n + 1) + 1
h4 : ∀ m ∈ Ico (p * n + 1) (p * (n + 1)), emultiplicity p m = 0
⊢ ∑ x ∈ Ico 1 (p * n + 1), emultiplicity p x ≠ ⊤ →
∑ x ∈ Ico 1 (p * n + 1), emultiplicity p x + ∑ x ∈ Ico (p * n + 1) ... | intro h | n p : ℕ
hp : Prime p
hp' : _root_.Prime p
h0 : 2 ≤ p
h1 : 1 ≤ p * n + 1
h2 : p * n + 1 ≤ p * (n + 1)
h3 : p * n + 1 ≤ p * (n + 1) + 1
h4 : ∀ m ∈ Ico (p * n + 1) (p * (n + 1)), emultiplicity p m = 0
h : ∑ x ∈ Ico 1 (p * n + 1), emultiplicity p x ≠ ⊤
⊢ ∑ x ∈ Ico 1 (p * n + 1), emultiplicity p x + ∑ x ∈ Ico (p * n + 1) (p... | 097a3d59a78795b9 |
Vector.toList_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem toList_append (a : Vector α m) (b : Vector α n) :
(a ++ b).toList = a.toList ++ b.toList | α : Type u_1
m n : Nat
a : Vector α m
b : Vector α n
⊢ (a ++ b).toList = a.toList ++ b.toList | simp | no goals | 98a8d33b9a407f33 |
riesz_lemma_of_norm_lt | Mathlib/Analysis/NormedSpace/RieszLemma.lean | theorem riesz_lemma_of_norm_lt {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) {F : Subspace 𝕜 E}
(hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) :
∃ x₀ : E, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖ | case intro.intro
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
⊢ ∃ x₀, ... | obtain ⟨d, d0, dxlt, ledx, -⟩ :
∃ d : 𝕜, d ≠ 0 ∧ ‖d • x‖ < R ∧ R / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹ ≤ R⁻¹ * ‖c‖ * ‖x‖ :=
rescale_to_shell hc Rpos x0 | case intro.intro.intro.intro.intro.intro
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x ... | c7ac2ec82577a867 |
Submodule.mem_smul_top_iff | Mathlib/RingTheory/Ideal/Operations.lean | theorem mem_smul_top_iff (N : Submodule R M) (x : N) :
x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N | R : Type u
M : Type v
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
I : Ideal R
N : Submodule R M
x : ↥N
this : map N.subtype (I • ⊤) = I • N
⊢ x ∈ I • ⊤ ↔ N.subtype x ∈ map N.subtype (I • ⊤) | exact (Function.Injective.mem_set_image N.injective_subtype).symm | no goals | 3b90af95b2b69d1a |
Polynomial.eval_mul_X_sub_C | Mathlib/Algebra/Polynomial/AlgebraMap.lean | theorem eval_mul_X_sub_C {p : R[X]} (r : R) : (p * (X - C r)).eval r = 0 | R : Type u
inst✝ : Ring R
p : R[X]
r : R
⊢ eval r (p * (X - C r)) = 0 | simp only [eval, eval₂_eq_sum, RingHom.id_apply] | R : Type u
inst✝ : Ring R
p : R[X]
r : R
⊢ ((p * (X - C r)).sum fun e a => a * r ^ e) = 0 | 4f55d1cdfcc846fa |
Fin.coe_eq_castSucc | Mathlib/Data/Fin/Basic.lean | theorem coe_eq_castSucc {a : Fin n} : (a : Fin (n + 1)) = castSucc a | n : ℕ
a : Fin n
⊢ ↑↑a = a.castSucc | ext | case h
n : ℕ
a : Fin n
⊢ ↑↑↑a = ↑a.castSucc | bb2da0fb02c782cc |
EuclideanGeometry.inner_pos_or_eq_of_dist_le_radius | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | theorem inner_pos_or_eq_of_dist_le_radius {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
(hp₂ : dist p₂ s.center ≤ s.radius) : 0 < ⟪p₁ -ᵥ p₂, p₁ -ᵥ s.center⟫ ∨ p₁ = p₂ | case neg.refine_2.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₁ p₂ : P
hp₁ : dist p₁ s.center = s.radius
hp₂ : dist p₂ s.center ≤ s.radius
h : ¬p₁ = p₂
hp₂' : ‖p₂ -ᵥ s.center‖ = ‖p₁ -ᵥ s.center‖
⊢ inner (p₂ -... | nth_rw 1 [← hp₂'] | case neg.refine_2.inr
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₁ p₂ : P
hp₁ : dist p₁ s.center = s.radius
hp₂ : dist p₂ s.center ≤ s.radius
h : ¬p₁ = p₂
hp₂' : ‖p₂ -ᵥ s.center‖ = ‖p₁ -ᵥ s.center‖
⊢ inner (p₂ -... | 886356a6641013a5 |
CategoryTheory.Subpresheaf.image_comp | Mathlib/CategoryTheory/Subpresheaf/Image.lean | lemma image_comp (g : F' ⟶ F'') :
G.image (f ≫ g) = (G.image f).image g | C : Type u
inst✝ : Category.{v, u} C
F F' F'' : Cᵒᵖ ⥤ Type w
G : Subpresheaf F
f : F ⟶ F'
g : F' ⟶ F''
⊢ G.image (f ≫ g) = (G.image f).image g | aesop | no goals | c021998aabd097e8 |
mellin_inversion | Mathlib/Analysis/MellinInversion.lean | theorem mellin_inversion (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f σ)
(hFf : VerticalIntegrable (mellin f) σ) (hfx : ContinuousAt f x) :
mellinInv σ (mellin f) x = f x | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑x ^ (-↑σ) • rexp (σ * Real.log x) • f (rexp (Real... | norm_cast | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑x ^ ↑(-σ) • rexp (σ * Real.log x) • f (rexp (Real... | eb76f337444b2548 |
Int.sub_left_lt_of_lt_add | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Order.lean | theorem sub_left_lt_of_lt_add {a b c : Int} (h : a < b + c) : a - b < c | a b c : Int
h : a < b + c
⊢ a - b < c | have h := Int.add_lt_add_right h (-b) | a b c : Int
h✝ : a < b + c
h : a + -b < b + c + -b
⊢ a - b < c | e9735ea8335e3e1d |
NormedField.completeSpace_iff_isComplete_closedBall | Mathlib/Analysis/Normed/Field/Lemmas.lean | lemma NormedField.completeSpace_iff_isComplete_closedBall {K : Type*} [NormedField K] :
CompleteSpace K ↔ IsComplete (Metric.closedBall 0 1 : Set K) | case mpr.inl
K : Type u_4
inst✝ : NormedField K
h : IsComplete (Metric.closedBall 0 1)
h✝ : DiscreteTopology K
⊢ CompleteSpace K | rwa [completeSpace_iff_isComplete_univ,
← NormedDivisionRing.unitClosedBall_eq_univ_of_discrete] | no goals | a36770b11a5c8086 |
tendsto_div_of_monotone_of_tendsto_div_floor_pow | Mathlib/Analysis/SpecificLimits/FloorPow.lean | theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
(c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
(hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) | case h.e'_3.h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k /... | field_simp [(zero_lt_one.trans (cone k)).ne', (H n).ne'] | case h.e'_3.h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
c : ℕ → ℝ
cone : ∀ (k : ℕ), 1 < c k
clim : Tendsto c atTop (𝓝 1)
hc : ∀ (k : ℕ), Tendsto (fun n => u ⌊c k ^ n⌋₊ / ↑⌊c k ^ n⌋₊) atTop (𝓝 l)
a : ℝ
ha : 1 < a
k : ℕ
hk : c k < a
H : ∀ (n : ℕ), 0 < ↑⌊c k ^ n⌋₊
A : Tendsto (fun n => ↑⌊c k ^ (n + 1)⌋₊ / c k ^ (n + 1) * c k /... | 39b6908bafcd74f7 |
LinearMap.splittingOfFunOnFintypeSurjective_splits | Mathlib/LinearAlgebra/Finsupp/Pi.lean | theorem splittingOfFunOnFintypeSurjective_splits [Finite α] (f : M →ₗ[R] α → R)
(s : Surjective f) : f.comp (splittingOfFunOnFintypeSurjective f s) = LinearMap.id | R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
α : Type u_4
inst✝ : Finite α
f : M →ₗ[R] α → R
s : Surjective ⇑f
⊢ f ∘ₗ f.splittingOfFunOnFintypeSurjective s = id | ext x y | case h.h.h
R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
α : Type u_4
inst✝ : Finite α
f : M →ₗ[R] α → R
s : Surjective ⇑f
x y : α
⊢ ((f ∘ₗ f.splittingOfFunOnFintypeSurjective s) ∘ₗ single R (fun i => R) x) 1 y = (id ∘ₗ single R (fun i => R) x) 1 y | 18adc5d0c902c2b5 |
IsLowerSet.nonMemberSubfamily | Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | theorem IsLowerSet.nonMemberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :
IsLowerSet (𝒜.nonMemberSubfamily a : Set (Finset α)) := fun s t hts => by
simp_rw [mem_coe, mem_nonMemberSubfamily]
exact And.imp (h hts) (mt <| @hts _)
| α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
a : α
h : IsLowerSet ↑𝒜
s t : Finset α
hts : t ≤ s
⊢ s ∈ 𝒜 ∧ a ∉ s → t ∈ 𝒜 ∧ a ∉ t | exact And.imp (h hts) (mt <| @hts _) | no goals | b585c6cfe2d5e301 |
Std.Range.size_step_1 | Mathlib/.lake/packages/batteries/Batteries/Data/Range/Lemmas.lean | theorem size_step_1 (start stop) : size ⟨start, stop, 1, by decide⟩ = stop - start | start stop : Nat
⊢ { start := start, stop := stop, step := 1, step_pos := ⋯ }.size = stop - start | simp [size] | no goals | 625d05ac89410e8e |
Nat.xor_trichotomy | Mathlib/Data/Nat/Bitwise.lean | theorem xor_trichotomy {a b c : ℕ} (h : a ^^^ b ^^^ c ≠ 0) :
b ^^^ c < a ∨ c ^^^ a < b ∨ a ^^^ b < c | a b c : ℕ
v : ℕ := a ^^^ b ^^^ c
h : v ≠ 0
hv : v = a ^^^ b ^^^ c
hab : a ^^^ b = c ^^^ v
⊢ b ^^^ c < a ∨ c ^^^ a < b ∨ a ^^^ b < c | have hbc : b ^^^ c = a ^^^ v := by
rw [← Nat.xor_assoc, xor_cancel_left] | a b c : ℕ
v : ℕ := a ^^^ b ^^^ c
h : v ≠ 0
hv : v = a ^^^ b ^^^ c
hab : a ^^^ b = c ^^^ v
hbc : b ^^^ c = a ^^^ v
⊢ b ^^^ c < a ∨ c ^^^ a < b ∨ a ^^^ b < c | acc65930ae06795b |
ReaderT.ext | Mathlib/.lake/packages/lean4/src/lean/Init/Control/Lawful/Instances.lean | theorem ext {x y : ReaderT ρ m α} (h : ∀ ctx, x.run ctx = y.run ctx) : x = y | ρ : Type u_1
m : Type u_1 → Type u_2
α : Type u_1
x y : ReaderT ρ m α
h : ∀ (ctx : ρ), x ctx = y ctx
⊢ x = y | exact funext h | no goals | d855592c0ca7598e |
Ideal.sInf_isPrime_of_isChain | Mathlib/RingTheory/Ideal/Maximal.lean | theorem sInf_isPrime_of_isChain {s : Set (Ideal α)} (hs : s.Nonempty) (hs' : IsChain (· ≤ ·) s)
(H : ∀ p ∈ s, Ideal.IsPrime p) : (sInf s).IsPrime :=
⟨fun e =>
let ⟨x, hx⟩ := hs
(H x hx).ne_top (eq_top_iff.mpr (e.symm.trans_le (sInf_le hx))),
fun e =>
or_iff_not_imp_left.mpr fun hx => by
rw [... | α : Type u
inst✝ : Semiring α
s : Set (Ideal α)
hs : s.Nonempty
hs' : IsChain (fun x1 x2 => x1 ≤ x2) s
H : ∀ p ∈ s, p.IsPrime
x✝ y✝ : α
e : ∀ ⦃I : Ideal α⦄, I ∈ s → x✝ * y✝ ∈ I
hx : ∃ I ∈ s, x✝ ∉ I
⊢ ∀ ⦃I : Ideal α⦄, I ∈ s → y✝ ∈ I | obtain ⟨I, hI, hI'⟩ := hx | case intro.intro
α : Type u
inst✝ : Semiring α
s : Set (Ideal α)
hs : s.Nonempty
hs' : IsChain (fun x1 x2 => x1 ≤ x2) s
H : ∀ p ∈ s, p.IsPrime
x✝ y✝ : α
e : ∀ ⦃I : Ideal α⦄, I ∈ s → x✝ * y✝ ∈ I
I : Ideal α
hI : I ∈ s
hI' : x✝ ∉ I
⊢ ∀ ⦃I : Ideal α⦄, I ∈ s → y✝ ∈ I | 2dc5df5a771c0933 |
Semiquot.mem_bind | Mathlib/Data/Semiquot.lean | theorem mem_bind (q : Semiquot α) (f : α → Semiquot β) (b : β) :
b ∈ bind q f ↔ ∃ a ∈ q, b ∈ f a | α : Type u_1
β : Type u_2
q : Semiquot α
f : α → Semiquot β
b : β
⊢ b ∈ q.bind f ↔ ∃ a, ∃ (_ : a ∈ q), b ∈ f a | exact Set.mem_iUnion₂ | no goals | a1b4c482ebb937bb |
PowerSeries.mk_one_pow_eq_mk_choose_add | Mathlib/RingTheory/PowerSeries/WellKnown.lean | theorem mk_one_pow_eq_mk_choose_add :
(mk 1 : S⟦X⟧) ^ (d + 1) = (mk fun n => Nat.choose (d + n) d : S⟦X⟧) | case succ.h
S : Type u_1
inst✝ : CommRing S
d✝ d : ℕ
hd : mk 1 ^ (d + 1) = mk fun n => ↑((d + n).choose d)
n : ℕ
⊢ ↑(∑ x ∈ Finset.antidiagonal n, (d + x.2).choose d) = ↑((d + 1 + n).choose (d + 1)) | rw [Finset.sum_antidiagonal_choose_add, add_right_comm] | no goals | 70d9cdde4192c8cc |
integral_rpow | Mathlib/Analysis/SpecialFunctions/Integrals.lean | theorem integral_rpow {r : ℝ} (h : -1 < r ∨ r ≠ -1 ∧ (0 : ℝ) ∉ [[a, b]]) :
∫ x in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) | a b r : ℝ
h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
h' : -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]
⊢ ∫ (x : ℝ) in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) | have :
(∫ x in a..b, (x : ℂ) ^ (r : ℂ)) = ((b : ℂ) ^ (r + 1 : ℂ) - (a : ℂ) ^ (r + 1 : ℂ)) / (r + 1) :=
integral_cpow h' | a b r : ℝ
h : -1 < r ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]
h' : -1 < (↑r).re ∨ ↑r ≠ -1 ∧ 0 ∉ [[a, b]]
this : ∫ (x : ℝ) in a..b, ↑x ^ ↑r = (↑b ^ (↑r + 1) - ↑a ^ (↑r + 1)) / (↑r + 1)
⊢ ∫ (x : ℝ) in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) | 4c1bda95f96e0fe4 |
Vector.foldl_subtype | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Attach.lean | theorem foldl_subtype {p : α → Prop} {l : Vector { x // p x } n}
{f : β → { x // p x } → β} {g : β → α → β} {x : β}
(hf : ∀ b x h, f b ⟨x, h⟩ = g b x) :
l.foldl f x = l.unattach.foldl g x | case mk
α : Type u_1
β : Type u_2
p : α → Prop
f : β → { x // p x } → β
g : β → α → β
x : β
hf : ∀ (b : β) (x : α) (h : p x), f b ⟨x, h⟩ = g b x
l : Array { x // p x }
⊢ foldl f x { toArray := l, size_toArray := ⋯ } = foldl g x { toArray := l, size_toArray := ⋯ }.unattach | simp [Array.foldl_subtype hf] | no goals | 7060d5cebb3c62a5 |
Equiv.Perm.disjoint_ofSubtype_of_memFixedPoints_self | Mathlib/GroupTheory/Perm/Finite.lean | lemma disjoint_ofSubtype_of_memFixedPoints_self {g : Perm α}
(u : Perm (Function.fixedPoints g)) :
Disjoint (ofSubtype u) g | α : Type u
inst✝ : DecidableEq α
g : Perm α
u : Perm ↑(fixedPoints ⇑g)
⊢ (ofSubtype u).Disjoint g | rw [disjoint_iff_eq_or_eq] | α : Type u
inst✝ : DecidableEq α
g : Perm α
u : Perm ↑(fixedPoints ⇑g)
⊢ ∀ (x : α), (ofSubtype u) x = x ∨ g x = x | 2505f366ac1245d8 |
FractionalIdeal.inv_le_dual | Mathlib/RingTheory/DedekindDomain/Different.lean | lemma inv_le_dual :
I⁻¹ ≤ dual A K I | A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁸ : CommRing A
inst✝¹⁷ : Field K
inst✝¹⁶ : CommRing B
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra A K
inst✝¹³ : Algebra B L
inst✝¹² : Algebra A B
inst✝¹¹ : Algebra K L
inst✝¹⁰ : Algebra A L
inst✝⁹ : IsScalarTower A K L
inst✝⁸ : IsScalarTower A B L
inst✝⁷ : IsDomain A
inst... | simp [hI] | no goals | 54545f38bb00b669 |
List.sum_map_count_dedup_filter_eq_countP | Mathlib/Algebra/BigOperators/Group/List/Lemmas.lean | theorem sum_map_count_dedup_filter_eq_countP (p : α → Bool) (l : List α) :
((l.dedup.filter p).map fun x => l.count x).sum = l.countP p | case neg
α : Type u_2
inst✝ : DecidableEq α
p : α → Bool
a : α
as : List α
h : (map (fun x => count x as) (filter p as.dedup)).sum = countP p as
ha : a ∉ as
⊢ (map (fun i => count i as)
(match p a with
| true => a :: filter p as.dedup
| false => filter p as.dedup)).sum =
(map (fun i => count... | match p a with
| true => simp only [List.map_cons, List.sum_cons, List.count_eq_zero.2 ha, zero_add]
| false => simp only | no goals | 26eedee1a0b217fa |
ContinuousLinearMap.norm_pi_le_of_le | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | lemma norm_pi_le_of_le {ι : Type*} [Fintype ι]
{M : ι → Type*} [∀ i, SeminormedAddCommGroup (M i)] [∀ i, NormedSpace 𝕜 (M i)] {C : ℝ}
{L : (i : ι) → (E →L[𝕜] M i)} (hL : ∀ i, ‖L i‖ ≤ C) (hC : 0 ≤ C) :
‖pi L‖ ≤ C | 𝕜 : Type u_1
E : Type u_4
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
ι : Type u_9
inst✝² : Fintype ι
M : ι → Type u_10
inst✝¹ : (i : ι) → SeminormedAddCommGroup (M i)
inst✝ : (i : ι) → NormedSpace 𝕜 (M i)
C : ℝ
L : (i : ι) → E →L[𝕜] M i
hL : ∀ (i : ι), ‖L i‖ ≤ C
h... | positivity | no goals | 13ad9a33e7bcc9f4 |
CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsSheafIff.fac | Mathlib/CategoryTheory/Sites/IsSheafOneHypercover.lean | lemma fac [H.IsGenerating] {Y : C} (f : Y ⟶ X) (hf : S f) :
lift hP hE le F ≫ P.map f.op = F.ι ⟨Y, f, hf⟩ | C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
hP : ∀ ⦃X : C⦄ (E : J.OneHypercover X), H E → Nonempty (IsLimit (E.multifork P))
X : C
S : Sieve X
E : J.OneHypercover X
hE : H E
le : E.sieve₀ ≤ S
F : Multifork (Cover.index ⟨... | intro Z g | C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
H : J.OneHypercoverFamily
P : Cᵒᵖ ⥤ A
hP : ∀ ⦃X : C⦄ (E : J.OneHypercover X), H E → Nonempty (IsLimit (E.multifork P))
X : C
S : Sieve X
E : J.OneHypercover X
hE : H E
le : E.sieve₀ ≤ S
F : Multifork (Cover.index ⟨... | 7312dc4cb01981b4 |
Set.toFinset_insert | Mathlib/Data/Fintype/Sets.lean | theorem toFinset_insert [DecidableEq α] {a : α} {s : Set α} [Fintype (insert a s : Set α)]
[Fintype s] : (insert a s).toFinset = insert a s.toFinset | α : Type u_1
inst✝² : DecidableEq α
a : α
s : Set α
inst✝¹ : Fintype ↑(insert a s)
inst✝ : Fintype ↑s
⊢ (insert a s).toFinset = insert a s.toFinset | ext | case h
α : Type u_1
inst✝² : DecidableEq α
a : α
s : Set α
inst✝¹ : Fintype ↑(insert a s)
inst✝ : Fintype ↑s
a✝ : α
⊢ a✝ ∈ (insert a s).toFinset ↔ a✝ ∈ insert a s.toFinset | 86f5b2aef32ba890 |
Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le | Mathlib/Analysis/Calculus/MeanValue.lean | theorem norm_image_sub_le_of_norm_hasFDerivWithin_le
(hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
(xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
C : ℝ
s : Set E
x y : E
f' : E → E →L[𝕜] G
hf : ∀ x ∈ s, HasFDeri... | have segm : MapsTo g (Icc 0 1 : Set ℝ) s := hs.mapsTo_lineMap xs ys | E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
𝕜 : Type u_3
G : Type u_4
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : E → G
C : ℝ
s : Set E
x y : E
f' : E → E →L[𝕜] G
hf : ∀ x ∈ s, HasFDeri... | d17bcea2b37bc830 |
sum_mul_Ico_le_integral_of_monotone_antitone | Mathlib/Analysis/SumIntegralComparisons.lean | lemma sum_mul_Ico_le_integral_of_monotone_antitone
(hab : a ≤ b) (hf : MonotoneOn f (Icc a b)) (hg : AntitoneOn g (Icc (a - 1) (b - 1)))
(fpos : 0 ≤ f a) (gpos : 0 ≤ g (b - 1)) :
∑ i ∈ Finset.Ico a b, f i * g i ≤ ∫ x in a..b, f x * g (x - 1) | case h.h₂.x
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc ↑a ↑b)
hg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑a
gpos : 0 ≤ g (↑b - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ ↑i + 1 ≤ ↑b | norm_cast | case h.h₂.x
a b : ℕ
f g : ℝ → ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc ↑a ↑b)
hg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))
fpos : 0 ≤ f ↑a
gpos : 0 ≤ g (↑b - 1)
i : ℕ
x : ℝ
hx : ↑i ≤ x ∧ x < ↑i + 1
hi : a ≤ i ∧ i < b
I0 : ↑i ≤ ↑b - 1
I1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)
I2 : x ∈ Icc ↑a ↑b
⊢ i + 1 ≤ b | cf3bc679c1354b33 |
Fin.foldr_rev | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Fin/Fold.lean | theorem foldr_rev (f : α → Fin n → α) (x) :
foldr n (fun i x => f x i.rev) x = foldl n f x | case succ
α : Sort u_1
n : Nat
ih : ∀ (f : α → Fin n → α) (x : α), foldr n (fun i x => f x i.rev) x = foldl n f x
f : α → Fin (n + 1) → α
x : α
⊢ f (foldr n (fun i x => f x i.succ.rev) x) (rev 0) = f (foldr n (fun i x => f x i.rev.castSucc) x) (last n) | simp [rev_succ] | no goals | 7860ca2be6251b5b |
SimpleGraph.connected_iff_exists_forall_reachable | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma connected_iff_exists_forall_reachable : G.Connected ↔ ∃ v, ∀ w, G.Reachable v w | case mpr
V : Type u
G : SimpleGraph V
⊢ (∃ v, ∀ (w : V), G.Reachable v w) → G.Preconnected ∧ Nonempty V | rintro ⟨v, h⟩ | case mpr.intro
V : Type u
G : SimpleGraph V
v : V
h : ∀ (w : V), G.Reachable v w
⊢ G.Preconnected ∧ Nonempty V | 1c445adc9e510c30 |
ModN.natCard_eq | Mathlib/LinearAlgebra/FreeModule/ModN.lean | @[simp] lemma natCard_eq : Nat.card (ModN G n) = n ^ Module.finrank ℤ G | G : Type u_1
inst✝³ : AddCommGroup G
n : ℕ
inst✝² : NeZero n
inst✝¹ : Module.Free ℤ G
inst✝ : Module.Finite ℤ G
⊢ Nat.card (ModN G n) = n ^ Module.finrank ℤ G | simp [Nat.card_congr (basis <| Module.Free.chooseBasis ℤ G).repr.toEquiv,
Nat.card_eq_fintype_card, Module.finrank_eq_card_chooseBasisIndex] | no goals | d116eabcb7440e6d |
Ideal.prime_of_isPrime | Mathlib/RingTheory/DedekindDomain/Ideal.lean | theorem Ideal.prime_of_isPrime {P : Ideal A} (hP : P ≠ ⊥) (h : IsPrime P) : Prime P | A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
P : Ideal A
hP : P ≠ ⊥
h : P.IsPrime
⊢ Prime P | refine ⟨hP, mt Ideal.isUnit_iff.mp h.ne_top, fun I J hIJ => ?_⟩ | A : Type u_2
inst✝¹ : CommRing A
inst✝ : IsDedekindDomain A
P : Ideal A
hP : P ≠ ⊥
h : P.IsPrime
I J : Ideal A
hIJ : P ∣ I * J
⊢ P ∣ I ∨ P ∣ J | ca3eba0aa13c81a9 |
Polynomial.leadingCoeff_add_of_degree_lt' | Mathlib/Algebra/Polynomial/Degree/Operations.lean | theorem leadingCoeff_add_of_degree_lt' (h : degree q < degree p) :
leadingCoeff (p + q) = leadingCoeff p | R : Type u
inst✝ : Semiring R
p q : R[X]
h : q.degree < p.degree
⊢ (q + p).leadingCoeff = p.leadingCoeff | exact leadingCoeff_add_of_degree_lt h | no goals | d41c71b12c174e95 |
Asymptotics.IsBigO.trans_tendsto_norm_atTop | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | theorem Asymptotics.IsBigO.trans_tendsto_norm_atTop {α : Type*} {u v : α → 𝕜} {l : Filter α}
(huv : u =O[l] v) (hu : Tendsto (fun x => ‖u x‖) l atTop) :
Tendsto (fun x => ‖v x‖) l atTop | case intro.intro
𝕜 : Type u_1
inst✝ : NormedLinearOrderedField 𝕜
α : Type u_2
u v : α → 𝕜
l : Filter α
huv : u =O[l] v
hu : Tendsto (fun x => ‖u x‖) l atTop
c : ℝ
hc : c > 0
hcuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ c * ‖v x‖
⊢ Tendsto (fun x => ‖v x‖) l atTop | convert Tendsto.atTop_div_const hc (tendsto_atTop_mono' l hcuv hu) | case h.e'_3.h
𝕜 : Type u_1
inst✝ : NormedLinearOrderedField 𝕜
α : Type u_2
u v : α → 𝕜
l : Filter α
huv : u =O[l] v
hu : Tendsto (fun x => ‖u x‖) l atTop
c : ℝ
hc : c > 0
hcuv : ∀ᶠ (x : α) in l, ‖u x‖ ≤ c * ‖v x‖
x✝ : α
⊢ ‖v x✝‖ = c * ‖v x✝‖ / c | 53ba852803eec157 |
RingQuot.sub_quot | Mathlib/Algebra/RingQuot.lean | theorem sub_quot {R : Type uR} [Ring R] (r : R → R → Prop) {a b} :
(⟨Quot.mk _ a⟩ - ⟨Quot.mk _ b⟩ : RingQuot r) = ⟨Quot.mk _ (a - b)⟩ | R : Type uR
inst✝ : Ring R
r : R → R → Prop
a b : R
⊢ { toQuot := Quot.mk (Rel r) a } - { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a - b) } | show sub r _ _ = _ | R : Type uR
inst✝ : Ring R
r : R → R → Prop
a b : R
⊢ RingQuot.sub r { toQuot := Quot.mk (Rel r) a } { toQuot := Quot.mk (Rel r) b } = { toQuot := Quot.mk (Rel r) (a - b) } | 8713bb8ced716ee6 |
ArithmeticFunction.cardDistinctFactors_apply_prime | Mathlib/NumberTheory/ArithmeticFunction.lean | theorem cardDistinctFactors_apply_prime {p : ℕ} (hp : p.Prime) : ω p = 1 | p : ℕ
hp : Nat.Prime p
⊢ ω p = 1 | rw [← pow_one p, cardDistinctFactors_apply_prime_pow hp one_ne_zero] | no goals | 80194d2a9c2ac76a |
Real.ediam_eq | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem ediam_eq {s : Set ℝ} (h : Bornology.IsBounded s) :
EMetric.diam s = ENNReal.ofReal (sSup s - sInf s) | case inr.refine_2.h
s : Set ℝ
h : Bornology.IsBounded s
hne : s.Nonempty
⊢ sSup s - sInf s ≤ Metric.diam (closure s) | calc sSup s - sInf s ≤ dist (sSup s) (sInf s) := le_abs_self _
_ ≤ Metric.diam (closure s) := dist_le_diam_of_mem h.closure (csSup_mem_closure hne h.bddAbove)
(csInf_mem_closure hne h.bddBelow) | no goals | d6382336badce924 |
OmegaCompletePartialOrder.ContinuousHom.ωScottContinuous.bind | Mathlib/Order/OmegaCompletePartialOrder.lean | lemma ωScottContinuous.bind {β γ} {f : α → Part β} {g : α → β → Part γ} (hf : ωScottContinuous f)
(hg : ωScottContinuous g) : ωScottContinuous fun x ↦ f x >>= g x :=
ωScottContinuous.of_monotone_map_ωSup
⟨hf.monotone.partBind hg.monotone, fun c ↦ by rw [hf.map_ωSup, hg.map_ωSup, ← ωSup_bind]; rfl⟩
| α : Type u_2
inst✝ : OmegaCompletePartialOrder α
β γ : Type u_6
f : α → Part β
g : α → β → Part γ
hf : ωScottContinuous f
hg : ωScottContinuous g
c : Chain α
⊢ ωSup (c.map ({ toFun := f, monotone' := ⋯ }.partBind { toFun := g, monotone' := ⋯ })) =
ωSup (c.map { toFun := fun x => f x >>= g x, monotone' := ⋯ }) | rfl | no goals | e5ce0d4585f86fae |
MeasureTheory.IsSetSemiring.subset_of_mem_disjointOfDiffUnion | Mathlib/MeasureTheory/SetSemiring.lean | lemma subset_of_mem_disjointOfDiffUnion (hC : IsSetSemiring C) {I : Finset (Set α)}
(hs : s ∈ C) (hI : ↑I ⊆ C) (t : Set α)
(ht : t ∈ (hC.disjointOfDiffUnion hs hI : Set (Set α))) :
t ⊆ s | α : Type u_1
C : Set (Set α)
s : Set α
hC : IsSetSemiring C
I : Finset (Set α)
hs : s ∈ C
hI : ↑I ⊆ C
t : Set α
ht : t ∈ ↑(hC.disjointOfDiffUnion hs hI)
⊢ s \ ⋃₀ ↑I ≤ s | exact sdiff_le (a := s) (b := ⋃₀ I) | no goals | 9db661540cbd5e01 |
Nat.bitwise_bit' | Mathlib/Data/Nat/Bitwise.lean | /-- An alternative for `bitwise_bit` which replaces the `f false false = false` assumption
with assumptions that neither `bit a m` nor `bit b n` are `0`
(albeit, phrased as the implications `m = 0 → a = true` and `n = 0 → b = true`) -/
lemma bitwise_bit' {f : Bool → Bool → Bool} (a : Bool) (m : Nat) (b : Bool) (n : Nat... | f : Bool → Bool → Bool
a : Bool
m : ℕ
b : Bool
n : ℕ
ham : bit a m ≠ 0
hbn : bit b n ≠ 0
⊢ (if f a b = true then bitwise f m n + bitwise f m n + 1 else bitwise f m n + bitwise f m n) =
bit (f a b) (bitwise f m n) | conv_rhs => simp only [bit, two_mul, Bool.cond_eq_ite] | f : Bool → Bool → Bool
a : Bool
m : ℕ
b : Bool
n : ℕ
ham : bit a m ≠ 0
hbn : bit b n ≠ 0
⊢ (if f a b = true then bitwise f m n + bitwise f m n + 1 else bitwise f m n + bitwise f m n) =
(if f a b = true then fun x => x + x + 1 else fun x => x + x) (bitwise f m n) | ae8208658e829741 |
Set.wellFoundedOn_iff | Mathlib/Order/WellFoundedSet.lean | theorem wellFoundedOn_iff :
s.WellFoundedOn r ↔ WellFounded fun a b : α => r a b ∧ a ∈ s ∧ b ∈ s | case neg.intro
α : Type u_2
r : α → α → Prop
s : Set α
f : (fun a b => r ↑a ↑b) ↪r fun a b => r a b ∧ a ∈ s ∧ b ∈ s
h : s.WellFoundedOn r
t : Set α
hst : ¬(s ∩ t).Nonempty
m : α
mt : m ∈ t
⊢ ∃ m ∈ t, ∀ x ∈ t, ¬(r x m ∧ x ∈ s ∧ m ∈ s) | exact ⟨m, mt, fun x _ ⟨_, _, ms⟩ => hst ⟨m, ⟨ms, mt⟩⟩⟩ | no goals | 5455ade8bb8a1262 |
Equiv.Perm.Basis.mem_fixedPoints_or_exists_zpow_eq | Mathlib/GroupTheory/Perm/Centralizer.lean | theorem mem_fixedPoints_or_exists_zpow_eq (x : α) :
x ∈ Function.fixedPoints g ∨
∃ (c : g.cycleFactorsFinset) (_ : x ∈ c.val.support) (m : ℤ), (g ^ m) (a c) = x | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
x : α
⊢ x ∉ Function.fixedPoints ⇑g → ∃ c, ∃ (_ : x ∈ (↑c).support), ∃ m, (g ^ m) (a c) = x | intro hx | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g : Perm α
a : g.Basis
x : α
hx : x ∉ Function.fixedPoints ⇑g
⊢ ∃ c, ∃ (_ : x ∈ (↑c).support), ∃ m, (g ^ m) (a c) = x | afc5319d6aace422 |
Nat.Partrec.Code.evaln_map | Mathlib/Computability/PartrecCode.lean | theorem evaln_map (k c n) :
((List.range k)[n]?.bind fun a ↦ evaln k c a) = evaln k c n | case neg
k : ℕ
c : Code
n : ℕ
kn : ¬n < k
⊢ (Option.none.bind fun a => evaln k c a) = evaln k c n | cases e : evaln k c n | case neg.none
k : ℕ
c : Code
n : ℕ
kn : ¬n < k
e : evaln k c n = Option.none
⊢ (Option.none.bind fun a => evaln k c a) = Option.none
case neg.some
k : ℕ
c : Code
n : ℕ
kn : ¬n < k
val✝ : ℕ
e : evaln k c n = some val✝
⊢ (Option.none.bind fun a => evaln k c a) = some val✝ | ac6268b93b88bdde |
toAlgHom_comp_sectionOfRetractionKerToTensorAux | Mathlib/RingTheory/Smooth/Kaehler.lean | lemma toAlgHom_comp_sectionOfRetractionKerToTensorAux :
(IsScalarTower.toAlgHom R P S).comp
(sectionOfRetractionKerToTensorAux l hl σ hσ hf') = AlgHom.id _ _ | R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
σ : S → P
hσ : ∀ (x : S), (algebraMap P S) (σ x) = x
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R P S
hf... | ext x | case H
R P S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommRing P
inst✝⁴ : CommRing S
inst✝³ : Algebra R P
inst✝² : Algebra P S
l : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))
hl : l ∘ₗ kerToTensor R P S = LinearMap.id
σ : S → P
hσ : ∀ (x : S), (algebraMap P S) (σ x) = x
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R... | 235d235fa9f329b5 |
Polynomial.nodup_aroots_iff_of_splits | Mathlib/FieldTheory/Separable.lean | theorem nodup_aroots_iff_of_splits [Algebra F K] {f : F[X]} (hf : f ≠ 0)
(h : f.Splits (algebraMap F K)) : (f.aroots K).Nodup ↔ f.Separable | F : Type u
inst✝² : Field F
K : Type v
inst✝¹ : Field K
inst✝ : Algebra F K
f : F[X]
hf : f ≠ 0
h : Splits (algebraMap F K) f
⊢ (f.aroots K).Nodup ↔ f.Separable | rw [← (algebraMap F K).id_comp, ← splits_map_iff] at h | F : Type u
inst✝² : Field F
K : Type v
inst✝¹ : Field K
inst✝ : Algebra F K
f : F[X]
hf : f ≠ 0
h : Splits (RingHom.id K) (map (algebraMap F K) f)
⊢ (f.aroots K).Nodup ↔ f.Separable | 51b06326b2ffa5a4 |
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