Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne, SΓ©bastien GouΓ«zel,
RΓ©my Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 473 | 475 | theorem rpow_neg_one (x : β) : x ^ (-1 : β) = xβ»ΒΉ := by |
suffices H : x ^ ((-1 : β€) : β) = xβ»ΒΉ by rwa [Int.cast_neg, Int.cast_one] at H
simp only [rpow_intCast, zpow_one, zpow_neg]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
impo... | Mathlib/RingTheory/Adjoin/Basic.lean | 173 | 202 | theorem adjoin_eq_span : Subalgebra.toSubmodule (adjoin R s) = span R (Submonoid.closure s) := by |
apply le_antisymm
Β· intro r hr
rcases Subsemiring.mem_closure_iff_exists_list.1 hr with β¨L, HL, rflβ©
clear hr
induction' L with hd tl ih
Β· exact zero_mem _
rw [List.forall_mem_cons] at HL
rw [List.map_cons, List.sum_cons]
refine Submodule.add_mem _ ?_ (ih HL.2)
replace HL := HL.1
... |
/-
Copyright (c) 2022 YaΓ«l Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies
-/
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 241 | 248 | theorem mem_compression : s β π a π β s β π β§ s.erase a β π β¨ s β π β§ insert a s β π := by |
simp_rw [compression, mem_disjUnion, mem_filter, mem_image, and_comm (a := (Β¬ s β π))]
refine
or_congr_right
(and_congr_left fun hs =>
β¨?_, fun h => β¨_, h, erase_insert <| insert_ne_self.1 <| ne_of_mem_of_not_mem h hsβ©β©)
rintro β¨t, ht, rflβ©
rwa [insert_erase (erase_ne_self.1 (ne_of_mem_of_no... |
/-
Copyright (c) 2022 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 463 | 545 | theorem _root_.ApproximatesLinearOn.norm_fderiv_sub_le {A : E βL[β] E} {Ξ΄ : ββ₯0}
(hf : ApproximatesLinearOn f A s Ξ΄) (hs : MeasurableSet s) (f' : E β E βL[β] E)
(hf' : β x β s, HasFDerivWithinAt f (f' x) s x) : βα΅ x βΞΌ.restrict s, βf' x - Aββ β€ Ξ΄ := by |
/- The conclusion will hold at the Lebesgue density points of `s` (which have full measure).
At such a point `x`, for any `z` and any `Ξ΅ > 0` one has for small `r`
that `{x} + r β’ closedBall z Ξ΅` intersects `s`. At a point `y` in the intersection,
`f y - f x` is close both to `f' x (r z)` (by differentia... |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez, Eric Wieser
-/
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
/-!
# D... | Mathlib/Data/List/Destutter.lean | 101 | 105 | theorem destutter'_of_chain (h : l.Chain R a) : l.destutter' R a = a :: l := by |
induction' l with b l hb generalizing a
Β· simp
obtain β¨h, hcβ© := chain_cons.mp h
rw [l.destutter'_cons_pos h, hb hc]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Floris van Doorn, Mario Carneiro, Martin Dvorak
-/
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac... | Mathlib/Data/List/Join.lean | 44 | 44 | theorem join_concat (L : List (List Ξ±)) (l : List Ξ±) : join (L.concat l) = join L ++ l := by | simp
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Winston Yin
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof fr... | Mathlib/Analysis/ODE/PicardLindelof.lean | 342 | 349 | theorem exists_contracting_iterate :
β (N : β) (K : _), ContractingWith K (FunSpace.next : v.FunSpace β v.FunSpace)^[N] := by |
rcases ((Real.tendsto_pow_div_factorial_atTop (v.L * v.tDist)).eventually
(gt_mem_nhds zero_lt_one)).exists with β¨N, hNβ©
have : (0 : β) β€ (v.L * v.tDist) ^ N / N ! :=
div_nonneg (pow_nonneg (mul_nonneg v.L.2 v.tDist_nonneg) _) (Nat.cast_nonneg _)
exact β¨N, β¨_, thisβ©, hN, LipschitzWith.of_dist_le_mul fun ... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
#align_import... | Mathlib/Topology/List.lean | 200 | 203 | theorem tendsto_cons {n : β} {a : Ξ±} {l : Vector Ξ± n} :
Tendsto (fun p : Ξ± Γ Vector Ξ± n => p.1 ::α΅₯ p.2) (π a ΓΛ’ π l) (π (a ::α΅₯ l)) := by |
rw [tendsto_subtype_rng, cons_val]
exact tendsto_fst.cons (Tendsto.comp continuousAt_subtype_val tendsto_snd)
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 788 | 790 | theorem div_lt_div_of_neg_of_lt (hc : c < 0) (h : b < a) : a / c < b / c := by |
rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]
exact mul_lt_mul_of_neg_right h (one_div_neg.2 hc)
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-communit... | Mathlib/SetTheory/Cardinal/Finite.lean | 291 | 293 | theorem _root_.Cardinal.toPartENat_lt_natCast_iff {n : β} {c : Cardinal} :
toPartENat c < βn β c < βn := by |
simp only [β not_le, Cardinal.natCast_le_toPartENat_iff]
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# LindelΓΆf sets and LindelΓΆf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 368 | 392 | theorem isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis (b : ΞΉ β Set X)
(hb : IsTopologicalBasis (Set.range b)) (hb' : β i, IsLindelof (b i)) (U : Set X) :
IsLindelof U β§ IsOpen U β β s : Set ΞΉ, s.Countable β§ U = β i β s, b i := by |
constructor
Β· rintro β¨hβ, hββ©
obtain β¨Y, f, rfl, hfβ© := hb.open_eq_iUnion hβ
choose f' hf' using hf
have : b β f' = f := funext hf'
subst this
obtain β¨t, htβ© :=
hβ.elim_countable_subcover (b β f') (fun i => hb.isOpen (Set.mem_range_self _)) Subset.rfl
refine β¨t.image f', Countable.ima... |
/-
Copyright (c) 2021 Kalle KytΓΆlΓ€. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle KytΓΆlΓ€
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 675 | 684 | theorem thickening_cthickening_subset (Ξ΅ : β) (hΞ΄ : 0 β€ Ξ΄) (s : Set Ξ±) :
thickening Ξ΅ (cthickening Ξ΄ s) β thickening (Ξ΅ + Ξ΄) s := by |
obtain hΞ΅ | hΞ΅ := le_total Ξ΅ 0
Β· simp only [thickening_of_nonpos hΞ΅, empty_subset]
intro x
simp_rw [mem_thickening_iff_exists_edist_lt, mem_cthickening_iff, β infEdist_lt_iff,
ENNReal.ofReal_add hΞ΅ hΞ΄]
rintro β¨y, hy, hxyβ©
exact infEdist_le_edist_add_infEdist.trans_lt
(ENNReal.add_lt_add_of_lt_of_le... |
/-
Copyright (c) 2020 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne, SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 382 | 389 | theorem norm_le_mul_norm_of_ae_le_mul {c : β} {f : Lp E p ΞΌ} {g : Lp F p ΞΌ}
(h : βα΅ x βΞΌ, βf xβ β€ c * βg xβ) : βfβ β€ c * βgβ := by |
rcases le_or_lt 0 c with hc | hc
Β· lift c to ββ₯0 using hc
exact NNReal.coe_le_coe.mpr (nnnorm_le_mul_nnnorm_of_ae_le_mul h)
Β· simp only [norm_def]
have := snorm_eq_zero_and_zero_of_ae_le_mul_neg h hc p
simp [this]
|
/-
Copyright (c) 2024 Jiecheng Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiecheng Zhao
-/
/-!
# Lemmas about `Array.extract`
Some useful lemmas about Array.extract
-/
set_option autoImplicit true
namespace Array
@[simp]
theorem extract_eq_nil_of_start_eq_... | Mathlib/Data/Array/ExtractLemmas.lean | 44 | 50 | theorem extract_extract {a : Array Ξ±} (h : s1 + e2 β€ e1) :
(a.extract s1 e1).extract s2 e2 = a.extract (s1 + s2) (s1 + e2) := by |
apply ext
Β· simp only [size_extract]
omega
Β· intro i h1 h2
simp only [get_extract, Nat.add_assoc]
|
/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere... | Mathlib/Geometry/Euclidean/Sphere/Power.lean | 84 | 96 | theorem mul_dist_eq_abs_sub_sq_dist {a b p q : P} (hp : β k : β, k β 1 β§ b -α΅₯ p = k β’ (a -α΅₯ p))
(hq : dist a q = dist b q) : dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2| := by |
let m : P := midpoint β a b
have h1 := vsub_sub_vsub_cancel_left a p m
have h2 := vsub_sub_vsub_cancel_left p q m
have h3 := vsub_sub_vsub_cancel_left a q m
have h : β r, b -α΅₯ r = m -α΅₯ r + (m -α΅₯ a) := fun r => by
rw [midpoint_vsub_left, β right_vsub_midpoint, add_comm, vsub_add_vsub_cancel]
iterate 4 r... |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,338 | 1,341 | theorem toTopologicalSpace_sInf {s : Set (UniformSpace Ξ±)} :
(sInf s).toTopologicalSpace = β¨
i β s, @UniformSpace.toTopologicalSpace Ξ± i := by |
rw [sInf_eq_iInf]
simp only [β toTopologicalSpace_iInf]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 251 | 255 | theorem degree_C_le : degree (C a) β€ 0 := by |
by_cases h : a = 0
Β· rw [h, C_0]
exact bot_le
Β· rw [degree_C h]
|
/-
Copyright (c) 2023 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Algebra.Module.Card
import Mathlib.SetTheory.Cardinal.CountableCover
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.Analysis.Specific... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 97 | 106 | theorem cardinal_eq_of_mem_nhds
{E : Type*} (π : Type*) [NontriviallyNormedField π] [AddCommGroup E] [Module π E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul π E]
{s : Set E} {x : E} (hs : s β π x) : #s = #E := by |
let g := Homeomorph.addLeft x
let t := g β»ΒΉ' s
have : t β π 0 := g.continuous.continuousAt.preimage_mem_nhds (by simpa [g] using hs)
have A : #t = #E := cardinal_eq_of_mem_nhds_zero π this
have B : #t = #s := Cardinal.mk_subtype_of_equiv s g.toEquiv
rwa [B] at A
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 582 | 584 | theorem tail_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :
(p.append p').support.tail = p.support.tail ++ p'.support.tail := by |
rw [support_append, List.tail_append_of_ne_nil _ _ (support_ne_nil _)]
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
impo... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 96 | 107 | theorem one_le_succ_nth_stream_b {ifp_succ_n : IntFractPair K}
(succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n) : 1 β€ ifp_succ_n.b := by |
obtain β¨ifp_n, nth_stream_eq, stream_nth_fr_ne_zero, β¨-β©β© :
β ifp_n, IntFractPair.stream v n = some ifp_n β§ ifp_n.fr β 0
β§ IntFractPair.of ifp_n.frβ»ΒΉ = ifp_succ_n :=
succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq
suffices 1 β€ ifp_n.frβ»ΒΉ by rwa [IntFractPair.of, le_floor, cast_one]
suffices if... |
/-
Copyright (c) 2019 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.f... | Mathlib/Topology/FiberBundle/Trivialization.lean | 566 | 572 | theorem continuousAt_of_comp_left {X : Type*} [TopologicalSpace X] {f : X β Z} {x : X}
(e : Trivialization F proj) (hf_proj : ContinuousAt (proj β f) x) (he : proj (f x) β e.baseSet)
(hf : ContinuousAt (e β f) x) : ContinuousAt f x := by |
rw [e.continuousAt_iff_continuousAt_comp_left]
Β· exact hf
rw [e.source_eq, β preimage_comp]
exact hf_proj.preimage_mem_nhds (e.open_baseSet.mem_nhds he)
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 1,562 | 1,564 | theorem insert'.valid [IsTotal Ξ± (Β· β€ Β·)] [@DecidableRel Ξ± (Β· β€ Β·)] (x : Ξ±) {t} (h : Valid t) :
Valid (insert' x t) := by |
rw [insert'_eq_insertWith]; exact insertWith.valid _ _ (fun _ => id) h
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 236 | 238 | theorem isClosed_imp {p q : X β Prop} (hp : IsOpen { x | p x }) (hq : IsClosed { x | q x }) :
IsClosed { x | p x β q x } := by |
simpa only [imp_iff_not_or] using hp.isClosed_compl.union hq
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f70... | Mathlib/Data/PFun.lean | 322 | 323 | theorem fix_fwd {f : Ξ± β. Sum Ξ² Ξ±} {b : Ξ²} {a a' : Ξ±} (hb : b β f.fix a) (ha' : Sum.inr a' β f a) :
b β f.fix a' := by | rwa [β fix_fwd_eq ha']
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, SΓ©bastien GouΓ«zel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,027 | 1,030 | theorem integral_congr_ae' (h : βα΅ x βΞΌ, x β Ioc a b β f x = g x)
(h' : βα΅ x βΞΌ, x β Ioc b a β f x = g x) : β« x in a..b, f x βΞΌ = β« x in a..b, g x βΞΌ := by |
simp only [intervalIntegral, setIntegral_congr_ae measurableSet_Ioc h,
setIntegral_congr_ae measurableSet_Ioc h']
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#a... | Mathlib/Geometry/Euclidean/Triangle.lean | 289 | 295 | theorem dist_eq_of_angle_eq_angle_of_angle_ne_pi {p1 p2 p3 : P} (h : β p1 p2 p3 = β p1 p3 p2)
(hpi : β p2 p1 p3 β Ο) : dist p1 p2 = dist p1 p3 := by |
unfold angle at h hpi
rw [dist_eq_norm_vsub V p1 p2, dist_eq_norm_vsub V p1 p3]
rw [β angle_neg_neg, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev] at hpi
rw [β vsub_sub_vsub_cancel_left p3 p2 p1, β vsub_sub_vsub_cancel_left p2 p3 p1] at h
exact norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi h hpi
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 45 | 50 | theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom β€ (FreeAbelianGroup X)).flip (of x) := by |
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Extensions of Groups
This file defines the HNN extension of a group `G`, `HNN... | Mathlib/GroupTheory/HNNExtension.lean | 490 | 492 | theorem prod_group_smul (g : G) (w : NormalWord d) :
(g β’ w).prod Ο = of g * (w.prod Ο) := by |
simp [ReducedWord.prod, smul_def, mul_assoc]
|
/-
Copyright (c) 2022 John Nicol. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: John Nicol
-/
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.wilson from "leanprover-community/mathlib"@"c471da714c044131b90c133701e51b877c246677"
/-!
# Wilson's the... | Mathlib/NumberTheory/Wilson.lean | 101 | 104 | theorem prime_iff_fac_equiv_neg_one (h : n β 1) : Prime n β ((n - 1)! : ZMod n) = -1 := by |
refine β¨fun h1 => ?_, fun h2 => prime_of_fac_equiv_neg_one h2 hβ©
haveI := Fact.mk h1
exact ZMod.wilsons_lemma n
|
/-
Copyright (c) 2017 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johannes HΓΆlzl, Chris Hughes, Jens Wagemaker, Jon Eugster
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Logic.Uni... | Mathlib/Algebra/Group/Units.lean | 238 | 238 | theorem val_eq_one {a : Ξ±Λ£} : (a : Ξ±) = 1 β a = 1 := by | rw [β Units.val_one, eq_iff]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, SΓ©bastien GouΓ«zel, RΓ©my Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | Mathlib/Analysis/MeanInequalitiesPow.lean | 299 | 309 | theorem add_rpow_le_rpow_add {p : β} (a b : ββ₯0β) (hp1 : 1 β€ p) : a ^ p + b ^ p β€ (a + b) ^ p := by |
have hp_pos : 0 < p := by positivity
by_cases h_top : a + b = β€
Β· rw [β @ENNReal.rpow_eq_top_iff_of_pos (a + b) p hp_pos] at h_top
rw [h_top]
exact le_top
obtain β¨ha_top, hb_topβ© := add_ne_top.mp h_top
lift a to ββ₯0 using ha_top
lift b to ββ₯0 using hb_top
simpa [β ENNReal.coe_rpow_of_nonneg _ hp_... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
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 f... | Mathlib/Data/List/Rotate.lean | 329 | 337 | theorem rotate_eq_iff {l l' : List Ξ±} {n : β} :
l.rotate n = l' β l = l'.rotate (l'.length - n % l'.length) := by |
rw [β @rotate_eq_rotate _ l _ n, rotate_rotate, β rotate_mod l', add_mod]
rcases l'.length.zero_le.eq_or_lt with hl | hl
Β· rw [eq_nil_of_length_eq_zero hl.symm, rotate_nil]
Β· rcases (Nat.zero_le (n % l'.length)).eq_or_lt with hn | hn
Β· simp [β hn]
Β· rw [mod_eq_of_lt (Nat.sub_lt hl hn), Nat.sub_add_canc... |
/-
Copyright (c) 2018 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 221 | 222 | theorem orderOf_pos_iff : 0 < orderOf x β IsOfFinOrder x := by |
rw [iff_not_comm.mp orderOf_eq_zero_iff, pos_iff_ne_zero]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 93 | 94 | theorem log_mul_ofReal (r : β) (hr : 0 < r) (x : β) (hx : x β 0) :
log (x * r) = Real.log r + log x := by | rw [mul_comm, log_ofReal_mul hr hx]
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 549 | 555 | theorem PosSemidef.fromBlocksββ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m π)
(B : Matrix m n π) {D : Matrix n n π} (hD : D.PosDef) [Invertible D] :
(fromBlocks A B Bα΄΄ D).PosSemidef β (A - B * Dβ»ΒΉ * Bα΄΄).PosSemidef := by |
rw [β posSemidef_submatrix_equiv (Equiv.sumComm n m), Equiv.sumComm_apply,
fromBlocks_submatrix_sum_swap_sum_swap]
convert PosSemidef.fromBlocksββ Bα΄΄ A hD <;>
simp
|
/-
Copyright (c) 2022 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 311 | 316 | theorem w_nonneg (D : β) (x : E) : 0 β€ w D x := by |
apply mul_nonneg _ (u_nonneg _)
apply inv_nonneg.2
apply mul_nonneg (u_int_pos E).le
norm_cast
apply pow_nonneg (abs_nonneg D)
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 62 | 70 | theorem coeff_det_X_add_C_zero (A B : Matrix n n Ξ±) :
coeff (det ((X : Ξ±[X]) β’ A.map C + B.map C)) 0 = det B := by |
rw [det_apply, finset_sum_coeff, det_apply]
refine Finset.sum_congr rfl ?_
rintro g -
convert coeff_smul (R := Ξ±) (sign g) _ 0
rw [coeff_zero_prod]
refine Finset.prod_congr rfl ?_
simp
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 3,051 | 3,052 | theorem takeWhile_idem : takeWhile p (takeWhile p l) = takeWhile p l := by |
simp_rw [takeWhile_takeWhile, and_self_iff, Bool.decide_coe]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lat... | Mathlib/Data/List/Perm.lean | 282 | 289 | theorem Perm.foldr_eq {f : Ξ± β Ξ² β Ξ²} {lβ lβ : List Ξ±} (lcomm : LeftCommutative f) (p : lβ ~ lβ) :
β b, foldr f b lβ = foldr f b lβ := by |
intro b
induction p using Perm.recOnSwap' generalizing b with
| nil => rfl
| cons _ _ r => simp; rw [r b]
| swap' _ _ _ r => simp; rw [lcomm, r b]
| trans _ _ rβ rβ => exact Eq.trans (rβ b) (rβ b)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | Mathlib/CategoryTheory/Abelian/Opposite.lean | 101 | 103 | theorem kernel.ΞΉ_op :
(kernel.ΞΉ f.op).unop = eqToHom (Opposite.unop_op _) β« cokernel.Ο f β« (kernelOpUnop f).inv := by |
simp [kernelOpUnop]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 1,239 | 1,244 | theorem support_update_ne_zero (f : Ξ β i, Ξ² i) (i : ΞΉ) {b : Ξ² i} (h : b β 0) :
support (f.update i b) = insert i f.support := by |
ext j
rcases eq_or_ne i j with (rfl | hi)
Β· simp [h]
Β· simp [hi.symm]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Separation
/-!
# Order-closed topologies
In this file we introduce 3 typeclass mixins that relate topology ... | Mathlib/Topology/Order/OrderClosed.lean | 900 | 903 | theorem Filter.tendsto_nhds_max_left {l : Filter Ξ²} {a : Ξ±} (h : Tendsto f l (π[>] a)) :
Tendsto (fun i => max (f i) a) l (π[>] a) := by |
simp_rw [max_comm _ a]
exact Filter.tendsto_nhds_max_right h
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Ralf Stephan, Neil Strickland, Ruben Van de Velde
-/
import Mathlib.Data.PNat.Defs
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Basic
import Mat... | Mathlib/Data/PNat/Basic.lean | 366 | 368 | theorem mod_add_div' (m k : β+) : (mod m k + div m k * k : β) = m := by |
rw [mul_comm]
exact mod_add_div _ _
|
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 187 | 202 | theorem isLittleO_of_lt_radius (h : βr < p.radius) :
β a β Ioo (0 : β) 1, (fun n => βp nβ * (r : β) ^ n) =o[atTop] (a ^ Β·) := by |
have := (TFAE_exists_lt_isLittleO_pow (fun n => βp nβ * (r : β) ^ n) 1).out 1 4
rw [this]
-- Porting note: was
-- rw [(TFAE_exists_lt_isLittleO_pow (fun n => βp nβ * (r : β) ^ n) 1).out 1 4]
simp only [radius, lt_iSup_iff] at h
rcases h with β¨t, C, hC, rtβ©
rw [ENNReal.coe_lt_coe, β NNReal.coe_lt_coe] at ... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PN... | Mathlib/NumberTheory/ADEInequality.lean | 259 | 268 | theorem admissible_of_one_lt_sumInv {p q r : β+} (H : 1 < sumInv {p, q, r}) :
Admissible {p, q, r} := by |
simp only [Admissible]
let S := sort ((Β· β€ Β·) : β+ β β+ β Prop) {p, q, r}
have hS : S.Sorted (Β· β€ Β·) := sort_sorted _ _
have hpqr : ({p, q, r} : Multiset β+) = S := (sort_eq LE.le {p, q, r}).symm
rw [hpqr]
rw [hpqr] at H
apply admissible_of_one_lt_sumInv_aux hS _ H
simp only [S, ge_iff_le, insert_eq_co... |
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl
-/
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 389 | 398 | theorem tprod_const [T2Space G] (a : G) : β' _ : Ξ², a = a ^ (Nat.card Ξ²) := by |
rcases finite_or_infinite Ξ² with hΞ²|hΞ²
Β· letI : Fintype Ξ² := Fintype.ofFinite Ξ²
rw [tprod_eq_prod (s := univ) (fun x hx β¦ (hx (mem_univ x)).elim)]
simp only [prod_const, Nat.card_eq_fintype_card, Fintype.card]
Β· simp only [Nat.card_eq_zero_of_infinite, pow_zero]
rcases eq_or_ne a 1 with rfl|ha
Β· ... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Grou... | Mathlib/RingTheory/Coprime/Basic.lean | 351 | 353 | theorem mul_add_left_right {x y : R} (h : IsCoprime x y) (z : R) : IsCoprime x (x * z + y) := by |
rw [add_comm]
exact h.add_mul_left_right z
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 127 | 131 | theorem monomial_def [DecidableEq Ο] (n : Ο ββ β) :
(monomial R n) = LinearMap.stdBasis R (fun _ β¦ R) n := by |
rw [monomial]
-- unify the `Decidable` arguments
convert rfl
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes HΓΆlzl, Mario Carneiro, SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 206 | 209 | theorem dist_triangle4_left (xβ yβ xβ yβ : Ξ±) :
dist xβ yβ β€ dist xβ yβ + (dist xβ xβ + dist yβ yβ) := by |
rw [add_left_comm, dist_comm xβ, β add_assoc]
apply dist_triangle4
|
/-
Copyright (c) 2021 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, SΓ©bastien GouΓ«zel, RΓ©my Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,286 | 1,291 | theorem setToFun_congr_left (hT : DominatedFinMeasAdditive ΞΌ T C)
(hT' : DominatedFinMeasAdditive ΞΌ T' C') (h : T = T') (f : Ξ± β E) :
setToFun ΞΌ T hT f = setToFun ΞΌ T' hT' f := by |
by_cases hf : Integrable f ΞΌ
Β· simp_rw [setToFun_eq _ hf, L1.setToL1_congr_left T T' hT hT' h]
Β· simp_rw [setToFun_undef _ hf]
|
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-commun... | Mathlib/Algebra/CharP/ExpChar.lean | 313 | 314 | theorem iterateFrobenius_zero_apply : iterateFrobenius R p 0 x = x := by |
rw [iterateFrobenius_def, pow_zero, pow_one]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 324 | 325 | theorem lt_mul_inv_iff_lt : 1 < a * bβ»ΒΉ β b < a := by |
rw [β mul_lt_mul_iff_right b, one_mul, inv_mul_cancel_right]
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510e... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 62 | 65 | theorem aeval_algebraMap_eq_zero_iff_of_injective {x : Ο β A} {p : MvPolynomial Ο R}
(h : Function.Injective (algebraMap A B)) :
aeval (algebraMap A B β x) p = 0 β aeval x p = 0 := by |
rw [aeval_algebraMap_apply, β (algebraMap A B).map_zero, h.eq_iff]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 455 | 457 | theorem isTheta_exp_comp_one {f : Ξ± β β} :
(fun x => exp (f x)) =Ξ[l] (fun _ => 1 : Ξ± β β) β IsBoundedUnder (Β· β€ Β·) l fun x => |f x| := by |
simp only [β exp_zero, isTheta_exp_comp_exp_comp, sub_zero]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 348 | 351 | theorem isometry_vertical_line (a : β) : Isometry fun y => mk β¨a, exp yβ© (exp_pos y) := by |
refine Isometry.of_dist_eq fun yβ yβ => ?_
rw [dist_of_re_eq]
exacts [congr_argβ _ (log_exp _) (log_exp _), rfl]
|
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael GeiΓer, Michael Stoll
-/
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
... | Mathlib/NumberTheory/Pell.lean | 226 | 230 | theorem y_ne_zero_of_one_lt_x {a : Solutionβ d} (ha : 1 < a.x) : a.y β 0 := by |
intro hy
have prop := a.prop
rw [hy, sq (0 : β€), zero_mul, mul_zero, sub_zero] at prop
exact lt_irrefl _ (((one_lt_sq_iff <| zero_le_one.trans ha.le).mpr ha).trans_eq prop)
|
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Nilpotent.lean | 704 | 715 | theorem lowerCentralSeries_prod (n : β) :
lowerCentralSeries (Gβ Γ Gβ) n = (lowerCentralSeries Gβ n).prod (lowerCentralSeries Gβ n) := by |
induction' n with n ih
Β· simp
Β· calc
lowerCentralSeries (Gβ Γ Gβ) n.succ = β
lowerCentralSeries (Gβ Γ Gβ) n, β€β := rfl
_ = β
(lowerCentralSeries Gβ n).prod (lowerCentralSeries Gβ n), β€β := by rw [ih]
_ = β
(lowerCentralSeries Gβ n).prod (lowerCentralSeries Gβ n), (β€ : Subgroup Gβ).prod β€β := by
... |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 229 | 232 | theorem le_caratheodory [MeasurableSpace X] [BorelSpace X] (hm : IsMetric ΞΌ) :
βΉMeasurableSpace XβΊ β€ ΞΌ.caratheodory := by |
rw [BorelSpace.measurable_eq (Ξ± := X)]
exact hm.borel_le_caratheodory
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 301 | 309 | theorem Subpresheaf.to_sheafifyLift (f : G.toPresheaf βΆ F') (h : Presieve.IsSheaf J F') :
Subpresheaf.homOfLe (G.le_sheafify J) β« G.sheafifyLift f h = f := by |
ext U s
apply (h _ ((Subpresheaf.homOfLe (G.le_sheafify J)).app U s).prop).isSeparatedFor.ext
intro V i hi
have := elementwise_of% f.naturality
-- Porting note: filled in some underscores where Lean3 could automatically fill.
exact (Presieve.IsSheafFor.valid_glue (h _ ((homOfLe (_ : G β€ sheafify J G)).app ... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 428 | 433 | theorem eval_C_mul : (C a * p).eval x = a * p.eval x := by |
induction p using Polynomial.induction_on' with
| h_add p q ph qh =>
simp only [mul_add, eval_add, ph, qh]
| h_monomial n b =>
simp only [mul_assoc, C_mul_monomial, eval_monomial]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes HΓΆlzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polyno... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 56 | 59 | theorem integralNormalization_support {f : R[X]} :
(integralNormalization f).support β f.support := by |
intro
simp (config := { contextual := true }) [integralNormalization, coeff_monomial, mem_support_iff]
|
/-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Data.FunLike... | Mathlib/Topology/MetricSpace/Dilation.lean | 519 | 520 | theorem diam_image (s : Set Ξ±) : Metric.diam ((f : Ξ± β Ξ²) '' s) = ratio f * Metric.diam s := by |
simp [Metric.diam, ediam_image, ENNReal.toReal_mul]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, SΓ©bastien GouΓ«zel, RΓ©my Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | Mathlib/Analysis/MeanInequalities.lean | 520 | 542 | theorem Lp_add_le_tsum {f g : ΞΉ β ββ₯0} {p : β} (hp : 1 β€ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) β§
(β' i, (f i + g i) ^ p) ^ (1 / p) β€
(β' i, f i ^ p) ^ (1 / p) + (β' i, g i ^ p) ^ (1 / p) := by |
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp
have Hβ : β s : Finset ΞΉ,
(β i β s, (f i + g i) ^ p) β€
((β' i, f i ^ p) ^ (1 / p) + (β' i, g i ^ p) ^ (1 / p)) ^ p := by
intro s
rw [β NNReal.rpow_one_div_le_iff pos]
refine le_trans (Lp_add_le s f g hp) (add_le_add ?_ ?_) <;>
rw [... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.P... | Mathlib/NumberTheory/Bernoulli.lean | 181 | 196 | theorem bernoulli'_odd_eq_zero {n : β} (h_odd : Odd n) (hlt : 1 < n) : bernoulli' n = 0 := by |
let B := mk fun n => bernoulli' n / (n ! : β)
suffices (B - evalNegHom B) * (exp β - 1) = X * (exp β - 1) by
cases' mul_eq_mul_right_iff.mp this with h h <;>
simp only [PowerSeries.ext_iff, evalNegHom, coeff_X] at h
Β· apply eq_zero_of_neg_eq
specialize h n
split_ifs at h <;> simp_all [B, ... |
/-
Copyright (c) 2022 Antoine Labelle, RΓ©mi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, RΓ©mi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 99 | 103 | theorem Path.cast_cast {u v u' v' u'' v'' : U} (p : Path u v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(p.cast hu hv).cast hu' hv' = p.cast (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,716 | 1,718 | theorem comap_top {f : Pβ βα΅[k] Pβ} : (β€ : AffineSubspace k Pβ).comap f = β€ := by |
rw [β ext_iff]
exact preimage_univ (f := f)
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 537 | 538 | theorem wellFoundedOn_insert : WellFoundedOn (insert a s) r β WellFoundedOn s r := by |
simp only [β singleton_union, wellFoundedOn_union, wellFoundedOn_singleton, true_and_iff]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.Cat... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 141 | 151 | theorem IsUniversalColimit.of_iso {F : J β₯€ C} {c c' : Cocone F} (hc : IsUniversalColimit c)
(e : c β
c') : IsUniversalColimit c' := by |
intro F' c'' Ξ± f h hΞ± H
have : c'.ΞΉ β« (Functor.const J).map e.inv.hom = c.ΞΉ := by
ext j
exact e.inv.2 j
apply hc c'' Ξ± (f β« e.inv.1) (by rw [Functor.map_comp, β reassoc_of% h, this]) hΞ±
intro j
rw [β Category.comp_id (Ξ±.app j)]
have : IsIso e.inv.hom := Functor.map_isIso (Cocones.forget _) e.inv
... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 558 | 562 | theorem affineCombination_filter_of_ne (w : ΞΉ β k) (p : ΞΉ β P) {pred : ΞΉ β Prop}
[DecidablePred pred] (h : β i β s, w i β 0 β pred i) :
(s.filter pred).affineCombination k p w = s.affineCombination k p w := by |
rw [affineCombination_apply, affineCombination_apply,
s.weightedVSubOfPoint_filter_of_ne _ _ _ h]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-communit... | Mathlib/AlgebraicGeometry/Gluing.lean | 319 | 322 | theorem glued_cover_cocycle_fst (x y z : π°.J) :
gluedCoverT' π° x y z β« gluedCoverT' π° y z x β« gluedCoverT' π° z x y β« pullback.fst =
pullback.fst := by |
apply pullback.hom_ext <;> simp
|
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Logic.Equiv.TransferInstance
import Ma... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 514 | 521 | theorem auxGroup_indep (i j : N) : (auxGroup i : Group (HomotopyGroup N X x)) = auxGroup j := by |
by_cases h : i = j; Β· rw [h]
refine Group.ext (EckmannHilton.mul (isUnital_auxGroup i) (isUnital_auxGroup j) ?_)
rintro β¨aβ© β¨bβ© β¨cβ© β¨dβ©
change Quotient.mk' _ = _
apply congr_arg Quotient.mk'
simp only [fromLoop_trans_toLoop, transAt_distrib h, coe_toEquiv, loopHomeo_apply,
coe_symm_toEquiv, loopHomeo_s... |
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 114 | 118 | theorem IsIntegrallyClosed.degree_le_of_ne_zero {s : S} (hs : IsIntegral R s) {p : R[X]}
(hp0 : p β 0) (hp : Polynomial.aeval s p = 0) : degree (minpoly R s) β€ degree p := by |
rw [degree_eq_natDegree (minpoly.ne_zero hs), degree_eq_natDegree hp0]
norm_cast
exact natDegree_le_of_dvd ((isIntegrallyClosed_dvd_iff hs _).mp hp) hp0
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.Homology
import Mathlib.Algebra.Homology.Single
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
#align_import algebra.homol... | Mathlib/Algebra/Homology/Additive.lean | 338 | 340 | theorem singleMapHomologicalComplex_inv_app_ne {i j : ΞΉ} (h : i β j) (X : Wβ) :
((singleMapHomologicalComplex F c j).inv.app X).f i = 0 := by |
simp [singleMapHomologicalComplex, h]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, FrΓ©dΓ©ric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 452 | 455 | theorem coe_sup : β(p β p') = (p + p' : Set M) := by |
ext
rw [SetLike.mem_coe, mem_sup, Set.mem_add]
simp
|
/-
Copyright (c) 2022 YaΓ«l Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: YaΓ«l Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 205 | 216 | theorem threeAPFree_image_sphere :
ThreeAPFree ((sphere n d k).image (map (2 * d - 1)) : Set β) := by |
rw [coe_image]
apply ThreeAPFree.image' (Ξ± := Fin n β β) (Ξ² := β) (s := sphere n d k) (map (2 * d - 1))
(map_injOn.mono _) threeAPFree_sphere
Β· rw [Set.add_subset_iff]
rintro a ha b hb i
have hai := mem_box.1 (sphere_subset_box ha) i
have hbi := mem_box.1 (sphere_subset_box hb) i
rw [lt_tsub_... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes HΓΆlzl, Sander Dahmen, Scott Morrison
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeMod... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 431 | 434 | theorem FiniteDimensional.finrank_eq_zero_iff :
finrank R M = 0 β β x : M, β a : R, a β 0 β§ a β’ x = 0 := by |
rw [β rank_eq_zero_iff (R := R), β finrank_eq_rank]
norm_cast
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 119 | 124 | theorem le_closure (f : E ββ.[R] F) : f β€ f.closure := by |
by_cases hf : f.IsClosable
Β· refine le_of_le_graph ?_
rw [β hf.graph_closure_eq_closure_graph]
exact (graph f).le_topologicalClosure
rw [closure_def' hf]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes HΓΆlzl
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canoni... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 143 | 144 | theorem add_lt_top [LT Ξ±] {a b : WithTop Ξ±} : a + b < β€ β a < β€ β§ b < β€ := by |
simp_rw [WithTop.lt_top_iff_ne_top, add_ne_top]
|
/-
Copyright (c) 2018 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Topology.Algebra.Order.Compact
import Mathlib.Topology.MetricSpace.PseudoMetric
/-! ## Proper spaces
## Main definitions and results
* `ProperS... | Mathlib/Topology/MetricSpace/ProperSpace.lean | 159 | 165 | theorem Metric.exists_isLocalMin_mem_ball [PseudoMetricSpace Ξ±] [ProperSpace Ξ±] [TopologicalSpace Ξ²]
[ConditionallyCompleteLinearOrder Ξ²] [OrderTopology Ξ²] {f : Ξ± β Ξ²} {a z : Ξ±} {r : β}
(hf : ContinuousOn f (closedBall a r)) (hz : z β closedBall a r)
(hf1 : β z' β sphere a r, f z < f z') : β z β ball a r, I... |
simp_rw [β closedBall_diff_ball] at hf1
exact (isCompact_closedBall a r).exists_isLocalMin_mem_open ball_subset_closedBall hf hz hf1
isOpen_ball
|
/-
Copyright (c) 2019 Calle SΓΆnne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle SΓΆnne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 986 | 989 | theorem sign_neg_coe_nonpos_of_nonneg_of_le_pi {ΞΈ : β} (h0 : 0 β€ ΞΈ) (hpi : ΞΈ β€ Ο) :
(-ΞΈ : Angle).sign β€ 0 := by |
rw [sign, sign_nonpos_iff, sin_neg, Left.neg_nonpos_iff]
exact sin_nonneg_of_nonneg_of_le_pi h0 hpi
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 1,116 | 1,118 | theorem Prod.noncompactSpace_iff :
NoncompactSpace (X Γ Y) β NoncompactSpace X β§ Nonempty Y β¨ Nonempty X β§ NoncompactSpace Y := by |
simp [β Filter.cocompact_neBot_iff, β Filter.coprod_cocompact, Filter.coprod_neBot_iff]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,836 | 1,845 | theorem bsup_succ_le_blsub {o : Ordinal.{u}} (f : β a < o, Ordinal.{max u v}) :
succ (bsup.{_, v} o f) β€ blsub.{_, v} o f β β i hi, f i hi = bsup.{_, v} o f := by |
refine β¨fun h => ?_, ?_β©
Β· by_contra! hf
exact
ne_of_lt (succ_le_iff.1 h)
(le_antisymm (bsup_le_blsub f) (blsub_le (lt_bsup_of_ne_bsup.1 hf)))
rintro β¨_, _, hfβ©
rw [succ_le_iff, β hf]
exact lt_blsub _ _ _
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 715 | 716 | theorem preimage_const_mul_Ico (a b : Ξ±) {c : Ξ±} (h : 0 < c) :
(c * Β·) β»ΒΉ' Ico a b = Ico (a / c) (b / c) := by | simp [β Ici_inter_Iio, h]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,210 | 1,215 | theorem neg_left {M : Type*} [AddCommGroup M] [TopologicalSpace M] [TopologicalAddGroup M]
{v : VectorMeasure Ξ± M} {w : VectorMeasure Ξ± N} (h : v βα΅₯ w) : -v βα΅₯ w := by |
obtain β¨u, hmu, huβ, huββ© := h
refine β¨u, hmu, fun s hs => ?_, huββ©
rw [neg_apply v s, neg_eq_zero]
exact huβ s hs
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,106 | 1,109 | theorem measurable_tProd_mk (l : List Ξ΄) : Measurable (@TProd.mk Ξ΄ Ο l) := by |
induction' l with i l ih
Β· exact measurable_const
Β· exact (measurable_pi_apply i).prod_mk ih
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle SΓΆnne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 496 | 502 | theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : β} (hr : 0 < r) (ΞΈ : β) :
arg (r * (cos ΞΈ + sin ΞΈ * I)) = toIocMod Real.two_pi_pos (-Ο) ΞΈ := by |
have hi : toIocMod Real.two_pi_pos (-Ο) ΞΈ β Set.Ioc (-Ο) Ο := by
convert toIocMod_mem_Ioc _ _ ΞΈ
ring
convert arg_mul_cos_add_sin_mul_I hr hi using 3
simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
/-!
# Induced Outer Measure
We can extend a function defined on a subset of `Set Ξ±` to an out... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 129 | 133 | theorem extend_mono' β¦sβ sβ : Set Ξ±β¦ (hβ : P sβ) (hs : sβ β sβ) : extend m sβ β€ extend m sβ := by |
refine le_iInf ?_
intro hβ
rw [extend_eq m hβ]
exact m_mono hβ hβ hs
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import M... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 275 | 277 | theorem Unit_of_divided_by_X_pow_order_nonzero {f : kβ¦Xβ§} (hf : f β 0) :
β(Unit_of_divided_by_X_pow_order f) = divided_by_X_pow_order hf := by |
simp only [Unit_of_divided_by_X_pow_order, dif_neg hf, Units.val_mk]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Init.Align
import Mathlib.Data.Fintype.Order
import Mathlib.Algebra.DirectLimit
import Mathlib.ModelTheory.Quotients
import Mathlib.ModelT... | Mathlib/ModelTheory/DirectLimit.lean | 324 | 327 | theorem of_f {i j : ΞΉ} {hij : i β€ j} {x : G i} : of L ΞΉ G f j (f i j hij x) = of L ΞΉ G f i x := by |
rw [of_apply, of_apply, Quotient.eq]
refine Setoid.symm β¨j, hij, refl j, ?_β©
simp only [DirectedSystem.map_self]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Logic.Function.Basic
#align_import logic.is_empty from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
/-!
# Types that... | Mathlib/Logic/IsEmpty.lean | 163 | 164 | theorem isEmpty_fun : IsEmpty (Ξ± β Ξ²) β Nonempty Ξ± β§ IsEmpty Ξ² := by |
rw [isEmpty_pi, β exists_true_iff_nonempty, β exists_and_right, true_and]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes HΓΆlzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.... | Mathlib/Data/ENNReal/Basic.lean | 723 | 724 | theorem iInter_Ioi_coe_nat : β n : β, Ioi (n : ββ₯0β) = {β} := by |
simp only [β compl_Iic, β compl_iUnion, iUnion_Iic_coe_nat, compl_compl]
|
/-
Copyright (c) 2020 SΓ©bastien GouΓ«zel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: SΓ©bastien GouΓ«zel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 405 | 413 | theorem mem_range_embedding_iff' {j : Fin n} {i : Fin c.length} :
j β Set.range (c.embedding i) β i = c.index j := by |
constructor
Β· rw [β not_imp_not]
intro h
exact Set.disjoint_right.1 (c.disjoint_range h) (c.mem_range_embedding j)
Β· intro h
rw [h]
exact c.mem_range_embedding j
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | Mathlib/Topology/Instances/AddCircle.lean | 444 | 449 | theorem addOrderOf_coe_rat {q : β} : addOrderOf (β(βq * p) : AddCircle p) = q.den := by |
have : (β(q.den : β€) : π) β 0 := by
norm_cast
exact q.pos.ne.symm
rw [β q.num_divInt_den, Rat.cast_divInt_of_ne_zero _ this, Int.cast_natCast, Rat.num_divInt_den,
addOrderOf_div_of_gcd_eq_one' q.pos q.reduced]
|
/-
Copyright (c) 2017 Johannes HΓΆlzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta HernΓ‘ndez Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,327 | 2,328 | theorem natCast_le {m n : β} : (m : Ordinal) β€ n β m β€ n := by |
rw [β Cardinal.ord_nat, β Cardinal.ord_nat, Cardinal.ord_le_ord, Cardinal.natCast_le]
|
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Logic.Equiv.TransferInstance
import Ma... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 292 | 305 | theorem homotopicTo (i : N) {p q : Ξ©^ N X x} :
Homotopic p q β (toLoop i p).Homotopic (toLoop i q) := by |
refine Nonempty.map fun H => β¨β¨β¨fun t => β¨homotopyTo i H t, ?_β©, ?_β©, ?_, ?_β©, ?_β©
Β· rintro y β¨i, iHβ©
rw [homotopyTo_apply, H.eq_fst, p.2]
all_goals apply Cube.insertAt_boundary; right; exact β¨i, iHβ©
Β· continuity
iterate 2 intro; ext; erw [homotopyTo_apply, toLoop_apply]; swap
Β· apply H.apply_zero
... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 663 | 675 | theorem prod_swap : map Prod.swap (ΞΌ.prod Ξ½) = Ξ½.prod ΞΌ := by |
have : sum (fun (i : β Γ β) β¦ map Prod.swap ((sFiniteSeq ΞΌ i.1).prod (sFiniteSeq Ξ½ i.2)))
= sum (fun (i : β Γ β) β¦ map Prod.swap ((sFiniteSeq ΞΌ i.2).prod (sFiniteSeq Ξ½ i.1))) := by
ext s hs
rw [sum_apply _ hs, sum_apply _ hs]
exact ((Equiv.prodComm β β).tsum_eq _).symm
rw [β sum_sFiniteSeq ΞΌ, β ... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 270 | 275 | theorem single_finset_sum [AddCommMonoid M] (s : Finset ΞΉ) (f : ΞΉ β M) (a : Ξ±) :
single a (β b β s, f b) = β b β s, single a (f b) := by |
trans
Β· apply single_multiset_sum
Β· rw [Multiset.map_map]
rfl
|
/-
Copyright (c) 2023 RΓ©my Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: RΓ©my Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 129 | 144 | theorem generateFrom_squareCylinders [β i, MeasurableSpace (Ξ± i)] :
MeasurableSpace.generateFrom (squareCylinders fun i β¦ {s : Set (Ξ± i) | MeasurableSet s}) =
MeasurableSpace.pi := by |
apply le_antisymm
Β· rw [MeasurableSpace.generateFrom_le_iff]
rintro S β¨s, t, h, rflβ©
simp only [mem_univ_pi, mem_setOf_eq] at h
exact MeasurableSet.pi (Finset.countable_toSet _) (fun i _ β¦ h i)
Β· refine iSup_le fun i β¦ ?_
refine (comap_eval_le_generateFrom_squareCylinders_singleton Ξ± i).trans ?_
... |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 346 | 349 | theorem HasStrictDerivAt.finset_prod (hf : β i β u, HasStrictDerivAt (f i) (f' i) x) :
HasStrictDerivAt (β i β u, f i Β·) (β i β u, (β j β u.erase i, f j x) β’ f' i) x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasStrictFDerivAt.finset_prod (fun i hi β¦ (hf i hi).hasStrictFDerivAt)).hasStrictDerivAt
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes HΓΆlzl, YaΓ«l Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,808 | 1,812 | theorem edist_mul_mul_le (aβ aβ bβ bβ : E) :
edist (aβ * aβ) (bβ * bβ) β€ edist aβ bβ + edist aβ bβ := by |
simp only [edist_nndist]
norm_cast
apply nndist_mul_mul_le
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.