Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... | @[simp]
theorem torsion'_torsion'_eq_top : torsion' R (torsion' R M S) S = ⊤ :=
(isTorsion'_iff_torsion'_eq_top S).mp <| torsion'_isTorsion' S
| Mathlib/Algebra/Module/Torsion.lean | 710 | 713 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | (EuclideanDomain.mul_div_cancel' (gcd_ne_zero_of_right hq) <| gcd_dvd_left _ _).symm
(EuclideanDomain.mul_div_cancel' (gcd_ne_zero_of_right hq) <| gcd_dvd_right _ _).symm <|
gcd_ne_zero_of_right hq
end GCDMonoid
| Mathlib/RingTheory/EuclideanDomain.lean | 50 | 55 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,668 | 3,688 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | · exact h₁
instance sub_nf (o₁ o₂) : ∀ [NF o₁] [NF o₂], NF (o₁ - o₂)
| ⟨⟨b₁, h₁⟩⟩, h₂ => ⟨⟨b₁, sub_nfBelow h₁ h₂⟩⟩
@[simp]
theorem repr_sub : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ - o₂) = repr o₁ - repr o₂
| 0, o, _, h₂ => by cases o <;> exact (Ordinal.zero_sub _).symm
| oadd _ _ _, 0, _, _ => (Ordinal.sub_ze... | Mathlib/SetTheory/Ordinal/Notation.lean | 462 | 475 |
/-
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.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Topology.Instances.AddCircle
/-!
# The additive circle as a norm... | simp only [Real.closedBall_eq_Icc, zero_sub, zero_add] at hs ⊢
rcases eq_or_ne z 0 with (rfl | hz)
· simp
simp only [hz, zsmul_eq_mul, if_false, eq_empty_iff_forall_not_mem]
| Mathlib/Analysis/Normed/Group/AddCircle.lean | 169 | 172 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | add_comm := by
rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩
simp only [add_quot, add_comm]
nsmul := (· • ·)
nsmul_zero := by
| Mathlib/Algebra/RingQuot.lean | 269 | 273 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]
dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢
abel
lemma ih4 (ih' : ∀ a, ArgsRel a (Args.P24 x₁ x₂ y) → P124 a) : IH4 x₁ x₂ y := by
| Mathlib/SetTheory/Surreal/Multiplication.lean | 303 | 307 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Eisenstein.Criterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# Eisenstein polynomials
Given an ideal `𝓟` of a com... | exact mul_mem_right _ _ this
rw [aeval_def, eval₂_eq_eval_map, ← IsRoot.def] at hx
exact pow_natDegree_le_of_root_of_monic_mem (hf.map _) hx (hmo.map _) _ rfl.le
end CommRing
end IsWeaklyEisensteinAt
section ScaleRoots
| Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 141 | 150 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Prod
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.Bases.Basic
/... | theorem mem_lift'_sets (hh : Monotone h) {s : Set β} : s ∈ f.lift' h ↔ ∃ t ∈ f, h t ⊆ s :=
mem_lift_sets <| monotone_principal.comp hh
theorem eventually_lift'_iff (hh : Monotone h) {p : β → Prop} :
(∀ᶠ y in f.lift' h, p y) ↔ ∃ t ∈ f, ∀ y ∈ h t, p y :=
| Mathlib/Order/Filter/Lift.lean | 216 | 220 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... |
/-- auxiliary definition for `FullyFaithful.hasShift` -/
def add (a b : A) : s (a + b) ≅ s a ⋙ s b :=
(hF.whiskeringRight C).preimageIso (i (a + b) ≪≫ isoWhiskerLeft _ (shiftFunctorAdd D a b) ≪≫
(Functor.associator _ _ _).symm ≪≫ (isoWhiskerRight (i a).symm _) ≪≫
| Mathlib/CategoryTheory/Shift/Basic.lean | 686 | 690 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... |
@[deprecated (since := "2024-12-01")] alias degreeOf_mul_X_ne := degreeOf_mul_X_of_ne
theorem degreeOf_mul_X_self (j : σ) (f : MvPolynomial σ R) :
degreeOf j (f * X j) ≤ degreeOf j f + 1 := by
classical
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 295 | 300 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | lemma hom_coe_pow {F : Type*} [Monoid F] (c : F → M → M) (h1 : c 1 = id)
(hmul : ∀ f g, c (f * g) = c f ∘ c g) (f : F) : ∀ n, c (f ^ n) = (c f)^[n]
| 0 => by
| Mathlib/Algebra/Group/Basic.lean | 1,052 | 1,054 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | by_cases h : p ∣ orderOf x
exacts [hdvd p hp h, (hp.coprime_iff_not_dvd.2 h).mul_dvd_of_dvd_of_dvd hpy hxy]
| Mathlib/GroupTheory/OrderOfElement.lean | 451 | 453 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 665 | 673 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... |
@[to_additive additive_of_symmetric_of_isTotal]
| Mathlib/Algebra/Group/Basic.lean | 1,014 | 1,015 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,062 | 1,066 | |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 998 | 999 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... |
end MeasureTheory
| Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 122 | 126 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | (gc_generateFrom α).u_iInf
theorem setOf_isOpen_iSup {t : ι → TopologicalSpace α} :
| Mathlib/Topology/Order.lean | 842 | 844 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... |
theorem cast_add (h : m ∣ n) (a b : ZMod n) : (cast (a + b : ZMod n) : R) = cast a + cast b := by
cases n
· apply Int.cast_add
symm
| Mathlib/Data/ZMod/Basic.lean | 284 | 288 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 382 | 389 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | refine le_iInf₂ fun i hi => ?_
calc
edist x y ≤ edist x i + edist i y := edist_triangle _ _ _
_ ≤ edist x i + diam s := add_le_add le_rfl (edist_le_diam_of_mem hi hy)
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 126 | 130 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | have := not_cliqueFree_of_le_card V f le_rfl
contrapose! this
exact hc.mono this
end completeMultipartiteGraph
end SimpleGraph
| Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 482 | 492 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Huỳnh Trần Khanh, Stuart Presnell
-/
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Math... | /-- Type **stars and bars** for the case `n = 2`. -/
protected theorem card {α} [Fintype α] : card (Sym2 α) = Nat.choose (card α + 1) 2 :=
Finset.card_sym2 _
end Sym2
| Mathlib/Data/Sym/Card.lean | 173 | 179 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | · exact b.elim0
rcases m with - | m
· rw [zero_pow n.succ_ne_zero] at a
exact a.elim0
· exact Nat.mod_lt _ m.succ_pos⟩)
fun a => by
dsimp
induction n with
| zero => subsingleton [(finCongr <| pow_zero _).subsingleton]
| succ n ih =>
simp_rw [Fin.forall... | Mathlib/Algebra/BigOperators/Fin.lean | 295 | 315 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,240 | 1,241 | |
/-
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.Ring.Int.Units
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
/-!
# fintype instanc... | Fintype.card αˣ = Fintype.card α - 1 := by
rw [← Nat.card_eq_fintype_card, Nat.card_units, Nat.card_eq_fintype_card]
| Mathlib/Data/Fintype/Units.lean | 48 | 50 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,340 | 1,352 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.Prime.Factorial
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# Lemmas of Gauss and Eisenstein
... | add_neg_eq_iff_eq_add.symm, add_assoc] using
Eq.symm (eisenstein_lemma_aux₁ p hap)
theorem div_eq_filter_card {a b c : ℕ} (hb0 : 0 < b) (hc : a / b ≤ c) :
a / b = #{x ∈ Ico 1 c.succ | x * b ≤ a} :=
calc
a / b = #(Ico 1 (a / b).succ) := by simp
_ = #{x ∈ Ico 1 c.succ | x * b ≤ a} :=
cong... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 146 | 154 |
/-
Copyright (c) 2018 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.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | Mathlib/Computability/Primrec.lean | 1,532 | 1,534 | |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
/-!
# Topological star (sub)algebras
A topological... | (hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ := by
refine StarAlgHomClass.ext_topologicalClosure hφ hψ fun x => ?_
refine adjoin_induction_subtype x ?_ ?_ ?_ ?_ ?_
exacts [fun y hy => by simpa only [Set.mem_singleton_iff.mp hy] using h, fun r => by
simp only [AlgHomClass.com... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 270 | 281 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Constructions
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
/-! ## Equivalence of smoothne... | hg.contDiffAt.comp_contMDiffWithinAt hf
| Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 81 | 82 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... |
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 136 | 137 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | simp only [g, Polynomial.Monic.def, ← Polynomial.leadingCoeff_det_X_one_add_C M, add_comm]
rw [← f'_adj, ← f'_adj, ← f'_adj, ← f'.map_mul, ←
adjugate_mul_distrib_aux _ _ (hu A).left (hu B).left, RingHom.map_adjugate,
RingHom.map_adjugate, f'_inv, f'_g_mul]
| Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 462 | 465 |
/-
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.Data.NNReal.Basic
import Mathlib.Order.Fin.Tuple
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.... | inf_le_right := fun I J ↦ by
rw [← withBotCoe_subset_iff, coe_inf]
exact inter_subset_right
| Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 325 | 327 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | Mathlib/Topology/PartialHomeomorph.lean | 1,517 | 1,523 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Properties of LUB and GLB in an order topology
-/
open Set Filter TopologicalSpa... | theorem IsLUB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
{s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=
haveI := ha.nhdsWithin_neBot hs
⟨ha.mem_upperBounds_of_tendsto hf ... | Mathlib/Topology/Order/IsLUB.lean | 93 | 100 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | @[simp, norm_cast]
theorem coe_inv {f : CauSeq β abv} (hf) : ⇑(inv f hf) = (f : ℕ → β)⁻¹ :=
rfl
@[simp, norm_cast]
theorem inv_apply {f : CauSeq β abv} (hf i) : inv f hf i = (f i)⁻¹ :=
rfl
theorem inv_mul_cancel {f : CauSeq β abv} (hf) : inv f hf * f ≈ 1 := fun ε ε0 =>
let ⟨K, K0, i, H⟩ := abv_pos_of_not_limZer... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 546 | 562 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Data.Nat.Factorization.Defs
/-!
# Roots of natural numbers, rounded up and down
This file defines the flooring and... | @[simp] lemma factorization_floorRoot (n a : ℕ) :
(floorRoot n a).factorization = a.factorization ⌊/⌋ n := by
rw [floorRoot_def]
split_ifs with h
· obtain rfl | rfl := h <;> simp
refine prod_pow_factorization_eq_self fun p hp ↦ ?_
have : p.Prime ∧ p ∣ a ∧ ¬a = 0 := by simpa using support_floorDiv_subset h... | Mathlib/Data/Nat/Factorization/Root.lean | 73 | 80 |
/-
Copyright (c) 2024 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Analysis.InnerProductSpace.NormP... | simpa using hpq
_ = eLpNorm u p' μ * t := by rw [eLpNorm_restrict_eq_of_support_subset h2u]
_ ≤ (C * eLpNorm (fderiv ℝ u) p μ) * t := by
have h2u' : HasCompactSupport u := by
apply HasCompactSupport.of_support_subset_isCompact hs.isCompact_closure
exact h2u.trans subset_clo... | Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | 702 | 709 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... |
This depends on `W`, and has argument order: `x₁`, `x₂`, `y₁`, `ℓ`. -/
@[simp]
def negAddY (x₁ x₂ y₁ ℓ : R) : R :=
ℓ * (W'.addX x₁ x₂ ℓ - x₁) + y₁
variable (W') in
/-- The `Y`-coordinate of `(x₁, y₁) + (x₂, y₂)` for two nonsingular affine points `(x₁, y₁)` and
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 390 | 397 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... |
theorem isOpen_iff_of_mem' (h : ∞ ∈ s) :
IsOpen s ↔ IsCompact ((↑) ⁻¹' s : Set X)ᶜ ∧ IsOpen ((↑) ⁻¹' s : Set X) := by
| Mathlib/Topology/Compactification/OnePoint.lean | 219 | 221 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.UniformConvergence
/-!
# Algebraic fact... | i.e., the pointwise scalar multiplication is continuous in both variables.
For convenience we require that `H` is a vector space over `𝕜`
with a topology induced by `UniformFun.ofFun ∘ φ`, where `φ : H →ₗ[𝕜] (α → E)`. -/
lemma UniformFun.continuousSMul_induced_of_range_bounded (φ : hom)
(hφ : IsInducing (ofFun ∘... | Mathlib/Topology/Algebra/Module/UniformConvergence.lean | 54 | 86 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | rwa [ContinuousAt, Metric.nhds_basis_closedBall.tendsto_right_iff, map_zero]
theorem continuousAt_zero' [TopologicalSpace E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E}
{r : ℝ} (hp : p.closedBall 0 r ∈ (𝓝 0 : Filter E)) : ContinuousAt p 0 := by
| Mathlib/Analysis/Seminorm.lean | 1,054 | 1,057 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,274 | 1,285 | |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.ChosenFiniteProducts
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
/-!
# Functor categories have chosen finite products
If `C` is a... |
@[reassoc (attr := simp)]
lemma whiskerRight_app_fst {F₁ F₁' : J ⥤ C} (f : F₁ ⟶ F₁') (F₂ : J ⥤ C) (j : J) :
(f ▷ F₂).app j ≫ fst _ _ = fst _ _ ≫ f.app j :=
(tensorHom_app_fst f (𝟙 F₂) j)
@[reassoc (attr := simp)]
lemma whiskerRight_app_snd {F₁ F₁' : J ⥤ C} (f : F₁ ⟶ F₁') (F₂ : J ⥤ C) (j : J) :
(f ▷ F₂).app... | Mathlib/CategoryTheory/ChosenFiniteProducts/FunctorCategory.lean | 131 | 147 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,197 | 1,200 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... |
instance thinkN_terminates (s : Computation α) : ∀ [Terminates s] (n), Terminates (thinkN s n)
| ⟨⟨a, h⟩⟩, n => ⟨⟨a, (thinkN_mem n).2 h⟩⟩
theorem of_thinkN_terminates (s : Computation α) (n) : Terminates (thinkN s n) → Terminates s
| ⟨⟨a, h⟩⟩ => ⟨⟨a, (thinkN_mem _).1 h⟩⟩
| Mathlib/Data/Seq/Computation.lean | 354 | 360 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | @[simp]
theorem rotation_zero : o.rotation 0 = LinearIsometryEquiv.refl ℝ V := by ext; simp [rotation]
/-- Rotation by π is negation. -/
@[simp]
theorem rotation_pi : o.rotation π = LinearIsometryEquiv.neg ℝ := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 124 | 129 |
/-
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 Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
/-!
# Limits and a... | simp [abs_of_nonpos (log_nonpos hx.1 hx.2)])
.rfl
@[deprecated (since := "2025-03-02")]
alias isLittleO_log_rpow_nhds_zero := isLittleO_log_rpow_nhdsGT_zero
theorem tendsto_log_div_rpow_nhdsGT_zero {r : ℝ} (hr : r < 0) :
| Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 366 | 372 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.IsTensorProduct
/-!
# Base change of polynomial algebras
Given `[CommSemiring R] [Semiring A] [Al... | /-- The `R`-algebra isomorphism `A[X] ≃ₐ[R] (A ⊗[R] R[X])`.
-/
def polyEquivTensor : A[X] ≃ₐ[R] A ⊗[R] R[X] :=
AlgEquiv.symm { PolyEquivTensor.toFunAlgHom R A, PolyEquivTensor.equiv R A with }
@[simp]
theorem polyEquivTensor_apply (p : A[X]) :
polyEquivTensor R A p =
p.eval₂ (includeLeft : A →ₐ[R] A ⊗[R] R... | Mathlib/RingTheory/PolynomialAlgebra.lean | 163 | 171 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | (wittStructureRat p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1) := by
rw [wittStructureRat]
intro x hx
simp only [Finset.mem_product, true_and, Finset.mem_univ, Finset.mem_range]
obtain ⟨k, hk, hx'⟩ := mem_vars_bind₁ _ _ hx
obtain ⟨i, -, hx''⟩ := mem_vars_bind₁ _ _ hx'
| Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 370 | 375 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Equiv.Opposite
import Mathlib.Algebra.... | Surjective f' := by
rw [← Nat.succ_pred_eq_of_pos (Nat.pos_iff_ne_zero.mpr hn), pow_succ', coe_mul] at h
exact Surjective.of_comp h
end
| Mathlib/Algebra/Module/LinearMap/End.lean | 178 | 182 |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | Mathlib/Data/List/Cycle.lean | 927 | 927 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,345 | 1,346 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | @[to_additive (attr := bound)]
theorem norm_sub_le_norm_mul (a b : E) : ‖a‖ - ‖b‖ ≤ ‖a * b‖ := by
simpa using norm_mul_le' (a * b) (b⁻¹)
| Mathlib/Analysis/Normed/Group/Basic.lean | 513 | 516 |
/-
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
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | obtain n0 | n0 := le_total 0 n
· obtain ⟨n, rfl⟩ := Int.eq_ofNat_of_zero_le n0
simp only [mabs_pow, zpow_natCast, Nat.abs_cast]
· obtain ⟨m, h⟩ := Int.eq_ofNat_of_zero_le (neg_nonneg.2 n0)
rw [← mabs_inv, ← zpow_neg, ← abs_neg, h, zpow_natCast, Nat.abs_cast, zpow_natCast]
exact mabs_pow m _
| Mathlib/Algebra/Order/Ring/Abs.lean | 24 | 30 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,028 | 2,029 | |
/-
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.Group.Units.Basic
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathl... | rw [H]
simp [coeff_mul, support_single_ne_zero, h]
else by
classical
have : ((0 : σ →₀ ℕ), n) ∈ antidiagonal n := by rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, ← Finset.insert_erase this,
Finset.sum_insert (Finset.not_mem_erase _ _), coeff_zero_eq_const... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 107 | 137 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... |
theorem map_comp_heq (hx : F.obj X = G.obj X) (hy : F.obj Y = G.obj Y) (hz : F.obj Z = G.obj Z)
(hf : HEq (F.map f) (G.map f)) (hg : HEq (F.map g) (G.map g)) :
| Mathlib/CategoryTheory/EqToHom.lean | 282 | 284 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation... | simp_rw [jacobiTheta₂_term, mul_add, Complex.exp_add, mul_one, mul_comm _ (n : ℂ),
exp_int_mul_two_pi_mul_I, mul_one]
/-- The two-variable Jacobi theta function is quasi-periodic in `z` with period `τ`. -/
lemma jacobiTheta₂_add_left' (z τ : ℂ) :
jacobiTheta₂ (z + τ) τ = cexp (-π * I * (τ + 2 * z)) * jacobiT... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 393 | 399 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.Portmanteau
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | · exact Eventually.of_forall f_nn
· positivity
· rw [ENNReal.ofReal_add (by positivity) (by positivity), ← add_zero (levyProkhorovEDist _ _)]
apply ENNReal.add_lt_add_of_le_of_lt (levyProkhorovEDist_ne_top _ _)
(le_of_eq ?_) (ofReal_pos.mpr εs_pos)
rw [LevyProkhorov.dist_def, levyP... | Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean | 463 | 493 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 893 | 897 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Logic.Equiv.Defs
/-!
# Theorems... | _ = a * ⅟ b := by simp [mul_assoc]
theorem commute_invOf {M : Type*} [One M] [Mul M] (m : M) [Invertible m] : Commute m (⅟ m) :=
calc
m * ⅟ m = 1 := mul_invOf_self m
_ = ⅟ m * m := (invOf_mul_self m).symm
| Mathlib/Algebra/Group/Invertible/Basic.lean | 69 | 74 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | ((1 + X) ^ n).coeff k = (n.choose k : R) := by rw [add_comm _ X, coeff_X_add_one_pow]
theorem C_dvd_iff_dvd_coeff (r : R) (φ : R[X]) : C r ∣ φ ↔ ∀ i, r ∣ φ.coeff i := by
| Mathlib/Algebra/Polynomial/Coeff.lean | 304 | 306 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
/-!
# N... | variable {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
theorem IsNilpotent.mapQ (hnp : IsNilpotent f) : IsNilpotent (p.mapQ p f hp) := by
| Mathlib/RingTheory/Nilpotent/Lemmas.lean | 123 | 126 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
/-!
# Orientations of modules
This file defines orientations of modules.
## Main definitions
... |
section Reindex
| Mathlib/LinearAlgebra/Orientation.lean | 74 | 75 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,127 | 1,129 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | obtain hε | hε := le_total ε 0
· simp only [thickening_of_nonpos hε, empty_subset]
obtain hδ | hδ := le_total δ 0
· simp only [thickening_of_nonpos hδ, thickening_empty, empty_subset]
| Mathlib/Topology/MetricSpace/Thickening.lean | 587 | 590 |
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
/-!
# Graph partitions
This module provides an interface for dealing with partitions on simple graphs... | Each vertex is colored by the part it's contained in. -/
def toColoring : G.Coloring P.parts :=
Coloring.mk (fun v ↦ ⟨P.partOfVertex v, P.partOfVertex_mem v⟩) fun hvw ↦ by
rw [Ne, Subtype.mk_eq_mk]
exact P.partOfVertex_ne_of_adj hvw
| Mathlib/Combinatorics/SimpleGraph/Partition.lean | 99 | 103 |
/-
Copyright (c) 2023 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.Data.Set.Function
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.EMetricSpace.BoundedVariation
/-!
# Constant speed
This file defines... | rcases le_total x y with (xy | yx)
· exact h xs ys xy
· rw [eVariationOn.subsingleton, ENNReal.ofReal_of_nonpos]
· exact mul_nonpos_of_nonneg_of_nonpos l.prop (sub_nonpos_of_le yx)
· rintro z ⟨zs, xz, zy⟩ w ⟨ws, xw, wy⟩
cases le_antisymm (zy.trans yx) xz
cases le_antisymm (wy.trans yx) xw
... | Mathlib/Analysis/ConstantSpeed.lean | 71 | 82 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... | by_cases H : Multipliable f
· intro e he
obtain ⟨s, hs⟩ := H.tprod_vanishing he
| Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 372 | 374 |
/-
Copyright (c) 2024 Lean FRO. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.InsertIdx
/-!
This is a stub file for importing `Mathlib.Data.List.InsertNth`,
which has been renamed to `Mathlib.Data.List.InsertIdx`.
This file c... | Mathlib/Data/List/InsertNth.lean | 197 | 201 | |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual.Lemmas
/-!
# Bilinear form
This file defines various properties of ... | Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 418 | 421 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | theorem orderOf_eq_zero_iff : orderOf x = 0 ↔ ¬IsOfFinOrder x :=
⟨fun h H ↦ H.orderOf_pos.ne' h, orderOf_eq_zero⟩
| Mathlib/GroupTheory/OrderOfElement.lean | 185 | 186 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideals of `R`
inside its f... | apply Subtype.coe_injective
simp only [val_eq_coe, hx, coe_spanSingleton]
refine ⟨Units.mk0 x ?_, ?_⟩
· intro x_eq; apply Units.ne_zero I; simp [hx', x_eq]
· simp [hx']
theorem ClassGroup.mk0_eq_one_iff [IsDedekindDomain R] {I : Ideal R} (hI : I ∈ (Ideal R)⁰) :
ClassGroup.mk0 ⟨I, hI⟩ = 1 ↔ I.IsPrinci... | Mathlib/RingTheory/ClassGroup.lean | 328 | 338 |
/-
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.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... |
theorem iff_not_exists_isMinBadSeq (rk : α → ℕ) {s : Set α} :
s.PartiallyWellOrderedOn r ↔ ¬∃ f, IsBadSeq r s f ∧ ∀ n, IsMinBadSeq r rk s n f := by
| Mathlib/Order/WellFoundedSet.lean | 735 | 737 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analy... | theorem ContDiffOn.dist (hf : ContDiffOn ℝ n f s) (hg : ContDiffOn ℝ n g s)
(hne : ∀ x ∈ s, f x ≠ g x) : ContDiffOn ℝ n (fun y => dist (f y) (g y)) s := fun x hx =>
(hf x hx).dist 𝕜 (hg x hx) (hne x hx)
| Mathlib/Analysis/InnerProductSpace/Calculus.lean | 179 | 182 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Connected.Clopen
/-!
# Totally disconnected and totally separated topological spaces
## Main definitions
We define th... | [TotallyDisconnectedSpace α] : TotallyDisconnectedSpace (Subtype p) :=
totallyDisconnectedSpace_subtype_iff.2 (isTotallyDisconnected_of_totallyDisconnectedSpace _)
end TotallyDisconnected
section TotallySeparated
| Mathlib/Topology/Connected/TotallyDisconnected.lean | 163 | 169 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | included in the `r`-neighborhood of the other. If there is no such `r`, it is defined to
be `0`, arbitrarily. -/
def hausdorffDist (s t : Set α) : ℝ :=
ENNReal.toReal (hausdorffEdist s t)
/-- The Hausdorff distance is nonnegative. -/
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 624 | 629 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,419 | 2,421 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Multiset.Antidiagona... |
end CommMonoidWithZero
| Mathlib/Algebra/BigOperators/Ring/Multiset.lean | 37 | 38 |
/-
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.SetLike.Basic
import Mathlib.ModelTheory.Semantics
/-!
# Definable Sets
This file defines what it means for a set over a first-order structure t... | exact Definable.mono hd hA0
/-- This lemma is only intended as a helper for `Definable.image_comp`. -/
theorem Definable.image_comp_sumInl_fin (m : ℕ) {s : Set (Sum α (Fin m) → M)}
(h : A.Definable L s) : A.Definable L ((fun g : Sum α (Fin m) → M => g ∘ Sum.inl) '' s) := by
obtain ⟨φ, rfl⟩ := h
refine ⟨(Bo... | Mathlib/ModelTheory/Definability.lean | 187 | 211 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Data.Complex.ExponentialBounds
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.SpecialF... | lemma tendsto_eulerMascheroniSeq' :
Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant) := by
suffices Tendsto (fun n ↦ eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0) by
simpa using this.add tendsto_eulerMascheroniSeq
suffices Tendsto (fun x : ℝ ↦ log (x + 1) - log x) atTop (𝓝 0) by
... | Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | 137 | 150 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,319 | 1,323 | |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.RingTheory.Ideal.BigOperators
import Mathlib.RingTheory.Ideal.Span
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Lemmas about ideals of `MonoidAlgebra` an... | {x : MonoidAlgebra k G} :
x ∈ Ideal.span (MonoidAlgebra.of k G '' s) ↔ ∀ m ∈ x.support, ∃ m' ∈ s, ∃ d, m = d * m' := by
let RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := { p | ∀ m : G, m ∈ p.support → ∃ m' ∈ s, ∃ d, m = d * m' }
add_mem' := fun {x y} hx hy m hm => by
classical exact (Finse... | Mathlib/Algebra/MonoidAlgebra/Ideal.lean | 23 | 58 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | conv =>
rhs
rw [← Nat.mod_add_div a n]
| Mathlib/Data/ZMod/Basic.lean | 729 | 731 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | simp [div_eq_mul_inv, mul_comm, hs.mul_closedBall hδ]
@[to_additive]
theorem IsCompact.closedBall_mul (hs : IsCompact s) (hδ : 0 ≤ δ) (x : E) :
closedBall x δ * s = x • cthickening δ s := by rw [mul_comm, hs.mul_closedBall hδ]
| Mathlib/Analysis/Normed/Group/Pointwise.lean | 222 | 227 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... |
theorem isTorsionBySet_iff_is_torsion_by_span :
IsTorsionBySet R M s ↔ IsTorsionBySet R M (Ideal.span s) := by
| Mathlib/Algebra/Module/Torsion.lean | 311 | 313 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,457 | 1,457 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.Normed.Operator.Banach
import Mathlib.LinearAlgebra.Sesqui... | theorem IsSymmetric.sub {T S : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hS : S.IsSymmetric) :
(T - S).IsSymmetric := by
intro x y
rw [sub_apply, inner_sub_left, hT x y, hS x y, ← inner_sub_right, sub_apply]
| Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 88 | 92 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.S... | /-- The topology on a linearly ordered group with zero element adjoined is compatible with the order
structure: the set `{p : Γ₀ × Γ₀ | p.1 ≤ p.2}` is closed. -/
@[nolint defLemma]
scoped instance (priority := 100) orderClosedTopology : OrderClosedTopology Γ₀ where
| Mathlib/Topology/Algebra/WithZeroTopology.lean | 136 | 139 |
/-
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.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | if ha : (⟨↑a, 1⟩ : ℤ√(d a1)) ≤ b then
let ⟨m, e⟩ :=
eq_pell_lem n (b * ⟨a, -1⟩) (by rw [← a1m]; exact mul_le_mul_of_nonneg_right ha am1p)
(isPell_mul hp (isPell_star.1 (isPell_one a1)))
(by
have t := mul_le_mul_of_nonneg_right h am1p
rwa [pellZd_succ, mul_as... | Mathlib/NumberTheory/PellMatiyasevic.lean | 240 | 246 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | {I₁ : R₁ →+* R} {I₂ : R₂ →+* R} {I₁' : R₁ →+* R}
/-- A bilinear map is called left-separating if
the only element that is left-orthogonal to every other element is `0`; i.e.,
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 608 | 611 |
/-
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 Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | iterate 2 rw [Real.rpow_def_of_nonneg]; split_ifs with h_ifs <;> simp_all
· rw [log_mul ‹_› ‹_›, add_mul, exp_add, rpow_def_of_pos (hy.lt_of_ne' ‹_›)]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 459 | 460 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | simp only [hF1, this]
rw [this]
apply Measurable.piecewise (m_meas j) (F0_meas j)
apply Measurable.find (fun n ↦ F0_meas (g n)) (fun n ↦ ?_)
exact (m_meas (g n)).union mc_meas
/-- `limsup` over a general filter is measurable. See `Measurable.limsup` for the version over `ℕ`.
-/
theorem Measurable.limsup'... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 889 | 938 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | /-- The canonical isomorphism `S.op.homology ≅ Opposite.op S.homology` when a short
complex `S` has homology. -/
noncomputable def homologyOpIso [S.HasHomology] :
S.op.homology ≅ Opposite.op S.homology :=
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 1,012 | 1,015 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | theorem Filter.Tendsto.finTail {f : Y → ∀ j : Fin (n + 1), π j} {l : Filter Y} {x : ∀ j, π j}
(hg : Tendsto f l (𝓝 x)) : Tendsto (fun a ↦ Fin.tail (f a)) l (𝓝 <| Fin.tail x) :=
tendsto_pi_nhds.2 fun j ↦ apply_nhds hg j.succ
| Mathlib/Topology/Constructions.lean | 839 | 841 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | intro h
apply mul_eq_zero_of_right
rw [ih _ _ (Nat.le_of_succ_le h), sub_zero]
convert ih _ _ (Nat.pred_le_pred h)
exact (Nat.succ_pred_eq_of_pos (k.succ_pos.trans_le h)).symm
@[simp]
theorem iterate_derivative_succ_at_0_eq_zero (n ν : ℕ) :
(Polynomial.derivative^[ν] (bernsteinPolynom... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 146 | 161 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | calc exp 1 ≤ ⌈exp 1⌉₊ := by exact Nat.le_ceil _
_ ≤ n := by exact_mod_cast h_exp n hn
_ = C * (2 * c₁) := by ring
_ ≤ (2 * c₁)⁻¹ * (2 * c₁) := by gcongr; exact min_le_left _ _
_ = c₁⁻¹ * c₁ := by ring
... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 1,452 | 1,460 |
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