path stringlengths 11 71 | content stringlengths 75 124k |
|---|---|
Data\Multiset\Sections.lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Multiset.Bind
/-!
# Sections of a multiset
-/
assert_not_exists Ring
namespace Multiset
variable {α : Type*}
section Sections
/-- The sectio... |
Data\Multiset\Sort.lean | /-
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.List.Sort
import Mathlib.Data.Multiset.Basic
/-!
# Construct a sorted list from a multiset.
-/
namespace Multiset
open List
variable {α : Typ... |
Data\Multiset\Sum.lean | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Multiset.Nodup
/-!
# Disjoint sum of multisets
This file defines the disjoint sum of two multisets as `Multiset (α ⊕ β)`. Beware not to confuse
with... |
Data\Multiset\Sym.lean | /-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.List.Sym
/-! # Unordered tuples of elements of a multiset
Defines `Multiset.sym` and the specialized `Multiset.sym2` for computing multisets of all
un... |
Data\Nat\BitIndices.lean | /-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.List.Sort
import Mathlib.Data.Nat.Bitwise
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Algebra.Order.BigOperators.Group.List
import Ma... |
Data\Nat\Bits.lean | /-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Tactic.Con... |
Data\Nat\Bitwise.lean | /-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
... |
Data\Nat\ChineseRemainder.lean | /-
Copyright (c) 2023 Shogo Saito. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shogo Saito. Adapted for mathlib by Hunter Monroe
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.GCD.BigOperators
/-!
# Chinese Re... |
Data\Nat\Count.lean | /-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Counting on ℕ
This file defines the `count` functio... |
Data\Nat\Defs.lean | /-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Nontrivial.Defs
import M... |
Data\Nat\Digits.lean | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... |
Data\Nat\Dist.lean | /-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Nat
/-!
# Distance function on ℕ
This file defines a simple distance function on naturals from truncated subtr... |
Data\Nat\EvenOddRec.lean | /-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.Bits
/-! # A recursion principle based on even and odd numbers. -/
namespace Nat
/-- Recursion principl... |
Data\Nat\Factors.lean | /-
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.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.List.Prime
import Mathli... |
Data\Nat\Find.lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Batteries.WF
/-!
# `Nat.find` and `Nat.findGreatest`
-/
variable {a b c d m n k : ℕ} ... |
Data\Nat\Hyperoperation.lean | /-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Tactic.Ring
/-!
# Hyperoperation sequence
This file defines the Hyperoperation sequence.
`hyperoperation 0 m k = k + ... |
Data\Nat\Lattice.lean | /-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov
-/
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
/-!
# Conditional... |
Data\Nat\Log.lean | /-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.... |
Data\Nat\MaxPowDiv.lean | /-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Algebra.Divisibility.Units
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural n... |
Data\Nat\ModEq.lean | /-
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.Regular
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Tactic.NormNum.Basic
/-!
# Congruences modulo a n... |
Data\Nat\Multiplicity.lean | /-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat... |
Data\Nat\Notation.lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura
-/
/-!
# Notation `ℕ` for the natural numbers.
-/
@[inherit_doc] notation "ℕ" => Nat
|
Data\Nat\Nth.lean | /-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import... |
Data\Nat\Pairing.lean | /-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
/-!
# Naturals pairing function
This file defines a pairing function for... |
Data\Nat\PartENat.lean | /-
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.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... |
Data\Nat\Periodic.lean | /-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey
-/
import Mathlib.Algebra.Periodic
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Interval.Finset.Nat
/-!
# Periodic Functions on ℕ
... |
Data\Nat\PrimeFin.lean | /-
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.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Set.Finite
import Mathlib.Data.Nat.Pri... |
Data\Nat\PSub.lean | /-
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.Group.Basic
import Mathlib.Algebra.Group.Nat
/-!
# Partial predecessor and partial subtraction on the natural numbers
The usual definition of... |
Data\Nat\Set.lean | /-
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.Data.Set.Image
/-!
### Recursion on the natural numbers and `Set.range`
-/
namespace Nat
section Set
open Set
theorem zero_union_range_succ : {... |
Data\Nat\Size.lean | /-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Nat.Bits
import Mathlib.Order.Lattice
/-! Lemmas about `size... |
Data\Nat\Squarefree.lean | /-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
/-!
# Lemmas about squarefreeness of natural numbers
A number is squarefree wh... |
Data\Nat\SuccPred.lean | /-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Fin.Basic
import Mathlib.Order.SuccPred.Basic
/-!
# Successors and predecessors of naturals
In this file, we s... |
Data\Nat\Totient.lean | /-
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.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mathlib.Data.Nat.Periodic
import Mathl... |
Data\Nat\Upto.lean | /-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Algebra.Order.Ring.Nat
/-!
# `Nat.Upto`
`Nat.Upto p`, with `p` a predicate on `ℕ`, is a subtype of elements `n : ℕ` such that no value
(strictly) below `n`... |
Data\Nat\WithBot.lean | /-
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.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.WithTop
/-!
# `WithBot ℕ`
Lemmas about the type of natur... |
Data\Nat\Cast\Basic.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Algebra.Ring.Nat
/-!
# Cast of natural numbers (additional theorems)
T... |
Data\Nat\Cast\Commute.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Ring.Commute
/-!
# Cast of natural numbers: lemmas about `Commute`
-/
variable {α β : Type*}
n... |
Data\Nat\Cast\Defs.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.SplitIfs
/-!
# Cast of natural numbers
This file defines the *canonical* homomorphism from th... |
Data\Nat\Cast\Field.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Patrick Stevens
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Tactic.Common
import Mathlib.Algebra.Field.Defs
... |
Data\Nat\Cast\NeZero.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Algebra.NeZero
/-!
# Lemmas about nonzero elements of an `AddMonoidWithOne`
-/
open Nat
namespace N... |
Data\Nat\Cast\Prod.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
/-!
# The product of two `AddMonoidWithOne`s.
-/
assert_not_exists MonoidWithZero
variable {α β : Type*}
namespace Prod
variabl... |
Data\Nat\Cast\SetInterval.lean | /-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Order.UpperLower.Basic
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Nat.Cast.Order.Basic
/-!
# Im... |
Data\Nat\Cast\Synonym.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Order.Synonym
/-!
# Cast of natural numbers (additional theorems)
This file proves additional properties about the ... |
Data\Nat\Cast\WithTop.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Nat
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
/-!
# Lemma about the coercion `ℕ → WithBot ℕ`.
An orphaned lemma about castin... |
Data\Nat\Cast\Order\Basic.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.ZeroLEOne
impo... |
Data\Nat\Cast\Order\Field.lean | /-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Patrick Stevens
-/
import Mathlib.Algebra.Order.Field.Basic
/-!
# Cast of naturals into ordered fields
This file concerns the canonical homomorphism `ℕ ... |
Data\Nat\Cast\Order\Ring.lean | /-
Copyright (c) 2014 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.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Data.Nat.Cast.Order.Basic
/-!
# Cast of natural numbers: lemmas... |
Data\Nat\Choose\Basic.lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
/-!
# Binomial coefficients
This file defines binomial coeffi... |
Data\Nat\Choose\Bounds.lean | /-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Rodriguez
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order.Basic
import Mathlib.Data.Nat.Choose.Basic
/-!
# Inequalities for binomial co... |
Data\Nat\Choose\Cast.lean | /-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Factorial.Cast
/-!
# Cast of binomial coefficients
This file allows calculating the binomial coefficient `a... |
Data\Nat\Choose\Central.lean | /-
Copyright (c) 2021 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Central bin... |
Data\Nat\Choose\Dvd.lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Prime.Basic
/-!
# Divisibility properties of binomial coefficients
-/
namespace Nat
open... |
Data\Nat\Choose\Factorization.lean | /-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
/-!
# Factori... |
Data\Nat\Choose\Lucas.lean | /-
Copyright (c) 2023 Gareth Ma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gareth Ma
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Polynomial.Basic
/-!
# Lucas's theorem
This file contains a proof of [Lucas's theorem](https://en.wikipedia.org/wiki/... |
Data\Nat\Choose\Multinomial.lean | /-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Antidiag.Pi
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
im... |
Data\Nat\Choose\Sum.lean | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring
import Mathlib... |
Data\Nat\Choose\Vandermonde.lean | /-
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.Polynomial.Coeff
import Mathlib.Data.Nat.Choose.Basic
/-!
# Vandermonde's identity
In this file we prove Vandermonde's identity (`Nat.add_ch... |
Data\Nat\Factorial\Basic.lean | /-
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, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monoto... |
Data\Nat\Factorial\BigOperators.lean | /-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
/-!
# Factorial with big operators
This file contains some lemmas on ... |
Data\Nat\Factorial\Cast.lean | /-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.RingTheory.Polynomial.Pochhammer
/-!
# Cast of factorials
This file allows calculating factorials (including ascending and descending ones) as elements o... |
Data\Nat\Factorial\DoubleFactorial.lean | /-
Copyright (c) 2023 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
/-!
# Double ... |
Data\Nat\Factorial\SuperFactorial.lean | /-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vanderm... |
Data\Nat\Factorization\Basic.lean | /-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
/-!
#... |
Data\Nat\Factorization\Defs.lean | /-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.NumberTheory.Padics.PadicVal
/-!
# Prime factorizations
`n.factorization` is the finitely supported function ... |
Data\Nat\Factorization\Induction.lean | /-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.Factorization.Defs
/-!
# Induction principles involving factorizations
-/
open Nat Finset List Finsupp
namespace Nat
variable {a b m n p ... |
Data\Nat\Factorization\PrimePow.lean | /-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
/-!
# Prime powers and factorizations
This file deals with factorizations of prime powers.... |
Data\Nat\Factorization\Root.lean | /-
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... |
Data\Nat\Fib\Basic.lean | /-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import ... |
Data\Nat\Fib\Zeckendorf.lean | /-
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.Data.Nat.Fib.Basic
/-!
# Zeckendorf's Theorem
This file proves Zeckendorf's theorem: Every natural number can be written uniquely as a sum of
distinct no... |
Data\Nat\GCD\Basic.lean | /-
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.Order.Lattice
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Nat
import Mathlib.Init.Data.Nat.Lemmas
/... |
Data\Nat\GCD\BigOperators.lean | /-
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.Algebra.BigOperators.Group.Finset
/-! # Lemmas about coprimality with big products.
These lemmas are kept separate from `Data.Nat.GC... |
Data\Nat\Order\Lemmas.lean | /-
Copyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Nat.Find
import Mathlib.Data.Set... |
Data\Nat\Prime\Basic.lean | /-
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.Algebra.Ring.Int
import Mathlib.Order.Bounds.Basic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data... |
Data\Nat\Prime\Defs.lean | /-
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.Algebra.Associated.Basic
import Mathlib.Algebra.Ring.Parity
import Mathlib.Data.Nat.GCD.Basic
/-!
# Prime num... |
Data\NNRat\BigOperators.lean | /-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.NNRat.Defs
/-! # Casting lemmas for non-negative rational numbers i... |
Data\NNRat\Defs.lean | /-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Nonneg.Ring
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Nat.Cast.Order.Ring
/-!
# Nonnegative rati... |
Data\NNRat\Lemmas.lean | /-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Order.Field.Rat
/-!
# Field and action structu... |
Data\NNReal\Basic.lean | /-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical.Basic
import Mathlib.Algebra.Or... |
Data\NNReal\Star.lean | /-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Data.Real.Star
import Mathlib.Data.NNReal.Basic
/-!
# The non-negative real numbers are a `*`-ring, with the trivial `*`-structure
-/
open scoped NNRea... |
Data\Num\Basic.lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Lean.Linter.Deprecated
import Mathlib.Data.Int.Notation
import Mathlib.Algebra.Group.ZeroOne
import Mathlib.Data.Nat.Bits
/-!
# Binar... |
Data\Num\Bitwise.lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Num.Basic
import Mathlib.Data.Vector.Basic
/-!
# Bitwise operations using binary representation of integers
## Definitions
* bitwise ope... |
Data\Num\Lemmas.lean | /-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... |
Data\Num\Prime.lean | /-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Tactic.Ring
/-!
# Primality for binary natural numbers
This file defines versions o... |
Data\Option\Basic.lean | /-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... |
Data\Option\Defs.lean | /-
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.Tactic.Lemma
import Mathlib.Tactic.TypeStar
/-!
# Extra definitions on `Option`
This file defines more operations involving `Option α`. Lemmas about ... |
Data\Option\NAry.lean | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to define pointwise operations
on int... |
Data\Ordering\Basic.lean | /-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
/-!
# Helper definitions and instances for `Ordering`
-/
universe u
deriving instance Repr for Ordering
namespace Ordering
/-- Combine two `Ordering`s... |
Data\Ordering\Lemmas.lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Algebra.Classes
import Mathlib.Data.Ordering.Basic
/-!
# Some `Ordering` lemmas
-/
universe u
namespace Ordering
@[simp]
theorem ite... |
Data\Ordmap\Ordnode.lean | /-
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.Order.Compare
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Option.Basic
/-!
# Ordered sets
This file defines a data... |
Data\Ordmap\Ordset.lean | /-
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... |
Data\PFunctor\Multivariate\Basic.lean | /-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# Multivariate polynomial functors.
Multivariate polynomial... |
Data\PFunctor\Multivariate\M.lean | /-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
/-!
# The M construction as a multivariate polyn... |
Data\PFunctor\Multivariate\W.lean | /-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
/-!
# The W construction as a multivariate polynomial functor.
W types are well-founded tree-like structu... |
Data\PFunctor\Univariate\Basic.lean | /-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.W.Basic
/-!
# Polynomial functors
This file defines polynomial functors and the W-type construction as a
polynomial functor. (For the M-type cons... |
Data\PFunctor\Univariate\M.lean | /-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... |
Data\Pi\Interval.lean | /-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
/-!
# Intervals in a pi type
This file shows that (dependent) functions to locally f... |
Data\PNat\Basic.lean | /-
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.Equiv
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Basic
import Ma... |
Data\PNat\Defs.lean | /-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Basic
import Mathlib.Tactic.Coe
import Mathlib.Tactic.Lift
imp... |
Data\PNat\Equiv.lean | /-
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.Logic.Equiv.Defs
/-!
# The equivalence between `ℕ+` and `ℕ`
-/... |
Data\PNat\Factors.lean | /-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
/-!
# Prime fac... |
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