Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
section
open CategoryTheory Opposite
namespace CategoryTheory.Limits
-- attribute [local tid... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 929 | 932 | theorem equalizer.isoSourceOfSelf_inv :
(equalizer.isoSourceOfSelf f).inv = equalizer.lift (𝟙 X) (by simp) := by |
ext
simp [equalizer.isoSourceOfSelf]
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 88 | 89 | theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by |
rw [← coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff]
|
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.Lattice
import Mathlib.Order.CompleteLattice
#align_import category_theory.sites.sieves from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef... | Mathlib/CategoryTheory/Sites/Sieves.lean | 246 | 254 | theorem functorPushforward_comp (R : Presieve X) :
R.functorPushforward (F ⋙ G) = (R.functorPushforward F).functorPushforward G := by |
funext x
ext f
constructor
· rintro ⟨X, f₁, g₁, h₁, rfl⟩
exact ⟨F.obj X, F.map f₁, g₁, ⟨X, f₁, 𝟙 _, h₁, by simp⟩, rfl⟩
· rintro ⟨X, f₁, g₁, ⟨X', f₂, g₂, h₁, rfl⟩, rfl⟩
exact ⟨X', f₂, g₁ ≫ G.map g₂, h₁, by simp⟩
|
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 354 | 359 | theorem ι_gluedIso_hom (i : D.J) : F.map (D.ι i) ≫ (D.gluedIso F).hom = (D.mapGlueData F).ι i := by |
haveI : HasColimit (MultispanIndex.multispan (diagram (mapGlueData D F))) := inferInstance
erw [ι_preservesColimitsIso_hom_assoc]
rw [HasColimit.isoOfNatIso_ι_hom]
erw [Category.id_comp]
rfl
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 761 | 764 | theorem Integrable.sup {β} [NormedLatticeAddCommGroup β] {f g : α → β} (hf : Integrable f μ)
(hg : Integrable g μ) : Integrable (f ⊔ g) μ := by |
rw [← memℒp_one_iff_integrable] at hf hg ⊢
exact hf.sup hg
|
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 126 | 126 | theorem mk_toNat_of_infinite [h : Infinite α] : toNat #α = 0 := by | simp
|
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 95 | 98 | theorem rel_join {r : α → β → Prop} {s t} (h : Rel (Rel r) s t) : Rel r s.join t.join := by |
induction h with
| zero => simp
| cons hab hst ih => simpa using hab.add ih
|
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 112 | 113 | theorem wellFoundedOn_image {s : Set β} : (f '' s).WellFoundedOn r ↔ s.WellFoundedOn (r on f) := by |
rw [image_eq_range]; exact wellFoundedOn_range
|
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α R : Type*}
open Filter Set Function
@[simp]
theorem Nat.comap_cast_atTop [S... | Mathlib/Order/Filter/Archimedean.lean | 212 | 224 | theorem Tendsto.atTop_mul_const' (hr : 0 < r) (hf : Tendsto f l atTop) :
Tendsto (fun x => f x * r) l atTop := by |
refine tendsto_atTop.2 fun b => ?_
obtain ⟨n : ℕ, hn : 1 ≤ n • r⟩ := Archimedean.arch 1 hr
have hn' : 1 ≤ (n : R) * r := by rwa [nsmul_eq_mul] at hn
filter_upwards [tendsto_atTop.1 hf (max b 0 * n)] with x hx
calc
b ≤ max b 0 * 1 := by
{ rw [mul_one]
exact le_max_left _ _ }
_ ≤ max b 0 * (n... |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s... | Mathlib/Data/List/InsertNth.lean | 177 | 185 | theorem get_insertNth_add_succ (l : List α) (x : α) (n k : ℕ) (hk' : n + k < l.length)
(hk : n + k + 1 < (insertNth n x l).length := (by
rwa [length_insertNth _ _ (by omega), Nat.succ_lt_succ_iff])):
(insertNth n x l).get ⟨n + k + 1, hk⟩ = get l ⟨n + k, hk'⟩ := by |
induction' l with hd tl IH generalizing n k
· simp at hk'
· cases n
· simp
· simpa [Nat.add_right_comm] using IH _ _ _
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 89 | 91 | theorem prodZeroIso_iso_inv_snd (X : C) : (prodZeroIso X).inv ≫ prod.fst = 𝟙 X := by |
dsimp [prodZeroIso, binaryFanZeroRight]
simp
|
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 39 | 44 | theorem Coprime.gcd_mul_left_cancel (m : Nat) (H : Coprime k n) : gcd (k * m) n = gcd m n :=
have H1 : Coprime (gcd (k * m) n) k := by |
rw [Coprime, Nat.gcd_assoc, H.symm.gcd_eq_one, gcd_one_right]
Nat.dvd_antisymm
(dvd_gcd (H1.dvd_of_dvd_mul_left (gcd_dvd_left _ _)) (gcd_dvd_right _ _))
(gcd_dvd_gcd_mul_left _ _ _)
|
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
noncomputable section
open scoped Topology ENNReal MeasureTheory ProbabilityTheory
op... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 78 | 82 | theorem _root_.MeasureTheory.AEStronglyMeasurable.compProd_mk_left {δ : Type*} [TopologicalSpace δ]
{f : β × γ → δ} (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
∀ᵐ x ∂κ a, AEStronglyMeasurable (fun y => f (x, y)) (η (a, x)) := by |
filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with x hx using
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩
|
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 298 | 307 | theorem pwFilter_eq_self [DecidableRel (α := α) R] {l : List α} :
pwFilter R l = l ↔ Pairwise R l := by |
refine ⟨fun e => e ▸ pairwise_pwFilter l, fun p => ?_⟩
induction l with
| nil => rfl
| cons x l IH =>
let ⟨al, p⟩ := pairwise_cons.1 p
rw [pwFilter_cons_of_pos fun b hb => ?_, IH p]
rw [IH p] at hb
exact al _ hb
|
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 108 | 110 | theorem int_valuation_zero_le (x : nonZeroDivisors R) : 0 < v.intValuationDef x := by |
rw [v.intValuationDef_if_neg (nonZeroDivisors.coe_ne_zero x)]
exact WithZero.zero_lt_coe _
|
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.zmod from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
| Mathlib/RingTheory/ZMod.lean | 25 | 29 | theorem ZMod.ker_intCastRingHom (n : ℕ) :
RingHom.ker (Int.castRingHom (ZMod n)) = Ideal.span ({(n : ℤ)} : Set ℤ) := by |
ext
rw [Ideal.mem_span_singleton, RingHom.mem_ker, Int.coe_castRingHom,
ZMod.intCast_zmod_eq_zero_iff_dvd]
|
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 523 | 525 | theorem card_dvd_card_image₂_left (hf : ∀ b ∈ t, Injective fun a => f a b)
(ht : ((fun b => s.image fun a => f a b) '' t).PairwiseDisjoint id) :
s.card ∣ (image₂ f s t).card := by | rw [← image₂_swap]; exact card_dvd_card_image₂_right hf ht
|
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import Mathlib.Analysis.Complex.Polynomial
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 91 | 96 | theorem IsNontrivial.isPrimitive {F : Type u} [Field F] {ψ : AddChar F R'} (hψ : IsNontrivial ψ) :
IsPrimitive ψ := by |
intro a ha
cases' hψ with x h
use a⁻¹ * x
rwa [mulShift_apply, mul_inv_cancel_left₀ ha]
|
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace TopologicalSpace
open ... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean | 177 | 181 | theorem tendsto_measure_thickening_of_isClosed {μ : Measure α} {s : Set α}
(hs : ∃ R > 0, μ (thickening R s) ≠ ∞) (h's : IsClosed s) :
Tendsto (fun r => μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ s)) := by |
convert tendsto_measure_thickening hs
exact h's.closure_eq.symm
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Set Fin Filter Function
open scoped NNReal Topology
section Real
variab... | Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 116 | 127 | theorem ContDiffWithinAt.exists_lipschitzOnWith {E F : Type*} [NormedAddCommGroup E]
[NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {s : Set E} {x : E}
(hf : ContDiffWithinAt ℝ 1 f s x) (hs : Convex ℝ s) :
∃ K : ℝ≥0, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by |
rcases hf 1 le_rfl with ⟨t, hst, p, hp⟩
rcases Metric.mem_nhdsWithin_iff.mp hst with ⟨ε, ε0, hε⟩
replace hp : HasFTaylorSeriesUpToOn 1 f p (Metric.ball x ε ∩ insert x s) := hp.mono hε
clear hst hε t
rw [← insert_eq_of_mem (Metric.mem_ball_self ε0), ← insert_inter_distrib] at hp
rcases hp.exists_lipschitzOn... |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.of_norm from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
open RCLike
open scoped ComplexConjugate
variable {𝕜 : Type*} [RCLike 𝕜] (E : Type*) [Normed... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 105 | 117 | theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by |
intro x y
simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
Int.cast_neg, neg_smul, neg_one_mul]
rw [neg_mul_comm]
congr 1
have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_... |
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.split_simplicial_object from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Categ... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 47 | 49 | theorem cofan_inj_πSummand_eq_id [HasZeroMorphisms C] {Δ : SimplexCategoryᵒᵖ} (A : IndexSet Δ) :
(s.cofan Δ).inj A ≫ s.πSummand A = 𝟙 _ := by |
simp [πSummand]
|
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 85 | 87 | theorem of_mul_inv_t (a : A) :
(of (a : G) : HNNExtension G A B φ) * t⁻¹ = t⁻¹ * of (φ a : G) := by |
rw [equiv_eq_conj]; simp [mul_assoc]
|
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
open Function
namespace Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v... | Mathlib/Data/Set/NAry.lean | 96 | 98 | theorem image2_swap (s : Set α) (t : Set β) : image2 f s t = image2 (fun a b => f b a) t s := by |
ext
constructor <;> rintro ⟨a, ha, b, hb, rfl⟩ <;> exact ⟨b, hb, a, ha, rfl⟩
|
import Mathlib.Order.WellFounded
import Mathlib.Tactic.Common
#align_import data.pi.lex from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
assert_not_exists Monoid
variable {ι : Type*} {β : ι → Type*} (r : ι → ι → Prop) (s : ∀ {i}, β i → β i → Prop)
namespace Pi
protected def Lex (x... | Mathlib/Order/PiLex.lean | 160 | 161 | theorem le_toLex_update_self_iff : toLex x ≤ toLex (update x i a) ↔ x i ≤ a := by |
simp_rw [le_iff_lt_or_eq, lt_toLex_update_self_iff, toLex_inj, eq_update_self_iff]
|
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 262 | 264 | theorem tendsto_normSq_cocompact_atTop : Tendsto normSq (cocompact ℂ) atTop := by |
simpa [mul_self_abs]
using tendsto_abs_cocompact_atTop.atTop_mul_atTop tendsto_abs_cocompact_atTop
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover-community/mathlib"@"b1abe23ae96fef89ad30d9f4362c307f72a55010"
open Function Set
open OrderDual (toDual ofDual)
universe u v w x
variable {α : Type u} {β : Type v}... | Mathlib/Order/Bounds/Basic.lean | 126 | 127 | theorem not_bddAbove_iff' : ¬BddAbove s ↔ ∀ x, ∃ y ∈ s, ¬y ≤ x := by |
simp [BddAbove, upperBounds, Set.Nonempty]
|
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
variable {K n : Type*} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K}
theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
∃ k,... | Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 69 | 72 | theorem det_ne_zero_of_sum_col_lt_diag (h : ∀ k, ∑ i ∈ Finset.univ.erase k, ‖A i k‖ < ‖A k k‖) :
A.det ≠ 0 := by |
rw [← Matrix.det_transpose]
exact det_ne_zero_of_sum_row_lt_diag (by simp_rw [Matrix.transpose_apply]; exact h)
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 177 | 178 | theorem lt_add_right (ha : a ≠ ∞) (hb : b ≠ 0) : a < a + b := by |
rwa [← pos_iff_ne_zero, ← ENNReal.add_lt_add_iff_left ha, add_zero] at hb
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (μ : Measure M) (ν : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 50 | 55 | theorem mconv_dirac_one [MeasurableMul₂ M]
(μ : Measure M) [SFinite μ] : μ ∗ (Measure.dirac 1) = μ := by |
unfold mconv
rw [MeasureTheory.Measure.prod_dirac, map_map]
· simp only [Function.comp_def, mul_one, map_id']
all_goals { measurability }
|
import Mathlib.Algebra.Ring.Int
import Mathlib.SetTheory.Game.PGame
import Mathlib.Tactic.Abel
#align_import set_theory.game.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
-- Porting note: many definitions here are noncomputable as the compiler does not support PGame.rec
nonco... | Mathlib/SetTheory/Game/Basic.lean | 118 | 120 | theorem not_lf : ∀ {x y : Game}, ¬x ⧏ y ↔ y ≤ x := by |
rintro ⟨x⟩ ⟨y⟩
exact PGame.not_lf
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,669 | 1,674 | theorem lsub_eq_zero_iff {ι : Type u} (f : ι → Ordinal.{max u v}) :
lsub.{_, v} f = 0 ↔ IsEmpty ι := by |
refine ⟨fun h => ⟨fun i => ?_⟩, fun h => @lsub_empty _ h _⟩
have := @lsub_pos.{_, v} _ ⟨i⟩ f
rw [h] at this
exact this.false
|
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 442 | 496 | theorem radius_rightInv_pos_of_radius_pos_aux2 {n : ℕ} (hn : 2 ≤ n + 1)
(p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) {r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
(hC : 0 ≤ C) (hp : ∀ n, ‖p n‖ ≤ C * r ^ n) :
∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ ≤
‖(i.symm : F →L[𝕜] E)‖ * a +
‖(i.sym... |
simp only [LinearIsometryEquiv.norm_map, pow_one, rightInv_coeff_one,
show Ico (1 : ℕ) 2 = {1} from Nat.Ico_succ_singleton 1,
sum_singleton, ← sum_Ico_consecutive _ one_le_two hn]
_ =
a * I +
∑ k ∈ Ico 2 (n + 1),
a ^ k *
‖(i.symm : F →L[𝕜] E).compC... |
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.absolute_value from "leanprover-community/mathlib"@"ab0a2959c83b06280ef576bc830d4aa5fe8c8e61"
open Matrix
namespace Matrix
open Equiv Finset
variable {R S : Type*} [CommRing R] [Nontr... | Mathlib/LinearAlgebra/Matrix/AbsoluteValue.lean | 52 | 61 | theorem det_sum_le {ι : Type*} (s : Finset ι) {A : ι → Matrix n n R} {abv : AbsoluteValue R S}
{x : S} (hx : ∀ k i j, abv (A k i j) ≤ x) :
abv (det (∑ k ∈ s, A k)) ≤
Nat.factorial (Fintype.card n) • (Finset.card s • x) ^ Fintype.card n :=
det_le fun i j =>
calc
abv ((∑ k ∈ s, A k) i j) = abv (... | simp only [sum_apply]
_ ≤ ∑ k ∈ s, abv (A k i j) := abv.sum_le _ _
_ ≤ ∑ _k ∈ s, x := sum_le_sum fun k _ => hx k i j
_ = s.card • x := sum_const _
|
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 143 | 146 | theorem ofPowerSeries_powerSeriesPart (x : LaurentSeries R) :
ofPowerSeries ℤ R x.powerSeriesPart = single (-x.order) 1 * x := by |
refine Eq.trans ?_ (congr rfl x.single_order_mul_powerSeriesPart)
rw [← mul_assoc, single_mul_single, neg_add_self, mul_one, ← C_apply, C_one, one_mul]
|
import Mathlib.Algebra.Order.Group.TypeTags
import Mathlib.FieldTheory.RatFunc.Degree
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.Topology.Algebra.ValuedField
#align_import number_theory.function_field from "leanprover-community/mathlib"@"70fd9563... | Mathlib/NumberTheory/FunctionField.lean | 83 | 86 | theorem algebraMap_injective [Algebra Fq[X] F] [Algebra (RatFunc Fq) F]
[IsScalarTower Fq[X] (RatFunc Fq) F] : Function.Injective (⇑(algebraMap Fq[X] F)) := by |
rw [IsScalarTower.algebraMap_eq Fq[X] (RatFunc Fq) F]
exact (algebraMap (RatFunc Fq) F).injective.comp (IsFractionRing.injective Fq[X] (RatFunc Fq))
|
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryThe... | Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean | 102 | 106 | theorem sheafificationWhiskerRightIso_hom_app :
(J.sheafificationWhiskerRightIso F).hom.app P = (J.sheafifyCompIso F P).hom := by |
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
simp only [Category.id_comp, Category.comp_id]
erw [Category.id_comp]
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Analysis.Normed.Field.UnitBall
#align_import analysis.complex.circle from "leanprover-community/mathlib"@"ad3dfaca9ea2465198bcf58aa114401c324e29d1"
noncomputable section
open Complex Metric
open ComplexC... | Mathlib/Analysis/Complex/Circle.lean | 62 | 62 | theorem mem_circle_iff_normSq {z : ℂ} : z ∈ circle ↔ normSq z = 1 := by | simp [Complex.abs]
|
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 80 | 80 | theorem expand_zero (f : R[X]) : expand R 0 f = C (eval 1 f) := by | simp [expand]
|
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.LocallyFinite
import Mathlib.Topology.ProperMap
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.compact_convergence from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
univer... | Mathlib/Topology/UniformSpace/CompactConvergence.lean | 340 | 343 | theorem tendsto_iff_tendstoUniformly :
Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a => F i a) f p := by |
rw [tendsto_iff_forall_compact_tendstoUniformlyOn, ← tendstoUniformlyOn_univ]
exact ⟨fun h => h univ isCompact_univ, fun h K _hK => h.mono (subset_univ K)⟩
|
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 161 | 170 | theorem count_injective_image' {f : β → α} (hf : Function.Injective f) {s : Set β}
(s_mble : MeasurableSet s) (fs_mble : MeasurableSet (f '' s)) : count (f '' s) = count s := by |
by_cases hs : s.Finite
· lift s to Finset β using hs
rw [← Finset.coe_image, count_apply_finset' _, count_apply_finset' s_mble,
s.card_image_of_injective hf]
simpa only [Finset.coe_image] using fs_mble
· rw [count_apply_infinite hs]
rw [← finite_image_iff hf.injOn] at hs
rw [count_apply_inf... |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.InsertNth
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import data.vector.basic from "leanprover-community/mathlib"... | Mathlib/Data/Vector/Basic.lean | 658 | 661 | theorem prod_set' [CommGroup α] (v : Vector α n) (i : Fin n) (a : α) :
(v.set i a).toList.prod = v.toList.prod * (v.get i)⁻¹ * a := by |
refine (List.prod_set' v.toList i a).trans ?_
simp [get_eq_get, mul_assoc]; rfl
|
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 251 | 258 | theorem ofNat_le_ofNat {n m : Nat} (h : n ≤ m) : ofNat n ≤ ofNat m := by |
simp only [ofNat, ne_eq, _root_.decide_not]
cases Nat.decEq n 0 with
| isTrue hn => rw [_root_.decide_eq_true hn]; exact Bool.false_le _
| isFalse hn =>
cases Nat.decEq m 0 with
| isFalse hm => rw [_root_.decide_eq_false hm]; exact Bool.le_true _
| isTrue hm => subst hm; have h := Nat.le_antisymm h... |
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.stream.init from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
set_option autoImplicit true
open Nat Function Option
namespace Stre... | Mathlib/Data/Stream/Init.lean | 76 | 76 | theorem tail_drop (n : Nat) (s : Stream' α) : tail (drop n s) = drop n (tail s) := by | simp
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import data.nat.choose.central from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
namespace Nat
def centralBinom (n : ℕ) :=
(2 * n).choose n
#alig... | Mathlib/Data/Nat/Choose/Central.lean | 57 | 60 | theorem choose_le_centralBinom (r n : ℕ) : choose (2 * n) r ≤ centralBinom n :=
calc
(2 * n).choose r ≤ (2 * n).choose (2 * n / 2) := choose_le_middle r (2 * n)
_ = (2 * n).choose n := by | rw [Nat.mul_div_cancel_left n zero_lt_two]
|
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 121 | 124 | theorem discr_of_matrix_mulVec (b : ι → B) (P : Matrix ι ι A) :
discr A (P.map (algebraMap A B) *ᵥ b) = P.det ^ 2 * discr A b := by |
rw [discr_def, traceMatrix_of_matrix_mulVec, det_mul, det_mul, det_transpose, mul_comm, ←
mul_assoc, discr_def, pow_two]
|
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.comm_ring from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/CommRing.lean | 155 | 166 | theorem eval₂Hom_X {R : Type u} (c : ℤ →+* S) (f : MvPolynomial R ℤ →+* S) (x : MvPolynomial R ℤ) :
eval₂ c (f ∘ X) x = f x := by |
apply MvPolynomial.induction_on x
(fun n => by
rw [hom_C f, eval₂_C]
exact eq_intCast c n)
(fun p q hp hq => by
rw [eval₂_add, hp, hq]
exact (f.map_add _ _).symm)
(fun p n hp => by
rw [eval₂_mul, eval₂_X, hp]
exact (f.map_mul _ _).symm)
|
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Tactic.NormNum.Basic
set_option autoImplicit true
namespace Mathlib
open Lean hiding Rat mkRat
open Meta
namespace Meta.NormNum
open Qq
theorem natPow_zero : Nat.pow a (nat_lit 0) = nat_lit 1 := rfl
theorem natPow_one : Nat.pow a (nat_lit 1) = a := Nat.pow_one _... | Mathlib/Tactic/NormNum/Pow.lean | 196 | 199 | theorem isNat_zpow_pos {α : Type*} [DivisionSemiring α] {a : α} {b : ℤ} {nb ne : ℕ}
(pb : IsNat b nb) (pe' : IsNat (a ^ nb) ne) :
IsNat (a ^ b) ne := by |
rwa [pb.out, zpow_natCast]
|
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 48 | 62 | theorem exists_hasSum_smul_of_apply_eq_zero (hs : HasSum (fun m => z ^ m • a m) s)
(ha : ∀ k < n, a k = 0) : ∃ t : E, z ^ n • t = s ∧ HasSum (fun m => z ^ m • a (m + n)) t := by |
obtain rfl | hn := n.eq_zero_or_pos
· simpa
by_cases h : z = 0
· have : s = 0 := hs.unique (by simpa [ha 0 hn, h] using hasSum_at_zero a)
exact ⟨a n, by simp [h, hn.ne', this], by simpa [h] using hasSum_at_zero fun m => a (m + n)⟩
· refine ⟨(z ^ n)⁻¹ • s, by field_simp [smul_smul], ?_⟩
have h1 : ∑ i ... |
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {α : Type*} {r : α → α → Prop} {s t : Set α}
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounde... | Mathlib/Order/Bounded.lean | 188 | 189 | theorem bounded_lt_Iic [Preorder α] [NoMaxOrder α] (a : α) : Bounded (· < ·) (Iic a) := by |
simp only [← bounded_le_iff_bounded_lt, bounded_le_Iic]
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.IntervalCases
#align_import number_the... | Mathlib/NumberTheory/LucasLehmer.lean | 433 | 444 | theorem closed_form (i : ℕ) : (s i : X q) = (ω : X q) ^ 2 ^ i + (ωb : X q) ^ 2 ^ i := by |
induction' i with i ih
· dsimp [s, ω, ωb]
ext <;> norm_num
· calc
(s (i + 1) : X q) = (s i ^ 2 - 2 : ℤ) := rfl
_ = (s i : X q) ^ 2 - 2 := by push_cast; rfl
_ = (ω ^ 2 ^ i + ωb ^ 2 ^ i) ^ 2 - 2 := by rw [ih]
_ = (ω ^ 2 ^ i) ^ 2 + (ωb ^ 2 ^ i) ^ 2 + 2 * (ωb ^ 2 ^ i * ω ^ 2 ^ i) - 2 := b... |
import Mathlib.Order.Filter.Bases
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.filter_basis from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Set TopologicalSpace Function
open Topology Filter Pointwise
universe u
class GroupFilterBasis (... | Mathlib/Topology/Algebra/FilterBasis.lean | 213 | 216 | theorem mem_nhds_one (B : GroupFilterBasis G) {U : Set G} (hU : U ∈ B) :
U ∈ @nhds G B.topology 1 := by |
rw [B.nhds_one_hasBasis.mem_iff]
exact ⟨U, hU, rfl.subset⟩
|
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 76 | 78 | theorem cobounded_eq : Bornology.cobounded ℤ = atBot ⊔ atTop := by |
simp_rw [← comap_dist_right_atTop (0 : ℤ), dist_eq', sub_zero,
← comap_abs_atTop, ← @Int.comap_cast_atTop ℝ, comap_comap]; rfl
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 162 | 165 | theorem mk_coe_def (p : K[X]) (q : K[X]⁰) :
-- Porting note: filled in `(FractionRing K[X])` that was an underscore.
RatFunc.mk p q = ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p q) := by |
simp only [mk_eq_div', ← Localization.mk_eq_mk', FractionRing.mk_eq_div]
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.StrongTopology
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Tactic.SuppressCompilation
#align_import analysis.nor... | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 147 | 151 | theorem isLeast_opNorm [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
IsLeast {c | 0 ≤ c ∧ ∀ x, ‖f x‖ ≤ c * ‖x‖} ‖f‖ := by |
refine IsClosed.isLeast_csInf ?_ bounds_nonempty bounds_bddBelow
simp only [setOf_and, setOf_forall]
refine isClosed_Ici.inter <| isClosed_iInter fun _ ↦ isClosed_le ?_ ?_ <;> continuity
|
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.monad from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace MvPolynomial
open Finsupp
variable {σ : Type*} {τ : Type*}
variable {R S... | Mathlib/Algebra/MvPolynomial/Monad.lean | 346 | 347 | theorem bind₂_monomial_one (f : R →+* MvPolynomial σ S) (d : σ →₀ ℕ) :
bind₂ f (monomial d 1) = monomial d 1 := by | rw [bind₂_monomial, f.map_one, one_mul]
|
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 266 | 269 | theorem ClassGroup.mk_mk0 [IsDedekindDomain R] (I : (Ideal R)⁰) :
ClassGroup.mk (FractionalIdeal.mk0 K I) = ClassGroup.mk0 I := by |
rw [ClassGroup.mk0, MonoidHom.comp_apply, ← ClassGroup.mk_canonicalEquiv K (FractionRing R),
FractionalIdeal.map_canonicalEquiv_mk0]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 588 | 590 | theorem map_reverseAux (f : α → β) (l₁ l₂ : List α) :
map f (reverseAux l₁ l₂) = reverseAux (map f l₁) (map f l₂) := by |
simp only [reverseAux_eq, map_append, map_reverse]
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 874 | 889 | theorem nhdset_of_mem_uniformity {d : Set (α × α)} (s : Set (α × α)) (hd : d ∈ 𝓤 α) :
∃ t : Set (α × α), IsOpen t ∧ s ⊆ t ∧
t ⊆ { p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d } := by |
let cl_d := { p : α × α | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d }
have : ∀ p ∈ s, ∃ t, t ⊆ cl_d ∧ IsOpen t ∧ p ∈ t := fun ⟨x, y⟩ hp =>
mem_nhds_iff.mp <|
show cl_d ∈ 𝓝 (x, y) by
rw [nhds_eq_uniformity_prod, mem_lift'_sets]
· exact ⟨d, hd, fun ⟨a, b⟩ ⟨ha, hb⟩ => ⟨x, y, ha, hp, h... |
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
open List hiding le_antisymm
open OrderDual
universe u v
variable {α β : Type*}
namespace Set
section LT
varia... | Mathlib/Order/Height.lean | 187 | 195 | theorem chainHeight_le_chainHeight_TFAE (s : Set α) (t : Set β) :
TFAE [s.chainHeight ≤ t.chainHeight, ∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l = length l',
∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l ≤ length l'] := by |
tfae_have 1 ↔ 3
· convert ← chainHeight_add_le_chainHeight_add s t 0 0 <;> apply add_zero
tfae_have 2 ↔ 3
· refine forall₂_congr fun l hl ↦ ?_
simp_rw [← (le_chainHeight_TFAE t l.length).out 1 2, eq_comm]
tfae_finish
|
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section FoldrIdx
-- Porting... | Mathlib/Data/List/Indexes.lean | 246 | 250 | theorem foldrIdx_eq_foldrIdxSpec (f : ℕ → α → β → β) (b as start) :
foldrIdx f b as start = foldrIdxSpec f b as start := by |
induction as generalizing start
· rfl
· simp only [foldrIdx, foldrIdxSpec_cons, *]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 245 | 250 | theorem rpow_sum_of_nonneg {ι : Type*} {a : ℝ} (ha : 0 ≤ a) {s : Finset ι} {f : ι → ℝ}
(h : ∀ x ∈ s, 0 ≤ f x) : (a ^ ∑ x ∈ s, f x) = ∏ x ∈ s, a ^ f x := by |
induction' s using Finset.cons_induction with i s hi ihs
· rw [sum_empty, Finset.prod_empty, rpow_zero]
· rw [forall_mem_cons] at h
rw [sum_cons, prod_cons, ← ihs h.2, rpow_add_of_nonneg ha h.1 (sum_nonneg h.2)]
|
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 59 | 61 | theorem intDegree_X : intDegree (X : RatFunc K) = 1 := by |
rw [intDegree, num_X, Polynomial.natDegree_X, denom_X, Polynomial.natDegree_one,
Int.ofNat_one, Int.ofNat_zero, sub_zero]
|
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology.uniform_space.abstract_completion from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
attribute [local instance] Classical.propDecidable
open F... | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 136 | 138 | theorem extend_coe [T2Space β] (hf : UniformContinuous f) (a : α) : (pkg.extend f) (ι a) = f a := by |
rw [pkg.extend_def hf]
exact pkg.denseInducing.extend_eq hf.continuous a
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 753 | 757 | theorem le_rootMultiplicity_map {p : A[X]} {f : A →+* B} (hmap : map f p ≠ 0) (a : A) :
rootMultiplicity a p ≤ rootMultiplicity (f a) (p.map f) := by |
rw [le_rootMultiplicity_iff hmap]
refine _root_.trans ?_ ((mapRingHom f).map_dvd (pow_rootMultiplicity_dvd p a))
rw [map_pow, map_sub, coe_mapRingHom, map_X, map_C]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 596 | 598 | theorem mem_connectedComponentIn {x : α} {F : Set α} (hx : x ∈ F) :
x ∈ connectedComponentIn F x := by |
simp [connectedComponentIn_eq_image hx, mem_connectedComponent, hx]
|
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.Pointwise.SMul
#align_import data.set.pointwise.support from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Pointwise
open Function Set
section Group
variable {α β γ : Type*} [Group α] [MulAction α β]
| Mathlib/Data/Set/Pointwise/Support.lean | 26 | 29 | theorem mulSupport_comp_inv_smul [One γ] (c : α) (f : β → γ) :
(mulSupport fun x ↦ f (c⁻¹ • x)) = c • mulSupport f := by |
ext x
simp only [mem_smul_set_iff_inv_smul_mem, mem_mulSupport]
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 116 | 118 | theorem card_eq_zero_iff [Finite α] : Nat.card α = 0 ↔ IsEmpty α := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.card_eq_zero_iff]
|
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Logic.Function.Conjugate
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.OrdContinuous
import Mathlib.Order.RelIso.Group
#align_import order.semiconj_Sup from "leanprover-community/mathlib"@"422e7... | Mathlib/Order/SemiconjSup.lean | 91 | 96 | theorem Semiconj.symm_adjoint [PartialOrder α] [Preorder β] {fa : α ≃o α} {fb : β ↪o β} {g : α → β}
(h : Function.Semiconj g fa fb) {g' : β → α} (hg' : IsOrderRightAdjoint g g') :
Function.Semiconj g' fb fa := by |
refine fun y => (hg' _).unique ?_
rw [← fa.surjective.image_preimage { x | g x ≤ fb y }, preimage_setOf_eq]
simp only [h.eq, fb.le_iff_le, fa.leftOrdContinuous (hg' _)]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finite.Card
#align_import group_theory.subgroup.finite from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
variable {G : Type*} [Group G]
variable {A : Type*} [AddGroup A]
n... | Mathlib/Algebra/Group/Subgroup/Finite.lean | 231 | 234 | theorem pi_mem_of_mulSingle_mem [Finite η] [DecidableEq η] {H : Subgroup (∀ i, f i)} (x : ∀ i, f i)
(h : ∀ i, Pi.mulSingle i (x i) ∈ H) : x ∈ H := by |
cases nonempty_fintype η
exact pi_mem_of_mulSingle_mem_aux Finset.univ x (by simp) fun i _ => h i
|
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 79 | 79 | theorem compl_mem_ae_iff {s : Set α} : sᶜ ∈ ae μ ↔ μ s = 0 := by | simp only [mem_ae_iff, compl_compl]
|
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_theory.minpoly.field from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
open scoped Classical
open Polynomial Set Function minpoly
namespace... | Mathlib/FieldTheory/Minpoly/Field.lean | 53 | 62 | theorem unique {p : A[X]} (pmonic : p.Monic) (hp : Polynomial.aeval x p = 0)
(pmin : ∀ q : A[X], q.Monic → Polynomial.aeval x q = 0 → degree p ≤ degree q) :
p = minpoly A x := by |
have hx : IsIntegral A x := ⟨p, pmonic, hp⟩
symm; apply eq_of_sub_eq_zero
by_contra hnz
apply degree_le_of_ne_zero A x hnz (by simp [hp]) |>.not_lt
apply degree_sub_lt _ (minpoly.ne_zero hx)
· rw [(monic hx).leadingCoeff, pmonic.leadingCoeff]
· exact le_antisymm (min A x pmonic hp) (pmin (minpoly A x) (m... |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 273 | 274 | theorem measure_diff_le_iff_le_add (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ∞) {ε : ℝ≥0∞} :
μ (t \ s) ≤ ε ↔ μ t ≤ μ s + ε := by | rw [measure_diff hst hs hs', tsub_le_iff_left]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 168 | 169 | theorem eraseLead_X_pow (n : ℕ) : eraseLead (X ^ n : R[X]) = 0 := by |
rw [X_pow_eq_monomial, eraseLead_monomial]
|
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, S... | Mathlib/Order/BooleanAlgebra.lean | 340 | 344 | theorem sup_inf_inf_sdiff : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z :=
calc
x ⊓ y ⊓ z ⊔ y \ z = x ⊓ (y ⊓ z) ⊔ y \ z := by | rw [inf_assoc]
_ = (x ⊔ y \ z) ⊓ y := by rw [sup_inf_right, sup_inf_sdiff]
_ = x ⊓ y ⊔ y \ z := by rw [inf_sup_right, inf_sdiff_left]
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 225 | 227 | theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ := by |
rw [← coe_subset, coe_Ioc, coe_Ioo]
exact Set.Ioc_subset_Ioo_right h
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 112 | 115 | theorem right_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f n x) :
IsPeriodicPt f m x := by |
rw [add_comm] at hn
exact hn.left_of_add hm
|
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 289 | 291 | theorem norm_eq_neg_one_pow (hζ : IsPrimitiveRoot ζ 2) [IsDomain L] :
norm K ζ = (-1 : K) ^ finrank K L := by |
rw [hζ.eq_neg_one_of_two_right, show -1 = algebraMap K L (-1) by simp, Algebra.norm_algebraMap]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 217 | 219 | theorem IsCoprime.of_mul_add_left_left (h : IsCoprime (y * z + x) y) : IsCoprime x y := by |
rw [add_comm] at h
exact h.of_add_mul_left_left
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 154 | 155 | theorem set_lintegral_const (s : Set α) (c : ℝ≥0∞) : ∫⁻ _ in s, c ∂μ = c * μ s := by |
rw [lintegral_const, Measure.restrict_apply_univ]
|
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section Fix
#align list.prefix_append List.prefix_append
#align list.... | Mathlib/Data/List/Infix.lean | 186 | 188 | theorem drop_sublist_drop_left (l : List α) {m n : ℕ} (h : m ≤ n) : drop n l <+ drop m l := by |
rw [← Nat.sub_add_cancel h, drop_add]
apply drop_sublist
|
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v₁ v₂ u₁ u₂
namespace CategoryTheory
open Opposite CategoryTheory Category Limits Sieve
namespace Presieve
variable {C : Type ... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 742 | 746 | theorem exists_familyOfElements :
∃ (x' : FamilyOfElements P (ofArrows X π)), ∀ (i : I), x' _ (ofArrows.mk i) = x i := by |
choose i h h' using @ofArrows_surj _ _ _ _ _ π
exact ⟨fun Y f hf ↦ P.map (eqToHom (h f hf).symm).op (x _),
fun j ↦ (hx _ j (X j) _ (𝟙 _) <| by rw [← h', id_comp]).trans <| by simp⟩
|
import Mathlib.CategoryTheory.Category.Basic
import Mathlib.CategoryTheory.Functor.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Tactic.NthRewrite
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import category_theory... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 168 | 171 | theorem lift_spec (φ : V ⥤q V') : of V ⋙q (lift φ).toPrefunctor = φ := by |
rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]
dsimp [lift]
rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec]
|
import Mathlib.Data.ZMod.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Tactic.IntervalCases
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import group_theory.specific_groups.quaternion from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915... | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | 256 | 259 | theorem orderOf_a [NeZero n] (i : ZMod (2 * n)) :
orderOf (a i) = 2 * n / Nat.gcd (2 * n) i.val := by |
conv_lhs => rw [← ZMod.natCast_zmod_val i]
rw [← a_one_pow, orderOf_pow, orderOf_a_one]
|
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_import set_theory.ordinal.basic from "leanprover-community/mathlib"@"8ea5598db6caeddde6cb734aa179cc2408dbd345"
assert_not_exists Module
assert_not_exists Field
noncomputabl... | Mathlib/SetTheory/Ordinal/Basic.lean | 137 | 139 | theorem eta (o : WellOrder) : mk o.α o.r o.wo = o := by |
cases o
rfl
|
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 129 | 138 | theorem isCodetecting_op_iff (𝒢 : Set C) : IsCodetecting 𝒢.op ↔ IsDetecting 𝒢 := by |
refine ⟨fun h𝒢 X Y f hf => ?_, fun h𝒢 X Y f hf => ?_⟩
· refine (isIso_op_iff _).1 (h𝒢 _ fun G hG h => ?_)
obtain ⟨t, ht, ht'⟩ := hf (unop G) (Set.mem_op.1 hG) h.unop
exact
⟨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))⟩
· refine (isIso_unop_iff _).1... |
import Mathlib.Data.List.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v w
open List
variable {α : Type u} {β : α → Type v}
structure AList (β : α → Type v) : Type max u v where
entries : List (Sigma β)
nodupKeys : entri... | Mathlib/Data/List/AList.lean | 360 | 362 | theorem mk_cons_eq_insert (c : Sigma β) (l : List (Sigma β)) (h : (c :: l).NodupKeys) :
(⟨c :: l, h⟩ : AList β) = insert c.1 c.2 ⟨l, nodupKeys_of_nodupKeys_cons h⟩ := by |
simpa [insert] using (kerase_of_not_mem_keys <| not_mem_keys_of_nodupKeys_cons h).symm
|
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 242 | 246 | theorem Function.Bijective.cauchySeq_comp_iff {f : ℕ → ℕ} (hf : Bijective f) (u : ℕ → α) :
CauchySeq (u ∘ f) ↔ CauchySeq u := by |
refine ⟨fun H => ?_, fun H => H.comp_injective hf.injective⟩
lift f to ℕ ≃ ℕ using hf
simpa only [(· ∘ ·), f.apply_symm_apply] using H.comp_injective f.symm.injective
|
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
open Function
universe u v w
def Computation (α : Type u) : Type u :=
{ f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = ... | Mathlib/Data/Seq/Computation.lean | 400 | 403 | theorem eq_empty_of_not_terminates {s} (H : ¬Terminates s) : s = empty α := by |
apply Subtype.eq; funext n
induction' h : s.val n with _; · rfl
refine absurd ?_ H; exact ⟨⟨_, _, h.symm⟩⟩
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 220 | 225 | theorem log_natCast_nonneg (n : ℕ) : 0 ≤ log n := by |
if hn : n = 0 then
simp [hn]
else
have : (1 : ℝ) ≤ n := mod_cast Nat.one_le_of_lt <| Nat.pos_of_ne_zero hn
exact log_nonneg this
|
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-community/mathlib"@"c3216069e5f9369e6be586ccbfcde2592b3cec92"
universe u v w
namespace Module
namespace End
open Polynomial FiniteDimensional
open scoped Poly... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 65 | 69 | theorem isRoot_of_hasEigenvalue {f : End K V} {μ : K} (h : f.HasEigenvalue μ) :
(minpoly K f).IsRoot μ := by |
rcases (Submodule.ne_bot_iff _).1 h with ⟨w, ⟨H, ne0⟩⟩
refine Or.resolve_right (smul_eq_zero.1 ?_) ne0
simp [← aeval_apply_of_hasEigenvector ⟨H, ne0⟩, minpoly.aeval K f]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 281 | 285 | theorem DifferentiableWithinAt.dist (hf : DifferentiableWithinAt ℝ f s x)
(hg : DifferentiableWithinAt ℝ g s x) (hne : f x ≠ g x) :
DifferentiableWithinAt ℝ (fun y => dist (f y) (g y)) s x := by |
simp only [dist_eq_norm]
exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne)
|
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 436 | 441 | theorem whiskerLeft_rightUnitor (f : a ⟶ b) (g : b ⟶ c) :
f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom := by |
rw [← whiskerRight_iff, comp_whiskerRight, ← cancel_epi (α_ _ _ _).inv, ←
cancel_epi (f ◁ (α_ _ _ _).inv), pentagon_inv_assoc, triangle_assoc_comp_right, ←
associator_inv_naturality_middle, ← whiskerLeft_comp_assoc, triangle_assoc_comp_right,
associator_inv_naturality_right]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | Mathlib/Data/Nat/Choose/Multinomial.lean | 88 | 92 | theorem multinomial_congr {f g : α → ℕ} (h : ∀ a ∈ s, f a = g a) :
multinomial s f = multinomial s g := by |
simp only [multinomial]; congr 1
· rw [Finset.sum_congr rfl h]
· exact Finset.prod_congr rfl fun a ha => by rw [h a ha]
|
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#align_import algebraic_topology.dold_kan.p_infty from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Category... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 160 | 162 | theorem PInfty_add_QInfty : (PInfty : K[X] ⟶ _) + QInfty = 𝟙 _ := by |
dsimp only [QInfty]
simp only [add_sub_cancel]
|
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 214 | 214 | theorem isLeft_right (h : LiftRel r s (inl a) y) : y.isLeft := by | cases h; rfl
|
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 625 | 626 | theorem isLindelof_iff_isLindelof_univ : IsLindelof s ↔ IsLindelof (univ : Set s) := by |
rw [Subtype.isLindelof_iff, image_univ, Subtype.range_coe]
|
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 46 | 46 | theorem mem_map {f : α → β} {y : β} {o : Option α} : y ∈ o.map f ↔ ∃ x ∈ o, f x = y := by | simp
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.Perm
#align_import data.list.prime from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
open List
section CommMonoidWithZero
variable {M : Type*} [CommMonoidWithZero M]
| Mathlib/Data/List/Prime.lean | 27 | 38 | theorem Prime.dvd_prod_iff {p : M} {L : List M} (pp : Prime p) : p ∣ L.prod ↔ ∃ a ∈ L, p ∣ a := by |
constructor
· intro h
induction' L with L_hd L_tl L_ih
· rw [prod_nil] at h
exact absurd h pp.not_dvd_one
· rw [prod_cons] at h
cases' pp.dvd_or_dvd h with hd hd
· exact ⟨L_hd, mem_cons_self L_hd L_tl, hd⟩
· obtain ⟨x, hx1, hx2⟩ := L_ih hd
exact ⟨x, mem_cons_of_mem L_hd ... |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 295 | 305 | theorem smul_map_volume_mul_left {a : ℝ} (h : a ≠ 0) :
ENNReal.ofReal |a| • Measure.map (a * ·) volume = volume := by |
refine (Real.measure_ext_Ioo_rat fun p q => ?_).symm
cases' lt_or_gt_of_ne h with h h
· simp only [Real.volume_Ioo, Measure.smul_apply, ← ENNReal.ofReal_mul (le_of_lt <| neg_pos.2 h),
Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, neg_sub_neg, neg_mul,
preimage_const_mul_Ioo_of_neg _ _... |
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