Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
namespace Iso
variable {C : Type u} [Category.{v} C]
def homCongr {X Y Xβ... | Mathlib/CategoryTheory/Conj.lean | 50 | 52 | theorem homCongr_apply {X Y Xβ Yβ : C} (Ξ± : X β
Xβ) (Ξ² : Y β
Yβ) (f : X βΆ Y) :
Ξ±.homCongr Ξ² f = Ξ±.inv β« f β« Ξ².hom := by |
rfl
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import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part β
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 175 | 175 | theorem add_top (x : PartENat) : x + β€ = β€ := by | rw [add_comm, top_add]
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import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ΞΉ : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 84 | 87 | theorem natDegree_add_le_iff_right {n : β} (p q : R[X]) (pn : p.natDegree β€ n) :
(p + q).natDegree β€ n β q.natDegree β€ n := by |
rw [add_comm]
exact natDegree_add_le_iff_left _ _ pn
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import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 46 | 47 | theorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0 := by |
rwa [zero_opow', Ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero]
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import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : β)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 142 | 143 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [β card_Ioc, Fintype.card_ofFinset]
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import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 64 | 66 | theorem lift_lt_continuum {c : Cardinal.{u}} : lift.{v} c < π β c < π := by |
-- Porting note: added explicit universes
rw [β lift_continuum.{u,v}, lift_lt]
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import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S β]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 61 | 63 | theorem smeval_monomial (n : β) :
(monomial n r).smeval x = r β’ x ^ n := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
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import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
#align_import category_theory.bicategory.strict from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
open Bicategory
universe w v u
variable (B : Type u) [Bicategory.{w, v} B]... | Mathlib/CategoryTheory/Bicategory/Strict.lean | 78 | 81 | theorem whiskerLeft_eqToHom {a b c : B} (f : a βΆ b) {g h : b βΆ c} (Ξ· : g = h) :
f β eqToHom Ξ· = eqToHom (congr_argβ (Β· β« Β·) rfl Ξ·) := by |
cases Ξ·
simp only [whiskerLeft_id, eqToHom_refl]
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import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N Ξ± E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace Ξ±] [NormedAddCommGroup E] {ΞΌ : Me... | Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 78 | 79 | theorem smul_Lp_neg (c : Mα΅α΅α΅) (f : Lp E p ΞΌ) : c β’ (-f) = -(c β’ f) := by |
rcases f with β¨β¨_β©, _β©; rfl
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import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V Γ V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 33 | 34 | theorem Dart.ext_iff (dβ dβ : G.Dart) : dβ = dβ β dβ.toProd = dβ.toProd := by |
cases dβ; cases dβ; simp
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import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 106 | 106 | theorem content_one : content (1 : R[X]) = 1 := by | rw [β C_1, content_C, normalize_one]
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import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.Tactic.FinCases
namespace PMF
open ENNReal
noncomputable
def binomial (p : ββ₯0β) (h : p β€ 1) (n : β) : PMF (Fin (n + 1)) :=
.ofFintype (fun i => p^(i : β) * (1-p)^((Fin.last n - i) : β) * (n.choose i : β)) (by
convert (add_pow ... | Mathlib/Probability/ProbabilityMassFunction/Binomial.lean | 45 | 47 | theorem binomial_apply_last (p : ββ₯0β) (h : p β€ 1) (n : β) :
binomial p h n (.last n) = p^n := by |
simp [binomial_apply]
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import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 79 | 81 | theorem edist_le_of_fixedPoint (hf : ContractingWith K f) {x y} (h : edist x y β β)
(hy : IsFixedPt f y) : edist x y β€ edist x (f x) / (1 - K) := by |
simpa only [hy.eq, edist_self, add_zero] using hf.edist_inequality h
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import Mathlib.Topology.Category.LightProfinite.Basic
import Mathlib.Topology.Category.Profinite.Limits
namespace LightProfinite
universe u w
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
section Pullbacks
variable {X Y B : LightProfinite.{u}} (f : X βΆ B) (g ... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 128 | 131 | theorem pullback_snd_eq :
LightProfinite.pullback.snd f g = (pullbackIsoPullback f g).hom β« Limits.pullback.snd := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_Ο]
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import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 127 | 129 | theorem fourier_coe_apply' {n : β€} {x : β} :
toCircle (n β’ (x : AddCircle T) :) = Complex.exp (2 * Ο * Complex.I * n * x / T) := by |
rw [β fourier_apply]; exact fourier_coe_apply
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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 | 69 | 73 | theorem card_roots (hp0 : p β 0) : (Multiset.card (roots p) : WithBot β) β€ degree p := by |
classical
unfold roots
rw [dif_neg hp0]
exact (Classical.choose_spec (exists_multiset_roots hp0)).1
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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 {Ξ± Ξ² : Type*}
namespace Set
section LT
varia... | Mathlib/Order/Height.lean | 77 | 77 | theorem singleton_mem_subchain_iff : [a] β s.subchain β a β s := by | simp [cons_mem_subchain_iff]
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import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 99 | 107 | theorem opNorm_zero_iff [RingHomIsometric Οββ] : βfβ = 0 β f = 0 :=
Iff.intro
(fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1
(calc
_ β€ βfβ * βxβ := le_opNorm _ _
_ = _ := by | rw [hn, zero_mul]))
(by
rintro rfl
exact opNorm_zero)
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import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {Ξ± Ξ² : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist Ξ±] (s : Set Ξ±) : ββ₯0β :=
β¨
(x... | Mathlib/Topology/MetricSpace/Infsep.lean | 84 | 86 | theorem nontrivial_of_einfsep_lt_top (hs : s.einfsep < β) : s.Nontrivial := by |
rcases einfsep_lt_top.1 hs with β¨_, hx, _, hy, hxy, _β©
exact β¨_, hx, _, hy, hxyβ©
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import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 118 | 120 | theorem HasStrictDerivAt.scomp (hg : HasStrictDerivAt gβ gβ' (h x)) (hh : HasStrictDerivAt h h' x) :
HasStrictDerivAt (gβ β h) (h' β’ gβ') x := by |
simpa using ((hg.restrictScalars π).comp x hh).hasStrictDerivAt
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import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 171 | 172 | theorem weightedSMul_apply {m : MeasurableSpace Ξ±} (ΞΌ : Measure Ξ±) (s : Set Ξ±) (x : F) :
weightedSMul ΞΌ s x = (ΞΌ s).toReal β’ x := by | simp [weightedSMul]
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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 | 46 | 48 | theorem eraseLead_coeff (i : β) :
f.eraseLead.coeff i = if i = f.natDegree then 0 else f.coeff i := by |
simp only [eraseLead, coeff_erase]
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import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V Vβ : Type v} {V' V'β : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 46 | 47 | theorem rank_zero [Nontrivial K] : rank (0 : V ββ[K] V') = 0 := by |
rw [rank, LinearMap.range_zero, rank_bot]
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import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.AdjoinRoot
#align_import ring_theory.adjoin.field from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
noncomputable section
open Polynomial
variable {R K L M : Type*} [CommRing R]... | Mathlib/RingTheory/Adjoin/Field.lean | 106 | 110 | theorem IsIntegral.minpoly_splits_tower_top [Algebra K L] [IsScalarTower R K L]
(h : Splits (algebraMap R L) (minpoly R x)) :
Splits (algebraMap K L) (minpoly K x) := by |
rw [IsScalarTower.algebraMap_eq R K L] at h
exact int.minpoly_splits_tower_top' h
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import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Order.Monoid.Defs
#align_import algebra.order.monoid.canonical.defs from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
universe u
variable {Ξ± : Type u}
class ExistsMulOf... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 56 | 58 | theorem exists_one_lt_mul_of_lt' (h : a < b) : β c, 1 < c β§ a * c = b := by |
obtain β¨c, rflβ© := exists_mul_of_le h.le
exact β¨c, one_lt_of_lt_mul_right h, rflβ©
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import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 143 | 143 | theorem succ_eval : succ.eval = fun v => pure [v.headI.succ] := by | simp [eval]
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import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Bornology.Absorbs
#align_import analysis.locally_convex.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set
open Pointwise Topology
... | Mathlib/Analysis/LocallyConvex/Basic.lean | 81 | 82 | theorem balanced_iff_closedBall_smul : Balanced π s β Metric.closedBall (0 : π) 1 β’ s β s := by |
simp [balanced_iff_smul_mem, smul_subset_iff]
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import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 71 | 77 | theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X βΆ Y) [ReflectsColimit (span f f) F]
[Epi (F.map f)] : Epi f := by |
have := PushoutCocone.isColimitMkIdId (F.map f)
simp_rw [β F.map_id] at this
apply
PushoutCocone.epi_of_isColimitMkIdId _
(isColimitOfIsColimitPushoutCoconeMap F _ this)
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import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 71 | 75 | theorem add [Add Ξ²] [ContinuousAdd Ξ²] (hf : AEStronglyMeasurable' m f ΞΌ)
(hg : AEStronglyMeasurable' m g ΞΌ) : AEStronglyMeasurable' m (f + g) ΞΌ := by |
rcases hf with β¨f', h_f'_meas, hff'β©
rcases hg with β¨g', h_g'_meas, hgg'β©
exact β¨f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'β©
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import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.CategoryTheory.DifferentialObject
#align_import algebra.homology.differential_object from "leanprover-community/mathlib"@"b535c2d5d996acd9b0554b76395d9c920e186f4f"
open CategoryTheory CategoryTheory.Limits
open scoped Classical
noncomputable secti... | Mathlib/Algebra/Homology/DifferentialObject.lean | 78 | 79 | theorem d_eqToHom (X : HomologicalComplex V (ComplexShape.up' b)) {x y z : Ξ²} (h : y = z) :
X.d x y β« eqToHom (congr_arg X.X h) = X.d x z := by | cases h; simp
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import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 64 | 65 | theorem cantorFunctionAux_true (h : f n = true) : cantorFunctionAux c f n = c ^ n := by |
simp [cantorFunctionAux, h]
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import Mathlib.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
#align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
open Finset
variable {Ξ± : Type*}
namespace Set
section SemilatticeInf
variable [SemilatticeInf Ξ±] [OrderBot ... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 61 | 61 | theorem intersecting_singleton : ({a} : Set Ξ±).Intersecting β a β β₯ := by | simp [Intersecting]
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import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {Ξ± Ξ² A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 93 | 95 | theorem mulSupport_update_one [DecidableEq Ξ±] (f : Ξ± β M) (x : Ξ±) :
mulSupport (update f x 1) = mulSupport f \ {x} := by |
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
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import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 210 | 210 | theorem add_zero (n : Num) : n + 0 = n := by | cases n <;> rfl
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import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 36 | 37 | theorem le_of_le_of_eq {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a β€ 0) (hb : b = 0) : a + b β€ 0 := by |
simp [*]
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import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 105 | 106 | theorem circleMap_sub_center (c : β) (R : β) (ΞΈ : β) : circleMap c R ΞΈ - c = circleMap 0 R ΞΈ := by |
simp [circleMap]
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import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 262 | 262 | theorem smooth_one [One M'] : Smooth I I' (1 : M β M') := by | simp only [Pi.one_def, smooth_const]
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import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 341 | 344 | theorem HasDerivWithinAt.finset_prod (hf : β i β u, HasDerivWithinAt (f i) (f' i) s x) :
HasDerivWithinAt (β i β u, f i Β·) (β i β u, (β j β u.erase i, f j x) β’ f' i) s x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasFDerivWithinAt.finset_prod (fun i hi β¦ (hf i hi).hasFDerivWithinAt)).hasDerivWithinAt
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import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w wβ
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 44 | 45 | theorem lieCharacter_apply_lie (Ο : LieCharacter R L) (x y : L) : Ο β
x, yβ = 0 := by |
rw [LieHom.map_lie, LieRing.of_associative_ring_bracket, mul_comm, sub_self]
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import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Opposites
import Mathlib.Data.Rel
#align_import category_theory.category.Rel from "leanprover-community/mathlib"@"afad8e438d03f9d89da2914aa06cb4964ba87a18"
namespace Cate... | Mathlib/CategoryTheory/Category/RelCat.lean | 65 | 66 | theorem rel_comp_applyβ {X Y Z : RelCat} (f : X βΆ Y) (g : Y βΆ Z) (x : X) (z : Z) :
(f β« g) x z β β y, f x y β§ g y z := by | rfl
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import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.function.ae_measurable_sequence from "leanprover-community/mathlib"@"d003c55042c3cd08aefd1ae9a42ef89441cdaaf3"
open MeasureTheory
open scoped Classical
variable {ΞΉ : Sort*} {Ξ± Ξ² Ξ³... | Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 59 | 61 | theorem aeSeq_eq_mk_of_mem_aeSeqSet (hf : β i, AEMeasurable (f i) ΞΌ) {x : Ξ±}
(hx : x β aeSeqSet hf p) (i : ΞΉ) : aeSeq hf p i x = (hf i).mk (f i) x := by |
simp only [aeSeq, hx, if_true]
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import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 128 | 130 | theorem HasDerivWithinAt.complexToReal_fderiv {f : β β β} {s : Set β} {f' x : β}
(h : HasDerivWithinAt f f' s x) : HasFDerivWithinAt f (f' β’ (1 : β βL[β] β)) s x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasFDerivWithinAt.restrictScalars β
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import Mathlib.Data.Set.Finite
#align_import data.finset.preimage from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
assert_not_exists Finset.sum
open Set Function
universe u v w x
variable {Ξ± : Type u} {Ξ² : Type v} {ΞΉ : Sort w} {Ξ³ : Type x}
namespace Finset
section Preimage
nonc... | Mathlib/Data/Finset/Preimage.lean | 92 | 94 | theorem map_subset_iff_subset_preimage {f : Ξ± βͺ Ξ²} {s : Finset Ξ±} {t : Finset Ξ²} :
s.map f β t β s β t.preimage f f.injective.injOn := by |
classical rw [map_eq_image, image_subset_iff_subset_preimage]
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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 | 113 | 114 | theorem volume_closedBall (a r : β) : volume (Metric.closedBall a r) = ofReal (2 * r) := by |
rw [closedBall_eq_Icc, volume_Icc, β sub_add, add_sub_cancel_left, two_mul]
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import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 83 | 87 | theorem ascPochhammer_map (f : S β+* T) (n : β) :
(ascPochhammer S n).map f = ascPochhammer T n := by |
induction' n with n ih
Β· simp
Β· simp [ih, ascPochhammer_succ_left, map_comp]
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import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ΞΉ Ξ± Ξ² : Type*} {Ο : ΞΉ β Type*}
def symmDiff [Sup Ξ±] [SDiff Ξ±] (a b : Ξ±) : Ξ± :=
a \ b β b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 137 | 138 | theorem symmDiff_of_le {a b : Ξ±} (h : a β€ b) : a β b = b \ a := by |
rw [symmDiff, sdiff_eq_bot_iff.2 h, bot_sup_eq]
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import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 58 | 58 | theorem stepSet_empty (a : Ξ±) : M.stepSet β
a = β
:= by | simp [stepSet]
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import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 276 | 280 | theorem descPochhammer_map (f : R β+* T) (n : β) :
(descPochhammer R n).map f = descPochhammer T n := by |
induction' n with n ih
Β· simp
Β· simp [ih, descPochhammer_succ_left, map_comp]
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import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 66 | 67 | theorem interedges_empty_left (t : Finset Ξ²) : interedges r β
t = β
:= by |
rw [interedges, Finset.empty_product, filter_empty]
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import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {π V W Q P : Type*}
section AddTorsor
variable (π) [Ring π] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 116 | 116 | theorem intrinsicFrontier_empty : intrinsicFrontier π (β
: Set P) = β
:= by | simp [intrinsicFrontier]
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import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ β΅β
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 58 | 60 | theorem continuum_lt_lift {c : Cardinal.{u}} : π < lift.{v} c β π < c := by |
-- Porting note: added explicit universes
rw [β lift_continuum.{u,v}, lift_lt]
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import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 90 | 92 | theorem baseChange_zero : baseChange A (0 : M ββ[R] N) = 0 := by |
ext
simp [baseChange_eq_ltensor]
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import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 53 | 55 | theorem convexOn_dist (z : E) (hs : Convex β s) : ConvexOn β s fun z' => dist z' z := by |
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id β E - AffineMap.const β E z)
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import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 304 | 306 | theorem coe_support_eq_set_support (f : Perm Ξ±) : (f.support : Set Ξ±) = { x | f x β x } := by |
ext
simp
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import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 74 | 77 | theorem Ico_mul_Icc_subset' (a b c d : Ξ±) : Ico a b * Icc c d β Ico (a * c) (b * d) := by |
haveI := covariantClass_le_of_lt
rintro x β¨y, β¨hya, hybβ©, z, β¨hzc, hzdβ©, rflβ©
exact β¨mul_le_mul' hya hzc, mul_lt_mul_of_lt_of_le hyb hzdβ©
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import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 75 | 76 | theorem ofFractionRing_zero : (ofFractionRing 0 : RatFunc K) = 0 := by |
simp only [Zero.zero, OfNat.ofNat, RatFunc.zero]
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import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 81 | 81 | theorem toComplex_def' (x y : β€) : ((β¨x, yβ© : β€[i]) : β) = x + y * I := by | simp [toComplex_def]
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import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 78 | 80 | theorem mem_graph_iff {c : Ξ± Γ M} {f : Ξ± ββ M} : c β f.graph β f c.1 = c.2 β§ c.2 β 0 := by |
cases c
exact mk_mem_graph_iff
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import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 98 | 100 | theorem kernelSubobject_arrow :
(kernelSubobjectIso f).hom β« kernel.ΞΉ f = (kernelSubobject f).arrow := by |
simp [kernelSubobjectIso]
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import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.SuccPred
#align_import data.int.succ_pred from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Order
namespace Int
-- so that Lean reads `Int.succ` through `SuccOrder.succ`
@[instance] abbrev instSuccOrder : Su... | Mathlib/Data/Int/SuccPred.lean | 79 | 79 | theorem sub_one_covBy (z : β€) : z - 1 β z := by | rw [Int.covBy_iff_succ_eq, sub_add_cancel]
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import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 46 | 46 | theorem zero_of_nonpos {x : β} (hx : x β€ 0) : expNegInvGlue x = 0 := by | simp [expNegInvGlue, hx]
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import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 624 | 625 | theorem preimage_mul_const_Ioc (a b : Ξ±) {c : Ξ±} (h : 0 < c) :
(fun x => x * c) β»ΒΉ' Ioc a b = Ioc (a / c) (b / c) := by | simp [β Ioi_inter_Iic, h]
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import Mathlib.Topology.Sets.Opens
#align_import topology.sets.closeds from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Order OrderDual Set
variable {ΞΉ Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²]
namespace TopologicalSpace
structure Closeds (Ξ± : Type*) [Topolog... | Mathlib/Topology/Sets/Closeds.lean | 110 | 111 | theorem coe_sup (s t : Closeds Ξ±) : (β(s β t) : Set Ξ±) = βs βͺ βt := by |
rfl
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import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 172 | 175 | theorem eq_zero_of_dvd_of_natDegree_lt {p q : R[X]} (hβ : p β£ q) (hβ : natDegree q < natDegree p) :
q = 0 := by |
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (natDegree_le_of_dvd hβ hc)
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import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 133 | 134 | theorem image_const_add_Ioc : (fun x => a + x) '' Ioc b c = Ioc (a + b) (a + c) := by |
simp only [add_comm a, image_add_const_Ioc]
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import Mathlib.Logic.Equiv.Option
import Mathlib.Order.RelIso.Basic
import Mathlib.Order.Disjoint
import Mathlib.Order.WithBot
import Mathlib.Tactic.Monotonicity.Attr
import Mathlib.Util.AssertExists
#align_import order.hom.basic from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
open ... | Mathlib/Order/Hom/Basic.lean | 1,235 | 1,239 | theorem OrderIso.map_bot' [LE Ξ±] [PartialOrder Ξ²] (f : Ξ± βo Ξ²) {x : Ξ±} {y : Ξ²} (hx : β x', x β€ x')
(hy : β y', y β€ y') : f x = y := by |
refine le_antisymm ?_ (hy _)
rw [β f.apply_symm_apply y, f.map_rel_iff]
apply hx
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import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 166 | 169 | theorem eq_zero_of_dvd_of_degree_lt {p q : R[X]} (hβ : p β£ q) (hβ : degree q < degree p) :
q = 0 := by |
by_contra hc
exact (lt_iff_not_ge _ _).mp hβ (degree_le_of_dvd hβ hc)
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import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ΞΉ : Type*}
theorem coprime_list_prod_left_iff {l : List β} {k : β} :
Coprime l.prod k β β n β l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 28 | 30 | theorem coprime_multiset_prod_left_iff {m : Multiset β} {k : β} :
Coprime m.prod k β β n β m, Coprime n k := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_left_iff
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import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 66 | 67 | theorem coe_injective β¦f g : Ξ± ββ Ξ²β¦ (H : (f : Ξ± β Ξ²) = g) : f = g := by |
cases f; cases g; congr
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import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (π E : Type*) {ΞΉ : Type*} [OrderedRing π] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 86 | 87 | theorem mem_space_iff : x β K.space β β s β K.faces, x β convexHull π (s : Set E) := by |
simp [space]
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import Mathlib.Algebra.Order.Floor
import Mathlib.Topology.Algebra.Order.Group
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.floor from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Filter Function Int Set Topology
variable {Ξ± Ξ² Ξ³ : Type*} [LinearOrdere... | Mathlib/Topology/Algebra/Order/Floor.lean | 84 | 85 | theorem tendsto_ceil_left_pure (n : β€) : Tendsto (ceil : Ξ± β β€) (π[β€] n) (pure n) := by |
simpa only [ceil_intCast] using tendsto_ceil_left_pure_ceil (n : Ξ±)
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import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 73 | 73 | theorem evalβ_X : X.evalβ f x = x := by | simp [evalβ_eq_sum]
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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 | 119 | 120 | theorem add_left_cancel (a) {b c : Ordinal} : a + b = a + c β b = c := by |
simp only [le_antisymm_iff, add_le_add_iff_left]
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import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra
#align_import algebra.direct_sum.internal from "leanprover-community/mathlib"@"9936c3dfc04e5876f4368aeb2e60f8d8358d095a"
open DirectSum
variable {ΞΉ : Type*} {Ο S R : Type*}
instance... | Mathlib/Algebra/DirectSum/Internal.lean | 62 | 68 | theorem SetLike.natCast_mem_graded [Zero ΞΉ] [AddMonoidWithOne R] [SetLike Ο R]
[AddSubmonoidClass Ο R] (A : ΞΉ β Ο) [SetLike.GradedOne A] (n : β) : (n : R) β A 0 := by |
induction' n with _ n_ih
Β· rw [Nat.cast_zero]
exact zero_mem (A 0)
Β· rw [Nat.cast_succ]
exact add_mem n_ih (SetLike.one_mem_graded _)
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import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 72 | 74 | theorem Finsupp.toFreeAbelianGroup_toFinsupp {X} (x : FreeAbelianGroup X) :
Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x := by |
rw [β AddMonoidHom.comp_apply, Finsupp.toFreeAbelianGroup_comp_toFinsupp, AddMonoidHom.id_apply]
| 0.90625 |
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 53 | 54 | theorem mem_stepSet (s : Ο) (S : Set Ο) (a : Ξ±) : s β M.stepSet S a β β t β S, s β M.step t a := by |
simp [stepSet]
| 0.90625 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 176 | 178 | theorem iUnion_Ioc_eq_Ioi_self_iff {f : ΞΉ β Ξ±} {a : Ξ±} :
β i, Ioc a (f i) = Ioi a β β x, a < x β β i, x β€ f i := by |
simp [β Ioi_inter_Iic, β inter_iUnion, subset_def]
| 0.90625 |
import Mathlib.Init.Algebra.Classes
import Mathlib.Init.Data.Ordering.Basic
#align_import init.data.ordering.lemmas from "leanprover-community/lean"@"4bd314f7bd5e0c9e813fc201f1279a23f13f9f1d"
universe u
namespace Ordering
@[simp]
theorem ite_eq_lt_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c th... | Mathlib/Init/Data/Ordering/Lemmas.lean | 26 | 28 | theorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by |
by_cases c <;> simp [*]
| 0.90625 |
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib"@"347636a7a80595d55bedf6e6fbd996a3c39da69a"
universe u v w
open Cardinal Submodule Module Function
variable {R : Type u} {M : Type v} {N : Type w}
variable... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 58 | 61 | theorem finrank_eq_of_rank_eq {n : β} (h : Module.rank R M = βn) : finrank R M = n := by |
apply_fun toNat at h
rw [toNat_natCast] at h
exact mod_cast h
| 0.90625 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' : Sort*}
theorem powerset_insert (s : Set Ξ±) (a : Ξ±)... | Mathlib/Data/Set/Image.lean | 666 | 666 | theorem exists_range_iff {p : Ξ± β Prop} : (β a β range f, p a) β β i, p (f i) := by | simp
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import Mathlib.Order.PartialSups
#align_import order.disjointed from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {Ξ± Ξ² : Type*}
section GeneralizedBooleanAlgebra
variable [GeneralizedBooleanAlgebra Ξ±]
def disjointed (f : β β Ξ±) : β β Ξ±
| 0 => f 0
| n + 1 => f (n + 1) ... | Mathlib/Order/Disjointed.lean | 63 | 67 | theorem disjointed_le_id : disjointed β€ (id : (β β Ξ±) β β β Ξ±) := by |
rintro f n
cases n
Β· rfl
Β· exact sdiff_le
| 0.90625 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {Ξ± Ξ² Ξ³ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 96 | 96 | theorem coe_eq_univ : (s : Set Ξ±) = Set.univ β s = univ := by | rw [β coe_univ, coe_inj]
| 0.90625 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 81 | 82 | theorem op_geom_sum (x : Ξ±) (n : β) : op (β i β range n, x ^ i) = β i β range n, op x ^ i := by |
simp
| 0.90625 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Restrict
import Mathlib.Tactic.FailIfNoProgress
#align_import analysis.normed_space.affine_isomet... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 72 | 74 | theorem linear_eq_linearIsometry : f.linear = f.linearIsometry.toLinearMap := by |
ext
rfl
| 0.90625 |
import Mathlib.Data.Set.Basic
#align_import data.bundle from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Function Set
namespace Bundle
variable {B F : Type*} (E : B β Type*)
@[ext]
structure TotalSpace (F : Type*) (E : B β Type*) where
proj : B
snd : E proj
#align bund... | Mathlib/Data/Bundle.lean | 69 | 70 | theorem TotalSpace.mk_cast {x x' : B} (h : x = x') (b : E x) :
.mk' F x' (cast (congr_arg E h) b) = TotalSpace.mk x b := by | subst h; rfl
| 0.90625 |
import Mathlib.Algebra.Module.Equiv
#align_import linear_algebra.general_linear_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
variable (R M : Type*)
namespace LinearMap
variable [Semiring R] [AddCommMonoid M] [Module R M]
abbrev GeneralLinearGroup :=
(M ββ[R] M)Λ£
#alig... | Mathlib/LinearAlgebra/GeneralLinearGroup.lean | 68 | 69 | theorem generalLinearEquiv_to_linearMap (f : GeneralLinearGroup R M) :
(generalLinearEquiv R M f : M ββ[R] M) = f := by | ext; rfl
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import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
set_option autoImplicit true
-- In this file we would like to be able to use multi-character auto-implicits.
set_option relaxedAutoImplicit true
namespace Mathlib.Tactic
open Lean hiding Rat
open Qq Me... | Mathlib/Tactic/Ring/RingNF.lean | 120 | 120 | theorem nat_rawCast_0 : (Nat.rawCast 0 : R) = 0 := by | simp
| 0.90625 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ΞΉ : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 99 | 100 | theorem list_sum_mul_self (l : List R) : l.sum * l.sum = (List.map (fun x => x * x) l).sum := by |
simp_rw [β pow_two, list_sum_sq]
| 0.90625 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 59 | 63 | theorem ContMDiffWithinAt.prod_mk {f : M β M'} {g : M β N'} (hf : ContMDiffWithinAt I I' n f s x)
(hg : ContMDiffWithinAt I J' n g s x) :
ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x := by |
rw [contMDiffWithinAt_iff] at *
exact β¨hf.1.prod hg.1, hf.2.prod hg.2β©
| 0.90625 |
import Mathlib.Algebra.Group.Center
#align_import group_theory.subsemigroup.centralizer from "leanprover-community/mathlib"@"cc67cd75b4e54191e13c2e8d722289a89e67e4fa"
variable {M : Type*} {S T : Set M}
namespace Set
variable (S)
@[to_additive addCentralizer " The centralizer of a subset of an additive magma. ... | Mathlib/Algebra/Group/Centralizer.lean | 94 | 97 | theorem div_mem_centralizer [Group M] (ha : a β centralizer S) (hb : b β centralizer S) :
a / b β centralizer S := by |
rw [div_eq_mul_inv]
exact mul_mem_centralizer ha (inv_mem_centralizer hb)
| 0.90625 |
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : β) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 31 | 31 | theorem dist_self (n : β) : dist n n = 0 := by | simp [dist, tsub_self]
| 0.90625 |
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 102 | 106 | theorem measurable_rnDeriv (ΞΌ Ξ½ : Measure Ξ±) : Measurable <| ΞΌ.rnDeriv Ξ½ := by |
by_cases h : HaveLebesgueDecomposition ΞΌ Ξ½
Β· exact (haveLebesgueDecomposition_spec ΞΌ Ξ½).1
Β· rw [rnDeriv_of_not_haveLebesgueDecomposition h]
exact measurable_zero
| 0.90625 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 182 | 184 | theorem biUnion_Ico_eq_Iio_self_iff {p : ΞΉ β Prop} {f : β i, p i β Ξ±} {a : Ξ±} :
β (i) (hi : p i), Ico (f i hi) a = Iio a β β x < a, β i hi, f i hi β€ x := by |
simp [β Ici_inter_Iio, β iUnion_inter, subset_def]
| 0.90625 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 120 | 122 | theorem circleMap_mem_sphere (c : β) {R : β} (hR : 0 β€ R) (ΞΈ : β) :
circleMap c R ΞΈ β sphere c R := by |
simpa only [_root_.abs_of_nonneg hR] using circleMap_mem_sphere' c R ΞΈ
| 0.90625 |
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {Ξ± Ξ² Ξ³ : Type*} {ΞΉ ΞΉ' ΞΉ... | Mathlib/Data/Set/Lattice.lean | 207 | 211 | theorem exists_set_mem_of_union_eq_top {ΞΉ : Type*} (t : Set ΞΉ) (s : ΞΉ β Set Ξ²)
(w : β i β t, s i = β€) (x : Ξ²) : β i β t, x β s i := by |
have p : x β β€ := Set.mem_univ x
rw [β w, Set.mem_iUnion] at p
simpa using p
| 0.90625 |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 55 | 56 | theorem rank_zero [Nontrivial R] : rank (0 : Matrix m n R) = 0 := by |
rw [rank, mulVecLin_zero, LinearMap.range_zero, finrank_bot]
| 0.90625 |
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n β Fin n
rank : Nat β Nat
rank_lt : β i, (parent i).1 β i β rank i < rank (parent i)
structure UFNode (Ξ± : Type*) where
parent : Nat
value : Ξ±
rank : Nat
inductive... | Mathlib/Data/UnionFind.lean | 82 | 84 | theorem get_eq {arr : Array Ξ±} {n} {m : Fin n β Ξ²} (H : Agrees arr f m) :
β i hβ hβ, f (arr.get β¨i, hββ©) = m β¨i, hββ© := by |
cases H; exact fun i h _ β¦ rfl
| 0.90625 |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 42 | 42 | theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a := by | simp only [opow_def, if_true]
| 0.90625 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Rat.Init
#align_import algebra.field.defs from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
-- `NeZero` should not be needed in the basic algebraic hierarchy.
assert_not_exists NeZero
-- Check that we have not imported `Mathlib.Tact... | Mathlib/Algebra/Field/Defs.lean | 226 | 227 | theorem smul_one_eq_cast (A : Type*) [DivisionRing A] (m : β) : m β’ (1 : A) = βm := by |
rw [Rat.smul_def, mul_one]
| 0.90625 |
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 {Ξ± : Type u} {Ξ±' : Type w} {Ξ² : Type v} {Ξ²' : Type x} {Ξ³ Ξ΄ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 38 | 40 | theorem sum_rec_congr (P : Ξ± β Ξ² β Sort*) (f : β i, P (inl i)) (g : β i, P (inr i))
{x y : Ξ± β Ξ²} (h : x = y) :
@Sum.rec _ _ _ f g x = cast (congr_arg P h.symm) (@Sum.rec _ _ _ f g y) := by | cases h; rfl
| 0.90625 |
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