Index of /~nad/listings/lib-0.5

	Name	Last modified	Size	Description
	Parent Directory		-
	Function.Injection.html	2011-02-20 19:04	27K
	Data.Vec.Equality.html	2011-02-20 19:04	78K
	Data.Plus.html	2011-02-20 19:04	51K
	Data.Graph.Acyclic.html	2011-02-20 19:04	194K
	Relation.Nullary.Sum..>	2011-02-20 19:04	17K
	Relation.Binary.Inde..>	2011-02-20 19:04	17K
	Relation.Binary.Flip..>	2011-02-20 19:04	154K
	Function.Bijection.html	2011-02-20 19:04	41K
	Data.Star.List.html	2011-02-20 19:04	9.3K
	Data.Sign.Properties..>	2011-02-20 19:04	11K
	Data.List.Any.BagAnd..>	2011-02-20 19:04	289K
	Data.Fin.Dec.html	2011-02-20 19:04	139K
	Data.DifferenceList...>	2011-02-20 19:04	33K
	Data.Container.html	2011-02-20 19:04	192K
	Data.Cofin.html	2011-02-20 19:04	19K
	Coinduction.html	2011-02-20 19:04	13K
	Algebra.RingSolver.html	2011-02-20 19:04	195K
	Relation.Binary.Prop..>	2011-02-20 19:04	10K
	Relation.Binary.Prod..>	2011-02-20 19:04	228K
	Induction.html	2011-02-20 19:04	28K
	IO.html	2011-02-20 19:04	54K
	Data.Vec.N-ary.html	2011-02-20 19:04	163K
	Data.List.All.html	2011-02-20 19:04	49K
	Data.Integer.Propert..>	2011-02-20 19:04	80K
	Data.Conat.html	2011-02-20 19:04	25K
	Category.Monad.State..>	2011-02-20 19:04	84K
	Category.Monad.Parti..>	2011-02-20 19:04	695K
	Category.Monad.Index..>	2011-02-20 19:04	36K
	Algebra.FunctionProp..>	2011-02-20 19:04	60K
	Relation.Binary.Sigm..>	2011-02-20 19:04	237K
	Relation.Binary.Prop..>	2011-02-20 19:04	83K
	Data.Star.BoundedVec..>	2011-02-20 19:04	28K
	Data.Nat.GCD.html	2011-02-20 19:04	127K
	Data.Nat.Coprimality..>	2011-02-20 19:04	96K
	Data.Fin.Props.html	2011-02-20 19:04	134K
	Data.BoundedVec.Inef..>	2011-02-20 19:04	20K
	Algebra.Structures.html	2011-02-20 19:04	210K
	Agda.css	2011-02-20 19:04	1.1K
	Relation.Unary.html	2011-02-20 19:04	66K
	Relation.Binary.Simp..>	2011-02-20 19:04	14K
	Relation.Binary.Prop..>	2011-02-20 19:04	9.4K
	Relation.Binary.List..>	2011-02-20 19:04	108K
	Reflection.html	2011-02-20 19:04	174K
	Induction.Lexicograp..>	2011-02-20 19:04	51K
	Function.LeftInverse..>	2011-02-20 19:04	53K
	Function.Equivalence..>	2011-02-20 19:04	67K
	Function.Equality.html	2011-02-20 19:04	62K
	Data.List.html	2011-02-20 19:04	214K
	Data.Covec.html	2011-02-20 19:04	110K
	Data.Context.html	2011-02-20 19:04	76K
	Relation.Nullary.Dec..>	2011-02-20 19:04	43K
	Relation.Binary.Prod..>	2011-02-20 19:04	126K
	Relation.Binary.Hete..>	2011-02-20 19:04	153K
	Induction.WellFounde..>	2011-02-20 19:04	91K
	IO.Primitive.html	2011-02-20 19:04	16K
	Function.Surjection...>	2011-02-20 19:04	41K
	Data.Star.Nat.html	2011-02-20 19:04	19K
	Data.Integer.html	2011-02-20 19:04	156K
	Data.Fin.html	2011-02-20 19:04	109K
	Data.Fin.Substitutio..>	2011-02-20 19:04	499K
	Algebra.RingSolver.A..>	2011-02-20 19:04	61K
	Relation.Binary.Vec...>	2011-02-20 19:04	148K
	Relation.Binary.Sum...>	2011-02-20 19:04	382K
	Data.Star.Vec.html	2011-02-20 19:04	29K
	Data.Star.Fin.html	2011-02-20 19:04	8.8K
	Data.Nat.Properties...>	2011-02-20 19:04	399K
	Data.Integer.Divisib..>	2011-02-20 19:04	13K
	Data.Digit.html	2011-02-20 19:04	61K
	Data.Container.Combi..>	2011-02-20 19:04	159K
	Data.Colist.html	2011-02-20 19:04	178K
	Algebra.html	2011-02-20 19:04	155K
	Algebra.RingSolver.L..>	2011-02-20 19:04	143K
	Algebra.Operations.html	2011-02-20 19:04	32K
	Relation.Nullary.Cor..>	2011-02-20 19:04	7.9K
	Relation.Binary.List..>	2011-02-20 19:04	120K
	Relation.Binary.EqRe..>	2011-02-20 19:04	7.2K
	Relation.Binary.Core..>	2011-02-20 19:04	132K
	Function.html	2011-02-20 19:04	53K
	Data.Star.Properties..>	2011-02-20 19:04	100K
	Data.Star.Decoration..>	2011-02-20 19:04	74K
	Data.Rational.html	2011-02-20 19:04	63K
	Data.List.NonEmpty.html	2011-02-20 19:04	138K
	Data.List.Any.html	2011-02-20 19:04	134K
	Data.Fin.Subset.Prop..>	2011-02-20 19:04	118K
	Category.Applicative..>	2011-02-20 19:04	7.7K
	Algebra.Morphism.html	2011-02-20 19:04	45K
	Relation.Binary.html	2011-02-20 19:03	140K
	Relation.Binary.Stri..>	2011-02-20 19:03	69K
	Relation.Binary.Prop..>	2011-02-20 19:03	8.2K
	Relation.Binary.Inde..>	2011-02-20 19:03	42K
	Induction.Nat.html	2011-02-20 19:03	41K
	Function.Inverse.html	2011-02-20 19:03	157K
	Data.Vec.Properties...>	2011-02-20 19:03	244K
	Data.Sum.html	2011-02-20 19:03	36K
	Data.String.html	2011-02-20 19:03	29K
	Data.List.Properties..>	2011-02-20 19:03	487K
	Data.List.NonEmpty.P..>	2011-02-20 19:03	48K
	Category.Functor.html	2011-02-20 19:03	8.2K
	Algebra.Props.Boolea..>	2011-02-20 19:03	504K
	Algebra.FunctionProp..>	2011-02-20 19:03	6.7K
	Relation.Binary.Orde..>	2011-02-20 19:03	31K
	Relation.Binary.List..>	2011-02-20 19:03	243K
	Relation.Binary.Cons..>	2011-02-20 19:03	11K
	Data.Star.html	2011-02-20 19:03	99K
	Data.Nat.DivMod.html	2011-02-20 19:03	79K
	Data.Maybe.html	2011-02-20 19:03	106K
	Data.List.Reverse.html	2011-02-20 19:03	28K
	Data.List.Any.Proper..>	2011-02-20 19:03	625K
	Data.Fin.Subset.html	2011-02-20 19:03	42K
	Data.DifferenceNat.html	2011-02-20 19:03	12K
	Category.Monad.Ident..>	2011-02-20 19:03	6.6K
	Category.Monad.Conti..>	2011-02-20 19:03	44K
	Algebra.RingSolver.S..>	2011-02-20 19:03	5.1K
	Size.html	2011-02-20 19:03	3.4K
	Relation.Binary.Prop..>	2011-02-20 19:03	6.5K
	Level.html	2011-02-20 19:03	10K
	Function.Inverse.Typ..>	2011-02-20 19:03	235K
	Everything.html	2011-02-20 19:03	94K
	Data.Stream.html	2011-02-20 19:03	93K
	Data.Star.Environmen..>	2011-02-20 19:03	19K
	Data.ReflexiveClosur..>	2011-02-20 19:03	26K
	Data.Product.html	2011-02-20 19:03	91K
	Data.Nat.Show.html	2011-02-20 19:03	11K
	Data.List.Countdown...>	2011-02-20 19:03	197K
	Data.Empty.html	2011-02-20 19:03	3.8K
	Data.BoundedVec.html	2011-02-20 19:03	38K
	Data.Bool.html	2011-02-20 19:03	26K
	Algebra.Props.Group...>	2011-02-20 19:03	59K
	Relation.Nullary.html	2011-02-20 19:03	3.8K
	Relation.Binary.Prop..>	2011-02-20 19:03	18K
	Relation.Binary.Part..>	2011-02-20 19:03	5.3K
	Relation.Binary.Indu..>	2011-02-20 19:03	21K
	Data.W.html	2011-02-20 19:03	18K
	Data.Vec.html	2011-02-20 19:03	191K
	Data.Product.N-ary.html	2011-02-20 19:03	41K
	Data.Nat.html	2011-02-20 19:03	124K
	Data.Fin.Substitutio..>	2011-02-20 19:03	23K
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	Data.Container.Any.html	2011-02-20 19:03	256K
	Data.Char.html	2011-02-20 19:03	15K
	Data.Bool.Show.html	2011-02-20 19:03	3.4K
	Data.AVL.Sets.html	2011-02-20 19:03	26K
	Category.Monad.html	2011-02-20 19:03	17K
	Algebra.Props.Distri..>	2011-02-20 19:03	59K
	Universe.html	2011-02-20 19:03	4.2K
	Relation.Binary.Prop..>	2011-02-20 19:03	9.9K
	Relation.Binary.Cons..>	2011-02-20 19:03	89K
	Data.Star.Pointer.html	2011-02-20 19:03	73K
	Data.Sign.html	2011-02-20 19:03	11K
	Data.Nat.LCM.html	2011-02-20 19:03	90K
	Data.Fin.Substitutio..>	2011-02-20 19:03	83K
	Data.DifferenceVec.html	2011-02-20 19:03	36K
	Data.Bin.html	2011-02-20 19:03	173K
	Data.AVL.IndexedMap...>	2011-02-20 19:03	46K
	Category.Applicative..>	2011-02-20 19:03	62K
	Algebra.Props.Ring.html	2011-02-20 19:03	42K
	Algebra.Props.Boolea..>	2011-02-20 19:03	159K
	Algebra.Props.Abelia..>	2011-02-20 19:03	34K
	Relation.Nullary.Uni..>	2011-02-20 19:03	104K
	Relation.Binary.Stri..>	2011-02-20 19:03	6.3K
	Relation.Binary.Prop..>	2011-02-20 19:03	11K
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	Relation.Binary.On.html	2011-02-20 19:03	107K
	Relation.Binary.NonS..>	2011-02-20 19:03	67K
	Data.Nat.InfinitelyO..>	2011-02-20 19:03	57K
	Data.Nat.GCD.Lemmas...>	2011-02-20 19:03	225K
	Data.Maybe.Core.html	2011-02-20 19:03	4.6K
	Data.List.Any.Member..>	2011-02-20 19:03	245K
	Data.List.All.Proper..>	2011-02-20 19:03	54K
	Data.AVL.html	2011-02-20 19:03	215K
	Relation.Nullary.Pro..>	2011-02-20 19:03	9.4K
	Relation.Nullary.Neg..>	2011-02-20 19:03	123K
	Relation.Binary.Refl..>	2011-02-20 19:03	62K
	Relation.Binary.Prod..>	2011-02-20 19:03	254K
	Foreign.Haskell.html	2011-02-20 19:03	17K
	Data.Unit.html	2011-02-20 19:03	22K
	Data.Nat.Divisibilit..>	2011-02-20 19:03	133K
	Data.Bool.Properties..>	2011-02-20 19:03	142K
	Algebra.Props.Lattic..>	2011-02-20 19:03	66K

module README where

------------------------------------------------------------------------
-- The Agda standard library, version 0.5
--
-- Author: Nils Anders Danielsson, with contributions from
-- Jean-Philippe Bernardy, Peter Berry, Samuel Bronson, Daniel Brown,
-- Liang-Ting Chen, Dominique Devriese, Dan Doel, Simon Foster, Patrik
-- Jansson, Darin Morrison, Shin-Cheng Mu, Ulf Norell, Nicolas
-- Pouillard and Andrés Sicard-Ramírez
------------------------------------------------------------------------

-- This version of the library has been tested using Agda 2.2.10.

-- Note that no guarantees are currently made about forwards or
-- backwards compatibility, the library is still at an experimental
-- stage.

-- To make use of the library, add the path to the library’s root
-- directory (src) to the Agda search path, either using the
-- --include-path flag or by customising the Emacs mode variable
-- agda2-include-dirs (M-x customize-group RET agda2 RET).

-- Contributions to this library are welcome (but to avoid wasted work
-- it is suggested that you discuss large changes before implementing
-- them). Please send contributions in the form of darcs patches (run
-- darcs send --output <patch file> and attach the patch file to an
-- email), and include a statement saying that you agree to release
-- your library patches under the library's licence. It is appreciated
-- if every patch contains a single, complete change, and if the
-- coding style used in the library is adhered to.

------------------------------------------------------------------------
-- Module hierarchy
------------------------------------------------------------------------

-- The top-level module names of the library are currently allocated
-- as follows:
--
-- • Algebra
--     Abstract algebra (monoids, groups, rings etc.), along with
--     properties needed to specify these structures (associativity,
--     commutativity, etc.), and operations on and proofs about the
--     structures.
-- • Category
--     Category theory-inspired idioms used to structure functional
--     programs (functors and monads, for instance).
-- • Coinduction
--     Support for coinduction.
-- • Data
--     Data types and properties about data types.
-- • Function
--     Combinators and properties related to functions.
-- • Foreign
--     Related to the foreign function interface.
-- • Induction
--     A general framework for induction (includes lexicographic and
--     well-founded induction).
-- • IO
--     Input/output-related functions.
-- • Level
--     Universe levels.
-- • Reflection
--     Support for reflection.
-- • Relation
--     Properties of and proofs about relations (mostly homogeneous
--     binary relations).
-- • Size
--     Sizes used by the sized types mechanism.
-- • Universe
--     A definition of universes.

------------------------------------------------------------------------
-- A selection of useful library modules
------------------------------------------------------------------------

-- Note that module names in source code are often hyperlinked to the
-- corresponding module. In the Emacs mode you can follow these
-- hyperlinks by typing M-. or clicking with the middle mouse button.

-- • Some data types

import Data.Bool     -- Booleans.
import Data.Char     -- Characters.
import Data.Empty    -- The empty type.
import Data.Fin      -- Finite sets.
import Data.List     -- Lists.
import Data.Maybe    -- The maybe type.
import Data.Nat      -- Natural numbers.
import Data.Product  -- Products.
import Data.Stream   -- Streams.
import Data.String   -- Strings.
import Data.Sum      -- Disjoint sums.
import Data.Unit     -- The unit type.
import Data.Vec      -- Fixed-length vectors.

-- • Some types used to structure computations

import Category.Functor      -- Functors.
import Category.Applicative  -- Applicative functors.
import Category.Monad        -- Monads.

-- • Equality

-- Propositional equality:
import Relation.Binary.PropositionalEquality

-- Convenient syntax for "equational reasoning" using a preorder:
import Relation.Binary.PreorderReasoning

-- Solver for commutative ring or semiring equalities:
import Algebra.RingSolver

-- • Properties of functions, sets and relations

-- Monoids, rings and similar algebraic structures:
import Algebra

-- Negation, decidability, and similar operations on sets:
import Relation.Nullary

-- Properties of homogeneous binary relations:
import Relation.Binary

-- • Induction

-- An abstraction of various forms of recursion/induction:
import Induction

-- Well-founded induction:
import Induction.WellFounded

-- Various forms of induction for natural numbers:
import Induction.Nat

-- • Support for coinduction

import Coinduction

-- • IO

import IO

------------------------------------------------------------------------
-- Record hierarchies
------------------------------------------------------------------------

-- When an abstract hierarchy of some sort (for instance semigroup →
-- monoid → group) is included in the library the basic approach is to
-- specify the properties of every concept in terms of a record
-- containing just properties, parameterised on the underlying
-- operations, sets etc.:
--
--   record IsSemigroup {A} (≈ : Rel A) (∙ : Op₂ A) : Set where
--     open FunctionProperties ≈
--     field
--       isEquivalence : IsEquivalence ≈
--       assoc         : Associative ∙
--       ∙-cong        : ∙ Preserves₂ ≈ ⟶ ≈ ⟶ ≈
--
-- More specific concepts are then specified in terms of the simpler
-- ones:
--
--     record IsMonoid {A} (≈ : Rel A) (∙ : Op₂ A) (ε : A) : Set where
--       open FunctionProperties ≈
--       field
--         isSemigroup : IsSemigroup ≈ ∙
--         identity    : Identity ε ∙
--
--     open IsSemigroup isSemigroup public
--
-- Note here that open IsSemigroup isSemigroup public ensures that the
-- fields of the isSemigroup record can be accessed directly; this
-- technique enables the user of an IsMonoid record to use underlying
-- records without having to manually open an entire record hierarchy.
-- This is not always possible, though. Consider the following definition
-- of preorders:
--
--   record IsPreorder {A : Set}
--                     (_≈_ : Rel A) -- The underlying equality.
--                     (_∼_ : Rel A) -- The relation.
--                     : Set where
--     field
--       isEquivalence : IsEquivalence _≈_
--       -- Reflexivity is expressed in terms of an underlying equality:
--       reflexive     : _≈_ ⇒ _∼_
--       trans         : Transitive _∼_
--
--     module Eq = IsEquivalence isEquivalence
--
--     ...
--
-- The Eq module in IsPreorder is not opened publicly, because it
-- contains some fields which clash with fields or other definitions
-- in IsPreorder.

-- Records packing up properties with the corresponding operations,
-- sets, etc. are sometimes also defined:
--
--   record Semigroup : Set₁ where
--     infixl 7 _∙_
--     infix  4 _≈_
--     field
--       Carrier     : Set
--       _≈_         : Rel Carrier
--       _∙_         : Op₂ Carrier
--       isSemigroup : IsSemigroup _≈_ _∙_
--
--     open IsSemigroup isSemigroup public
--
--     setoid : Setoid
--     setoid = record { isEquivalence = isEquivalence }
--
--   record Monoid : Set₁ where
--     infixl 7 _∙_
--     infix  4 _≈_
--     field
--       Carrier  : Set
--       _≈_      : Rel Carrier
--       _∙_      : Op₂ Carrier
--       ε        : Carrier
--       isMonoid : IsMonoid _≈_ _∙_ ε
--
--     open IsMonoid isMonoid public
--
--     semigroup : Semigroup
--     semigroup = record { isSemigroup = isSemigroup }
--
--     open Semigroup semigroup public using (setoid)
--
-- Note that the Monoid record does not include a Semigroup field.
-- Instead the Monoid /module/ includes a "repackaging function"
-- semigroup which converts a Monoid to a Semigroup.

-- The above setup may seem a bit complicated, but we think it makes the
-- library quite easy to work with, while also providing enough
-- flexibility.

------------------------------------------------------------------------
-- More documentation
------------------------------------------------------------------------

-- Some examples showing where the natural numbers and some related
-- operations and properties are defined, and how they can be used:

import README.Nat

------------------------------------------------------------------------
-- Core modules
------------------------------------------------------------------------

-- Some modules have names ending in ".Core". These modules are
-- internal, and have (mostly) been created to avoid mutual recursion
-- between modules. They should not be imported directly; their
-- contents are reexported by other modules.

------------------------------------------------------------------------
-- All library modules
------------------------------------------------------------------------

-- For short descriptions of every library module, see Everything:

import Everything