added Everything

src/extra/Everything.agda

@@ -0,0 +1,789 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- All library modules, along with short descriptions
+------------------------------------------------------------------------
+
+-- Note that core modules are not included.
+
+module Everything where
+
+-- Definitions of algebraic structures like monoids and rings
+-- (packed in records together with sets, operations, etc.)
+open import Algebra
+
+-- Solver for equations in commutative monoids
+open import Algebra.CommutativeMonoidSolver
+
+-- An example of how Algebra.CommutativeMonoidSolver can be used
+open import Algebra.CommutativeMonoidSolver.Example
+
+-- Properties of functions, such as associativity and commutativity
+open import Algebra.FunctionProperties
+
+-- Relations between properties of functions, such as associativity and
+-- commutativity
+open import Algebra.FunctionProperties.Consequences
+
+-- Solver for equations in commutative monoids
+open import Algebra.IdempotentCommutativeMonoidSolver
+
+-- An example of how Algebra.IdempotentCommutativeMonoidSolver can be
+-- used
+open import Algebra.IdempotentCommutativeMonoidSolver.Example
+
+-- Solver for monoid equalities
+open import Algebra.Monoid-solver
+
+-- Morphisms between algebraic structures
+open import Algebra.Morphism
+
+-- Some defined operations (multiplication by natural number and
+-- exponentiation)
+open import Algebra.Operations
+
+-- Some derivable properties
+open import Algebra.Properties.AbelianGroup
+
+-- Some derivable properties
+open import Algebra.Properties.BooleanAlgebra
+
+-- Boolean algebra expressions
+open import Algebra.Properties.BooleanAlgebra.Expression
+
+-- Some derivable properties
+open import Algebra.Properties.DistributiveLattice
+
+-- Some derivable properties
+open import Algebra.Properties.Group
+
+-- Some derivable properties
+open import Algebra.Properties.Lattice
+
+-- Some derivable properties
+open import Algebra.Properties.Ring
+
+-- Solver for commutative ring or semiring equalities
+open import Algebra.RingSolver
+
+-- Commutative semirings with some additional structure ("almost"
+-- commutative rings), used by the ring solver
+open import Algebra.RingSolver.AlmostCommutativeRing
+
+-- Some boring lemmas used by the ring solver
+open import Algebra.RingSolver.Lemmas
+
+-- Instantiates the ring solver, using the natural numbers as the
+-- coefficient "ring"
+open import Algebra.RingSolver.Natural-coefficients
+
+-- Instantiates the ring solver with two copies of the same ring with
+-- decidable equality
+open import Algebra.RingSolver.Simple
+
+-- Some algebraic structures (not packed up with sets, operations,
+-- etc.)
+open import Algebra.Structures
+
+-- Applicative functors
+open import Category.Applicative
+
+-- Indexed applicative functors
+open import Category.Applicative.Indexed
+
+-- Applicative functors on indexed sets (predicates)
+open import Category.Applicative.Predicate
+
+-- Functors
+open import Category.Functor
+
+-- The identity functor
+open import Category.Functor.Identity
+
+-- Functors on indexed sets (predicates)
+open import Category.Functor.Predicate
+
+-- Monads
+open import Category.Monad
+
+-- A delimited continuation monad
+open import Category.Monad.Continuation
+
+-- The identity monad
+open import Category.Monad.Identity
+
+-- Indexed monads
+open import Category.Monad.Indexed
+
+-- The partiality monad
+open import Category.Monad.Partiality
+
+-- An All predicate for the partiality monad
+open import Category.Monad.Partiality.All
+
+-- Monads on indexed sets (predicates)
+open import Category.Monad.Predicate
+
+-- The state monad
+open import Category.Monad.State
+
+-- Basic types related to coinduction
+open import Coinduction
+
+-- AVL trees
+open import Data.AVL
+
+-- Types and functions which are used to keep track of height
+-- invariants in AVL Trees
+open import Data.AVL.Height
+
+-- Indexed AVL trees
+open import Data.AVL.Indexed
+
+-- Finite maps with indexed keys and values, based on AVL trees
+open import Data.AVL.IndexedMap
+
+-- Keys for AVL trees
+-- The key type extended with a new minimum and maximum.
+open import Data.AVL.Key
+
+-- Finite sets, based on AVL trees
+open import Data.AVL.Sets
+
+-- A binary representation of natural numbers
+open import Data.Bin
+
+-- Properties of the binary representation of natural numbers
+open import Data.Bin.Properties
+
+-- Booleans
+open import Data.Bool
+
+-- The type for booleans and some operations
+open import Data.Bool.Base
+
+-- A bunch of properties
+open import Data.Bool.Properties
+
+-- Showing booleans
+open import Data.Bool.Show
+
+-- Bounded vectors
+open import Data.BoundedVec
+
+-- Bounded vectors (inefficient, concrete implementation)
+open import Data.BoundedVec.Inefficient
+
+-- Characters
+open import Data.Char
+
+-- Basic definitions for Characters
+open import Data.Char.Base
+
+-- "Finite" sets indexed on coinductive "natural" numbers
+open import Data.Cofin
+
+-- Coinductive lists
+open import Data.Colist
+
+-- Infinite merge operation for coinductive lists
+open import Data.Colist.Infinite-merge
+
+-- Coinductive "natural" numbers
+open import Data.Conat
+
+-- Containers, based on the work of Abbott and others
+open import Data.Container
+
+-- Properties related to ◇
+open import Data.Container.Any
+
+-- Container combinators
+open import Data.Container.Combinator
+
+-- The free monad construction on containers
+open import Data.Container.FreeMonad
+
+-- Indexed containers aka interaction structures aka polynomial
+-- functors. The notation and presentation here is closest to that of
+-- Hancock and Hyvernat in "Programming interfaces and basic topology"
+-- (2006/9).
+open import Data.Container.Indexed
+
+-- Indexed container combinators
+open import Data.Container.Indexed.Combinator
+
+-- The free monad construction on indexed containers
+open import Data.Container.Indexed.FreeMonad
+
+-- Coinductive vectors
+open import Data.Covec
+
+-- Lists with fast append
+open import Data.DifferenceList
+
+-- Natural numbers with fast addition (for use together with
+-- DifferenceVec)
+open import Data.DifferenceNat
+
+-- Vectors with fast append
+open import Data.DifferenceVec
+
+-- Digits and digit expansions
+open import Data.Digit
+
+-- Empty type
+open import Data.Empty
+
+-- An irrelevant version of ⊥-elim
+open import Data.Empty.Irrelevant
+
+-- Finite sets
+open import Data.Fin
+
+-- Decision procedures for finite sets and subsets of finite sets
+open import Data.Fin.Dec
+
+-- Properties related to Fin, and operations making use of these
+-- properties (or other properties not available in Data.Fin)
+open import Data.Fin.Properties
+
+-- Subsets of finite sets
+open import Data.Fin.Subset
+
+-- Some properties about subsets
+open import Data.Fin.Subset.Properties
+
+-- Substitutions
+open import Data.Fin.Substitution
+
+-- An example of how Data.Fin.Substitution can be used: a definition
+-- of substitution for the untyped λ-calculus, along with some lemmas
+open import Data.Fin.Substitution.Example
+
+-- Substitution lemmas
+open import Data.Fin.Substitution.Lemmas
+
+-- Application of substitutions to lists, along with various lemmas
+open import Data.Fin.Substitution.List
+
+-- Floats
+open import Data.Float
+
+-- Directed acyclic multigraphs
+open import Data.Graph.Acyclic
+
+-- Integers
+open import Data.Integer
+
+-- Properties related to addition of integers
+open import Data.Integer.Addition.Properties
+
+-- Integers, basic types and operations
+open import Data.Integer.Base
+
+-- Divisibility and coprimality
+open import Data.Integer.Divisibility
+
+-- Properties related to multiplication of integers
+open import Data.Integer.Multiplication.Properties
+
+-- Some properties about integers
+open import Data.Integer.Properties
+
+-- Lists
+open import Data.List
+
+-- Lists where all elements satisfy a given property
+open import Data.List.All
+
+-- Properties related to All
+open import Data.List.All.Properties
+
+-- Lists where at least one element satisfies a given property
+open import Data.List.Any
+
+-- Properties related to bag and set equality
+open import Data.List.Any.BagAndSetEquality
+
+-- List membership and some related definitions
+open import Data.List.Any.Membership
+
+-- Properties related to propositional list membership
+open import Data.List.Any.Membership.Properties
+
+-- Data.List.Any.Membership instantiated with propositional equality,
+-- along with some additional definitions.
+open import Data.List.Any.Membership.Propositional
+
+-- Properties related to propositional list membership
+open import Data.List.Any.Membership.Propositional.Properties
+
+-- Properties related to Any
+open import Data.List.Any.Properties
+
+-- Lists, basic types and operations
+open import Data.List.Base
+
+-- A categorical view of List
+open import Data.List.Categorical
+
+-- A data structure which keeps track of an upper bound on the number
+-- of elements /not/ in a given list
+open import Data.List.Countdown
+
+-- Non-empty lists
+open import Data.List.NonEmpty
+
+-- Properties of non-empty lists
+open import Data.List.NonEmpty.Properties
+
+-- List-related properties
+open import Data.List.Properties
+
+-- Lexicographic ordering of lists
+open import Data.List.Relation.NonStrictLex
+
+-- Pointwise lifting of relations to lists
+open import Data.List.Relation.Pointwise
+
+-- Lexicographic ordering of lists
+open import Data.List.Relation.StrictLex
+
+-- Reverse view
+open import Data.List.Reverse
+
+-- M-types (the dual of W-types)
+open import Data.M
+
+-- Indexed M-types (the dual of indexed W-types aka Petersson-Synek
+-- trees).
+open import Data.M.Indexed
+
+-- The Maybe type
+open import Data.Maybe
+
+-- The Maybe type and some operations
+open import Data.Maybe.Base
+
+-- Natural numbers
+open import Data.Nat
+
+-- Natural numbers, basic types and operations
+open import Data.Nat.Base
+
+-- Coprimality
+open import Data.Nat.Coprimality
+
+-- Integer division
+open import Data.Nat.DivMod
+
+-- Divisibility
+open import Data.Nat.Divisibility
+
+-- Greatest common divisor
+open import Data.Nat.GCD
+
+-- Boring lemmas used in Data.Nat.GCD and Data.Nat.Coprimality
+open import Data.Nat.GCD.Lemmas
+
+-- A generalisation of the arithmetic operations
+open import Data.Nat.GeneralisedArithmetic
+
+-- Definition of and lemmas related to "true infinitely often"
+open import Data.Nat.InfinitelyOften
+
+-- Least common multiple
+open import Data.Nat.LCM
+
+-- Primality
+open import Data.Nat.Primality
+
+-- A bunch of properties about natural number operations
+open import Data.Nat.Properties
+
+-- A bunch of properties about natural number operations
+open import Data.Nat.Properties.Simple
+
+-- Showing natural numbers
+open import Data.Nat.Show
+
+-- Transitive closures
+open import Data.Plus
+
+-- Products
+open import Data.Product
+
+-- N-ary products
+open import Data.Product.N-ary
+
+-- Properties of products
+open import Data.Product.Properties
+
+-- Lexicographic products of binary relations
+open import Data.Product.Relation.NonStrictLex
+
+-- Pointwise products of binary relations
+open import Data.Product.Relation.Pointwise
+
+-- Pointwise lifting of binary relations to sigma types
+open import Data.Product.Relation.SigmaPointwise
+
+-- Lexicographic products of binary relations
+open import Data.Product.Relation.StrictLex
+
+-- Rational numbers
+open import Data.Rational
+
+-- Properties of Rational numbers
+open import Data.Rational.Properties
+
+-- Reflexive closures
+open import Data.ReflexiveClosure
+
+-- Signs
+open import Data.Sign
+
+-- Some properties about signs
+open import Data.Sign.Properties
+
+-- The reflexive transitive closures of McBride, Norell and Jansson
+open import Data.Star
+
+-- Bounded vectors (inefficient implementation)
+open import Data.Star.BoundedVec
+
+-- Decorated star-lists
+open import Data.Star.Decoration
+
+-- Environments (heterogeneous collections)
+open import Data.Star.Environment
+
+-- Finite sets defined in terms of Data.Star
+open import Data.Star.Fin
+
+-- Lists defined in terms of Data.Star
+open import Data.Star.List
+
+-- Natural numbers defined in terms of Data.Star
+open import Data.Star.Nat
+
+-- Pointers into star-lists
+open import Data.Star.Pointer
+
+-- Some properties related to Data.Star
+open import Data.Star.Properties
+
+-- Vectors defined in terms of Data.Star
+open import Data.Star.Vec
+
+-- Streams
+open import Data.Stream
+
+-- Strings
+open import Data.String
+
+-- Strings
+open import Data.String.Base
+
+-- Sums (disjoint unions)
+open import Data.Sum
+
+-- Properties of sums (disjoint unions)
+open import Data.Sum.Properties
+
+-- Sums of binary relations
+open import Data.Sum.Relation.General
+
+-- Fixed-size tables of values, implemented as functions from 'Fin n'.
+-- Similar to 'Data.Vec', but focusing on ease of retrieval instead of
+-- ease of adding and removing elements.
+open import Data.Table
+
+-- Tables, basic types and operations
+open import Data.Table.Base
+
+-- Table-related properties
+open import Data.Table.Properties
+
+-- Pointwise table equality
+open import Data.Table.Relation.Equality
+
+-- Some unit types
+open import Data.Unit
+
+-- The unit type and the total relation on unit
+open import Data.Unit.Base
+
+-- Some unit types
+open import Data.Unit.NonEta
+
+-- Vectors
+open import Data.Vec
+
+-- Vectors where all elements satisfy a given property
+open import Data.Vec.All
+
+-- Properties related to All
+open import Data.Vec.All.Properties
+
+-- A categorical view of Vec
+open import Data.Vec.Categorical
+
+-- Code for converting Vec A n → B to and from n-ary functions
+open import Data.Vec.N-ary
+
+-- Some Vec-related properties
+open import Data.Vec.Properties
+
+-- Semi-heterogeneous vector equality
+open import Data.Vec.Relation.Equality
+
+-- Extensional pointwise lifting of relations to vectors
+open import Data.Vec.Relation.ExtensionalPointwise
+
+-- Inductive pointwise lifting of relations to vectors
+open import Data.Vec.Relation.InductivePointwise
+
+-- W-types
+open import Data.W
+
+-- Indexed W-types aka Petersson-Synek trees
+open import Data.W.Indexed
+
+-- Machine words
+open import Data.Word
+
+-- Type(s) used (only) when calling out to Haskell via the FFI
+open import Foreign.Haskell
+
+-- Simple combinators working solely on and with functions
+open import Function
+
+-- Bijections
+open import Function.Bijection
+
+-- Function setoids and related constructions
+open import Function.Equality
+
+-- Equivalence (coinhabitance)
+open import Function.Equivalence
+
+-- Injections
+open import Function.Injection
+
+-- Inverses
+open import Function.Inverse
+
+-- Left inverses
+open import Function.LeftInverse
+
+-- A universe which includes several kinds of "relatedness" for sets,
+-- such as equivalences, surjections and bijections
+open import Function.Related
+
+-- Basic lemmas showing that various types are related (isomorphic or
+-- equivalent or…)
+open import Function.Related.TypeIsomorphisms
+
+-- Surjections
+open import Function.Surjection
+
+-- IO
+open import IO
+
+-- Primitive IO: simple bindings to Haskell types and functions
+open import IO.Primitive
+
+-- An abstraction of various forms of recursion/induction
+open import Induction
+
+-- Lexicographic induction
+open import Induction.Lexicographic
+
+-- Various forms of induction for natural numbers
+open import Induction.Nat
+
+-- Well-founded induction
+open import Induction.WellFounded
+
+-- Universe levels
+open import Level
+
+-- Record types with manifest fields and "with", based on Randy
+-- Pollack's "Dependently Typed Records in Type Theory"
+open import Record
+
+-- Support for reflection
+open import Reflection
+
+-- Properties of homogeneous binary relations
+open import Relation.Binary
+
+-- Some properties imply others
+open import Relation.Binary.Consequences
+
+-- Convenient syntax for equational reasoning
+open import Relation.Binary.EqReasoning
+
+-- Equivalence closures of binary relations
+open import Relation.Binary.EquivalenceClosure
+
+-- Many properties which hold for _∼_ also hold for flip _∼_
+open import Relation.Binary.Flip
+
+-- Heterogeneous equality
+open import Relation.Binary.HeterogeneousEquality
+
+-- Quotients for Heterogeneous equality
+open import Relation.Binary.HeterogeneousEquality.Quotients
+
+-- Example of a Quotient: ℤ as (ℕ × ℕ / ~)
+open import Relation.Binary.HeterogeneousEquality.Quotients.Examples
+
+-- Indexed binary relations
+open import Relation.Binary.Indexed
+
+-- Induced preorders
+open import Relation.Binary.InducedPreorders
+
+-- Order-theoretic lattices
+open import Relation.Binary.Lattice
+
+-- Lexicographic ordering of lists
+open import Relation.Binary.List.NonStrictLex
+
+-- Pointwise lifting of relations to lists
+open import Relation.Binary.List.Pointwise
+
+-- Lexicographic ordering of lists
+open import Relation.Binary.List.StrictLex
+
+-- Conversion of ≤ to <, along with a number of properties
+open import Relation.Binary.NonStrictToStrict
+
+-- Many properties which hold for _∼_ also hold for _∼_ on f
+open import Relation.Binary.On
+
+-- Order morphisms
+open import Relation.Binary.OrderMorphism
+
+-- Convenient syntax for "equational reasoning" using a partial order
+open import Relation.Binary.PartialOrderReasoning
+
+-- Convenient syntax for "equational reasoning" using a preorder
+open import Relation.Binary.PreorderReasoning
+
+-- Lexicographic products of binary relations
+open import Relation.Binary.Product.NonStrictLex
+
+-- Pointwise products of binary relations
+open import Relation.Binary.Product.Pointwise
+
+-- Lexicographic products of binary relations
+open import Relation.Binary.Product.StrictLex
+
+-- Properties satisfied by bounded join semilattices
+open import Relation.Binary.Properties.BoundedJoinSemilattice
+
+-- Properties satisfied by bounded meet semilattices
+open import Relation.Binary.Properties.BoundedMeetSemilattice
+
+-- Properties satisfied by decidable total orders
+open import Relation.Binary.Properties.DecTotalOrder
+
+-- Properties satisfied by join semilattices
+open import Relation.Binary.Properties.JoinSemilattice
+
+-- Properties satisfied by lattices
+open import Relation.Binary.Properties.Lattice
+
+-- Properties satisfied by meet semilattices
+open import Relation.Binary.Properties.MeetSemilattice
+
+-- Properties satisfied by posets
+open import Relation.Binary.Properties.Poset
+
+-- Properties satisfied by preorders
+open import Relation.Binary.Properties.Preorder
+
+-- Properties satisfied by strict partial orders
+open import Relation.Binary.Properties.StrictPartialOrder
+
+-- Properties satisfied by strict partial orders
+open import Relation.Binary.Properties.StrictTotalOrder
+
+-- Properties satisfied by total orders
+open import Relation.Binary.Properties.TotalOrder
+
+-- Propositional (intensional) equality
+open import Relation.Binary.PropositionalEquality
+
+-- An equality postulate which evaluates
+open import Relation.Binary.PropositionalEquality.TrustMe
+
+-- Helpers intended to ease the development of "tactics" which use
+-- proof by reflection
+open import Relation.Binary.Reflection
+
+-- Convenient syntax for "equational reasoning" in multiple Setoids
+open import Relation.Binary.SetoidReasoning
+
+-- Pointwise lifting of binary relations to sigma types
+open import Relation.Binary.Sigma.Pointwise
+
+-- Some simple binary relations
+open import Relation.Binary.Simple
+
+-- Convenient syntax for "equational reasoning" using a strict partial
+-- order
+open import Relation.Binary.StrictPartialOrderReasoning
+
+-- Conversion of < to ≤, along with a number of properties
+open import Relation.Binary.StrictToNonStrict
+
+-- Sums of binary relations
+open import Relation.Binary.Sum
+
+-- Symmetric closures of binary relations
+open import Relation.Binary.SymmetricClosure
+
+-- Pointwise lifting of relations to vectors
+open import Relation.Binary.Vec.Pointwise
+
+-- Operations on nullary relations (like negation and decidability)
+open import Relation.Nullary
+
+-- Operations on and properties of decidable relations
+open import Relation.Nullary.Decidable
+
+-- Implications of nullary relations
+open import Relation.Nullary.Implication
+
+-- Properties related to negation
+open import Relation.Nullary.Negation
+
+-- Products of nullary relations
+open import Relation.Nullary.Product
+
+-- Sums of nullary relations
+open import Relation.Nullary.Sum
+
+-- A universe of proposition functors, along with some properties
+open import Relation.Nullary.Universe
+
+-- Unary relations
+open import Relation.Unary
+
+-- Predicate transformers
+open import Relation.Unary.PredicateTransformer
+
+-- Sizes for Agda's sized types
+open import Size
+
+-- Strictness combinators
+open import Strict
+
+-- Universes
+open import Universe
+