name
stringlengths 3
112
| file
stringlengths 21
116
| statement
stringlengths 17
8.64k
| state
stringlengths 7
205k
| tactic
stringlengths 3
4.55k
| result
stringlengths 7
205k
| id
stringlengths 16
16
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|---|---|---|---|---|---|---|
List.getElem_insert
|
Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean
|
theorem getElem_insert (l : List α) (a : α) (i : Nat) (h : i < l.length) :
(l.insert a)[i]'(Nat.lt_of_lt_of_le h length_le_length_insert) =
if a ∈ l then l[i] else if i = 0 then a else l[i-1]'(Nat.lt_of_le_of_lt (Nat.pred_le _) h)
|
case x
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List α
a : α
i : Nat
h : i < l.length
⊢ (if a ∈ l then l[i]? else if i = 0 then some a else l[i - 1]?) =
some (if a ∈ l then l[i] else if i = 0 then a else l[i - 1])
|
split
|
case x.isTrue
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List α
a : α
i : Nat
h : i < l.length
h✝ : a ∈ l
⊢ l[i]? = some l[i]
case x.isFalse
α : Type u_1
inst✝¹ : BEq α
inst✝ : LawfulBEq α
l : List α
a : α
i : Nat
h : i < l.length
h✝ : ¬a ∈ l
⊢ (if i = 0 then some a else l[i - 1]?) = some (if i = 0 then a else l[i - 1])
|
5973fbc938ccf630
|
DFinsupp.lex_lt_of_lt_of_preorder
|
Mathlib/Data/DFinsupp/Lex.lean
|
theorem lex_lt_of_lt_of_preorder [∀ i, Preorder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i
|
case intro.intro
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → Preorder (α i)
r : ι → ι → Prop
inst✝ : IsStrictOrder ι r
x y : Π₀ (i : ι), α i
hlt✝ : x < y
hle : ⇑x ≤ ⇑y
j : ι
hlt : x j < y j
this : (↑(x.neLocus y)).WellFoundedOn r
⊢ ∃ i, (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i
|
obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩
|
case intro.intro.intro.intro
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → Preorder (α i)
r : ι → ι → Prop
inst✝ : IsStrictOrder ι r
x y : Π₀ (i : ι), α i
hlt✝ : x < y
hle : ⇑x ≤ ⇑y
j : ι
hlt : x j < y j
this : (↑(x.neLocus y)).WellFoundedOn r
i : ↑↑(x.neLocus y)
hi : i ∈ {i | x ↑i < y ↑i}
hl : ∀ x_1 ∈ {i | x ↑i < y ↑i}, ¬r ↑x_1 ↑i
⊢ ∃ i, (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i
|
b2be9adf1bae9d88
|
Real.exists_rat_abs_sub_lt_and_lt_of_irrational
|
Mathlib/NumberTheory/DiophantineApproximation/Basic.lean
|
theorem exists_rat_abs_sub_lt_and_lt_of_irrational {ξ : ℝ} (hξ : Irrational ξ) (q : ℚ) :
∃ q' : ℚ, |ξ - q'| < 1 / (q'.den : ℝ) ^ 2 ∧ |ξ - q'| < |ξ - q|
|
case intro.intro.intro
ξ : ℝ
hξ : Irrational ξ
q : ℚ
h : 0 < |ξ - ↑q|
m : ℕ
hm : 1 / |ξ - ↑q| < ↑m
m_pos : 0 < ↑m
q' : ℚ
hbd : |ξ - ↑q'| ≤ 1 / ((↑m + 1) * ↑q'.den)
hden : q'.den ≤ m
den_pos : 0 < ↑q'.den
md_pos : 0 < (↑m + 1) * ↑q'.den
⊢ ↑q'.den < ↑m + 1
|
exact lt_add_of_le_of_pos (Nat.cast_le.mpr hden) zero_lt_one
|
no goals
|
05e392f3a5eb1af5
|
NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis
|
Mathlib/NumberTheory/NumberField/Discriminant/Basic.lean
|
theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
volume (fundamentalDomain (latticeBasis K)) =
(2 : ℝ≥0∞)⁻¹ ^ nrComplexPlaces K * sqrt ‖discr K‖₊
|
case a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) RingHom.equivRatAlgHom
i✝ j✝ : index K
⊢ ((mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)).map (⇑ofRealHom) i✝ j✝ =
(matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ) i✝ j✝
|
rw [Matrix.map_apply, Basis.toMatrix_apply, Basis.coe_reindex, Function.comp_apply,
Equiv.symm_symm, latticeBasis_apply, ← commMap_canonical_eq_mixed, Complex.ofRealHom_eq_coe,
stdBasis_repr_eq_matrixToStdBasis_mul K _ (fun _ => rfl)]
|
case a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
f : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))
e : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
M : Matrix (index K) (index K) ℝ := (mixedEmbedding.stdBasis K).toMatrix ⇑((latticeBasis K).reindex e.symm)
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
Algebra.embeddingsMatrixReindex ℚ ℂ (⇑(integralBasis K) ∘ ⇑f.symm) RingHom.equivRatAlgHom
i✝ j✝ : index K
⊢ (matrixToStdBasis K *ᵥ (canonicalEmbedding K) ((integralBasis K) (e j✝)) ∘ ⇑(indexEquiv K)) i✝ =
(matrixToStdBasis K * ((reindex (indexEquiv K).symm (indexEquiv K).symm) N)ᵀ) i✝ j✝
|
0fabcb321045abbb
|
Polynomial.eval₂_mul_X
|
Mathlib/Algebra/Polynomial/Eval/Defs.lean
|
theorem eval₂_mul_X : eval₂ f x (p * X) = eval₂ f x p * x
|
R : Type u
S : Type v
inst✝¹ : Semiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
x : S
⊢ eval₂ f x (p * X) = eval₂ f x p * x
|
refine _root_.trans (eval₂_mul_noncomm _ _ fun k => ?_) (by rw [eval₂_X])
|
R : Type u
S : Type v
inst✝¹ : Semiring R
p : R[X]
inst✝ : Semiring S
f : R →+* S
x : S
k : ℕ
⊢ Commute (f (X.coeff k)) x
|
1b64714895ac341a
|
PrimeSpectrum.primeSpectrumProd_symm_inr
|
Mathlib/RingTheory/Spectrum/Prime/Topology.lean
|
lemma primeSpectrumProd_symm_inr (x) :
(primeSpectrumProd R S).symm (.inr x) = comap (RingHom.snd R S) x
|
case asIdeal.h
R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
x : PrimeSpectrum S
x✝ : R × S
⊢ x✝ ∈ ((primeSpectrumProd R S).symm (Sum.inr x)).asIdeal ↔ x✝ ∈ ((comap (RingHom.snd R S)) x).asIdeal
|
simp [Ideal.prod]
|
no goals
|
6e39d62f4f8df19f
|
ProbabilityTheory.integral_gaussianPDFReal_eq_one
|
Mathlib/Probability/Distributions/Gaussian.lean
|
/-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma integral_gaussianPDFReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
∫ x, gaussianPDFReal μ v x = 1
|
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
h : ENNReal.ofReal (∫ (x : ℝ), gaussianPDFReal μ v x) = ENNReal.ofReal 1
⊢ ∫ (x : ℝ), gaussianPDFReal μ v x = 1
|
rwa [← ENNReal.ofReal_eq_ofReal_iff (integral_nonneg (gaussianPDFReal_nonneg _ _)) zero_le_one]
|
no goals
|
65e8bfafed911366
|
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_invariant_helper
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean
|
theorem confirmRupHint_preserves_invariant_helper {n : Nat} (f : DefaultFormula n)
(f_assignments_size : f.assignments.size = n)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool) (hsize : acc.1.size = n)
(l : Literal (PosFin n)) (ih : DerivedLitsInvariant f f_assignments_size acc.1 hsize acc.2.1)
(h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true) :
have hsize' : (Array.modify acc.1 l.1.1 (addAssignment l.snd)).size = n
|
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
Eq.mpr (id (congrArg (fun _a => _a = n) (Array.size_modify acc.fst l.fst.val (addAssignment l.snd)))) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin (List.length acc.snd.fst)
j_eq_i : (List.get acc.snd.fst j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (List.get acc.snd.fst j).snd f.assignments[↑i]
h2 : ¬hasAssignment (List.get acc.snd.fst j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin (List.length acc.snd.fst)), k ≠ j → (List.get acc.snd.fst k).fst.val ≠ ↑i
l' : Literal (PosFin n) := List.get acc.snd.fst j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_eq_i : l.fst.val = ↑i
l_ne_l' : false ≠ l'.snd
l_eq_false : l.snd = false
l'_eq_true : l'.snd = true
k : Fin (List.length acc.snd.fst + 1)
k_ne_j_succ : ¬k = ⟨↑j + 1, j_succ_in_bounds⟩
k_ne_zero : ¬k = ⟨0, zero_in_bounds⟩
k' : Nat
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
k_eq_succ : k = ⟨k' + 1, k'_succ_in_bounds⟩
k'_in_bounds : k' < List.length acc.snd.fst
⊢ ⟨k', k'_in_bounds⟩ ≠ j
|
simp only [k_eq_succ, List.length_cons, Fin.mk.injEq, Nat.succ.injEq] at k_ne_j_succ
|
n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
l : Literal (PosFin n)
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
h : ¬hasAssignment l.snd acc.fst[l.fst.val]! = true
hsize'✝ : (acc.fst.modify l.fst.val (addAssignment l.snd)).size = n :=
Eq.mpr (id (congrArg (fun _a => _a = n) (Array.size_modify acc.fst l.fst.val (addAssignment l.snd)))) hsize
i : Fin n
i_in_bounds : ↑i < acc.fst.size
l_in_bounds : l.fst.val < acc.fst.size
j : Fin (List.length acc.snd.fst)
j_eq_i : (List.get acc.snd.fst j).fst.val = ↑i
h1 : acc.fst[↑i] = addAssignment (List.get acc.snd.fst j).snd f.assignments[↑i]
h2 : ¬hasAssignment (List.get acc.snd.fst j).snd f.assignments[↑i] = true
h3 : ∀ (k : Fin (List.length acc.snd.fst)), k ≠ j → (List.get acc.snd.fst k).fst.val ≠ ↑i
l' : Literal (PosFin n) := List.get acc.snd.fst j
zero_in_bounds : 0 < (l :: acc.snd.fst).length
j_succ_in_bounds : ↑j + 1 < (l :: acc.snd.fst).length
l_eq_i : l.fst.val = ↑i
l_ne_l' : false ≠ l'.snd
l_eq_false : l.snd = false
l'_eq_true : l'.snd = true
k : Fin (List.length acc.snd.fst + 1)
k_ne_zero : ¬k = ⟨0, zero_in_bounds⟩
k' : Nat
k'_succ_in_bounds : k' + 1 < (l :: acc.snd.fst).length
k_eq_succ : k = ⟨k' + 1, k'_succ_in_bounds⟩
k'_in_bounds : k' < List.length acc.snd.fst
k_ne_j_succ : ¬k' = ↑j
⊢ ⟨k', k'_in_bounds⟩ ≠ j
|
02ca1706b397b254
|
SimpleGraph.isAcyclic_of_path_unique
|
Mathlib/Combinatorics/SimpleGraph/Acyclic.lean
|
theorem isAcyclic_of_path_unique (h : ∀ (v w : V) (p q : G.Path v w), p = q) : G.IsAcyclic
|
case cons
V : Type u
G : SimpleGraph V
h : ∀ (v w : V) (p q : G.Path v w), p = q
v v✝ : V
ha : G.Adj v v✝
c' : G.Walk v✝ v
hc : (cons ha c').IsTrail ∧ ¬cons ha c' = nil ∧ (cons ha c').support.tail.Nodup
⊢ False
|
simp only [Walk.cons_isTrail_iff, Walk.support_cons, List.tail_cons] at hc
|
case cons
V : Type u
G : SimpleGraph V
h : ∀ (v w : V) (p q : G.Path v w), p = q
v v✝ : V
ha : G.Adj v v✝
c' : G.Walk v✝ v
hc : (c'.IsTrail ∧ s(v, v✝) ∉ c'.edges) ∧ ¬cons ha c' = nil ∧ c'.support.Nodup
⊢ False
|
0605fe545831c468
|
intervalIntegral.integral_add_adjacent_intervals_cancel
|
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
theorem integral_add_adjacent_intervals_cancel (hab : IntervalIntegrable f μ a b)
(hbc : IntervalIntegrable f μ b c) :
(((∫ x in a..b, f x ∂μ) + ∫ x in b..c, f x ∂μ) + ∫ x in c..a, f x ∂μ) = 0
|
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in Ioc a b, f x ∂μ + ∫ (x : ℝ) in Ioc b c, f x ∂μ + ∫ (x : ℝ) in Ioc c a, f x ∂μ =
∫ (x : ℝ) in Ioc b a, f x ∂μ + ∫ (x : ℝ) in Ioc c b, f x ∂μ + ∫ (x : ℝ) in Ioc a c, f x ∂μ
|
iterate 4 rw [← setIntegral_union]
|
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ ∫ (x : ℝ) in Ioc a b ∪ Ioc b c ∪ Ioc c a, f x ∂μ = ∫ (x : ℝ) in Ioc b a ∪ Ioc c b ∪ Ioc a c, f x ∂μ
case hst
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ Disjoint (Ioc b a ∪ Ioc c b) (Ioc a c)
case ht
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ MeasurableSet (Ioc a c)
case hfs
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc b a ∪ Ioc c b) μ
case hft
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc a c) μ
case hst
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ Disjoint (Ioc b a) (Ioc c b)
case ht
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ MeasurableSet (Ioc c b)
case hfs
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc b a) μ
case hft
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc c b) μ
case hst
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ Disjoint (Ioc a b ∪ Ioc b c) (Ioc c a)
case ht
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ MeasurableSet (Ioc c a)
case hfs
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc a b ∪ Ioc b c) μ
case hft
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc c a) μ
case hst
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ Disjoint (Ioc a b) (Ioc b c)
case ht
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ MeasurableSet (Ioc b c)
case hfs
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc a b) μ
case hft
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b c : ℝ
f : ℝ → E
μ : Measure ℝ
hab : IntervalIntegrable f μ a b
hbc : IntervalIntegrable f μ b c
hac : IntervalIntegrable f μ a c
⊢ IntegrableOn f (Ioc b c) μ
|
7a0784e26ee3c388
|
FirstOrder.Language.embedding_from_cg
|
Mathlib/ModelTheory/PartialEquiv.lean
|
theorem embedding_from_cg (M_cg : Structure.CG L M) (g : L.FGEquiv M N)
(H : L.IsExtensionPair M N) :
∃ f : M ↪[L] N, g ≤ f.toPartialEquiv
|
case mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
H : L.IsExtensionPair M N
X : Set M
left✝ : X.Countable
X_gen : (closure L).toFun X = ⊤
x✝ : Countable ↑X
⊢ ∃ f, ↑g ≤ f.toPartialEquiv
|
have _ : Encodable (↑X : Type _) := Encodable.ofCountable _
|
case mk.intro.intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
g : L.FGEquiv M N
H : L.IsExtensionPair M N
X : Set M
left✝ : X.Countable
X_gen : (closure L).toFun X = ⊤
x✝¹ : Countable ↑X
x✝ : Encodable ↑X
⊢ ∃ f, ↑g ≤ f.toPartialEquiv
|
31c3cb67a548b276
|
Ordinal.CNF_fst_le_log
|
Mathlib/SetTheory/Ordinal/CantorNormalForm.lean
|
theorem CNF_fst_le_log {b o : Ordinal.{u}} {x : Ordinal × Ordinal} :
x ∈ CNF b o → x.1 ≤ log b o
|
case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x ∈ CNF b o → x.1 ≤ log b o
|
rw [CNF_ne_zero ho, mem_cons]
|
case refine_2
b o✝ : Ordinal.{u}
x : Ordinal.{u} × Ordinal.{u}
o : Ordinal.{u}
ho : o ≠ 0
H : x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b (o % b ^ log b o)
⊢ x = (log b o, o / b ^ log b o) ∨ x ∈ CNF b (o % b ^ log b o) → x.1 ≤ log b o
|
bb723149b204c5e1
|
corners_theorem
|
Mathlib/Combinatorics/Additive/Corner/Roth.lean
|
theorem corners_theorem (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε ≤ card G)
(A : Finset (G × G)) (hAε : ε * card G ^ 2 ≤ #A) : ¬ IsCornerFree (A : Set (G × G))
|
G : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : Fintype G
ε : ℝ
hε : 0 < ε
hG : ⌊(triangleRemovalBound (ε / 9) * 27)⁻¹⌋₊ < Fintype.card G
A : Finset (G × G)
hAε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)
hA : IsCornerFree ↑A
this : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(Fintype.card (G × G))
⊢ ε ≤ 1
|
simp only [sq, Nat.cast_mul, Fintype.card_prod, Fintype.card_fin] at this
|
G : Type u_1
inst✝¹ : AddCommGroup G
inst✝ : Fintype G
ε : ℝ
hε : 0 < ε
hG : ⌊(triangleRemovalBound (ε / 9) * 27)⁻¹⌋₊ < Fintype.card G
A : Finset (G × G)
hAε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)
hA : IsCornerFree ↑A
this : ε * (↑(Fintype.card G) * ↑(Fintype.card G)) ≤ ↑(Fintype.card G) * ↑(Fintype.card G)
⊢ ε ≤ 1
|
76345cb9fe001f2e
|
YoungDiagram.row_eq_prod
|
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
theorem row_eq_prod {μ : YoungDiagram} {i : ℕ} : μ.row i = {i} ×ˢ Finset.range (μ.rowLen i)
|
case h.mk
μ : YoungDiagram
a b : ℕ
⊢ b < μ.rowLen a ↔ b < μ.rowLen a
|
rfl
|
no goals
|
abd38ed31320b3ef
|
CategoryTheory.Sheaf.isLocallySurjective_iff_epi
|
Mathlib/CategoryTheory/Sites/LocallySurjective.lean
|
lemma isLocallySurjective_iff_epi {F G : Sheaf J (Type w)} (φ : F ⟶ G)
[HasSheafify J (Type w)] :
IsLocallySurjective φ ↔ Epi φ
|
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
φ : F ⟶ G
inst✝ : HasSheafify J (Type w)
⊢ IsLocallySurjective φ ↔ Epi φ
|
constructor
|
case mp
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
φ : F ⟶ G
inst✝ : HasSheafify J (Type w)
⊢ IsLocallySurjective φ → Epi φ
case mpr
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type w)
φ : F ⟶ G
inst✝ : HasSheafify J (Type w)
⊢ Epi φ → IsLocallySurjective φ
|
1d3f917e21094b8e
|
MeasureTheory.LocallyIntegrableOn.exists_nat_integrableOn
|
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
theorem LocallyIntegrableOn.exists_nat_integrableOn [SecondCountableTopology X]
(hf : LocallyIntegrableOn f s μ) : ∃ u : ℕ → Set X,
(∀ n, IsOpen (u n)) ∧ (s ⊆ ⋃ n, u n) ∧ (∀ n, IntegrableOn f (u n ∩ s) μ)
|
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
|
rcases hf.exists_countable_integrableOn with ⟨T, T_count, T_open, sT, hT⟩
|
case intro.intro.intro.intro
X : Type u_1
E : Type u_3
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
f : X → E
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
⊢ ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ s ⊆ ⋃ n, u n ∧ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
|
7d50e417c306b9b4
|
Subalgebra.centralizer_coe_iSup
|
Mathlib/Algebra/Algebra/Subalgebra/Centralizer.lean
|
lemma centralizer_coe_iSup {ι : Sort*} (S : ι → Subalgebra R A) :
centralizer R ((⨆ i, S i : Subalgebra R A) : Set A) = ⨅ i, centralizer R (S i) :=
eq_of_forall_le_iff fun K ↦ by
simp_rw [le_centralizer_iff, iSup_le_iff, le_iInf_iff, K.le_centralizer_iff]
|
R : Type u_1
inst✝² : CommSemiring R
A : Type u_2
inst✝¹ : Semiring A
inst✝ : Algebra R A
ι : Sort u_3
S : ι → Subalgebra R A
K : Subalgebra R A
⊢ K ≤ centralizer R ↑(⨆ i, S i) ↔ K ≤ ⨅ i, centralizer R ↑(S i)
|
simp_rw [le_centralizer_iff, iSup_le_iff, le_iInf_iff, K.le_centralizer_iff]
|
no goals
|
be698f42a1aee23e
|
MultilinearMap.bound_of_shell
|
Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
|
theorem bound_of_shell (f : MultilinearMap 𝕜 E G) {ε : ι → ℝ} {C : ℝ} {c : ι → 𝕜}
(hε : ∀ i, 0 < ε i) (hc : ∀ i, 1 < ‖c i‖)
(hf : ∀ m : ∀ i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
(m : ∀ i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ :=
bound_of_shell_of_norm_map_coord_zero f
(fun h ↦ by rw [map_coord_zero f _ (norm_eq_zero.1 h), norm_zero]) hε hc hf m
|
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : Fintype ι
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : (i : ι) → NormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹ : SeminormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : MultilinearMap 𝕜 E G
ε : ι → ℝ
C : ℝ
c : ι → 𝕜
hε : ∀ (i : ι), 0 < ε i
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖
m m✝ : (i : ι) → E i
i✝ : ι
h : ‖m✝ i✝‖ = 0
⊢ ‖f m✝‖ = 0
|
rw [map_coord_zero f _ (norm_eq_zero.1 h), norm_zero]
|
no goals
|
8ffee075cd434aaa
|
MonoidAlgebra.prod_single
|
Mathlib/Algebra/MonoidAlgebra/Defs.lean
|
theorem prod_single [CommSemiring k] [CommMonoid G] {s : Finset ι} {a : ι → G} {b : ι → k} :
(∏ i ∈ s, single (a i) (b i)) = single (∏ i ∈ s, a i) (∏ i ∈ s, b i) :=
Finset.cons_induction_on s rfl fun a s has ih => by
rw [prod_cons has, ih, single_mul_single, prod_cons has, prod_cons has]
|
k : Type u₁
G : Type u₂
ι : Type ui
inst✝¹ : CommSemiring k
inst✝ : CommMonoid G
s✝ : Finset ι
a✝ : ι → G
b : ι → k
a : ι
s : Finset ι
has : a ∉ s
ih : ∏ i ∈ s, single (a✝ i) (b i) = single (∏ i ∈ s, a✝ i) (∏ i ∈ s, b i)
⊢ ∏ i ∈ Finset.cons a s has, single (a✝ i) (b i) =
single (∏ i ∈ Finset.cons a s has, a✝ i) (∏ i ∈ Finset.cons a s has, b i)
|
rw [prod_cons has, ih, single_mul_single, prod_cons has, prod_cons has]
|
no goals
|
4adb3807e3f96e25
|
IsFractionRing.num_mul_den_eq_num_iff_eq
|
Mathlib/RingTheory/Localization/NumDen.lean
|
theorem num_mul_den_eq_num_iff_eq {x y : K} :
x * algebraMap A K (den A y) = algebraMap A K (num A y) ↔ x = y :=
⟨fun h => by simpa only [mk'_num_den] using eq_mk'_iff_mul_eq.mpr h, fun h ↦
eq_mk'_iff_mul_eq.mp (by rw [h, mk'_num_den])⟩
|
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x y : K
h : x = y
⊢ x = mk' K (num A y) (den A y)
|
rw [h, mk'_num_den]
|
no goals
|
021c453a6bf35a8a
|
PiTensorProduct.injectiveSeminorm_le_projectiveSeminorm
|
Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
|
theorem injectiveSeminorm_le_projectiveSeminorm :
injectiveSeminorm (𝕜 := 𝕜) (E := E) ≤ projectiveSeminorm
|
case refine_1
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ 0 ∈
{p |
∃ G x x_1, p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)}
|
simp only [Set.mem_setOf_eq]
|
case refine_1
ι : Type uι
inst✝³ : Fintype ι
𝕜 : Type u𝕜
inst✝² : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace 𝕜 (E i)
⊢ ∃ G x x_1, 0 = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)
|
8de3922266918b06
|
ConvexBody.zero_mem_of_symmetric
|
Mathlib/Analysis/Convex/Body.lean
|
theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : ∀ x ∈ K, - x ∈ K) : 0 ∈ K
|
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K : ConvexBody V
h_symm : ∀ x ∈ K, -x ∈ K
⊢ 0 ∈ K
|
obtain ⟨x, hx⟩ := K.nonempty
|
case intro
V : Type u_1
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module ℝ V
K : ConvexBody V
h_symm : ∀ x ∈ K, -x ∈ K
x : V
hx : x ∈ ↑K
⊢ 0 ∈ K
|
b2b52a546c22c779
|
Function.semiconj_of_isLUB
|
Mathlib/Order/SemiconjSup.lean
|
theorem semiconj_of_isLUB [PartialOrder α] [Group G] (f₁ f₂ : G →* α ≃o α) {h : α → α}
(H : ∀ x, IsLUB (range fun g' => (f₁ g')⁻¹ (f₂ g' x)) (h x)) (g : G) :
Function.Semiconj h (f₂ g) (f₁ g)
|
α : Type u_1
G : Type u_4
inst✝¹ : PartialOrder α
inst✝ : Group G
f₁ f₂ : G →* α ≃o α
h : α → α
H : ∀ (x : α), IsLUB (range fun g' => (f₁ g')⁻¹ ((f₂ g') x)) (h x)
g : G
y : α
this : IsLUB (range ((⇑(f₁ g) ∘ fun g' => (f₁ g')⁻¹ ((f₂ g') y)) ∘ ⇑(Equiv.mulRight g))) ((f₁ g) (h y))
⊢ IsLUB (range fun g' => (f₁ g')⁻¹ ((f₂ g') ((f₂ g) y))) ((f₁ g) (h y))
|
simpa [comp_def] using this
|
no goals
|
e6a3d83bbf194959
|
AlgebraicGeometry.Scheme.Hom.residueFieldMap_congr
|
Mathlib/AlgebraicGeometry/ResidueField.lean
|
lemma Hom.residueFieldMap_congr {f g : X ⟶ Y} (e : f = g) (x : X) :
f.residueFieldMap x = (Y.residueFieldCongr (by subst e; rfl)).hom ≫ g.residueFieldMap x
|
X Y : Scheme
f g : X ⟶ Y
e : f = g
x : ↑↑X.toPresheafedSpace
⊢ residueFieldMap f x = (residueFieldCongr ⋯).hom ≫ residueFieldMap g x
|
subst e
|
X Y : Scheme
f : X ⟶ Y
x : ↑↑X.toPresheafedSpace
⊢ residueFieldMap f x = (residueFieldCongr ⋯).hom ≫ residueFieldMap f x
|
6910f3d84dfbcdd2
|
HomologicalComplex.inl_biprodXIso_inv
|
Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean
|
@[reassoc (attr := simp)]
lemma inl_biprodXIso_inv (i : ι) :
biprod.inl ≫ (biprodXIso K L i).inv = (biprod.inl : K ⟶ K ⊞ L).f i
|
C : Type u_1
ι : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
inst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)
i : ι
⊢ biprod.inl ≫ (K.biprodXIso L i).inv = biprod.inl.f i
|
simp [biprodXIso]
|
no goals
|
a8f9485fad92a25b
|
PiNat.apply_eq_of_dist_lt
|
Mathlib/Topology/MetricSpace/PiNat.lean
|
theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2) ^ n) {i : ℕ}
(hi : i ≤ n) : x i = y i
|
case inr
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
h : dist x y < (1 / 2) ^ n
i : ℕ
hi : i ≤ n
hne : x ≠ y
this : n < firstDiff x y
⊢ x i = y i
|
exact apply_eq_of_lt_firstDiff (hi.trans_lt this)
|
no goals
|
d5560e1fbb39ebfa
|
hasProd_subtype_iff_of_mulSupport_subset
|
Mathlib/Topology/Algebra/InfiniteSum/Defs.lean
|
theorem hasProd_subtype_iff_of_mulSupport_subset {s : Set β} (hf : mulSupport f ⊆ s) :
HasProd (f ∘ (↑) : s → α) a ↔ HasProd f a :=
Subtype.coe_injective.hasProd_iff <| by simpa using mulSupport_subset_iff'.1 hf
|
α : Type u_1
β : Type u_2
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f : β → α
a : α
s : Set β
hf : mulSupport f ⊆ s
⊢ ∀ x ∉ Set.range fun a => ↑a, f x = 1
|
simpa using mulSupport_subset_iff'.1 hf
|
no goals
|
ec00d0e2998a70a3
|
ContinuousMonoidHom.locallyCompactSpace_of_hasBasis
|
Mathlib/Topology/Algebra/Group/CompactOpen.lean
|
theorem locallyCompactSpace_of_hasBasis (V : ℕ → Set Y)
(hV : ∀ {n x}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1))
(hVo : Filter.HasBasis (nhds 1) (fun _ ↦ True) V) :
LocallyCompactSpace (ContinuousMonoidHom X Y)
|
case intro.intro
X : Type u_7
Y : Type u_8
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group X
inst✝⁶ : IsTopologicalGroup X
inst✝⁵ : UniformSpace Y
inst✝⁴ : CommGroup Y
inst✝³ : UniformGroup Y
inst✝² : T0Space Y
inst✝¹ : CompactSpace Y
inst✝ : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : (𝓝 1).HasBasis (fun x => True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : U0 ∈ 𝓝 1
U_aux : ℕ → ↑{S | S ∈ 𝓝 1} :=
fun t =>
Nat.rec ⟨U0, hU0o⟩
(fun x S =>
let h := ⋯;
⟨Classical.choose h, ⋯⟩)
t
U : ℕ → Set X := fun n => ↑(U_aux n)
hU1 : ∀ (n : ℕ), U n ∈ 𝓝 1
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f) (U n) (V n)
⊢ ∀ (k : ℕ), True → ∀ᶠ (x : X) in 𝓝 1, ∀ (i : ↑{f | Set.MapsTo (⇑f) (U 0) (V 0)}), (↑i 1, ↑i x) ∈ {x | x.2 / x.1 ∈ V k}
|
refine fun n _ ↦ Filter.eventually_iff_exists_mem.mpr ⟨U n, hU1 n, fun x hx ⟨f, hf⟩ ↦ ?_⟩
|
case intro.intro
X : Type u_7
Y : Type u_8
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group X
inst✝⁶ : IsTopologicalGroup X
inst✝⁵ : UniformSpace Y
inst✝⁴ : CommGroup Y
inst✝³ : UniformGroup Y
inst✝² : T0Space Y
inst✝¹ : CompactSpace Y
inst✝ : LocallyCompactSpace X
V : ℕ → Set Y
hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)
hVo : (𝓝 1).HasBasis (fun x => True) V
U0 : Set X
hU0c : IsCompact U0
hU0o : U0 ∈ 𝓝 1
U_aux : ℕ → ↑{S | S ∈ 𝓝 1} :=
fun t =>
Nat.rec ⟨U0, hU0o⟩
(fun x S =>
let h := ⋯;
⟨Classical.choose h, ⋯⟩)
t
U : ℕ → Set X := fun n => ↑(U_aux n)
hU1 : ∀ (n : ℕ), U n ∈ 𝓝 1
hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n
hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n
hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f) (U n) (V n)
n : ℕ
x✝¹ : True
x : X
hx : x ∈ U n
x✝ : ↑{f | Set.MapsTo (⇑f) (U 0) (V 0)}
f : X →* Y
hf : f ∈ {f | Set.MapsTo (⇑f) (U 0) (V 0)}
⊢ (↑⟨f, hf⟩ 1, ↑⟨f, hf⟩ x) ∈ {x | x.2 / x.1 ∈ V n}
|
3c261961af7a0b99
|
Std.Tactic.BVDecide.Normalize.BitVec.udiv_ofNat_eq_of_lt
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Normalize/BitVec.lean
|
theorem BitVec.udiv_ofNat_eq_of_lt (w : Nat) (x : BitVec w) (n : Nat) (k : Nat) (hk : 2 ^ k = n) (hlt : k < w) :
x / (BitVec.ofNat w n) = x >>> k
|
w : Nat
x : BitVec w
n k : Nat
hk : 2 ^ k = n
hlt : k < w
⊢ x / BitVec.ofNat w n = x >>> k
|
have : BitVec.ofNat w n = BitVec.twoPow w k := by simp [bv_toNat, hk]
|
w : Nat
x : BitVec w
n k : Nat
hk : 2 ^ k = n
hlt : k < w
this : BitVec.ofNat w n = BitVec.twoPow w k
⊢ x / BitVec.ofNat w n = x >>> k
|
59cd4912b9dd9868
|
MeasureTheory.OuterMeasure.top_caratheodory
|
Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean
|
theorem top_caratheodory : (⊤ : OuterMeasure α).caratheodory = ⊤ :=
top_unique fun s _ =>
(isCaratheodory_iff_le _).2 fun t =>
t.eq_empty_or_nonempty.elim (fun ht => by simp [ht]) fun ht => by
simp only [ht, top_apply, le_top]
|
α : Type u_1
s : Set α
x✝ : MeasurableSet s
t : Set α
ht : t.Nonempty
⊢ ⊤ (t ∩ s) + ⊤ (t \ s) ≤ ⊤ t
|
simp only [ht, top_apply, le_top]
|
no goals
|
b8a5c7a7a66e5617
|
Monotone.le_leftLim
|
Mathlib/Topology/Order/LeftRightLim.lean
|
theorem le_leftLim (h : x < y) : f x ≤ leftLim f y
|
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
⊢ f x ≤ leftLim f y
|
rcases eq_or_ne (𝓝[<] y) ⊥ with (h' | h')
|
case inl
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y = ⊥
⊢ f x ≤ leftLim f y
case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : 𝓝[<] y ≠ ⊥
⊢ f x ≤ leftLim f y
|
55e2cef9a9579060
|
MeasureTheory.compl_mem_measurableCylinders
|
Mathlib/MeasureTheory/Constructions/Cylinders.lean
|
theorem compl_mem_measurableCylinders (hs : s ∈ measurableCylinders α) :
sᶜ ∈ measurableCylinders α
|
case intro.intro.intro
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Finset ι
S : Set ((i : { x // x ∈ s }) → α ↑i)
hS : MeasurableSet S
⊢ (cylinder s S)ᶜ = cylinder s Sᶜ
|
rw [compl_cylinder]
|
no goals
|
e220a0270436ffee
|
TopologicalSpace.productOfMemOpens_isInducing
|
Mathlib/Topology/ContinuousMap/T0Sierpinski.lean
|
theorem productOfMemOpens_isInducing : IsInducing (productOfMemOpens X)
|
case h.e'_3
X : Type u_1
inst✝ : TopologicalSpace X
⊢ inst✝ = ⨅ i, induced (fun x => x ∈ i) inferInstance
|
apply eq_induced_by_maps_to_sierpinski
|
no goals
|
302b702956e75666
|
Irreducible.isRelPrime_iff_not_dvd
|
Mathlib/Algebra/GroupWithZero/Associated.lean
|
/-- See also `Irreducible.coprime_iff_not_dvd`. -/
lemma Irreducible.isRelPrime_iff_not_dvd [Monoid M] {p n : M} (hp : Irreducible p) :
IsRelPrime p n ↔ ¬ p ∣ n
|
M : Type u_1
inst✝ : Monoid M
p n : M
hp : Irreducible p
d : M
hdp : d ∣ p
hdn : d ∣ n
hpn : ¬IsUnit d
⊢ p ∣ n
|
suffices Associated p d from this.dvd.trans hdn
|
M : Type u_1
inst✝ : Monoid M
p n : M
hp : Irreducible p
d : M
hdp : d ∣ p
hdn : d ∣ n
hpn : ¬IsUnit d
⊢ p ~ᵤ d
|
7649a8e0b6517ae9
|
Submodule.fg_iff_compact
|
Mathlib/RingTheory/Finiteness/Basic.lean
|
theorem fg_iff_compact (s : Submodule R M) : s.FG ↔ CompleteLattice.IsCompactElement s
|
case mp.intro.h
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
sp : M → Submodule R M := fun a => span R {a}
supr_rw : ∀ (t : Finset M), ⨆ x ∈ t, sp x = ⨆ x ∈ ↑t, sp x
t : Finset M
⊢ ∀ x ∈ t, CompleteLattice.IsCompactElement (sp x)
|
exact fun n _ => singleton_span_isCompactElement n
|
no goals
|
caae8d7d512db78b
|
Antitone.tendsto_setIntegral
|
Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
|
theorem _root_.Antitone.tendsto_setIntegral (hsm : ∀ i, MeasurableSet (s i)) (h_anti : Antitone s)
(hfi : IntegrableOn f (s 0) μ) :
Tendsto (fun i => ∫ a in s i, f a ∂μ) atTop (𝓝 (∫ a in ⋂ n, s n, f a ∂μ))
|
case refine_2
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a => ‖f a‖
h_int_eq : (fun i => ∫ (a : α) in s i, f a ∂μ) = fun i => ∫ (a : α), (s i).indicator f a ∂μ
⊢ Integrable bound μ
|
rw [integrable_indicator_iff (hsm 0)]
|
case refine_2
α : Type u_1
E : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : IntegrableOn f (s 0) μ
bound : α → ℝ := (s 0).indicator fun a => ‖f a‖
h_int_eq : (fun i => ∫ (a : α) in s i, f a ∂μ) = fun i => ∫ (a : α), (s i).indicator f a ∂μ
⊢ IntegrableOn (fun a => ‖f a‖) (s 0) μ
|
28378f6f1986a1f8
|
Lean.Omega.Fin.not_lt
|
Mathlib/.lake/packages/lean4/src/lean/Init/Omega/Int.lean
|
theorem not_lt {i j : Fin n} : ¬ i < j ↔ j ≤ i
|
case mk.mk
n val✝¹ : Nat
isLt✝¹ : val✝¹ < n
val✝ : Nat
isLt✝ : val✝ < n
⊢ ¬⟨val✝¹, isLt✝¹⟩ < ⟨val✝, isLt✝⟩ ↔ ⟨val✝, isLt✝⟩ ≤ ⟨val✝¹, isLt✝¹⟩
|
exact Nat.not_lt
|
no goals
|
f7cc2ed2266fbd31
|
Std.DHashMap.Internal.Raw₀.Const.getKey?_insertMany_list_of_contains_eq_false
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean
|
theorem getKey?_insertMany_list_of_contains_eq_false [EquivBEq α] [LawfulHashable α] (h : m.1.WF)
{l : List (α × β)} {k : α}
(h' : (l.map Prod.fst).contains k = false) :
(insertMany m l).1.getKey? k = m.getKey? k
|
α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw₀ α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
l : List (α × β)
k : α
h' : (List.map Prod.fst l).contains k = false
⊢ (insertMany m l).val.getKey? k = m.getKey? k
|
simp_to_model [Const.insertMany] using List.getKey?_insertListConst_of_contains_eq_false
|
no goals
|
c071507cbbf10f14
|
UniformConvexOn.add
|
Mathlib/Analysis/Convex/Strong.lean
|
lemma UniformConvexOn.add (hf : UniformConvexOn s φ f) (hg : UniformConvexOn s ψ g) :
UniformConvexOn s (φ + ψ) (f + g)
|
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
φ ψ : ℝ → ℝ
s : Set E
f g : E → ℝ
hf : UniformConvexOn s φ f
hg : UniformConvexOn s ψ g
x : E
hx : x ∈ s
y : E
hy : y ∈ s
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (f + g) (a • x + b • y) ≤ a • (f + g) x + b • (f + g) y - a * b * (φ + ψ) ‖x - y‖
|
simpa [mul_add, add_add_add_comm, sub_add_sub_comm]
using add_le_add (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab)
|
no goals
|
bf2ce77815b18108
|
cfcₙ_tsub
|
Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
|
theorem cfcₙ_tsub {A : Type*} [TopologicalSpace A] [NonUnitalRing A] [PartialOrder A] [StarRing A]
[StarOrderedRing A] [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A]
[IsTopologicalRing A] [T2Space A] [NonUnitalContinuousFunctionalCalculus ℝ
(IsSelfAdjoint : A → Prop)] [NonnegSpectrumClass ℝ A] (f g : ℝ≥0 → ℝ≥0)
(a : A) (hfg : ∀ x ∈ σₙ ℝ≥0 a, g x ≤ f x) (ha : 0 ≤ a
|
A : Type u_1
inst✝¹¹ : TopologicalSpace A
inst✝¹⁰ : NonUnitalRing A
inst✝⁹ : PartialOrder A
inst✝⁸ : StarRing A
inst✝⁷ : StarOrderedRing A
inst✝⁶ : Module ℝ A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : SMulCommClass ℝ A A
inst✝³ : IsTopologicalRing A
inst✝² : T2Space A
inst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝ : NonnegSpectrumClass ℝ A
f g : ℝ≥0 → ℝ≥0
a : A
hfg : ∀ x ∈ σₙ ℝ≥0 a, g x ≤ f x
ha : autoParam (0 ≤ a) _auto✝
hf : autoParam (ContinuousOn f (σₙ ℝ≥0 a)) _auto✝
hf0 : autoParam (f 0 = 0) _auto✝
hg : autoParam (ContinuousOn g (σₙ ℝ≥0 a)) _auto✝
hg0 : autoParam (g 0 = 0) _auto✝
ha' : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal
this : Set.EqOn (fun x => ↑(f x.toNNReal - g x.toNNReal)) (fun x => ↑(f x.toNNReal) - ↑(g x.toNNReal)) (σₙ ℝ a)
⊢ ↑(f (Real.toNNReal 0)) = 0
|
simpa
|
no goals
|
886df411d08cec67
|
AlgebraicGeometry.Scheme.Hom.image_le_image_iff
|
Mathlib/AlgebraicGeometry/OpenImmersion.lean
|
lemma image_le_image_iff (f : X ⟶ Y) [IsOpenImmersion f] (U U' : X.Opens) :
f ''ᵁ U ≤ f ''ᵁ U' ↔ U ≤ U'
|
X Y : Scheme
f : X ⟶ Y
inst✝ : IsOpenImmersion f
U U' : X.Opens
h : f ''ᵁ U ≤ f ''ᵁ U'
⊢ f ⁻¹ᵁ f ''ᵁ U ≤ f ⁻¹ᵁ f ''ᵁ U'
|
apply preimage_le_preimage_of_le f h
|
no goals
|
de71456d8e8ba2da
|
CategoryTheory.Limits.biprod.map_eq_map'
|
Mathlib/CategoryTheory/Limits/Shapes/BinaryBiproducts.lean
|
theorem biprod.map_eq_map' {W X Y Z : C} [HasBinaryBiproduct W X] [HasBinaryBiproduct Y Z]
(f : W ⟶ Y) (g : X ⟶ Z) : biprod.map f g = biprod.map' f g
|
case h₁.h₁
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
W X Y Z : C
inst✝¹ : HasBinaryBiproduct W X
inst✝ : HasBinaryBiproduct Y Z
f : W ⟶ Y
g : X ⟶ Z
⊢ (inr ≫ map f g) ≫ snd = (inr ≫ map' f g) ≫ snd
|
simp only [mapPair_right, IsColimit.ι_map, IsLimit.map_π, biprod.inr_snd_assoc,
Category.assoc, ← BinaryBicone.toCone_π_app_right, ← BinaryBiproduct.bicone_snd, ←
BinaryBicone.toCocone_ι_app_right, ← BinaryBiproduct.bicone_inr]
|
case h₁.h₁
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
W X Y Z : C
inst✝¹ : HasBinaryBiproduct W X
inst✝ : HasBinaryBiproduct Y Z
f : W ⟶ Y
g : X ⟶ Z
⊢ (BinaryBiproduct.bicone W X).toCocone.ι.app { as := WalkingPair.right } ≫
(BinaryBiproduct.bicone W X).toCone.π.app { as := WalkingPair.right } ≫ g =
g ≫
(BinaryBiproduct.bicone Y Z).toCocone.ι.app { as := WalkingPair.right } ≫
(BinaryBiproduct.bicone Y Z).toCone.π.app { as := WalkingPair.right }
|
e830a66cbb8ace18
|
Function.Periodic.map_mod_nat
|
Mathlib/Data/Nat/Periodic.lean
|
theorem _root_.Function.Periodic.map_mod_nat {α : Type*} {f : ℕ → α} {a : ℕ} (hf : Periodic f a) :
∀ n, f (n % a) = f n := fun n => by
conv_rhs => rw [← Nat.mod_add_div n a, mul_comm, ← Nat.nsmul_eq_mul, hf.nsmul]
|
α : Type u_1
f : ℕ → α
a : ℕ
hf : Periodic f a
n : ℕ
⊢ f (n % a) = f n
|
conv_rhs => rw [← Nat.mod_add_div n a, mul_comm, ← Nat.nsmul_eq_mul, hf.nsmul]
|
no goals
|
46646ae98dab5a3f
|
CHSH_inequality_of_comm
|
Mathlib/Algebra/Star/CHSH.lean
|
theorem CHSH_inequality_of_comm [OrderedCommRing R] [StarRing R] [StarOrderedRing R] [Algebra ℝ R]
[OrderedSMul ℝ R] (A₀ A₁ B₀ B₁ : R) (T : IsCHSHTuple A₀ A₁ B₀ B₁) :
A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2
|
R : Type u
inst✝⁴ : OrderedCommRing R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : Algebra ℝ R
inst✝ : OrderedSMul ℝ R
A₀ A₁ B₀ B₁ : R
T : IsCHSHTuple A₀ A₁ B₀ B₁
P : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁
idem : P * P = 4 * P
idem' : P = (1 / 4) • (P * P)
⊢ star P = P
|
dsimp [P]
|
R : Type u
inst✝⁴ : OrderedCommRing R
inst✝³ : StarRing R
inst✝² : StarOrderedRing R
inst✝¹ : Algebra ℝ R
inst✝ : OrderedSMul ℝ R
A₀ A₁ B₀ B₁ : R
T : IsCHSHTuple A₀ A₁ B₀ B₁
P : R := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁
idem : P * P = 4 * P
idem' : P = (1 / 4) • (P * P)
⊢ star (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) = 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁
|
69323f291cd142bc
|
AlgebraicGeometry.RingedSpace.basicOpen_res_eq
|
Mathlib/Geometry/RingedSpace/Basic.lean
|
theorem basicOpen_res_eq {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) [IsIso i] (f : X.presheaf.obj U) :
@basicOpen X (unop V) (X.presheaf.map i f) = @RingedSpace.basicOpen X (unop U) f
|
case a
X : RingedSpace
U V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ
i : U ⟶ V
inst✝ : IsIso i
f : ↑(X.presheaf.obj U)
this :
X.basicOpen ((ConcreteCategory.hom (X.presheaf.map (inv i))) ((ConcreteCategory.hom (X.presheaf.map i)) f)) =
unop U ⊓ X.basicOpen ((ConcreteCategory.hom (X.presheaf.map i)) f)
⊢ X.basicOpen f ≤ X.basicOpen ((ConcreteCategory.hom (X.presheaf.map i)) f)
|
rw [← CommRingCat.comp_apply, ← X.presheaf.map_comp, IsIso.hom_inv_id, X.presheaf.map_id,
CommRingCat.id_apply] at this
|
case a
X : RingedSpace
U V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ
i : U ⟶ V
inst✝ : IsIso i
f : ↑(X.presheaf.obj U)
this : X.basicOpen f = unop U ⊓ X.basicOpen ((ConcreteCategory.hom (X.presheaf.map i)) f)
⊢ X.basicOpen f ≤ X.basicOpen ((ConcreteCategory.hom (X.presheaf.map i)) f)
|
0fa1508b89eed3cf
|
zpow_induction_right
|
Mathlib/Algebra/Group/Basic.lean
|
/-- To show a property of all powers of `g` it suffices to show it is closed under multiplication
by `g` and `g⁻¹` on the right. For subgroups generated by more than one element, see
`Subgroup.closure_induction_right`. -/
@[to_additive "To show a property of all multiples of `g` it suffices to show it is closed under
addition by `g` and `-g` on the right. For additive subgroups generated by more than one element,
see `AddSubgroup.closure_induction_right`."]
lemma zpow_induction_right {g : G} {P : G → Prop} (h_one : P (1 : G))
(h_mul : ∀ a, P a → P (a * g)) (h_inv : ∀ a, P a → P (a * g⁻¹)) (n : ℤ) : P (g ^ n)
|
case hn
G : Type u_3
inst✝ : Group G
g : G
P : G → Prop
h_one : P 1
h_mul : ∀ (a : G), P a → P (a * g)
h_inv : ∀ (a : G), P a → P (a * g⁻¹)
n : ℕ
ih : P (g ^ (-↑n))
⊢ P (g ^ (-↑n - 1))
|
rw [zpow_sub_one]
|
case hn
G : Type u_3
inst✝ : Group G
g : G
P : G → Prop
h_one : P 1
h_mul : ∀ (a : G), P a → P (a * g)
h_inv : ∀ (a : G), P a → P (a * g⁻¹)
n : ℕ
ih : P (g ^ (-↑n))
⊢ P (g ^ (-↑n) * g⁻¹)
|
ada9e9579b0aa862
|
HallMarriageTheorem.hall_hard_inductive_step_B
|
Mathlib/Combinatorics/Hall/Finite.lean
|
theorem hall_hard_inductive_step_B {n : ℕ} (hn : Fintype.card ι = n + 1)
(ht : ∀ s : Finset ι, #s ≤ #(s.biUnion t))
(ih :
∀ {ι' : Type u} [Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ s' : Finset ι', #s' ≤ #(s'.biUnion t')) →
∃ f : ι' → α, Function.Injective f ∧ ∀ x, f x ∈ t' x)
(s : Finset ι) (hs : s.Nonempty) (hns : s ≠ univ) (hus : #s = #(s.biUnion t)) :
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x
|
ι : Type u
α : Type v
inst✝¹ : DecidableEq α
t : ι → Finset α
inst✝ : Fintype ι
n : ℕ
hn : Fintype.card ι = n.succ
ht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)
ih :
∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),
Fintype.card ι' ≤ n →
(∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x
s : Finset ι
hs : s.Nonempty
hns : s ≠ univ
hus : #s = #(s.biUnion t)
this : DecidableEq ι
card_ι'_le : Fintype.card { x // x ∈ s } ≤ n
t' : { x // x ∈ s } → Finset α := fun x' => t ↑x'
f' : { x // x ∈ s } → α
hf' : Function.Injective f'
hsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x
ι'' : Set ι := (↑s)ᶜ
t'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \ s.biUnion t
card_ι''_le : Fintype.card ↑ι'' ≤ n
f'' : ↑ι'' → α
hf'' : Function.Injective f''
hsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x
f'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' ⟨x', hx'⟩ ∈ s.biUnion t
f''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : x'' ∉ s), f'' ⟨x'', hx''⟩ ∉ s.biUnion t
x x' : ι
hx' : x ∈ s
hx'' : x' ∉ s
h : f' ⟨x, hx'⟩ = f'' ⟨x', hx''⟩
⊢ f' ⟨x, hx'⟩ ∈ s.biUnion t
|
apply f'_mem_biUnion x
|
no goals
|
de2e47a85ff6c58c
|
CochainComplex.isStrictlyGE_of_ge
|
Mathlib/Algebra/Homology/Embedding/CochainComplex.lean
|
lemma isStrictlyGE_of_ge (p q : ℤ) (hpq : p ≤ q) [K.IsStrictlyGE q] :
K.IsStrictlyGE p
|
case hi
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
K : CochainComplex C ℤ
p q : ℤ
hpq : p ≤ q
inst✝ : K.IsStrictlyGE q
i : ℤ
hi : i < p
⊢ i < q
|
omega
|
no goals
|
92c2380a683aaa81
|
CategoryTheory.Limits.colimitHomIsoLimitYoneda'_hom_comp_π
|
Mathlib/CategoryTheory/Limits/IndYoneda.lean
|
@[reassoc (attr := simp)]
lemma colimitHomIsoLimitYoneda'_hom_comp_π [HasLimitsOfShape I (Type u₂)] (A : C) (i : I) :
(colimitHomIsoLimitYoneda' F A).hom ≫ limit.π (F.rightOp ⋙ yoneda.obj A) i =
(coyoneda.map (colimit.ι F ⟨i⟩).op).app A
|
C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : Iᵒᵖ ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape I (Type u₂)
A : C
i : I
⊢ (colimitHomIsoLimitYoneda' F A).hom ≫ limit.π (F.rightOp ⋙ yoneda.obj A) i =
(coyoneda.map (colimit.ι F (op i)).op).app A
|
simp only [yoneda_obj_obj, colimitHomIsoLimitYoneda', Iso.trans_hom,
Iso.app_hom, Category.assoc]
|
C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : Iᵒᵖ ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape I (Type u₂)
A : C
i : I
⊢ (coyonedaOpColimitIsoLimitCoyoneda' F).hom.app A ≫
(limitObjIsoLimitCompEvaluation (F.rightOp ⋙ coyoneda) A).hom ≫ limit.π (F.rightOp ⋙ yoneda.obj A) i =
(coyoneda.map (colimit.ι F (op i)).op).app A
|
12f261e51b49fa46
|
Turing.TM1to1.tr_supports
|
Mathlib/Computability/PostTuringMachine.lean
|
theorem tr_supports [Inhabited Λ] {S : Finset Λ} (ss : Supports M S) :
Supports (tr enc dec M) (trSupp M S) :=
⟨Finset.mem_biUnion.2 ⟨_, ss.1, Finset.mem_insert_self _ _⟩, fun q h ↦ by
suffices ∀ q, SupportsStmt S q → (∀ q' ∈ writes q, q' ∈ trSupp M S) →
SupportsStmt (trSupp M S) (trNormal dec q) ∧
∀ q' ∈ writes q, SupportsStmt (trSupp M S) (tr enc dec M q') by
rcases Finset.mem_biUnion.1 h with ⟨l, hl, h⟩
have :=
this _ (ss.2 _ hl) fun q' hq ↦ Finset.mem_biUnion.2 ⟨_, hl, Finset.mem_insert_of_mem hq⟩
rcases Finset.mem_insert.1 h with (rfl | h)
exacts [this.1, this.2 _ h]
intro q hs hw
induction q with
| move d q IH =>
unfold writes at hw ⊢
replace IH := IH hs hw; refine ⟨?_, IH.2⟩
cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]
| write f q IH =>
unfold writes at hw ⊢
simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and]
at hw ⊢
replace IH := IH hs fun q hq ↦ hw q (Or.inr hq)
refine ⟨supportsStmt_read _ fun a _ s ↦ hw _ (Or.inl ⟨_, rfl⟩), fun q' hq ↦ ?_⟩
rcases hq with (⟨a, q₂, rfl⟩ | hq)
· simp only [tr, supportsStmt_write, supportsStmt_move, IH.1]
· exact IH.2 _ hq
| load a q IH =>
unfold writes at hw ⊢
replace IH := IH hs hw
exact ⟨supportsStmt_read _ fun _ ↦ IH.1, IH.2⟩
| branch p q₁ q₂ IH₁ IH₂ =>
unfold writes at hw ⊢
simp only [Finset.mem_union] at hw ⊢
replace IH₁ := IH₁ hs.1 fun q hq ↦ hw q (Or.inl hq)
replace IH₂ := IH₂ hs.2 fun q hq ↦ hw q (Or.inr hq)
exact ⟨supportsStmt_read _ fun _ ↦ ⟨IH₁.1, IH₂.1⟩, fun q ↦ Or.rec (IH₁.2 _) (IH₂.2 _)⟩
| goto l =>
simp only [writes, Finset.not_mem_empty]; refine ⟨?_, fun _ ↦ False.elim⟩
refine supportsStmt_read _ fun a _ s ↦ ?_
exact Finset.mem_biUnion.2 ⟨_, hs _ _, Finset.mem_insert_self _ _⟩
| halt =>
simp only [writes, Finset.not_mem_empty]; refine ⟨?_, fun _ ↦ False.elim⟩
simp only [SupportsStmt, supportsStmt_move, trNormal]⟩
|
case branch
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝¹ : Fintype Γ
inst✝ : Inhabited Λ
S : Finset Λ
ss : Supports M S
q : Λ' Γ Λ σ
h : q ∈ trSupp M S
p : Γ → σ → Bool
q₁ q₂ : Stmt Γ Λ σ
IH₁ :
SupportsStmt S q₁ →
(∀ q' ∈ writes q₁, q' ∈ trSupp M S) →
SupportsStmt (trSupp M S) (trNormal dec q₁) ∧ ∀ q' ∈ writes q₁, SupportsStmt (trSupp M S) (tr enc dec M q')
IH₂ :
SupportsStmt S q₂ →
(∀ q' ∈ writes q₂, q' ∈ trSupp M S) →
SupportsStmt (trSupp M S) (trNormal dec q₂) ∧ ∀ q' ∈ writes q₂, SupportsStmt (trSupp M S) (tr enc dec M q')
hs : SupportsStmt S (Stmt.branch p q₁ q₂)
hw : ∀ (q' : Λ' Γ Λ σ), q' ∈ writes q₁ ∨ q' ∈ writes q₂ → q' ∈ trSupp M S
⊢ SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q₁ q₂)) ∧
∀ (q' : Λ' Γ Λ σ), q' ∈ writes q₁ ∨ q' ∈ writes q₂ → SupportsStmt (trSupp M S) (tr enc dec M q')
|
replace IH₁ := IH₁ hs.1 fun q hq ↦ hw q (Or.inl hq)
|
case branch
Γ : Type u_1
Λ : Type u_2
σ : Type u_3
n : ℕ
enc : Γ → List.Vector Bool n
dec : List.Vector Bool n → Γ
M : Λ → Stmt Γ Λ σ
inst✝¹ : Fintype Γ
inst✝ : Inhabited Λ
S : Finset Λ
ss : Supports M S
q : Λ' Γ Λ σ
h : q ∈ trSupp M S
p : Γ → σ → Bool
q₁ q₂ : Stmt Γ Λ σ
IH₂ :
SupportsStmt S q₂ →
(∀ q' ∈ writes q₂, q' ∈ trSupp M S) →
SupportsStmt (trSupp M S) (trNormal dec q₂) ∧ ∀ q' ∈ writes q₂, SupportsStmt (trSupp M S) (tr enc dec M q')
hs : SupportsStmt S (Stmt.branch p q₁ q₂)
hw : ∀ (q' : Λ' Γ Λ σ), q' ∈ writes q₁ ∨ q' ∈ writes q₂ → q' ∈ trSupp M S
IH₁ : SupportsStmt (trSupp M S) (trNormal dec q₁) ∧ ∀ q' ∈ writes q₁, SupportsStmt (trSupp M S) (tr enc dec M q')
⊢ SupportsStmt (trSupp M S) (trNormal dec (Stmt.branch p q₁ q₂)) ∧
∀ (q' : Λ' Γ Λ σ), q' ∈ writes q₁ ∨ q' ∈ writes q₂ → SupportsStmt (trSupp M S) (tr enc dec M q')
|
bea148d15df2302d
|
rel_sup_mul
|
Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean
|
theorem rel_sup_mul [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R (m (⨆ i, s i)) (∏' i, m (s i))) (s₁ s₂ : α) :
R (m (s₁ ⊔ s₂)) (m s₁ * m s₂)
|
case h.e'_2
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
α : Type u_3
inst✝ : CompleteLattice α
m : α → M
m0 : m ⊥ = 1
R : M → M → Prop
m_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))
s₁ s₂ : α
⊢ m s₁ * m s₂ = ∏' (b : Bool), m (bif b then s₁ else s₂)
|
rw [tprod_fintype, Fintype.prod_bool, cond, cond]
|
no goals
|
f084f100f9af625d
|
Matrix.isAddUnit_detp_smul_mul_adjp
|
Mathlib/LinearAlgebra/Matrix/SemiringInverse.lean
|
theorem isAddUnit_detp_smul_mul_adjp (hAB : A * B = 1) :
IsAddUnit (detp 1 A • (B * adjp (-1) B) + detp (-1) A • (B * adjp 1 B))
|
n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
hAB : A * B = 1
s t : ℤˣ
h : s ≠ t
i j k : n
hk : k ∈ univ
σ : Perm n
hσ : σ ∈ filter (fun x => x j = k) (ofSign t)
τ : Perm n
hτ : τ ∈ ofSign s
⊢ IsAddUnit ((∏ k : n, A k (τ k)) * (B i k * ∏ k ∈ {j}ᶜ, B k (σ k)))
|
rw [mem_filter] at hσ
|
n : Type u_1
R : Type u_3
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommSemiring R
A B : Matrix n n R
hAB : A * B = 1
s t : ℤˣ
h : s ≠ t
i j k : n
hk : k ∈ univ
σ : Perm n
hσ : σ ∈ ofSign t ∧ σ j = k
τ : Perm n
hτ : τ ∈ ofSign s
⊢ IsAddUnit ((∏ k : n, A k (τ k)) * (B i k * ∏ k ∈ {j}ᶜ, B k (σ k)))
|
cfc0571f345e1ba6
|
Subalgebra.toSubsemiring_injective
|
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
theorem toSubsemiring_injective :
Function.Injective (toSubsemiring : Subalgebra R A → Subsemiring A) := fun S T h =>
ext fun x => by rw [← mem_toSubsemiring, ← mem_toSubsemiring, h]
|
R : Type u
A : Type v
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
S T : Subalgebra R A
h : S.toSubsemiring = T.toSubsemiring
x : A
⊢ x ∈ S ↔ x ∈ T
|
rw [← mem_toSubsemiring, ← mem_toSubsemiring, h]
|
no goals
|
ac9dadd49afb1dfc
|
HasFPowerSeriesAt.isBigO_sub_partialSum_pow
|
Mathlib/Analysis/Analytic/Basic.lean
|
theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow
(hf : HasFPowerSeriesAt f p x) (n : ℕ) :
(fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
hf : HasFPowerSeriesAt f p x
n : ℕ
⊢ (fun y => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n
|
rw [← hasFPowerSeriesWithinAt_univ] at hf
|
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
x : E
hf : HasFPowerSeriesWithinAt f p univ x
n : ℕ
⊢ (fun y => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n
|
515dadc5b5c5ce3c
|
Algebra.FormallyUnramified.of_comp
|
Mathlib/RingTheory/Unramified/Basic.lean
|
theorem of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
R : Type u_1
inst✝⁹ : CommRing R
A : Type u_2
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u_3
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : (Ideal.Quotient.mkₐ A I).comp f₁ = (Ideal.Quotient.mkₐ A I).comp f₂
⊢ f₁ = f₂
|
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
|
R : Type u_1
inst✝⁹ : CommRing R
A : Type u_2
inst✝⁸ : CommRing A
inst✝⁷ : Algebra R A
B : Type u_3
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u_3
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : (Ideal.Quotient.mkₐ A I).comp f₁ = (Ideal.Quotient.mkₐ A I).comp f₂
this : Algebra R Q := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
⊢ f₁ = f₂
|
9d134edf4c888af2
|
List.dropLast_append_getLast
|
Mathlib/Data/List/Basic.lean
|
theorem dropLast_append_getLast : ∀ {l : List α} (h : l ≠ []), dropLast l ++ [getLast l h] = l
| [], h => absurd rfl h
| [_], _ => rfl
| a :: b :: l, h => by
rw [dropLast_cons₂, cons_append, getLast_cons (cons_ne_nil _ _)]
congr
exact dropLast_append_getLast (cons_ne_nil b l)
|
case e_tail
α : Type u
a b : α
l : List α
h : a :: b :: l ≠ []
⊢ (b :: l).dropLast ++ [(b :: l).getLast ⋯] = b :: l
|
exact dropLast_append_getLast (cons_ne_nil b l)
|
no goals
|
205445815c6fa96b
|
Sym2.out_fst_mem
|
Mathlib/Data/Sym/Sym2.lean
|
theorem out_fst_mem (e : Sym2 α) : e.out.1 ∈ e :=
⟨e.out.2, by rw [Sym2.mk, e.out_eq]⟩
|
α : Type u_1
e : Sym2 α
⊢ e = s((Quot.out e).1, (Quot.out e).2)
|
rw [Sym2.mk, e.out_eq]
|
no goals
|
78e1aa11a1a1940a
|
CategoryTheory.Center.whiskerRight_comm
|
Mathlib/CategoryTheory/Monoidal/Center.lean
|
theorem whiskerRight_comm {X₁ X₂ : Center C} (f : X₁ ⟶ X₂) (Y : Center C) (U : C) :
f.f ▷ Y.1 ▷ U ≫ ((tensorObj X₂ Y).2.β U).hom =
((tensorObj X₁ Y).2.β U).hom ≫ U ◁ f.f ▷ Y.1
|
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ X₂ : Center C
f : X₁ ⟶ X₂
Y : Center C
U : C
⊢ 𝟙 ((X₁.fst ⊗ Y.fst) ⊗ U) ⊗≫
X₁.fst ◁ (Y.snd.β U).hom ⊗≫ ((X₁.snd.β U).hom ≫ U ◁ f.f) ▷ Y.fst ⊗≫ 𝟙 (U ⊗ X₂.fst ⊗ Y.fst) =
((α_ X₁.fst Y.fst U).hom ≫
X₁.fst ◁ (Y.snd.β U).hom ≫ (α_ X₁.fst U Y.fst).inv ≫ (X₁.snd.β U).hom ▷ Y.fst ≫ (α_ U X₁.fst Y.fst).hom) ≫
U ◁ f.f ▷ Y.fst
|
monoidal
|
no goals
|
16840298f1f65760
|
ArithmeticFunction.moebius_ne_zero_iff_squarefree
|
Mathlib/NumberTheory/ArithmeticFunction.lean
|
theorem moebius_ne_zero_iff_squarefree {n : ℕ} : μ n ≠ 0 ↔ Squarefree n
|
case mp
n : ℕ
h : ¬Squarefree n
⊢ μ n = 0
|
simp [h]
|
no goals
|
b72155ce994a85f1
|
Complex.arg_lt_pi_div_two_iff
|
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
lemma arg_lt_pi_div_two_iff {z : ℂ} : arg z < π / 2 ↔ 0 < re z ∨ im z < 0 ∨ z = 0
|
case inr.inl
z : ℂ
hre : z.re = 0
⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0
|
have : z = 0 ↔ z.im = 0 := by simp [Complex.ext_iff, hre]
|
case inr.inl
z : ℂ
hre : z.re = 0
this : z = 0 ↔ z.im = 0
⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0
|
93df07ab0a2d15fa
|
tangentMap_prod_snd
|
Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean
|
theorem tangentMap_prod_snd {p : TangentBundle (I.prod I') (M × M')} :
tangentMap (I.prod I') I' Prod.snd p = ⟨p.proj.2, p.2.2⟩
|
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
p : TangentBundle (I.prod I') (M × M')
⊢ tangentMap (I.prod I') I' Prod.snd p = { proj := p.proj.2, snd := p.snd.2 }
|
simp [tangentMap]
|
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝¹ : TopologicalSpace M'
inst✝ : ChartedSpace H' M'
p : TangentBundle (I.prod I') (M × M')
⊢ (ContinuousLinearMap.snd 𝕜 (TangentSpace I p.proj.1) (TangentSpace I' p.proj.2)) p.snd = p.snd.2
|
f6a3eb949f78663b
|
AffineIndependent.existsUnique_dist_eq
|
Mathlib/Geometry/Euclidean/Circumcenter.lean
|
theorem _root_.AffineIndependent.existsUnique_dist_eq {ι : Type*} [hne : Nonempty ι] [Finite ι]
{p : ι → P} (ha : AffineIndependent ℝ p) :
∃! cs : Sphere P, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ (cs : Set P)
|
case h.left
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
hm :
∀ {ι : Type u_3} [hne : Nonempty ι] [inst : Finite ι] {p : ι → P},
AffineIndependent ℝ p →
∀ (val : Fintype ι),
Fintype.card ι = 0 →
∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
i : ι
hi : ∀ (y : ι), y = i
this : Unique ι
⊢ p i = p default ∧ {p default} ⊆ Metric.sphere (p i) 0
|
simp_rw [hi default, Set.singleton_subset_iff]
|
case h.left
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
hm :
∀ {ι : Type u_3} [hne : Nonempty ι] [inst : Finite ι] {p : ι → P},
AffineIndependent ℝ p →
∀ (val : Fintype ι),
Fintype.card ι = 0 →
∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius
i : ι
hi : ∀ (y : ι), y = i
this : Unique ι
⊢ True ∧ p i ∈ Metric.sphere (p i) 0
|
c2eb4c938c177320
|
isOpen_pi_iff
|
Mathlib/Topology/Constructions.lean
|
theorem isOpen_pi_iff {s : Set (∀ a, π a)} :
IsOpen s ↔
∀ f, f ∈ s → ∃ (I : Finset ι) (u : ∀ a, Set (π a)),
(∀ a, a ∈ I → IsOpen (u a) ∧ f a ∈ u a) ∧ (I : Set ι).pi u ⊆ s
|
case refine_1
ι : Type u_5
π : ι → Type u_6
T : (i : ι) → TopologicalSpace (π i)
s : Set ((a : ι) → π a)
a : (a : ι) → π a
x✝ : a ∈ s
⊢ (∃ I t, (∀ (i : ι), ∃ t_1 ⊆ t i, IsOpen t_1 ∧ a i ∈ t_1) ∧ (↑I).pi t ⊆ s) →
∃ I u, (∀ a_2 ∈ I, IsOpen (u a_2) ∧ a a_2 ∈ u a_2) ∧ (↑I).pi u ⊆ s
|
rintro ⟨I, t, ⟨h1, h2⟩⟩
|
case refine_1.intro.intro.intro
ι : Type u_5
π : ι → Type u_6
T : (i : ι) → TopologicalSpace (π i)
s : Set ((a : ι) → π a)
a : (a : ι) → π a
x✝ : a ∈ s
I : Finset ι
t : (i : ι) → Set (π i)
h1 : ∀ (i : ι), ∃ t_1 ⊆ t i, IsOpen t_1 ∧ a i ∈ t_1
h2 : (↑I).pi t ⊆ s
⊢ ∃ I u, (∀ a_1 ∈ I, IsOpen (u a_1) ∧ a a_1 ∈ u a_1) ∧ (↑I).pi u ⊆ s
|
aadda838155adaf1
|
Finset.prod_subtype_map_embedding
|
Mathlib/Algebra/BigOperators/Group/Finset/Basic.lean
|
theorem prod_subtype_map_embedding {p : α → Prop} {s : Finset { x // p x }} {f : { x // p x } → β}
{g : α → β} (h : ∀ x : { x // p x }, x ∈ s → g x = f x) :
(∏ x ∈ s.map (Function.Embedding.subtype _), g x) = ∏ x ∈ s, f x
|
α : Type u_3
β : Type u_4
inst✝ : CommMonoid β
p : α → Prop
s : Finset { x // p x }
f : { x // p x } → β
g : α → β
h : ∀ x ∈ s, g ↑x = f x
⊢ ∏ x ∈ s, g ((Embedding.subtype fun x => p x) x) = ∏ x ∈ s, f x
|
exact Finset.prod_congr rfl h
|
no goals
|
43a3f39759200391
|
Nat.multinomial_empty
|
Mathlib/Data/Nat/Choose/Multinomial.lean
|
@[simp] lemma multinomial_empty : multinomial ∅ f = 1
|
α : Type u_1
f : α → ℕ
⊢ multinomial ∅ f = 1
|
simp [multinomial]
|
no goals
|
6d7d216b864da1a5
|
ENNReal.HolderConjugate.sub_one_mul_inv
|
Mathlib/Data/ENNReal/Holder.lean
|
lemma sub_one_mul_inv (hp : p ≠ ⊤) : (p - 1) * p⁻¹ = q⁻¹
|
p q : ℝ≥0∞
inst✝ : p.HolderConjugate q
hp : p ≠ ⊤
this : p ≠ 0
⊢ p ≠ ⊤
|
aesop
|
no goals
|
66d815affc70c838
|
IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime
|
Mathlib/RingTheory/DedekindDomain/PID.lean
|
theorem IsLocalization.OverPrime.mem_normalizedFactors_of_isPrime [IsDomain S]
{P : Ideal Sₚ} (hP : IsPrime P) (hP0 : P ≠ ⊥) :
P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)
|
case intro.intro.intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)
|
have : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥ := by
rw [Submodule.ne_bot_iff] at hp0 ⊢
obtain ⟨x, x_mem, x_ne⟩ := hp0
exact
⟨algebraMap _ _ x, (IsLocalization.AtPrime.to_map_mem_maximal_iff _ _ _).mpr x_mem,
IsLocalization.to_map_ne_zero_of_mem_nonZeroDivisors _ p.primeCompl_le_nonZeroDivisors
(mem_nonZeroDivisors_of_ne_zero x_ne)⟩
|
case intro.intro.intro.intro
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDedekindDomain R
S : Type u_2
inst✝¹¹ : CommRing S
inst✝¹⁰ : Algebra R S
inst✝⁹ : NoZeroSMulDivisors R S
inst✝⁸ : Module.Finite R S
p : Ideal R
hp0 : p ≠ ⊥
inst✝⁷ : p.IsPrime
Sₚ : Type u_3
inst✝⁶ : CommRing Sₚ
inst✝⁵ : Algebra S Sₚ
inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ
inst✝³ : Algebra R Sₚ
inst✝² : IsScalarTower R S Sₚ
inst✝¹ : IsDedekindDomain Sₚ
inst✝ : IsDomain S
P : Ideal Sₚ
hP : P.IsPrime
hP0 : P ≠ ⊥
non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰
this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S
this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ
pid : IsPrincipalIdealRing (Localization.AtPrime p)
p' : Ideal (Localization.AtPrime p)
hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'
hp'0 : p' ≠ ⊥
hp'p : p'.IsPrime
this : IsLocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥
⊢ P ∈ normalizedFactors (Ideal.map (algebraMap R Sₚ) p)
|
57d1628e40ba8911
|
Set.exists_chain_of_le_chainHeight
|
Mathlib/Order/Height.lean
|
theorem exists_chain_of_le_chainHeight {n : ℕ} (hn : ↑n ≤ s.chainHeight) :
∃ l ∈ s.subchain, length l = n
|
case inl.intro.intro.intro.mk.intro
α : Type u_1
inst✝ : LT α
s : Set α
n : ℕ
hn : ↑n ≤ s.chainHeight
ha : ⨆ l, ↑(↑l).length = ⊤
l : List α
h₁ : Chain' (fun x1 x2 => x1 < x2) l
h₂ : ∀ i ∈ l, i ∈ s
h₃ : ¬(fun x => (↑x).length) ⟨l, ⋯⟩ ≤ n
⊢ ∃ l ∈ s.subchain, l.length = n
|
exact ⟨l.take n, ⟨h₁.take _, fun x h ↦ h₂ _ <| take_subset _ _ h⟩,
(l.length_take n).trans <| min_eq_left <| le_of_not_ge h₃⟩
|
no goals
|
3760f2ef9d2f3ddc
|
ENNReal.HolderTriple.inv_sub_inv_eq_inv'
|
Mathlib/Data/ENNReal/Holder.lean
|
/-- assumes `q ≠ 0` instead of `r ≠ 0`. -/
lemma inv_sub_inv_eq_inv' (hq : q ≠ 0) : r⁻¹ - q⁻¹ = p⁻¹
|
p q : ℝ≥0∞
hq : q ≠ 0
inst✝ : p.HolderTriple q 0
hp : p ≠ 0
⊢ ⊤ + ⊤ = ⊤
|
simp
|
no goals
|
41de0a2858038f73
|
MeasureTheory.mem_generateSetAlgebra_elim
|
Mathlib/MeasureTheory/SetAlgebra.lean
|
theorem mem_generateSetAlgebra_elim (s_mem : s ∈ generateSetAlgebra 𝒜) :
∃ A : Set (Set (Set α)), A.Finite ∧ (∀ a ∈ A, a.Finite) ∧
(∀ᵉ (a ∈ A) (t ∈ a), t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ s = ⋃ a ∈ A, ⋂ t ∈ a, t
|
case compl.intro.intro.intro.intro
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
⊢ ∃ A, A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ uᶜ = ⋃ a ∈ A, ⋂ t ∈ a, t
|
have := finite_coe_iff.2 A_fin
|
case compl.intro.intro.intro.intro
α : Type u_1
𝒜 : Set (Set α)
s u : Set α
hs✝ : generateSetAlgebra 𝒜 u
A : Set (Set (Set α))
A_fin : A.Finite
mem_A : ∀ a ∈ A, a.Finite
hA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜
u_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t
this : Finite ↑A
⊢ ∃ A, A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ uᶜ = ⋃ a ∈ A, ⋂ t ∈ a, t
|
06f0f1876d6d12a0
|
Std.DHashMap.Internal.Raw.Const.get_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/Raw.lean
|
theorem Const.get_eq [BEq α] [Hashable α] {m : Raw α (fun _ => β)} {a : α} {h : a ∈ m} :
Raw.Const.get m a h = Raw₀.Const.get
⟨m, by change dite .. = true at h; split at h <;> simp_all⟩ a
(by change dite .. = true at h; split at h <;> simp_all) :=
rfl
|
α : Type u
β✝ : α → Type v
γ : Type w
δ : α → Type w
β : Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw α fun x => β
a : α
h : a ∈ m
⊢ 0 < m.buckets.size
|
change dite .. = true at h
|
α : Type u
β✝ : α → Type v
γ : Type w
δ : α → Type w
β : Type v
inst✝¹ : BEq α
inst✝ : Hashable α
m : Raw α fun x => β
a : α
h : (if h : 0 < m.buckets.size then Raw₀.contains ⟨m, h⟩ a else false) = true
⊢ 0 < m.buckets.size
|
aae82082bc678c71
|
lowerSemicontinuous_iff_isClosed_preimage
|
Mathlib/Topology/Semicontinuous.lean
|
theorem lowerSemicontinuous_iff_isClosed_preimage {f : α → γ} :
LowerSemicontinuous f ↔ ∀ y, IsClosed (f ⁻¹' Iic y)
|
α : Type u_1
inst✝¹ : TopologicalSpace α
γ : Type u_3
inst✝ : LinearOrder γ
f : α → γ
⊢ (∀ (y : γ), IsOpen (f ⁻¹' Ioi y)) ↔ ∀ (y : γ), IsClosed (f ⁻¹' Iic y)
|
simp only [← isOpen_compl_iff, ← preimage_compl, compl_Iic]
|
no goals
|
9af45dca5cfccec4
|
MulAction.automorphize_smul_left
|
Mathlib/Topology/Algebra/InfiniteSum/Module.lean
|
/-- Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
`R`-scalar multiplication. -/
lemma MulAction.automorphize_smul_left [Group α] [MulAction α β] (f : β → M)
(g : Quotient (MulAction.orbitRel α β) → R) :
MulAction.automorphize ((g ∘ (@Quotient.mk' _ (_))) • f)
= g • (MulAction.automorphize f : Quotient (MulAction.orbitRel α β) → M)
|
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : Group α
inst✝ : MulAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
b : β
π : β → Quotient (orbitRel α β) := Quotient.mk (orbitRel α β)
H₁ : ∀ (a : α), π (a • b) = π b
⊢ Quotient.lift (fun b => ∑' (a : α), g (Quotient.mk' (a • b)) • f (a • b)) ⋯ (Quotient.mk'' b) =
g (Quotient.mk'' b) • Quotient.lift (fun b => ∑' (a : α), f (a • b)) ⋯ (Quotient.mk'' b)
|
change ∑' a : α, g (π (a • b)) • f (a • b) = g (π b) • ∑' a : α, f (a • b)
|
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : Group α
inst✝ : MulAction α β
f : β → M
g : Quotient (orbitRel α β) → R
x : Quotient (orbitRel α β)
b : β
π : β → Quotient (orbitRel α β) := Quotient.mk (orbitRel α β)
H₁ : ∀ (a : α), π (a • b) = π b
⊢ ∑' (a : α), g (π (a • b)) • f (a • b) = g (π b) • ∑' (a : α), f (a • b)
|
45d0295fcade3edf
|
NonUnitalRingHom.map_srange
|
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
theorem map_srange (g : S →ₙ+* T) (f : R →ₙ+* S) : map g (srange f) = srange (g.comp f)
|
R : Type u
S : Type v
T : Type w
inst✝² : NonUnitalNonAssocSemiring R
inst✝¹ : NonUnitalNonAssocSemiring S
inst✝ : NonUnitalNonAssocSemiring T
g : S →ₙ+* T
f : R →ₙ+* S
⊢ map g (srange f) = srange (g.comp f)
|
simpa only [srange_eq_map] using (⊤ : NonUnitalSubsemiring R).map_map g f
|
no goals
|
b7b8044bc367fe61
|
ContMDiffFiberwiseLinear.locality_aux₁
|
Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
|
theorem ContMDiffFiberwiseLinear.locality_aux₁
(n : WithTop ℕ∞) (e : PartialHomeomorph (B × F) (B × F))
(h : ∀ p ∈ e.source, ∃ s : Set (B × F), IsOpen s ∧ p ∈ s ∧
∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u)
(hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (φ x : F →L[𝕜] F)) u)
(h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ((φ x).symm : F →L[𝕜] F)) u),
(e.restr s).EqOnSource
(FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)) :
∃ U : Set B, e.source = U ×ˢ univ ∧ ∀ x ∈ U,
∃ (φ : B → F ≃L[𝕜] F) (u : Set B) (hu : IsOpen u) (_huU : u ⊆ U) (_hux : x ∈ u),
∃ (hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => (φ x : F →L[𝕜] F)) u)
(h2φ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ((φ x).symm : F →L[𝕜] F)) u),
(e.restr (u ×ˢ univ)).EqOnSource
(FiberwiseLinear.partialHomeomorph φ hu hφ.continuousOn h2φ.continuousOn)
|
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
heu : ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source
he : e.source = (Prod.fst '' e.source) ×ˢ univ
⊢ ∃ U,
e.source = U ×ˢ univ ∧
∀ x ∈ U,
∃ φ u,
∃ (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(φ x)) u) (h2φ :
ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(φ x).symm) u),
(e.restr (u ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)
|
refine ⟨Prod.fst '' e.source, he, ?_⟩
|
𝕜 : Type u_1
B : Type u_2
F : Type u_3
inst✝⁷ : TopologicalSpace B
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
EB : Type u_4
inst✝³ : NormedAddCommGroup EB
inst✝² : NormedSpace 𝕜 EB
HB : Type u_5
inst✝¹ : TopologicalSpace HB
inst✝ : ChartedSpace HB B
IB : ModelWithCorners 𝕜 EB HB
n : WithTop ℕ∞
e : PartialHomeomorph (B × F) (B × F)
s : ↑e.source → Set (B × F)
hs : ∀ (x : ↑e.source), IsOpen (s x)
hsp : ∀ (x : ↑e.source), ↑x ∈ s x
φ : ↑e.source → B → F ≃L[𝕜] F
u : ↑e.source → Set B
hu : ∀ (x : ↑e.source), IsOpen (u x)
hφ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1)) (u x)
h2φ : ∀ (x : ↑e.source), ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x_1 => ↑(φ x x_1).symm) (u x)
heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)
hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ
hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p
heu : ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source
he : e.source = (Prod.fst '' e.source) ×ˢ univ
⊢ ∀ x ∈ Prod.fst '' e.source,
∃ φ u,
∃ (hu : IsOpen u) (_ : u ⊆ Prod.fst '' e.source) (_ : x ∈ u) (hφ :
ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(φ x)) u) (h2φ :
ContMDiffOn IB 𝓘(𝕜, F →L[𝕜] F) n (fun x => ↑(φ x).symm) u),
(e.restr (u ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)
|
70cae96477ef715a
|
RingHom.IsStandardSmoothOfRelativeDimension.comp
|
Mathlib/RingTheory/RingHom/StandardSmooth.lean
|
lemma IsStandardSmoothOfRelativeDimension.comp {g : S →+* T} {f : R →+* S}
(hg : IsStandardSmoothOfRelativeDimension.{t', w'} n g)
(hf : IsStandardSmoothOfRelativeDimension.{t, w} m f) :
IsStandardSmoothOfRelativeDimension.{max t t', max w w'} (n + m) (g.comp f)
|
n m : ℕ
R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
T : Type u_1
inst✝ : CommRing T
g : S →+* T
f : R →+* S
hg : IsStandardSmoothOfRelativeDimension n g
hf : IsStandardSmoothOfRelativeDimension m f
algInst✝² : Algebra R S := f.toAlgebra
algInst✝¹ : Algebra S T := g.toAlgebra
algInst✝ : Algebra R T := (g.comp f).toAlgebra
scalarTowerInst✝ : IsScalarTower R S T := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R T))
algebraizeInst✝¹ : Algebra.IsStandardSmoothOfRelativeDimension n S T
algebraizeInst✝ : Algebra.IsStandardSmoothOfRelativeDimension m R S
⊢ Algebra.IsStandardSmoothOfRelativeDimension (n + m) R T
|
exact Algebra.IsStandardSmoothOfRelativeDimension.trans m n R S T
|
no goals
|
dfb7a4e9f1e77569
|
Complex.cosh_mul_I
|
Mathlib/Data/Complex/Trigonometric.lean
|
theorem cosh_mul_I : cosh (x * I) = cos x
|
x : ℂ
⊢ cosh (x * I) = cos x
|
rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, two_cos, neg_mul_eq_neg_mul]
|
no goals
|
e4ca26df349dd293
|
norm_jacobiTheta₂'_term_le
|
Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
|
/-- A uniform upper bound for `jacobiTheta₂'_term` on compact subsets. -/
lemma norm_jacobiTheta₂'_term_le {S T : ℝ} (hT : 0 < T) {z τ : ℂ}
(hz : |im z| ≤ S) (hτ : T ≤ im τ) (n : ℤ) :
‖jacobiTheta₂'_term n z τ‖ ≤ 2 * π * |n| * rexp (-π * (T * n ^ 2 - 2 * S * |n|))
|
S T : ℝ
hT : 0 < T
z τ : ℂ
hz : |z.im| ≤ S
hτ : T ≤ τ.im
n : ℤ
⊢ ‖2 * ↑π * I * ↑n‖ * ‖jacobiTheta₂_term n z τ‖ ≤ 2 * π * ↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))
|
refine mul_le_mul (le_of_eq ?_) (norm_jacobiTheta₂_term_le hT hz hτ n)
(norm_nonneg _) (by positivity)
|
S T : ℝ
hT : 0 < T
z τ : ℂ
hz : |z.im| ≤ S
hτ : T ≤ τ.im
n : ℤ
⊢ ‖2 * ↑π * I * ↑n‖ = 2 * π * ↑|n|
|
2632210bfa95d22f
|
ConvexOn.continuousOn_tfae
|
Mathlib/Analysis/Convex/Continuous.lean
|
lemma ConvexOn.continuousOn_tfae (hC : IsOpen C) (hC' : C.Nonempty) (hf : ConvexOn ℝ C f) : TFAE [
LocallyLipschitzOn C f,
ContinuousOn f C,
∃ x₀ ∈ C, ContinuousAt f x₀,
∃ x₀ ∈ C, (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f|]
|
case intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
hr : ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x₀), (fun x1 x2 => x1 ≤ x2) x r
x : E
hx : x ∈ C
this : ∀ᶠ (δ : ℝ) in 𝓝 0, (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C
δ : ℝ
hδ₀ : δ > 0
hδ₁ : δ < 1
y : E := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
hy : y ∈ C
⊢ ∀ᶠ (x : ℝ) in Filter.map f (𝓝 x), (fun x1 x2 => x1 ≤ x2) x (r ⊔ f y)
|
simp only [Filter.eventually_map, Pi.abs_apply] at hr ⊢
|
case intro.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
C : Set E
f : E → ℝ
hC : IsOpen C
hC' : C.Nonempty
hf : ConvexOn ℝ C f
tfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C
tfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, ContinuousAt f x₀
tfae_3_to_4 : (∃ x₀ ∈ C, ContinuousAt f x₀) → ∃ x₀ ∈ C, Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (𝓝 x₀) f
x₀ : E
hx₀ : x₀ ∈ C
r : ℝ
x : E
hx : x ∈ C
this : ∀ᶠ (δ : ℝ) in 𝓝 0, (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C
δ : ℝ
hδ₀ : δ > 0
hδ₁ : δ < 1
y : E := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
hy : y ∈ C
hr : ∀ᶠ (a : E) in 𝓝 x₀, f a ≤ r
⊢ ∀ᶠ (a : E) in 𝓝 x, f a ≤ r ⊔ f y
|
d7a82928a7cc2bf5
|
Polynomial.Monic.natDegree_pow
|
Mathlib/Algebra/Polynomial/Monic.lean
|
theorem natDegree_pow (hp : p.Monic) (n : ℕ) : (p ^ n).natDegree = n * p.natDegree
|
R : Type u
inst✝ : Semiring R
p : R[X]
hp : p.Monic
n : ℕ
⊢ (p ^ n).natDegree = n * p.natDegree
|
induction n with
| zero => simp
| succ n hn => rw [pow_succ, (hp.pow n).natDegree_mul hp, hn, Nat.succ_mul, add_comm]
|
no goals
|
bdded36dd8fa5c98
|
Std.Tactic.BVDecide.BVExpr.bitblast.blastConst.go_decl_eq
|
Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Const.lean
|
theorem blastConst.go_decl_eq {aig : AIG α} (i : Nat) (s : AIG.RefVec aig i) (val : BitVec w)
(hi : i ≤ w) :
∀ (curr : Nat) (h1) (h2),
(go aig val i s hi).aig.decls[curr]'h2 = aig.decls[curr]'h1
|
case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
i : Nat
s : aig.RefVec i
val : BitVec w
hi : i ≤ w
res : AIG.RefVecEntry α w
h✝ : ¬i < w
hgo : { aig := aig, vec := ⋯ ▸ s } = res
⊢ ∀ (curr : Nat) (h1 : curr < aig.decls.size) (h2 : curr < res.aig.decls.size), res.aig.decls[curr] = aig.decls[curr]
|
rw [← hgo]
|
case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
i : Nat
s : aig.RefVec i
val : BitVec w
hi : i ≤ w
res : AIG.RefVecEntry α w
h✝ : ¬i < w
hgo : { aig := aig, vec := ⋯ ▸ s } = res
⊢ ∀ (curr : Nat) (h1 : curr < aig.decls.size) (h2 : curr < { aig := aig, vec := ⋯ ▸ s }.aig.decls.size),
{ aig := aig, vec := ⋯ ▸ s }.aig.decls[curr] = aig.decls[curr]
|
a8695d661d5b3e87
|
MeasureTheory.Measure.exists_measure_inter_spanningSets_pos
|
Mathlib/MeasureTheory/Measure/Typeclasses.lean
|
theorem exists_measure_inter_spanningSets_pos [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ]
(s : Set α) : (∃ n, 0 < μ (s ∩ spanningSets μ n)) ↔ 0 < μ s
|
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (¬∃ n, 0 < μ (s ∩ spanningSets μ n)) ↔ ¬0 < μ s
|
simp only [not_exists, not_lt, nonpos_iff_eq_zero]
|
α : Type u_1
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
⊢ (∀ (x : ℕ), μ (s ∩ spanningSets μ x) = 0) ↔ μ s = 0
|
3135169d69515cc1
|
RatFunc.intDegree_add_le
|
Mathlib/FieldTheory/RatFunc/Degree.lean
|
theorem intDegree_add_le {x y : RatFunc K} (hy : y ≠ 0) (hxy : x + y ≠ 0) :
intDegree (x + y) ≤ max (intDegree x) (intDegree y)
|
case neg
K : Type u
inst✝ : Field K
x y : RatFunc K
hy : y ≠ 0
hxy : x + y ≠ 0
hx : ¬x = 0
⊢ (x + y).intDegree ≤ x.intDegree ⊔ y.intDegree
|
rw [intDegree_add hxy, ←
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hx y.denom_ne_zero,
mul_comm y.denom, ←
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hy x.denom_ne_zero,
le_max_iff, sub_le_sub_iff_right, Int.ofNat_le, sub_le_sub_iff_right, Int.ofNat_le, ←
le_max_iff, mul_comm y.num]
|
case neg
K : Type u
inst✝ : Field K
x y : RatFunc K
hy : y ≠ 0
hxy : x + y ≠ 0
hx : ¬x = 0
⊢ (x.num * y.denom + x.denom * y.num).natDegree ≤ (x.num * y.denom).natDegree ⊔ (x.denom * y.num).natDegree
|
df522905dbc696e6
|
MeasureTheory.dist_convolution_le'
|
Mathlib/Analysis/Convolution.lean
|
theorem dist_convolution_le' {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hif : Integrable f μ)
(hf : support f ⊆ ball (0 : G) R) (hmg : AEStronglyMeasurable g μ)
(hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[L, μ] g : G → F) x₀) (∫ t, L (f t) z₀ ∂μ) ≤ (‖L‖ * ∫ x, ‖f x‖ ∂μ) * ε
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : μ.IsAddLeftInvariant
inst✝ : SFinite μ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hif : Integrable f μ
hf : support f ⊆ ball 0 R
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
hfg : ConvolutionExistsAt f g x₀ L μ
⊢ ∀ (t : G), dist ((L (f t)) (g (x₀ - t))) ((L (f t)) z₀) ≤ ‖L (f t)‖ * ε
|
intro t
|
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : μ.IsAddLeftInvariant
inst✝ : SFinite μ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hif : Integrable f μ
hf : support f ⊆ ball 0 R
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
hfg : ConvolutionExistsAt f g x₀ L μ
t : G
⊢ dist ((L (f t)) (g (x₀ - t))) ((L (f t)) z₀) ≤ ‖L (f t)‖ * ε
|
fddd5e43ba4662a2
|
LightCondensed.lanPresheafExt_inv
|
Mathlib/Condensed/Discrete/Colimit.lean
|
@[simp]
lemma lanPresheafExt_inv {F G : LightProfinite.{u}ᵒᵖ ⥤ Type u} (S : LightProfinite.{u}ᵒᵖ)
(i : toLightProfinite.op ⋙ F ≅ toLightProfinite.op ⋙ G) : (lanPresheafExt i).inv.app S =
colimMap (whiskerLeft (CostructuredArrow.proj toLightProfinite.op S) i.inv)
|
case w
F G : LightProfiniteᵒᵖ ⥤ Type u
S : LightProfiniteᵒᵖ
i : toLightProfinite.op ⋙ F ≅ toLightProfinite.op ⋙ G
⊢ ∀ (j : CostructuredArrow toLightProfinite.op S),
colimit.ι (CostructuredArrow.proj toLightProfinite.op S ⋙ toLightProfinite.op ⋙ G) j ≫
colimit.desc (CostructuredArrow.proj toLightProfinite.op S ⋙ toLightProfinite.op ⋙ G)
(toLightProfinite.op.costructuredArrowMapCocone (toLightProfinite.op ⋙ G)
(toLightProfinite.op.pointwiseLeftKanExtension (toLightProfinite.op ⋙ F))
(i.inv ≫ toLightProfinite.op.pointwiseLeftKanExtensionUnit (toLightProfinite.op ⋙ F)) S) =
colimit.ι (CostructuredArrow.proj toLightProfinite.op S ⋙ toLightProfinite.op ⋙ G) j ≫
colimMap (whiskerLeft (CostructuredArrow.proj toLightProfinite.op S) i.inv)
|
aesop
|
no goals
|
29ff65135ebe0cf2
|
ModularGroup.abs_c_le_one
|
Mathlib/NumberTheory/Modular.lean
|
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |g 1 0| ≤ 1
|
g : SL(2, ℤ)
z : ℍ
hz : z ∈ 𝒟ᵒ
hg : g • z ∈ 𝒟ᵒ
c' : ℤ := ↑g 1 0
c : ℝ := ↑c'
hc : c ≠ 0
h₁ : 3 * 3 * c ^ 4 < 4 * (g • z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4
h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1
nsq : ℝ := normSq (denom (↑g) z)
⊢ 9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g • z).im ^ 2 * 16
|
linarith
|
no goals
|
63ab86d6ad3b518f
|
Set.isUnit_iff_singleton
|
Mathlib/Algebra/Group/Pointwise/Set/Basic.lean
|
theorem isUnit_iff_singleton : IsUnit s ↔ ∃ a, s = {a}
|
α : Type u_2
inst✝ : Group α
s : Set α
⊢ IsUnit s ↔ ∃ a, s = {a}
|
simp only [isUnit_iff, Group.isUnit, and_true]
|
no goals
|
3e1f5745bae36986
|
SetTheory.PGame.le_grundyValue_of_Iio_subset_moveRight
|
Mathlib/SetTheory/Game/Nim.lean
|
theorem le_grundyValue_of_Iio_subset_moveRight {G : PGame} [G.Impartial] {o : Nimber}
(h : Set.Iio o ⊆ Set.range (grundyValue ∘ G.moveRight)) : o ≤ grundyValue G
|
case intro
G : PGame
inst✝ : G.Impartial
o : Nimber
h : Set.Iio o ⊆ Set.range (grundyValue ∘ G.moveRight)
ho : G.grundyValue < o
i : G.RightMoves
hi : (grundyValue ∘ G.moveRight) i = G.grundyValue
⊢ False
|
exact grundyValue_ne_moveRight i hi
|
no goals
|
d64e2ffaaf3ff491
|
VitaliFamily.exists_measurable_supersets_limRatio
|
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
∃ a b, MeasurableSet a ∧ MeasurableSet b ∧
{x | v.limRatio ρ x < p} ⊆ a ∧ {x | (q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧ μ (a ∩ b) = 0
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
⊢ ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ (p • μ) a
|
have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) :=
tendsto_nhds_limUnder hx.1.1.1
|
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α := {x | ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : ℕ → Set α := fun n => s ∩ {x | v.limRatio ρ x < ↑p} ∩ o n
w : ℕ → Set α := fun n => s ∩ {x | ↑q < v.limRatio ρ x} ∩ o n
m n : ℕ
I : (ρ + μ) (u m) ≠ ⊤
J : (ρ + μ) (w n) ≠ ⊤
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L : Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x))
⊢ ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ (p • μ) a
|
f2acb3dc8c3a0a55
|
MeasureTheory.Measure.prod_prod
|
Mathlib/MeasureTheory/Measure/Prod.lean
|
theorem prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t
|
case a
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set α
t : Set β
ST : Set (α × β) := toMeasurable (μ.prod ν) (s ×ˢ t)
⊢ μ s * ν t ≤ (μ.prod ν) (s ×ˢ t)
|
have hSTm : MeasurableSet ST := measurableSet_toMeasurable _ _
|
case a
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝ : SFinite ν
s : Set α
t : Set β
ST : Set (α × β) := toMeasurable (μ.prod ν) (s ×ˢ t)
hSTm : MeasurableSet ST
⊢ μ s * ν t ≤ (μ.prod ν) (s ×ˢ t)
|
3a4325941a637749
|
circleIntegral.integral_eq_zero_of_hasDerivWithinAt'
|
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
theorem integral_eq_zero_of_hasDerivWithinAt' [CompleteSpace E] {f f' : ℂ → E} {c : ℂ} {R : ℝ}
(h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z) :
(∮ z in C(c, R), f' z) = 0
|
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
⊢ (∮ (z : ℂ) in C(c, R), f' z) = 0
|
by_cases hi : CircleIntegrable f' c R
|
case pos
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
hi : CircleIntegrable f' c R
⊢ (∮ (z : ℂ) in C(c, R), f' z) = 0
case neg
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
f f' : ℂ → E
c : ℂ
R : ℝ
h : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z
hi : ¬CircleIntegrable f' c R
⊢ (∮ (z : ℂ) in C(c, R), f' z) = 0
|
cf51c2c4528e6df9
|
List.get?_range
|
Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
|
theorem get?_range {m n : Nat} (h : m < n) : get? (range n) m = some m
|
m n : Nat
h : m < n
⊢ (range n).get? m = some m
|
simp [getElem?_range, h]
|
no goals
|
f507b93a830daef9
|
HurwitzZeta.isBigO_atTop_cosKernel_sub
|
Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
|
/-- The function `cosKernel a - 1` has exponential decay at `+∞`, for any `a`. -/
lemma isBigO_atTop_cosKernel_sub (a : UnitAddCircle) :
∃ p, 0 < p ∧ IsBigO atTop (cosKernel a · - 1) (fun x ↦ Real.exp (-p * x))
|
case H.intro.intro
a p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_nat 0 1 t - if 1 = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
⊢ ∀ᶠ (x : ℝ) in atTop, ‖cosKernel (↑a) x - 1‖ ≤ 2 * (HurwitzKernelBounds.F_nat 0 1 x - if 1 = 0 then 1 else 0)
|
filter_upwards [eventually_gt_atTop 0] with t ht
|
case h
a p : ℝ
hp : 0 < p
hp' : (fun t => HurwitzKernelBounds.F_nat 0 1 t - if 1 = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)
t : ℝ
ht : 0 < t
⊢ ‖cosKernel (↑a) t - 1‖ ≤ 2 * (HurwitzKernelBounds.F_nat 0 1 t - if 1 = 0 then 1 else 0)
|
58d473e972d0ef4b
|
DirectSum.linearEquivFunOnFintype_symm_single
|
Mathlib/Algebra/DirectSum/Module.lean
|
theorem linearEquivFunOnFintype_symm_single [Fintype ι] (i : ι) (m : M i) :
(linearEquivFunOnFintype R ι M).symm (Pi.single i m) = lof R ι M i m
|
R : Type u
inst✝⁴ : Semiring R
ι : Type v
M : ι → Type w
inst✝³ : (i : ι) → AddCommMonoid (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
m : M i
⊢ DFinsupp.equivFunOnFintype.symm (Pi.single i m) = (lof R ι M i) m
|
rw [DFinsupp.equivFunOnFintype_symm_single i m]
|
R : Type u
inst✝⁴ : Semiring R
ι : Type v
M : ι → Type w
inst✝³ : (i : ι) → AddCommMonoid (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
m : M i
⊢ DFinsupp.single i m = (lof R ι M i) m
|
5f436913eabe37d9
|
Subgroup.isComplement_univ_left
|
Mathlib/GroupTheory/Complement.lean
|
theorem isComplement_univ_left : IsComplement univ S ↔ ∃ g : G, S = {g}
|
case refine_3.intro
G : Type u_1
inst✝ : Group G
g : G
⊢ IsComplement univ {g}
|
exact isComplement_univ_singleton
|
no goals
|
c88f532257310c2f
|
NormedSpace.isBounded_iff_subset_smul_closedBall
|
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
theorem isBounded_iff_subset_smul_closedBall {s : Set E} :
Bornology.IsBounded s ↔ ∃ a : 𝕜, s ⊆ a • Metric.closedBall (0 : E) 1
|
case mpr
𝕜 : Type u_1
E : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
⊢ (∃ a, s ⊆ a • Metric.closedBall 0 1) → Bornology.IsVonNBounded 𝕜 s
|
rintro ⟨a, ha⟩
|
case mpr.intro
𝕜 : Type u_1
E : Type u_3
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
s : Set E
a : 𝕜
ha : s ⊆ a • Metric.closedBall 0 1
⊢ Bornology.IsVonNBounded 𝕜 s
|
e27d56235c432417
|
SlashInvariantFormClass.periodic_comp_ofComplex
|
Mathlib/NumberTheory/ModularForms/QExpansion.lean
|
theorem periodic_comp_ofComplex [SlashInvariantFormClass F Γ(n) k] :
Periodic (f ∘ ofComplex) n
|
case neg
k : ℤ
F : Type u_1
inst✝¹ : FunLike F ℍ ℂ
n : ℕ
f : F
inst✝ : SlashInvariantFormClass F Γ(n) k
w : ℂ
hw : ¬0 < w.im
this : (w + ↑n).im ≤ 0
⊢ (⇑f ∘ ↑ofComplex) (w + ↑n) = (⇑f ∘ ↑ofComplex) w
|
simp only [comp_apply, ofComplex_apply_of_im_nonpos this,
ofComplex_apply_of_im_nonpos (not_lt.mp hw)]
|
no goals
|
83c0aedb6c75ba2f
|
CategoryTheory.MorphismProperty.TransfiniteCompositionOfShape.isIso
|
Mathlib/CategoryTheory/MorphismProperty/TransfiniteComposition.lean
|
/-- A transfinite composition of isomorphisms is an isomorphism. -/
lemma isIso : IsIso f
|
C : Type u
inst✝⁴ : Category.{v, u} C
J : Type w
inst✝³ : LinearOrder J
inst✝² : SuccOrder J
inst✝¹ : OrderBot J
inst✝ : WellFoundedLT J
X Y : C
f : X ⟶ Y
h : (isomorphisms C).TransfiniteCompositionOfShape J f
⊢ IsIso (h.isoBot.inv ≫ h.incl.app ⊥)
|
infer_instance
|
no goals
|
d72a75bb37d7ce00
|
Matrix.kroneckerMap_diagonal_left
|
Mathlib/Data/Matrix/Kronecker.lean
|
theorem kroneckerMap_diagonal_left [Zero α] [Zero γ] [DecidableEq l] (f : α → β → γ)
(hf : ∀ b, f 0 b = 0) (a : l → α) (B : Matrix m n β) :
kroneckerMap f (diagonal a) B =
Matrix.reindex (Equiv.prodComm _ _) (Equiv.prodComm _ _)
(blockDiagonal fun i => B.map fun b => f (a i) b)
|
case a.mk.mk
α : Type u_2
β : Type u_4
γ : Type u_6
l : Type u_8
m : Type u_9
n : Type u_10
inst✝² : Zero α
inst✝¹ : Zero γ
inst✝ : DecidableEq l
f : α → β → γ
hf : ∀ (b : β), f 0 b = 0
a : l → α
B : Matrix m n β
i₁ : l
i₂ : m
j₁ : l
j₂ : n
⊢ kroneckerMap f (diagonal a) B (i₁, i₂) (j₁, j₂) =
(reindex (Equiv.prodComm m l) (Equiv.prodComm n l)) (blockDiagonal fun i => B.map fun b => f (a i) b) (i₁, i₂)
(j₁, j₂)
|
simp [diagonal, blockDiagonal, apply_ite f, ite_apply, hf]
|
no goals
|
17bda97819d8cef6
|
Equiv.cast_eq_iff_heq
|
Mathlib/Logic/Equiv/Defs.lean
|
theorem cast_eq_iff_heq {α β} (h : α = β) {a : α} {b : β} : Equiv.cast h a = b ↔ HEq a b
|
α : Sort u_1
a b : α
⊢ (Equiv.cast ⋯) a = b ↔ HEq a b
|
simp [coe_refl]
|
no goals
|
ebab05b59f7f50c3
|
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