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 | num_lines int64 1 150 |
|---|---|---|---|---|---|---|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {Ξ± Ξ² : Type*} {s t : Set Ξ±}
noncomputable def encard (s : Set Ξ±) : ββ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 69 | 71 | theorem encard_univ (Ξ± : Type*) :
encard (univ : Set Ξ±) = PartENat.withTopEquiv (PartENat.card Ξ±) := by |
rw [encard, PartENat.card_congr (Equiv.Set.univ Ξ±)]
| 1 |
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 | 80 | 83 | theorem HasDerivAtFilter.scomp_of_eq (hg : HasDerivAtFilter gβ gβ' y L')
(hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Tendsto h L L') :
HasDerivAtFilter (gβ β h) (h' β’ gβ') x L := by |
rw [hy] at hg; exact hg.scomp x hh hL
| 1 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {π E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 85 | 87 | theorem adjointAux_inner_right (A : E βL[π] F) (x : E) (y : F) :
βͺx, adjointAux A yβ« = βͺA x, yβ« := by |
rw [β inner_conj_symm, adjointAux_inner_left, inner_conj_symm]
| 1 |
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite Topolog... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 117 | 118 | theorem quasiSeparated_eq_diagonal_is_quasiCompact :
@QuasiSeparated = MorphismProperty.diagonal @QuasiCompact := by | ext; exact quasiSeparated_iff _
| 1 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 74 | 75 | theorem mem_annihilator {r} : r β N.annihilator β β n β N, r β’ n = (0 : M) := by |
simp_rw [annihilator, Module.mem_annihilator, Subtype.forall, Subtype.ext_iff]; rfl
| 1 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {Ξ± Ξ² R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 69 | 69 | theorem card_mono : Monotone (@card Ξ±) := by | apply card_le_card
| 1 |
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]
| 1 |
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 | 90 | 93 | theorem HasDerivWithinAt.scomp_hasDerivAt_of_eq (hg : HasDerivWithinAt gβ gβ' s' y)
(hh : HasDerivAt h h' x) (hs : β x, h x β s') (hy : y = h x) :
HasDerivAt (gβ β h) (h' β’ gβ') x := by |
rw [hy] at hg; exact hg.scomp_hasDerivAt x hh hs
| 1 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {Ξ± : Type*}
@[simps]
protected def sym2 (s : Finset Ξ±) : Finset (Sym2 Ξ±) :... | Mathlib/Data/Finset/Sym.lean | 96 | 97 | theorem sym2_eq_empty : s.sym2 = β
β s = β
:= by |
rw [β val_eq_zero, sym2_val, Multiset.sym2_eq_zero_iff, val_eq_zero]
| 1 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 104 | 106 | theorem smul_closedBall' {c : π} (hc : c β 0) (x : E) (r : β) :
c β’ closedBall x r = closedBall (c β’ x) (βcβ * r) := by |
simp only [β ball_union_sphere, Set.smul_set_union, _root_.smul_ball hc, smul_sphere' hc]
| 1 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 88 | 90 | theorem _root_.FiniteDimensional.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
finrank R M = Fintype.card (ChooseBasisIndex R M) := by |
simp [finrank, rank_eq_card_chooseBasisIndex]
| 1 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Dynamics.FixedPoints.Basic
open Finset Function
section AddCommMonoid
variable {Ξ± M : Type*} [AddCommMonoid M]
def birkhoffSum (f : Ξ± β Ξ±) (g : Ξ± β M) (n : β) (x : Ξ±) : M := β k β range n, g (f^[k] x)
theorem birkhoffSum_zero (f : Ξ± β Ξ±) (g : Ξ± β ... | Mathlib/Dynamics/BirkhoffSum/Basic.lean | 51 | 53 | theorem birkhoffSum_add (f : Ξ± β Ξ±) (g : Ξ± β M) (m n : β) (x : Ξ±) :
birkhoffSum f g (m + n) x = birkhoffSum f g m x + birkhoffSum f g n (f^[m] x) := by |
simp_rw [birkhoffSum, sum_range_add, add_comm m, iterate_add_apply]
| 1 |
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 | 74 | 76 | theorem einfsep_ne_top :
s.einfsep β β β β x β s, β y β s, x β y β§ edist x y β β := by |
simp_rw [β lt_top_iff_ne_top, einfsep_lt_top]
| 1 |
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 | 49 | 50 | theorem dist_eq_sub_of_le_right {n m : β} (h : m β€ n) : dist n m = n - m := by |
rw [dist_comm]; apply dist_eq_sub_of_le h
| 1 |
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 | 109 | 112 | theorem HasDerivWithinAt.inner {f g : β β E} {f' g' : E} {s : Set β} {x : β}
(hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) :
HasDerivWithinAt (fun t => βͺf t, g tβ«) (βͺf x, g'β« + βͺf', g xβ«) s x := by |
simpa using (hf.hasFDerivWithinAt.inner π hg.hasFDerivWithinAt).hasDerivWithinAt
| 1 |
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversable.instances from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
universe u v
section Option
open Functor
variab... | Mathlib/Control/Traversable/Instances.lean | 35 | 38 | theorem Option.comp_traverse {Ξ± Ξ² Ξ³} (f : Ξ² β F Ξ³) (g : Ξ± β G Ξ²) (x : Option Ξ±) :
Option.traverse (Comp.mk β (f <$> Β·) β g) x =
Comp.mk (Option.traverse f <$> Option.traverse g x) := by |
cases x <;> simp! [functor_norm] <;> rfl
| 1 |
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 | 202 | 203 | theorem preimage_add_const_Ioc : (fun x => x + a) β»ΒΉ' Ioc b c = Ioc (b - a) (c - a) := by |
simp [β Ioi_inter_Iic]
| 1 |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 528 | 530 | theorem map_one_fst_of_isMulTwoCocycle {f : G Γ G β M} (hf : IsMulTwoCocycle f) (g : G) :
f (1, g) = f (1, 1) := by |
simpa only [one_smul, one_mul, mul_one, mul_right_inj] using (hf 1 1 g).symm
| 1 |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 49 | 49 | theorem coeff_bit0 (p : R[X]) (n : β) : coeff (bit0 p) n = bit0 (coeff p n) := by | simp [bit0]
| 1 |
import Batteries.Tactic.SeqFocus
namespace Batteries
class TotalBLE (le : Ξ± β Ξ± β Bool) : Prop where
total : le a b β¨ le b a
class OrientedCmp (cmp : Ξ± β Ξ± β Ordering) : Prop where
symm (x y) : (cmp x y).swap = cmp y x
class TransCmp (cmp : Ξ± β Ξ± β Ordering) extends OrientedCmp cmp : Prop where
... | .lake/packages/batteries/Batteries/Classes/Order.lean | 121 | 122 | theorem BEqCmp.cmp_iff_eq [BEq Ξ±] [LawfulBEq Ξ±] [BEqCmp (Ξ± := Ξ±) cmp] : cmp x y = .eq β x = y := by |
simp [BEqCmp.cmp_iff_beq]
| 1 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 127 | 129 | theorem WithTop.coe_iSup [SupSet Ξ±] (f : ΞΉ β Ξ±) (h : BddAbove (Set.range f)) :
β(β¨ i, f i) = (β¨ i, f i : WithTop Ξ±) := by |
rw [iSup, iSup, WithTop.coe_sSup' h, β range_comp]; rfl
| 1 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 78 | 79 | theorem eq_of_slope_eq_zero {f : k β PE} {a b : k} (h : slope f a b = (0 : E)) : f a = f b := by |
rw [β sub_smul_slope_vadd f a b, h, smul_zero, zero_vadd]
| 1 |
import Batteries.Tactic.Alias
import Batteries.Data.Nat.Basic
namespace Nat
@[simp] theorem recAux_zero {motive : Nat β Sort _} (zero : motive 0)
(succ : β n, motive n β motive (n+1)) :
Nat.recAux zero succ 0 = zero := rfl
theorem recAux_succ {motive : Nat β Sort _} (zero : motive 0)
(succ : β n, mo... | .lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean | 74 | 79 | theorem recDiag_zero_succ {motive : Nat β Nat β Sort _} (zero_zero : motive 0 0)
(zero_succ : β n, motive 0 n β motive 0 (n+1)) (succ_zero : β m, motive m 0 β motive (m+1) 0)
(succ_succ : β m n, motive m n β motive (m+1) (n+1)) (n) :
Nat.recDiag zero_zero zero_succ succ_zero succ_succ 0 (n+1)
= zero_s... |
simp [Nat.recDiag]; rfl
| 1 |
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 | 118 | 120 | theorem HasStrictDerivAt.complexToReal_fderiv {f : β β β} {f' x : β} (h : HasStrictDerivAt f f' x) :
HasStrictFDerivAt f (f' β’ (1 : β βL[β] β)) x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasStrictFDerivAt.restrictScalars β
| 1 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Operations
namespace Submodule
open Pointwise
variable {R M M' F G : Type*} [CommRing R] [AddCommGroup M] [Module R M]
variable {N Nβ Nβ P Pβ Pβ : Submodule R M}
def colon (N P : Submodule R M) : Ideal R :=
annihilator (P.map N.mkQ)
#align ... | Mathlib/RingTheory/Ideal/Colon.lean | 76 | 78 | theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} :
r β I.colon (Ideal.span {x}) β r * x β I := by |
simp only [β Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul]
| 1 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 140 | 141 | theorem inf_relindex_left : (H β K).relindex H = K.relindex H := by |
rw [inf_comm, inf_relindex_right]
| 1 |
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]
| 1 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 57 | 60 | theorem pentagon_inv_inv_hom (W X Y Z : C) :
(Ξ±_ W (X β Y) Z).inv β« ((Ξ±_ W X Y).inv β π Z) β« (Ξ±_ (W β X) Y Z).hom =
(π W β (Ξ±_ X Y Z).hom) β« (Ξ±_ W X (Y β Z)).inv := by |
coherence
| 1 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 101 | 103 | theorem withDensityα΅₯_sub (hf : Integrable f ΞΌ) (hg : Integrable g ΞΌ) :
ΞΌ.withDensityα΅₯ (f - g) = ΞΌ.withDensityα΅₯ f - ΞΌ.withDensityα΅₯ g := by |
rw [sub_eq_add_neg, sub_eq_add_neg, withDensityα΅₯_add hf hg.neg, withDensityα΅₯_neg]
| 1 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 44 | 44 | theorem sub_mem : x β‘ y [SMOD U] β x - y β U := by | rw [SModEq.def, Submodule.Quotient.eq]
| 1 |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Logic.Pairwise
#align_import data.set.intervals.group from "lean... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 226 | 228 | theorem pairwise_disjoint_Ioo_zpow :
Pairwise (Disjoint on fun n : β€ => Ioo (b ^ n) (b ^ (n + 1))) := by |
simpa only [one_mul] using pairwise_disjoint_Ioo_mul_zpow 1 b
| 1 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 68 | 69 | theorem catalan_succ (n : β) : catalan (n + 1) = β i : Fin n.succ, catalan i * catalan (n - i) := by |
rw [catalan]
| 1 |
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 237 | 238 | theorem lapply_comp_lsingle_of_ne (a a' : Ξ±) (h : a β a') :
lapply a ββ lsingle a' = (0 : M ββ[R] M) := by | ext; simp [h.symm]
| 1 |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {Rβ : Type*} {Rβ : Type*} {Rβ : Type*}
variable {K : Type*}
variable {M : Type*} {Mβ : Type*} {Mβ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 125 | 126 | theorem ker_codRestrict {Οββ : Rβ β+* R} (p : Submodule R M) (f : Mβ βββ[Οββ] M) (hf) :
ker (codRestrict p f hf) = ker f := by | rw [ker, comap_codRestrict, Submodule.map_bot]; rfl
| 1 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {Ξ± Ξ² : Type*} [Preorder Ξ±] [Preorder Ξ²]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 78 | 79 | theorem image_Iio (e : Ξ± βo Ξ²) (a : Ξ±) : e '' Iio a = Iio (e a) := by |
rw [e.image_eq_preimage, e.symm.preimage_Iio, e.symm_symm]
| 1 |
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 | 332 | 333 | theorem infsep_zero : s.infsep = 0 β s.einfsep = 0 β¨ s.einfsep = β := by |
rw [infsep, ENNReal.toReal_eq_zero_iff]
| 1 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {Ξ± : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 248 | 249 | theorem toList_get_zero (h : x β p.support) :
(toList p x).get β¨0, (length_toList_pos_of_mem_support _ _ h)β© = x := by | simp [toList]
| 1 |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {Rβ : Type*} {Rβ : Type*} {Rβ : Type*}
variable {K : Type*}
variable {M : Type*} {Mβ : Type*} {Mβ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 112 | 113 | theorem ker_eq_bot' {f : F} : ker f = β₯ β β m, f m = 0 β m = 0 := by |
simpa [disjoint_iff_inf_le] using disjoint_ker (f := f) (p := β€)
| 1 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 344 | 344 | theorem descFactorial_one (n : β) : n.descFactorial 1 = n := by | simp
| 1 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.normed.ring.seminorm from "leanprover-community/mathlib"@"7ea604785a41a0681eac70c5a82372493dbefc68"
open NNReal
variable {F R S : Type*} (x y : R) (r : β)
structure RingSeminorm (R : Type*) [NonU... | Mathlib/Analysis/Normed/Ring/Seminorm.lean | 116 | 116 | theorem ne_zero_iff {p : RingSeminorm R} : p β 0 β β x, p x β 0 := by | simp [eq_zero_iff]
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 72 | 73 | theorem logb_mul (hx : x β 0) (hy : y β 0) : logb b (x * y) = logb b x + logb b y := by |
simp_rw [logb, log_mul hx hy, add_div]
| 1 |
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 | 82 | 83 | theorem cantorFunctionAux_zero (f : β β Bool) : cantorFunctionAux c f 0 = cond (f 0) 1 0 := by |
cases h : f 0 <;> simp [h]
| 1 |
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 85 | 85 | theorem p_comp {P Q : Karoubi C} (f : Hom P Q) : P.p β« f.f = f.f := by | rw [f.comm, β assoc, P.idem]
| 1 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 67 | 68 | theorem prod_val [CommMonoid Ξ±] (s : Finset Ξ±) : s.1.prod = s.prod id := by |
rw [Finset.prod, Multiset.map_id]
| 1 |
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 | 674 | 676 | theorem preimage_mul_const_Ico_of_neg (a b : Ξ±) {c : Ξ±} (h : c < 0) :
(fun x => x * c) β»ΒΉ' Ico a b = Ioc (b / c) (a / c) := by |
simp [β Ici_inter_Iio, β Ioi_inter_Iic, h, inter_comm]
| 1 |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (cβ cβ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 84 | 85 | theorem i_mul_k : q.i * q.k = cβ β’ q.j := by |
rw [β i_mul_j, β mul_assoc, i_mul_i, smul_mul_assoc, one_mul]
| 1 |
import Mathlib.Data.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
#align_import data.analysis.topology from "leanprover-community/mathlib"@"55d771df074d0dd020139ee1cd4b95521422df9f"
open Set
open Filter hiding Realizer
open Topology
structure Ctop (Ξ± Ο : Type*) where
f ... | Mathlib/Data/Analysis/Topology.lean | 79 | 80 | theorem ofEquiv_val (E : Ο β Ο) (F : Ctop Ξ± Ο) (a : Ο) : F.ofEquiv E a = F (E.symm a) := by |
cases F; rfl
| 1 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 361 | 362 | theorem average_congr {f g : Ξ± β E} (h : f =α΅[ΞΌ] g) : β¨ x, f x βΞΌ = β¨ x, g x βΞΌ := by |
simp only [average_eq, integral_congr_ae h]
| 1 |
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 [*]
| 1 |
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 | 192 | 194 | theorem coe_add_get {x : β} {y : PartENat} (h : ((x : PartENat) + y).Dom) :
get ((x : PartENat) + y) h = x + get y h.2 := by |
rfl
| 1 |
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 {Ξ± Ξ±' Ξ² Ξ²' Ξ³ Ξ³' Ξ΄ Ξ΄' Ξ΅ Ξ΅' ΞΆ ΞΆ' Ξ½ : Type*}
namespace Finset
variable [DecidableEq Ξ±'] [DecidableEq Ξ²'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 98 | 100 | theorem forall_imageβ_iff {p : Ξ³ β Prop} :
(β z β imageβ f s t, p z) β β x β s, β y β t, p (f x y) := by |
simp_rw [β mem_coe, coe_imageβ, forall_image2_iff]
| 1 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 164 | 164 | theorem arsinh_nonneg_iff : 0 β€ arsinh x β 0 β€ x := by | rw [β sinh_le_sinh, sinh_zero, sinh_arsinh]
| 1 |
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 | 57 | 58 | theorem mem_interedges_iff {x : Ξ± Γ Ξ²} : x β interedges r s t β x.1 β s β§ x.2 β t β§ r x.1 x.2 := by |
rw [interedges, mem_filter, Finset.mem_product, and_assoc]
| 1 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Factorial.Cast
#align_import data.nat.choose.cast from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Nat
variable (K : Type*) [DivisionRing K] [CharZero K]
namespace Nat
theorem cast_choose {a b : β} (h : a β€ b) : (b.... | Mathlib/Data/Nat/Choose/Cast.lean | 31 | 32 | theorem cast_add_choose {a b : β} : ((a + b).choose a : K) = (a + b)! / (a ! * b !) := by |
rw [cast_choose K (_root_.le_add_right le_rfl), add_tsub_cancel_left]
| 1 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : β β β
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 340 | 341 | theorem zero_descFactorial_succ (k : β) : (0 : β).descFactorial (k + 1) = 0 := by |
rw [descFactorial_succ, Nat.zero_sub, Nat.zero_mul]
| 1 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {Ξ± : Type u}
class OrderedAddCommGroup (Ξ± : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 120 | 121 | theorem inv_mul_le_iff_le_mul : bβ»ΒΉ * a β€ c β a β€ b * c := by |
rw [β mul_le_mul_iff_left b, mul_inv_cancel_left]
| 1 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem Β«forallΒ» {p : Ξ± β Ξ² β Prop} :
(β x, p x) β (β a, p (inl a)) β§ β b, p (inr b) :=
β¨fun h => β¨fun _ => h _, fun _ => h _β©, fun β¨hβ, hββ© => Sum.rec hβ hββ©
@[simp] protected theorem Β«existsΒ» {p : Ξ± β Ξ² ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 81 | 81 | theorem not_isRight {x : Ξ± β Ξ²} : Β¬x.isRight β x.isLeft := by | simp
| 1 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : β β β β β
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 74 | 74 | theorem ack_succ_zero (m : β) : ack (m + 1) 0 = ack m 1 := by | rw [ack]
| 1 |
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 | 98 | 100 | theorem mulSupport_update_eq_ite [DecidableEq Ξ±] [DecidableEq M] (f : Ξ± β M) (x : Ξ±) (y : M) :
mulSupport (update f x y) = if y = 1 then mulSupport f \ {x} else insert x (mulSupport f) := by |
rcases eq_or_ne y 1 with rfl | hy <;> simp [mulSupport_update_one, mulSupport_update_of_ne_one, *]
| 1 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 99 | 100 | theorem Rel.subRight {a b : lib R X} (c : lib R X) (h : Rel R X a b) : Rel R X (a - c) (b - c) := by |
simpa only [sub_eq_add_neg] using h.add_right (-c)
| 1 |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (Ξ± : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 283 | 283 | theorem self_act_act_eq {x y : R} : (x β x) β y = x β y := by | rw [β right_inv x y, β self_distrib]
| 1 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 136 | 137 | theorem map_inv (I : FractionalIdeal Rββ° K) (h : K ββ[Rβ] K') :
Iβ»ΒΉ.map (h : K ββ[Rβ] K') = (I.map h)β»ΒΉ := by | rw [inv_eq, map_div, map_one, inv_eq]
| 1 |
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 | 241 | 242 | theorem preimage_neg_Ico : -Ico a b = Ioc (-b) (-a) := by |
simp [β Ici_inter_Iio, β Ioi_inter_Iic, inter_comm]
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : β}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 64 | 64 | theorem logb_abs (x : β) : logb b |x| = logb b x := by | rw [logb, logb, log_abs]
| 1 |
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w uβ uβ vβ vβ
structure MonadCont.Label (Ξ± : Type w) (m : Type u β Type v) (Ξ² : Typ... | Mathlib/Control/Monad/Cont.lean | 193 | 194 | theorem WriterT.goto_mkLabel {Ξ± Ξ² Ο : Type _} [EmptyCollection Ο] (x : Label (Ξ± Γ Ο) m Ξ²) (i : Ξ±) :
goto (WriterT.mkLabel x) i = monadLift (goto x (i, β
)) := by | cases x; rfl
| 1 |
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 | 104 | 106 | theorem kernelSubobject_arrow' :
(kernelSubobjectIso f).inv β« (kernelSubobject f).arrow = kernel.ΞΉ f := by |
simp [kernelSubobjectIso]
| 1 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 293 | 294 | theorem map_smulβββ (f : M βSL[Οββ] F βSL[Οββ] G') (c : R) (x : M) (y : F) :
f (c β’ x) y = Οββ c β’ f x y := by | rw [f.map_smulββ, smul_apply]
| 1 |
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 | 183 | 183 | theorem id_eval (v) : id.eval v = pure v := by | simp [id]
| 1 |
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 | 169 | 171 | theorem zmodChar_apply' {n : β+} {ΞΆ : C} (hΞΆ : ΞΆ ^ (n : β) = 1) (a : β) :
zmodChar n hΞΆ a = ΞΆ ^ a := by |
rw [pow_eq_pow_mod a hΞΆ, zmodChar_apply, ZMod.val_natCast a]
| 1 |
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
open Set Filter Bornology
open scoped ENNReal Uniformity Topology Pointwise
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {X ΞΉ : Type*}
variable [PseudoMetricSpace Ξ±]
namespace Metric
#align metric.bounded Bornology.I... | Mathlib/Topology/MetricSpace/Bounded.lean | 142 | 144 | theorem tendsto_dist_left_atTop_iff (c : Ξ±) {f : Ξ² β Ξ±} {l : Filter Ξ²} :
Tendsto (fun x β¦ dist c (f x)) l atTop β Tendsto f l (cobounded Ξ±) := by |
simp only [dist_comm c, tendsto_dist_right_atTop_iff]
| 1 |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {Ξ± : Type*} [DecidableEq Ξ±]
namespace Finset
section Group
variable [Group Ξ±] (e : Ξ±) (x : Finset... | Mathlib/Combinatorics/Additive/ETransform.lean | 137 | 137 | theorem mulETransformRight_one : mulETransformRight 1 x = x := by | simp [mulETransformRight]
| 1 |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Data.Real.Sqrt
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set Metric Pointwise
var... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 149 | 150 | theorem continuous_univBall (c : P) (r : β) : Continuous (univBall c r) := by |
simpa [continuous_iff_continuousOn_univ] using (univBall c r).continuousOn
| 1 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {Ξ± Ξ² : Type*}
section WeaklyCovers
section LT
variable [LT Ξ±] {a b : Ξ±}
def CovBy (a b :... | Mathlib/Order/Cover.lean | 233 | 234 | theorem not_covBy_iff (h : a < b) : Β¬a β b β β c, a < c β§ c < b := by |
simp_rw [CovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| 1 |
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db816233fb790e4fe9"
universe v u
namespace CategoryTheory
variable (C : Type u) [Category.{v} C]
def endofunctorMonoidalCategory : MonoidalCategory (C β₯€ C) where... | Mathlib/CategoryTheory/Monoidal/End.lean | 129 | 131 | theorem Ξ΅_inv_naturality {X Y : C} (f : X βΆ Y) :
(MonoidalFunctor.Ξ΅Iso F).inv.app X β« (π_ (C β₯€ C)).map f = F.Ξ΅Iso.inv.app X β« f := by |
aesop_cat
| 1 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 127 | 129 | theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite Ξ½] β¦f : Ξ± Γ Ξ² β Eβ¦
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => β« y, f (x, y) βΞ½ := by |
rw [β uncurry_curry f] at hf; exact hf.integral_prod_right
| 1 |
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 | 119 | 120 | theorem card_Ioo : (Ioo a b).card = b - a - 1 := by |
rw [β Nat.card_Ioo, β map_valEmbedding_Ioo, card_map]
| 1 |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem Β«forallΒ» {p : Ξ± β Ξ² β Prop} :
(β x, p x) β (β a, p (inl a)) β§ β b, p (inr b) :=
β¨fun h => β¨fun _ => h _, fun _ => h _β©, fun β¨hβ, hββ© => Sum.rec hβ hββ©
@[simp] protected theorem Β«existsΒ» {p : Ξ± β Ξ² ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 116 | 118 | theorem elim_eq_iff {u u' : Ξ± β Ξ³} {v v' : Ξ² β Ξ³} :
Sum.elim u v = Sum.elim u' v' β u = u' β§ v = v' := by |
simp [funext_iff]
| 1 |
import Batteries.Data.List.Basic
namespace Batteries
inductive AssocList (Ξ± : Type u) (Ξ² : Type v) where
| nil
| cons (key : Ξ±) (value : Ξ²) (tail : AssocList Ξ± Ξ²)
deriving Inhabited
namespace AssocList
@[simp] def toList : AssocList Ξ± Ξ² β List (Ξ± Γ Ξ²)
| nil => []
| cons a b es => (a, b) :: es.toL... | .lake/packages/batteries/Batteries/Data/AssocList.lean | 55 | 56 | theorem length_toList (l : AssocList Ξ± Ξ²) : l.toList.length = l.length := by |
induction l <;> simp_all
| 1 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {Ξ± : ... | Mathlib/Algebra/Ring/Defs.lean | 164 | 165 | theorem one_add_mul [RightDistribClass Ξ±] (a b : Ξ±) : (1 + a) * b = b + a * b := by |
rw [add_mul, one_mul]
| 1 |
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 126 | 128 | theorem pullbackIsoProdSubtype_hom_fst (f : X βΆ Z) (g : Y βΆ Z) :
(pullbackIsoProdSubtype f g).hom β« pullbackFst f g = pullback.fst := by |
rw [β Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_fst]
| 1 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open scoped Classical
open Set
variable {ΞΉ : ... | Mathlib/Order/CompleteLatticeIntervals.lean | 102 | 104 | theorem subset_sInf_emptyset [Inhabited s] :
sInf (β
: Set s) = default := by |
simp [sInf]
| 1 |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 87 | 88 | theorem Rel.addLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a + b) (a + c) := by |
rw [add_comm _ b, add_comm _ c]; exact h.add_right _
| 1 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 290 | 292 | theorem multipliable_iff_nat_tprod_vanishing {f : β β G} : Multipliable f β
β e β π 1, β N : β, β t β {n | N β€ n}, (β' n : t, f n) β e := by |
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_nat_tprod_vanishing]
| 1 |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (β Γ β)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 321 | 322 | theorem rowLen_eq_card (ΞΌ : YoungDiagram) {i : β} : ΞΌ.rowLen i = (ΞΌ.row i).card := by |
simp [row_eq_prod]
| 1 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iSup
@[simp]
theorem iSup_eq_zero {ΞΉ : Sort*} {f : ΞΉ β ββ₯0β} : β¨ i, f i = 0 β β i, f i = 0 :=
iSup_eq_bot
#align ennr... | Mathlib/Data/ENNReal/Real.lean | 676 | 676 | theorem iSup_zero_eq_zero {ΞΉ : Sort*} : β¨ _ : ΞΉ, (0 : ββ₯0β) = 0 := by | simp
| 1 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Finset
open Function Multiset
variable {Ξ± : Type*} {s t : Finset Ξ±}
section Powerset
def powerset (s : Finset... | Mathlib/Data/Finset/Powerset.lean | 83 | 84 | theorem powerset_eq_singleton_empty : s.powerset = {β
} β s = β
:= by |
rw [β powerset_empty, powerset_inj]
| 1 |
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 | 37 | 49 | theorem det_le {A : Matrix n n R} {abv : AbsoluteValue R S} {x : S} (hx : β i j, abv (A i j) β€ x) :
abv A.det β€ Nat.factorial (Fintype.card n) β’ x ^ Fintype.card n :=
calc
abv A.det = abv (β Ο : Perm n, Perm.sign Ο β’ β i, A (Ο i) i) := congr_arg abv (det_apply _)
_ β€ β Ο : Perm n, abv (Perm.sign Ο β’ β i, ... |
rw [sum_const, Finset.card_univ, Fintype.card_perm]
| 1 |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v} {X ΞΉ : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 96 | 97 | theorem eq_of_nndist_eq_zero {x y : Ξ³} : nndist x y = 0 β x = y := by |
simp only [β NNReal.eq_iff, β dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
| 1 |
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 290 | 291 | theorem hom_inv_id_assoc (f : X βΆ Y) [I : IsIso f] {Z} (g : X βΆ Z) : f β« inv f β« g = g := by |
simp [β Category.assoc]
| 1 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {Ξ± : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 132 | 133 | theorem length_rotate (l : List Ξ±) (n : β) : (l.rotate n).length = l.length := by |
rw [rotate_eq_rotate', length_rotate']
| 1 |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : β) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 151 | 153 | theorem frobeniusEquiv_symm_comp_frobenius :
((frobeniusEquiv R p).symm : R β+* R).comp (frobenius R p) = RingHom.id R := by |
ext; simp
| 1 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 147 | 147 | theorem map_zero (d : β) (a : Fin 0 β β) : map d a = 0 := by | simp [map]
| 1 |
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 | 52 | 53 | theorem mul_eq {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a = 0) (_ : 0 < b) : b * a = 0 := by |
simp [*]
| 1 |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 95 | 96 | theorem mapβ_map_left (f : Ξ³ β Ξ² β Ξ΄) (g : Ξ± β Ξ³) :
mapβ f (a.map g) b = mapβ (fun a b => f (g a) b) a b := by | cases a <;> rfl
| 1 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 218 | 219 | theorem units_apply_inv_apply (f : CircleDeg1LiftΛ£) (x : β) :
f ((fβ»ΒΉ : CircleDeg1LiftΛ£) x) = x := by | simp only [β mul_apply, f.mul_inv, coe_one, id]
| 1 |
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 | 52 | 54 | theorem coprime_fintype_prod_left_iff [Fintype ΞΉ] {s : ΞΉ β β} {x : β} :
Coprime (β i, s i) x β β i, Coprime (s i) x := by |
simp [coprime_prod_left_iff]
| 1 |
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 | 132 | 134 | theorem card_support_eraseLead' {c : β} (fc : f.support.card = c + 1) :
f.eraseLead.support.card = c := by |
rw [card_support_eraseLead, fc, add_tsub_cancel_right]
| 1 |
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed.mul_action from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
variable {Ξ± Ξ² : Type*}
section SeminormedAddGroup
variable [SeminormedAddGroup Ξ±] [SeminormedAddGroup Ξ²] ... | Mathlib/Analysis/Normed/MulAction.lean | 37 | 38 | theorem dist_smul_le (s : Ξ±) (x y : Ξ²) : dist (s β’ x) (s β’ y) β€ βsβ * dist x y := by |
simpa only [dist_eq_norm, sub_zero] using dist_smul_pair s x y
| 1 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {π E ΞΉ : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 81 | 82 | theorem mem_balancedCore_iff : x β balancedCore π s β β t, Balanced π t β§ t β s β§ x β t := by |
simp_rw [balancedCore, mem_sUnion, mem_setOf_eq, and_assoc]
| 1 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ΞΉ K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ΞΉ β ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 103 | 104 | theorem independent_iff_not_dependent : Independent f β Β¬Dependent f := by |
rw [dependent_iff_not_independent, Classical.not_not]
| 1 |
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