# Agda User Manual

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Agda User Manual Release 2.6.0 The Agda Team Oct 18, 2018 Contents 1 Overview 1 2 Getting Started 2.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Quick Guide to Editing, Type Checking and Compiling Agda Code . . . . . . . . . . . . . . . . . . 3 3 4 6 3 Language Reference 3.1 Abstract definitions . . . . . . . 3.2 Built-ins . . . . . . . . . . . . 3.3 Coinduction . . . . . . . . . . 3.4 Copatterns . . . . . . . . . . . 3.5 Core language . . . . . . . . . 3.6 Cubical Type Theory in Agda . 3.7 Data Types . . . . . . . . . . . 3.8 Foreign Function Interface . . . 3.9 Function Definitions . . . . . . 3.10 Function Types . . . . . . . . . 3.11 Implicit Arguments . . . . . . . 3.12 Instance Arguments . . . . . . 3.13 Irrelevance . . . . . . . . . . . 3.14 Lambda Abstraction . . . . . . 3.15 Local Definitions: let and where 3.16 Lexical Structure . . . . . . . . 3.17 Literal Overloading . . . . . . 3.18 Mixfix Operators . . . . . . . . 3.19 Module System . . . . . . . . . 3.20 Mutual Recursion . . . . . . . 3.21 Pattern Synonyms . . . . . . . 3.22 Positivity Checking . . . . . . . 3.23 Postulates . . . . . . . . . . . . 3.24 Pragmas . . . . . . . . . . . . 3.25 Record Types . . . . . . . . . . 3.26 Reflection . . . . . . . . . . . . 3.27 Rewriting . . . . . . . . . . . . 3.28 Safe Agda . . . . . . . . . . . 3.29 Sized Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 12 22 24 27 27 28 31 36 39 40 42 47 52 53 57 60 62 64 69 70 71 73 74 75 82 90 90 91 i 4 3.30 3.31 3.32 3.33 3.34 3.35 3.36 Syntactic Sugar . . . . Syntax Declarations . Telescopes . . . . . . Termination Checking Universe Levels . . . With-Abstraction . . . Without K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 . 98 . 99 . 99 . 100 . 100 . 110 Tools 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Automatic Proof Search (Auto) Command-line options . . . . . Compilers . . . . . . . . . . . Emacs Mode . . . . . . . . . . Literate Programming . . . . . Generating HTML . . . . . . . Generating LaTeX . . . . . . . Library Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 113 116 121 123 127 129 129 140 5 Contribute 143 5.1 Documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6 The Agda License 147 7 The Agda Team 149 8 Indices and tables 151 Bibliography ii 153 CHAPTER 1 Overview Note: The Agda User Manual is a work-in-progress and is still incomplete. Contributions, additions and corrections to the Agda manual are greatly appreciated. To do so, please open a pull request or issue on the Github Agda page. This is the manual for the Agda programming language, its type checking, compilation and editing system and related tools. A description of the Agda language is given in chapter Language Reference. Guidance on how the Agda editing and compilation system can be used can be found in chapter Tools. 1 Agda User Manual, Release 2.6.0 2 Chapter 1. Overview CHAPTER 2 Getting Started 2.1 Prerequisites You need recent versions of the following programs to compile Agda: • GHC: https://www.haskell.org/ghc/ • cabal-install: https://www.haskell.org/cabal/ • Alex: https://www.haskell.org/alex/ • Happy: https://www.haskell.org/happy/ • GNU Emacs: http://www.gnu.org/software/emacs/ You should also make sure that programs installed by cabal-install are on your shell’s search path. For instructions on installing a suitable version of Emacs under Windows, see Installing Emacs under Windows. Non-Windows users need to ensure that the development files for the C libraries zlib* and ncurses* are installed (see http://zlib.net and http://www.gnu.org/software/ncurses/). Your package manager may be able to install these files for you. For instance, on Debian or Ubuntu it should suffice to run apt-get install zlib1g-dev libncurses5-dev as root to get the correct files installed. Optionally one can also install the ICU library, which is used to implement the --count-clusters flag. Under Debian or Ubuntu it may suffice to install libicu-dev. Once the ICU library is installed one can hopefully enable the --count-clusters flag by giving the -fenable-cluster-counting flag to cabal install. 2.1.1 Installing Emacs under Windows A precompiled version of Emacs 24.3, with the necessary mathematical fonts, is available at http://homepage.cs.uiowa. edu/~astump/agda/ . 3 Agda User Manual, Release 2.6.0 2.2 Installation There are several ways to install Agda: • Using a released source package from Hackage • Using a binary package prepared for your platform • Using the development version from the Git repository Agda can be installed using different flags (see Installation Flags). 2.2.1 Installation from Hackage You can install the latest released version of Agda from Hackage. Install the prerequisites and then run the following commands: cabal update cabal install Agda agda-mode setup The last command tries to set up Emacs for use with Agda via the Emacs mode. As an alternative you can copy the following text to your .emacs file: (load-file (let ((coding-system-for-read 'utf-8)) (shell-command-to-string "agda-mode locate"))) It is also possible (but not necessary) to compile the Emacs mode’s files: agda-mode compile This can, in some cases, give a noticeable speedup. Warning: If you reinstall the Agda mode without recompiling the Emacs Lisp files, then Emacs may continue using the old, compiled files. 2.2.2 Prebuilt Packages and System-Specific Instructions Arch Linux The following prebuilt packages are available: • Agda • Agda standard library Debian / Ubuntu Prebuilt packages are available for Debian testing/unstable and Ubuntu from Karmic onwards. To install: apt-get install agda-mode This should install Agda and the Emacs mode. The standard library is available in Debian testing/unstable and Ubuntu from Lucid onwards. To install: 4 Chapter 2. Getting Started Agda User Manual, Release 2.6.0 apt-get install agda-stdlib More information: • Agda (Debian) • Agda standard library (Debian) • Agda (Ubuntu) • Agda standard library (Ubuntu) Reporting bugs: Please report any bugs to Debian, using: reportbug -B debian agda reportbug -B debian agda-stdlib Fedora Agda is packaged in Fedora (since before Fedora 18). yum install Agda will pull in emacs-agda-mode and ghc-Agda-devel. FreBSD Packages are available from FreshPorts for Agda and Agda standard library. NixOS Agda is part of the Nixpkgs collection that is used by https://nixos.org/nixos. To install Agda and agda-mode for Emacs, type: nix-env -f "" -iA haskellPackages.Agda If you’re just interested in the library, you can also install the library without the executable. The Agda standard library is currently not installed automatically. OS X Homebrew provides prebuilt packages for OS X. To install: brew install agda This should take less than a minute, and install Agda together with the Emacs mode and the standard library. By default, the standard library is installed in /usr/local/lib/agda/. To use the standard library, it is convenient to add /usr/local/lib/agda/standard-library.agda-lib to ~/.agda/libraries, and specify standard-library in ~/.agda/defaults. Note this is not performed automatically. It is also possible to install --without-stdlib, --without-ghc, or from --HEAD. Note this will require building Agda from source. 2.2. Installation 5 Agda User Manual, Release 2.6.0 For more information, refer to the Homebrew documentation. Note: If Emacs cannot find the agda-mode executable, it might help to install the exec-path-from-shell package by doing M-x package-install RET exec-path-from-shell RET, and adding (exec-path-from-shell-initialize) to your .emacs file. 2.2.3 Installation of the Development Version After getting the development version following the instructions in the Agda wiki: • Install the prerequisites • In the top-level directory of the Agda source tree – Follow the instructions for installing Agda from Hackage (except run cabal install instead of cabal install Agda) or – You can try to install Agda (including a compiled Emacs mode) by running the following command: make install Note that on a Mac, because ICU is installed in a non-standard location, you need to specify this location on the command line: make install-bin CABAL_OPTS='--extra-lib-dirs=/usr/local/opt/icu4c/lib -˓→extra-include-dirs=/usr/local/opt/icu4c/include' 2.2.4 Installation Flags When installing Agda the following flags can be used: cpphs Use cpphs instead of cpp. Default: off. debug Enable debugging features that may slow Agda down. Default: off. flag enable-cluster-counting Enable the --count-clusters flag (see Counting Extended Grapheme Clusters). Note that if enable-cluster-counting is False, then the --count-clusters flag triggers an error message. Default: off. 2.3 Quick Guide to Editing, Type Checking and Compiling Agda Code 2.3.1 Introduction Agda programs are commonly edited using Emacs or Atom. To edit a module (assuming you have installed Agda and its Emacs mode (or Atom’s) properly), start the editor and open a file ending in .agda. Programs are developed interactively, which means that one can type check code which is not yet complete: if a question mark (?) is used as a placeholder for an expression, and the buffer is then checked, Agda will replace the question mark with a “hole” which can be filled in later. One can also do various other things in the context of a hole: listing the context, inferring the type of an expression, and even evaluating an open term which mentions variables bound in the surrounding context. 6 Chapter 2. Getting Started Agda User Manual, Release 2.6.0 The following commands are the most common (see Notation for key combinations): C-c C-l Load. Type-checks the contents of the file. C-c C-, Shows the goal type, i.e. the type expected in the current hole, along with the types of locally defined identifiers. C-c C-. A variant of C-c C-, that also tries to infer the type of the current hole’s contents. C-c C-SPC Give. Checks whether the term written in the current hole has the right type and, if it does, replaces the hole with that term. C-c C-r Refine. Checks whether the return type of the expression e in the hole matches the expected type. If so, the hole is replaced by e { }1 ... { }n, where a sufficient number of new holes have been inserted. If the hole is empty, then the refine command instead inserts a lambda or constructor (if there is a unique type-correct choice). C-c C-c Case split. If the cursor is positioned in a hole which denotes the right hand side of a definition, then this command automatically performs pattern matching on variables of your choice. C-c C-n Normalise. The system asks for a term which is then evaluated. M-. Go to definition. Goes to the definition site of the identifier under the cursor (if known). M-* Go back (Emacs < 25.1) M-, Go back (Emacs 25.1) For information related to the Emacs mode (configuration, keybindings, Unicode input, etc.) see Emacs Mode. 2.3.2 Menus There are two main menus in the system: • A main menu called Agda2 which is used for global commands. • A context sensitive menu which appears if you right-click in a hole. The menus contain more commands than the ones listed above. See global and context sensitive commands. 2.3.3 Writing mathematical symbols in source code Agda uses Unicode characters in source files (more specifically: the UTF-8 character encoding). Almost any character can be used in an identifier (like , 𝛼, , or , for example). It is therefore necessary to have spaces between most lexical units. Many mathematical symbols can be typed using the corresponding LaTeX command names. For instance, you type \forall to input . A more detailed description of how to write various characters is available. (Note that if you try to read Agda code using another program, then you have to make sure that it uses the right character encoding when decoding the source files.) 2.3.4 Errors If a file does not type check Agda will complain. Often the cursor will jump to the position of the error, and the error will (by default) be underlined. Some errors are treated a bit differently, though. If Agda cannot see that a definition is terminating/productive it will highlight it in light salmon, and if some meta-variable other than the goals cannot be solved the code will be highlighted in yellow (the highlighting may not appear until after you have reloaded the file). 2.3. Quick Guide to Editing, Type Checking and Compiling Agda Code 7 Agda User Manual, Release 2.6.0 In case of the latter kinds of errors you can still work with the file, but Agda will (by default) refuse to import it into another module, and if your functions are not terminating Agda may hang. If you do not like the way errors are highlighted (if you are colour-blind, for instance), then you can tweak the settings by typing M-x customize-group RET agda2-highlight RET in Emacs (after loading an Agda file) and following the instructions. 2.3.5 Compiling Agda programs To compile a module containing a function main :: IO A for some A (where IO can be found in the Primitive.agda), use C-c C-x C-c. If the module is named A.B.C the resulting binary will be called C (located in the project’s top-level directory, the one containing the A directory). 2.3.6 Batch-mode command There is also a batch-mode command line tool: agda. To find out more about this command, use agda --help. 8 Chapter 2. Getting Started CHAPTER 3 Language Reference 3.1 Abstract definitions Definitions can be marked as abstract, for the purpose of hiding implementation details, or to speed up type-checking of other parts. In essence, abstract definitions behave like postulates, thus, do not reduce/compute. For instance, proofs whose content does not matter could be marked abstract, to prevent Agda from unfolding them (which might slow down type-checking). As a guiding principle, all the rules concerning abstract are designed to prevent the leaking of implementation details of abstract definitions. Similar concepts of other programming language include (non-representative sample): UCSD Pascal’s and Java’s interfaces and ML’s signatures. (Especially when abstract definitions are used in combination with modules.) 3.1.1 Synopsis • Declarations can be marked as abstract using the block keyword abstract. • Outside of abstract blocks, abstract definitions do not reduce, they are treated as postulates, in particular: – Abstract functions never match, thus, do not reduce. – Abstract data types do not expose their constructors. – Abstract record types do not expose their fields nor constructor. – Other declarations cannot be abstract. • Inside abstract blocks, abstract definitions reduce while type checking definitions, but not while checking their type signatures. Otherwise, due to dependent types, one could leak implementation details (e.g. expose reduction behavior by using propositional equality). • Inside private type signatures in abstract blocks, abstract definitions do reduce. However, there are some problems with this. See Issue #418. • The reach of the abstract keyword block extends recursively to the where-blocks of a function and the declarations inside of a record declaration, but not inside modules declared in an abstract block. 9 Agda User Manual, Release 2.6.0 3.1.2 Examples Integers can be implemented in various ways, e.g. as difference of two natural numbers: module Integer where abstract = Nat × Nat 0 : 0 = 0 , 0 1 : 1 = 1 , 0 _+_ : (x y : ) → (p , n) + (p' , n') = (p + p') , (n + n') -_ : → - (p , n) = (n , p) __ : (x y : ) → Set (p , n) (p' , n') = (p + n') private postulate +comm : n m → (n + m) (p' + n) (m + n) inv : x → (x + (- x)) 0 inv (p , n) rewrite +comm (p + n) 0 | +comm p n = refl Using abstract we do not give away the actual representation of integers, nor the implementation of the operations. We can construct them from 0, 1, _+_, and -, but only reason about equality with the provided lemma inv. The following property shape-of-0 of the integer zero exposes the representation of integers as pairs. As such, it is rejected by Agda: when checking its type signature, proj1 x fails to type check since x is of abstract type . Remember that the abstract definition of does not unfold in type signatures, even when in an abstract block! However, if we make shape-of- private, unfolding of abstract definitions like is enabled, and we succeed: -- A property about the representation of zero integers: abstract private shape-of-0 : (x : ) (is0 : x 0) → proj1 x proj2 x shape-of-0 (p , n) refl rewrite +comm p 0 = refl By requiring shape-of-0 to be private to type-check, leaking of representation details is prevented. 3.1.3 Scope of abstraction In child modules, when checking an abstract definition, the abstract definitions of the parent module are transparent: module M1 where abstract x = 0 (continues on next page) 10 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) module M2 where abstract x-is-0 : x 0 x-is-0 = refl Thus, child modules can see into the representation choices of their parent modules. However, parent modules cannot see like this into child modules, nor can sibling modules see through each others abstract definitions. An exception to this is anonymous modules, which share abstract scope with their parent module, allowing parent or sibling modules to see inside their abstract definitions. The reach of the abstract keyword does not extend into modules: module Parent where abstract module Child where y = 0 x = 0 -- to avoid "useless abstract" error y-is-0 : Child.y y-is-0 = refl 0 The declarations in module Child are not abstract! 3.1.4 Abstract definitions with where-blocks Definitions in a where block of an abstract definition are abstract as well. This means, they can see through the abstractions of their uncles: module Where where abstract x : Nat x = 0 y : Nat y = x where xy : x 0 xy = refl Type signatures in where blocks are private, so it is fine to make type abbreviations in where blocks of abstract definitions: module WherePrivate where abstract x : Nat x = proj1 t where T = Nat × Nat t : T t = 0 , 1 p : proj1 t 0 p = refl Note that if p was not private, application proj1 t in its type would be ill-formed, due to the abstract definition of T. 3.1. Abstract definitions 11 Agda User Manual, Release 2.6.0 Named where-modules do not make their declarations private, thus this example will fail if you replace x’s where by module M where. 3.2 Built-ins • Using the built-in types • The unit type • Booleans • Natural numbers • Machine words • Integers • Floats • Lists • Characters • Strings • Equality • Universe levels • Sized types • Coinduction • IO • Literal overloading • Reflection • Rewriting • Static values • Strictness The Agda type checker knows about, and has special treatment for, a number of different concepts. The most prominent is natural numbers, which has a special representation as Haskell integers and support for fast arithmetic. The surface syntax of these concepts are not fixed, however, so in order to use the special treatment of natural numbers (say) you define an appropriate data type and then bind that type to the natural number concept using a BUILTIN pragma. Some built-in types support primitive functions that have no corresponding Agda definition. These functions are declared using the primitive keyword by giving their type signature. 3.2.1 Using the built-in types While it is possible to define your own versions of the built-in types and bind them using BUILTIN pragmas, it is recommended to use the definitions in the Agda.Builtin modules. These modules are installed when you install Agda and so are always available. For instance, built-in natural numbers are defined in Agda.Builtin.Nat. The standard library and the agda-prelude reexport the definitions from these modules. 12 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 3.2.2 The unit type module Agda.Builtin.Unit The unit type is bound to the built-in UNIT as follows: record : Set where {-# BUILTIN UNIT #-} Agda needs to know about the unit type since some of the primitive operations in the reflected type checking monad return values in the unit type. 3.2.3 Booleans module Agda.Builtin.Bool where Built-in booleans are bound using the BOOL, TRUE and FALSE built-ins: data Bool : Set where false true : Bool {-# BUILTIN BOOL Bool #-} {-# BUILTIN TRUE true #-} {-# BUILTIN FALSE false #-} Note that unlike for natural numbers, you need to bind the constructors separately. The reason for this is that Agda cannot tell which constructor should correspond to true and which to false, since you are free to name them whatever you like. The effect of binding the boolean type is that you can then use primitive functions returning booleans, such as built-in NATEQUALS, and letting the GHC backend know to compile the type to Haskell Bool. 3.2.4 Natural numbers module Agda.Builtin.Nat Built-in natural numbers are bound using the NATURAL built-in as follows: data Nat : Set where zero : Nat suc : Nat → Nat {-# BUILTIN NATURAL Nat #-} The names of the data type and the constructors can be chosen freely, but the shape of the datatype needs to match the one given above (modulo the order of the constructors). Note that the constructors need not be bound explicitly. Binding the built-in natural numbers as above has the following effects: • The use of natural number literals is enabled. By default the type of a natural number literal will be Nat, but it can be overloaded to include other types as well. • Closed natural numbers are represented as Haskell integers at compile-time. • The compiler backends compile natural numbers to the appropriate number type in the target language. • Enabled binding the built-in natural number functions described below. 3.2. Built-ins 13 Agda User Manual, Release 2.6.0 Functions on natural numbers There are a number of built-in functions on natural numbers. These are special in that they have both an Agda definition and a primitive implementation. The primitive implementation is used to evaluate applications to closed terms, and the Agda definition is used otherwise. This lets you prove things about the functions while still enjoying good performance of compile-time evaluation. The built-in functions are the following: _+_ : Nat → Nat → Nat zero + m = m suc n + m = suc (n + m) {-# BUILTIN NATPLUS _+_ #-} _-_ : Nat → Nat → Nat n - zero = n zero - suc m = zero suc n - suc m = n - m {-# BUILTIN NATMINUS _-_ #-} _*_ : Nat → Nat → Nat zero * m = zero suc n * m = (n * m) + m {-# BUILTIN NATTIMES _*_ #-} _==_ : Nat → Nat → Bool zero == zero = true suc n == suc m = n == m _ == _ = false {-# BUILTIN NATEQUALS _==_ #-} _<_ : Nat → Nat → Bool _ < zero = false zero < suc _ = true suc n < suc m = n < m {-# BUILTIN NATLESS _<_ #-} div-helper : Nat → Nat → Nat → Nat → Nat div-helper k m zero j = k div-helper k m (suc n) zero = div-helper (suc k) m n m div-helper k m (suc n) (suc j) = div-helper k m n j {-# BUILTIN NATDIVSUCAUX div-helper #-} mod-helper : Nat → Nat → Nat → Nat → Nat mod-helper k m zero j = k mod-helper k m (suc n) zero = mod-helper 0 m n m mod-helper k m (suc n) (suc j) = mod-helper (suc k) m n j {-# BUILTIN NATMODSUCAUX mod-helper #-} The Agda definitions are checked to make sure that they really define the corresponding built-in function. The definitions are not required to be exactly those given above, for instance, addition and multiplication can be defined by recursion on either argument, and you can swap the arguments to the addition in the recursive case of multiplication. The NATDIVSUCAUX and NATMODSUCAUX are built-ins bind helper functions for defining natural number division and modulo operations, and satisfy the properties div n (suc m) mod n (suc m) 14 div-helper 0 m n m mod-helper 0 m n m Chapter 3. Language Reference Agda User Manual, Release 2.6.0 3.2.5 Machine words module Agda.Builtin.Word Agda supports built-in 64-bit machine words, bound with the WORD64 built-in: {-# BUILTIN WORD64 Word64 #-} Machine words can be converted to and from natural numbers using the following primitives: primitive primWord64ToNat : Word64 → Nat primWord64FromNat : Nat → Word64 Converting to a natural number is the trivial embedding, and converting from a natural number gives you the remainder modulo 264 . The proofs of these theorems are not primitive, but can be defined in a library using primTrustMe. Basic arithmetic operations can be defined on Word64 by converting to natural numbers, peforming the corresponding operation, and then converting back. The compiler will optimise these to use 64-bit arithmetic. For instance: addWord : Word64 → Word64 → Word64 addWord a b = primWord64FromNat (primWord64ToNat a + primWord64ToNat b) subWord : Word64 → Word64 → Word64 subWord a b = primWord64FromNat ((primWord64ToNat a + 18446744073709551616) ˓→primWord64ToNat b) These compile to primitive addition and subtraction on 64-bit words, which in the GHC backend map to operations on Haskell 64-bit words (Data.Word.Word64). 3.2.6 Integers module Agda.Builtin.Int Built-in integers are bound with the INTEGER built-in to a data type with two constructors: one for positive and one for negative numbers. The built-ins for the constructors are INTEGERPOS and INTEGERNEGSUC. data Int : Set where pos : Nat → Int negsuc : Nat → Int {-# BUILTIN INTEGER Int #-} {-# BUILTIN INTEGERPOS pos #-} {-# BUILTIN INTEGERNEGSUC negsuc #-} Here negsuc n represents the integer -n - 1. Unlike for natural numbers, there is no special representation of integers at compile-time since the overhead of using the data type compared to Haskell integers is not that big. Built-in integers support the following primitive operation (given a suitable binding for String): primitive primShowInteger : Int → String 3.2. Built-ins 15 Agda User Manual, Release 2.6.0 3.2.7 Floats module Agda.Builtin.Float Floating point numbers are bound with the FLOAT built-in: {-# BUILTIN FLOAT Float #-} This lets you use floating point literals. Floats are represented by the type checker as IEEE 754 binary64 double precision floats, with the restriction that there is exactly one NaN value. The following primitive functions are available (with suitable bindings for Nat, Bool, String and Int): primitive primNatToFloat primFloatPlus primFloatMinus primFloatTimes primFloatNegate primFloatDiv primFloatEquality primFloatLess primFloatNumericalEquality primFloatNumericalLess primRound primFloor primCeiling primExp primLog primSin primCos primTan primASin primACos primATan primATan2 primShowFloat : : : : : : : : : : : : : : : : : : : : : : : Nat → Float Float → Float → Float → Float → Float → Float → Float → Float Float → Float → Float → Float → Float → Float → Float → Float → Float → Float → Float → Int Float → Int Float → Int Float → Float Float → Float Float → Float Float → Float Float → Float Float → Float Float → Float Float → Float Float → Float → Float → String Float Float Float Float Bool Bool Bool Bool Float The primFloatEquality primitive is intended to be used for decidable propositional equality. To enable proof carrying comparisons while preserving consistency, the following laws apply: nan=nan : primFloatEquality NaN NaN nan=nan = refl true nan=-nan : primFloatEquality NaN (primFloatNegate NaN) nan=-nan = refl neg00 : primFloatEquality 0.0 -0.0 neg00 = refl true false Correspondingly, the primFloatLess can be used to provide a decidable total order, given by the following laws: _[<]_ : Float → Float → Set x [<] y = primFloatLess x y && not (primFloatLess y x) true -inf #-} 3.2.20 Strictness module Agda.Builtin.Strict There are two primitives for controlling evaluation order: primitive primForce : {a b} {A : Set a} {B : A → Set b} (x : A) → ( x → B x) → B x primForceLemma : {a b} {A : Set a} {B : A → Set b} (x : A) (f : x → B x) → ˓→primForce x f f x where __ is the built-in equality. At compile-time primForce x f evaluates to f x when x is in weak head normal form (whnf), i.e. one of the following: • a constructor application • a literal • a lambda abstraction • a type constructor application (data or record type) • a function type • a universe (Set _) Similarly primForceLemma x f, which lets you reason about programs using primForce, evaluates to refl when x is in whnf. At run-time, primForce e f is compiled (by the GHC backend) to let x = e in seq x (f x). For example, consider the following function: -- pow’ n a = a 2 pow’ : Nat → Nat → Nat pow’ zero a = a pow’ (suc n) a = pow’ n (a + a) There is a space leak here (both for compile-time and run-time evaluation), caused by unevaluated a + a thunks. This problem can be fixed with primForce: infixr 0 _$!_ _$!_ : {a b} {A : Set a} {B : A → Set b} → ( x → B x) → f $! x = primForce x f x → B x -- pow n a = a 2 pow : Nat → Nat → Nat (continues on next page) 3.2. Built-ins 21 Agda User Manual, Release 2.6.0 (continued from previous page) pow zero a = a pow (suc n) a = pow n $! a + a 3.3 Coinduction 3.3.1 Coinductive Records It is possible to define the type of infinite lists (or streams) of elements of some type A as follows, record Stream (A : Set) : Set where coinductive field hd : A tl : Stream A As opposed to inductive record types, we have to introduce the keyword coinductive before defining the fields that constitute the record. It is interesting to note that is not neccessary to give an explicit constructor to the record type Stream A. We can as well define bisimilarity (equivalence) of a pair of Stream A as a coinductive record. record __ {A : Set} (xs : Stream A) (ys : Stream A) : Set where coinductive field hd- : hd xs hd ys tl- : tl xs tl ys Using copatterns we can define a pair of functions on Stream such that one returns a Stream with the elements in the even positions and the other the elements in odd positions. even : {A} → Stream A → Stream A hd (even x) = hd x tl (even x) = even (tl (tl x)) odd : {A} → Stream A → Stream A odd x = even (tl x) split : {A } → Stream A → Stream A × Stream A split xs = even xs , odd xs And merge a pair of Stream by interleaving their elements. merge : {A} → Stream A × Stream A → Stream A hd (merge (fst , snd)) = hd fst tl (merge (fst , snd)) = merge (snd , tl fst) Finally, we can prove that split is the left inverse of merge. merge-split-id : {A} (xs : Stream A) → merge (split xs) hd- (merge-split-id _) = refl tl- (merge-split-id xs) = merge-split-id (tl xs) xs 22 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 3.3.2 Old Coinduction Note: This is the old way of coinduction support in Agda. You are advised to use Coinductive Records instead. Note: The type constructor ∞ can be used to prove absurdity! To use coinduction it is recommended that you import the module Coinduction from the standard library. Coinductive types can then be defined by labelling coinductive occurrences using the delay operator ∞: data Co : Set where zero : Co suc : ∞ Co → Co The type ∞ A can be seen as a suspended computation of type A. It comes with delay and force functions: _ : : {a} {A : Set a} → A → ∞ A {a} {A : Set a} → ∞ A → A Values of coinductive types can be constructed using corecursion, which does not need to terminate, but has to be productive. As an approximation to productivity the termination checker requires that corecursive definitions are guarded by coinductive constructors. As an example the infinite “natural number” can be defined as follows: inf : Co inf = suc ( inf) The check for guarded corecursion is integrated with the check for size-change termination, thus allowing interesting combinations of inductive and coinductive types. We can for instance define the type of stream processors, along with some functions: -- Infinite streams. data Stream (A : Set) : Set where __ : (x : A) (xs : ∞ (Stream A)) → Stream A -- A stream processor SP A B consumes elements of A and produces -- elements of B. It can only consume a finite number of A’s before -- producing a B. data SP (A B : Set) : Set where get : (f : A → SP A B) → SP A B put : (b : B) (sp : ∞ (SP A B)) → SP A B -- The function eat is defined by an outer corecursion into Stream B -- and an inner recursion on SP A B. eat : {A B} → SP A B → Stream A → Stream B eat (get f) (a as) = eat (f a) ( as) eat (put b sp) as = b eat ( sp) as -- Composition of stream processors. __ : {A B C} → SP B C → SP A B → SP A C get f1 put x sp2 = f1 x sp2 (continues on next page) 3.3. Coinduction 23 Agda User Manual, Release 2.6.0 (continued from previous page) put x sp1 sp1 sp2 get f2 = put x ( ( sp1 sp2 )) = get (𝜆 x → sp1 f2 x) It is also possible to define “coinductive families”. It is recommended not to use the delay constructor (_) in a constructor’s index expressions. The following definition of equality between coinductive “natural numbers” is discouraged: data _’_ : Co → Co → Set where zero : zero ’ zero suc : {m n} → ∞ (m ’ n) → suc ( m) ’ suc ( n) The recommended definition is the following one: data __ : Co → Co → Set where zero : zero zero suc : {m n} → ∞ ( m n) → suc m suc n The current implementation of coinductive types comes with some limitations. 3.4 Copatterns Consider the following record: record Enumeration (A : Set) : Set where constructor enumeration field start : A forward : A → A backward : A → A This gives an interfaces that allows us to move along the elements of a data type A. For example, we can get the “third” element of a type A: open Enumeration 3rd : {A : Set} → Enumeration A → A 3rd e = forward e (forward e (forward e (start e))) Or we can go back 2 positions starting from a given a: backward-2 : {A : Set} → Enumeration A → A → A backward-2 e a = backward (backward a) where open Enumeration e Now, we want to use these methods on natural numbers. For this, we need a record of type Enumeration Nat. Without copatterns, we would specify all the fields in a single expression: open Enumeration enum-Nat : Enumeration Nat enum-Nat = record { start = 0 ; forward = suc (continues on next page) 24 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) ; backward = pred } where pred : Nat → Nat pred zero = zero pred (suc x) = x test1 : 3rd enum-Nat test1 = refl 3 test2 : backward-2 enum-Nat 5 test2 = refl 3 Note that if we want to use automated case-splitting and pattern matching to implement one of the fields, we need to do so in a separate definition. With copatterns, we can define the fields of a record as separate declarations, in the same way that we would give different cases for a function: open Enumeration enum-Nat start forward backward backward : Enumeration Nat enum-Nat = 0 enum-Nat n = suc n enum-Nat zero = zero enum-Nat (suc n) = n The resulting behaviour is the same in both cases: test1 : 3rd enum-Nat test1 = refl 3 test2 : backward-2 enum-Nat 5 test2 = refl 3 3.4.1 Copatterns in function definitions In fact, we do not need to start at 0. We can allow the user to specify the starting element. Without copatterns, we just add the extra argument to the function declaration: open Enumeration enum-Nat : Nat → Enumeration Nat enum-Nat initial = record { start = initial ; forward = suc ; backward = pred } where pred : Nat → Nat pred zero = zero pred (suc x) = x test1 : 3rd (enum-Nat 10) test1 = refl 3.4. Copatterns 13 25 Agda User Manual, Release 2.6.0 With copatterns, the function argument must be repeated once for each field in the record: open Enumeration enum-Nat start forward backward backward : Nat → Enumeration (enum-Nat initial) = (enum-Nat _) n = suc (enum-Nat _) zero (enum-Nat _) (suc n) Nat initial n = zero = n 3.4.2 Mixing patterns and co-patterns Instead of allowing an arbitrary value, we want to limit the user to two choices: 0 or 42. Without copatterns, we would need an auxiliary definition to choose which value to start with based on the userprovided flag: open Enumeration if_then_else_ : {A : Set} → Bool → A → A → A if true then x else _ = x if false then _ else y = y enum-Nat : Bool → Enumeration Nat enum-Nat ahead = record { start = if ahead then 42 else 0 ; forward = suc ; backward = pred } where pred : Nat → Nat pred zero = zero pred (suc x) = x With copatterns, we can do the case analysis directly by pattern matching: open Enumeration enum-Nat start start forward backward backward : Bool → (enum-Nat (enum-Nat (enum-Nat (enum-Nat (enum-Nat Enumeration Nat true) = 42 false) = 0 _) n = suc n _) zero = zero _) (suc n) = n Tip: When using copatterns to define an element of a record type, the fields of the record must be in scope. In the examples above, we use open Enumeration to bring the fields of the record into scope. Consider the first example: enum-Nat start forward backward backward 26 : Enumeration Nat enum-Nat = 0 enum-Nat n = suc n enum-Nat zero = zero enum-Nat (suc n) = n Chapter 3. Language Reference Agda User Manual, Release 2.6.0 If the fields of the Enumeration record are not in scope (in particular, the start field), then Agda will not be able to figure out what the first copattern means: Could not parse the left-hand side start enum-Nat Operators used in the grammar: None when scope checking the left-hand side start enum-Nat in the definition of enum-Nat The solution is to open the record before using its fields: open Enumeration enum-Nat start forward backward backward : Enumeration Nat enum-Nat = 0 enum-Nat n = suc n enum-Nat zero = zero enum-Nat (suc n) = n 3.5 Core language Note: This is a stub data Term = | | | ˓→ignored | | ˓→space | | | | Var Def Con Lam Int Elims QName Elims ConHead Args ArgInfo (Abs Term) Lit Literal Pi (Dom Type) (Abs Type) -- ^ @f es@, possibly a delta/iota-redex -- ^ @c vs@ -- ^ Terms are beta normal. Relevance is -- ^ dependent or non-dependent function Sort Sort Level Level MetaV MetaId Elims DontCare Term -- ^ Irrelevant stuff in relevant position, but created -in an irrelevant context. 3.6 Cubical Type Theory in Agda Cubical Type Theory Cohen et al., Cubical is implemented in Agda (beta version). To use cubical type theory, you need to run Agda with the --cubical command-line-option. You can also write {-# OPTIONS --cubical #-} at the top of the file. To define paths, use 𝜆 abstractions. For example, this is the definition of the constant path: refl : {a} {A : Set a} {x : A} → Path x x refl {x = x} = 𝜆 i → x Although they use the same syntax, a path is not a function. For example, the following is not valid: 3.5. Core language 27 Agda User Manual, Release 2.6.0 refl : {a} {A : Set a} {x : A} → Path x x refl {x = x} = 𝜆 (i : _) → x 3.6.1 The interval type (I) The variable i in the definition of refl has type I, which topologically corresponds to the real unit interval. In a closed context, there are only two values of type I: the two ends of the interval, i0 and i1: a b : I a = i0 b = i1 Elements of the interval form a De Morgan algebra, with minimum (), maximum () and negation (~). max = i j min = i j neg = ~ i All the properties of de Morgan algebras hold definitionally. The ends of the interval i0 and i1 are the bottom and top elements, respectively: p1 p2 p3 p4 p5 p6 p7 p8 : : : : : : : : i0 i i i1 i j i j ~ (~ i) i0 ~ (i j) ~ (i j) i i1 j i j i i ~ i1 ~ i ~ j ~ i ~ j 3.6.2 References Cyril Cohen, Thierry Coquand, Simon Huber and Anders Mörtberg; “Cubical Type Theory: a constructive interpretation of the univalence axiom”. 3.7 Data Types 3.7.1 Simple datatypes Example datatypes In the introduction we already showed the definition of the data type of natural numbers (in unary notation): data Nat : Set where zero : Nat suc : Nat → Nat We give a few more examples. First the data type of truth values: 28 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 data Bool : Set where true : Bool false : Bool The True set represents the trivially true proposition: data True : Set where tt : True The False set has no constructor and hence no elements. It represent the trivially false proposition: data False : Set where Another example is the data type of non-empty binary trees with natural numbers in the leaves: data BinTree : Set where leaf : Nat → BinTree branch : BinTree → BinTree → BinTree Finally, the data type of Brouwer ordinals: data Ord : Set where zeroOrd : Ord sucOrd : Ord → Ord limOrd : (Nat → Ord) → Ord General form The general form of the definition of a simple datatype D is the following data D : Set where c1 : A1 ... c : A The name D of the data type and the names c1 , . . . , c of the constructors must be new w.r.t. the current signature and context, and the types A1 , . . . , A must be function types ending in D, i.e. they must be of the form (y1 : B1 ) → ... → (y : B) → D 3.7.2 Parametrized datatypes Datatypes can have parameters. They are declared after the name of the datatype but before the colon, for example: data List (A : Set) : Set where [] : List A __ : A → List A → List A 3.7.3 Indexed datatypes In addition to parameters, datatypes can also have indices. In contrast to parameters which are required to be the same for all constructors, indices can vary from constructor to constructor. They are declared after the colon as function arguments to Set. For example, fixed-length vectors can be defined by indexing them over their length of type Nat: 3.7. Data Types 29 Agda User Manual, Release 2.6.0 data Vector (A : Set) : Nat → Set where [] : Vector A zero __ : {n : Nat} → A → Vector A n → Vector A (suc n) Notice that the parameter A is bound once for all constructors, while the index {n : in the constructor __. Nat} must be bound locally Indexed datatypes can also be used to describe predicates, for example the predicate Even : defined as follows: Nat → Set can be data Even : Nat → Set where even-zero : Even zero even-plus2 : {n : Nat} → Even n → Even (suc (suc n)) General form The general form of the definition of a (parametrized, indexed) datatype D is the following data D (x1 : P1 ) ... (x : P) : (y1 : Q1 ) → ... → (y : Q) → Set c1 : A1 ... c : A where where the types A1 , . . . , A are function types of the form (z1 : B1 ) → ... → (z : B) → D x1 ... x t1 ... t 3.7.4 Strict positivity When defining a datatype D, Agda poses an additional requirement on the types of the constructors of D, namely that D may only occur strictly positively in the types of their arguments. Concretely, for a datatype with constructors c1 : A1 , . . . , c : A, Agda checks that each A has the form (y1 : B1 ) → ... → (y : B) → D where an argument types B of the constructors is either • non-inductive (a side condition) and does not mention D at all, • or inductive and has the form (z1 : C1 ) → ... → (z : C) → D where D must not occur in any C. The strict positivity condition rules out declarations such as data Bad : Set where bad : (Bad → Bad) → Bad -A B C -- A is in a negative position, B and C are OK since there is a negative occurrence of Bad in the type of the argument of the constructor. (Note that the corresponding data type declaration of Bad is allowed in standard functional languages such as Haskell and ML.). 30 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Non strictly-positive declarations are rejected because they admit non-terminating functions. If the positivity check is disabled, so that a similar declaration of Bad is allowed, it is possible to construct a term of the empty type, even without recursion. {-# OPTIONS --no-positivity-check #-} data : Set where data Bad : Set where bad : (Bad → ) → Bad self-app : Bad → self-app (bad f) = f (bad f) absurd : absurd = self-app (bad self-app) For more general information on termination see Termination Checking. 3.8 Foreign Function Interface • Compiler Pragmas • Haskell FFI – The FOREIGN pragma – The COMPILE pragma – Using Haskell Types from Agda – Using Haskell functions from Agda – Using Agda functions from Haskell – Polymorphic functions – Level-polymorphic types – Handling typeclass constraints • JavaScript FFI 3.8.1 Compiler Pragmas There are two backend-generic pragmas used for the FFI: {-# COMPILE #-} {-# FOREIGN #-} The COMPILE pragma associates some information with a name defined in the same module, and the FOREIGN pragma associates with the current top-level module. This information is interpreted by the specific backend during compilation (see below). These pragmas were added in Agda 2.5.3. 3.8. Foreign Function Interface 31 Agda User Manual, Release 2.6.0 3.8.2 Haskell FFI Note: This section applies to the GHC Backend. The FOREIGN pragma The GHC backend interprets FOREIGN pragmas as inline Haskell code and can contain arbitrary code (including import statements) that will be added to the compiled module. For instance: {-# FOREIGN GHC import Data.Maybe #-} {-# FOREIGN GHC data Foo = Foo | Bar Foo countBars :: Foo -> Integer countBars Foo = 0 countBars (Bar f) = 1 + countBars f #-} The COMPILE pragma There are four forms of COMPILE annotations recognized by the GHC backend {-# {-# {-# {-# COMPILE COMPILE COMPILE COMPILE GHC GHC GHC GHC = #-} = type #-} = data ( | .. | ) #-} as #-} The first three tells the compiler how to compile a given Agda definition and the last exposes an Agda definition under a particular Haskell name allowing Agda libraries to be used from Haskell. Using Haskell Types from Agda In order to use a Haskell function from Agda its type must be mapped to an Agda type. This mapping can be configured using the type and data forms of the COMPILE pragma. Opaque types Opaque Haskell types are exposed to Agda by postulating an Agda type and associating it to the Haskell type using the type form of the COMPILE pragma: {-# FOREIGN GHC import qualified System.IO #-} postulate FileHandle : Set {-# COMPILE GHC FileHandle = type System.IO.Handle #-} This tells the compiler that the Agda type FileHandle corresponds to the Haskell type System.IO.Handle and will enable functions using file handles to be used from Agda. 32 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Data types Non-opaque Haskell data types can be mapped to Agda datatypes using the data form of the COMPILED pragma: data Maybe (A : Set) : Set where nothing : Maybe A just : A → Maybe A {-# COMPILE GHC Maybe = data Maybe (Nothing | Just) #-} The compiler checks that the types of the Agda constructors match the types of the corresponding Haskell constructors and that no constructors have been left out (on either side). Built-in Types The GHC backend compiles certain Agda built-in types to special Haskell types. The mapping between Agda built-in types and Haskell types is as follows: Agda Built-in NAT INTEGER STRING CHAR BOOL FLOAT Haskell Type Integer Integer Data.Text.Text Char Bool Double Warning: Haskell code manipulating Agda natural numbers as integers must take care to avoid negative values. Warning: Agda FLOAT values have only one logical NaN value. At runtime, there might be multiple different NaN representations present. All such NaN values must be treated equal by FFI calls. Using Haskell functions from Agda Once a suitable mapping between Haskell types and Agda types has been set up, Haskell functions whose types map to an Agda type can be exposed to Agda code with a COMPILE pragma: open import Agda.Builtin.IO open import Agda.Builtin.String open import Agda.Builtin.Unit {-# FOREIGN GHC import qualified Data.Text.IO as Text import qualified System.IO as IO #-} postulate stdout : FileHandle hPutStrLn : FileHandle → String → IO (continues on next page) 3.8. Foreign Function Interface 33 Agda User Manual, Release 2.6.0 (continued from previous page) {-# COMPILE GHC stdout = IO.stdout #-} {-# COMPILE GHC hPutStrLn = Text.hPutStrLn #-} The compiler checks that the type of the given Haskell code matches the type of the Agda function. Note that the COMPILE pragma only affects the runtime behaviour–at type-checking time the functions are treated as postulates. Warning: It is possible to give Haskell definitions to defined (non-postulate) Agda functions. In this case the Agda definition will be used at type-checking time and the Haskell definition at runtime. However, there are no checks to ensure that the Agda code and the Haskell code behave the same and discrepancies may lead to undefined behaviour. This feature can be used to let you reason about code involving calls to Haskell functions under the assumption that you have a correct Agda model of the behaviour of the Haskell code. Using Agda functions from Haskell Since Agda 2.3.4 Agda functions can be exposed to Haskell code using the as form of the COMPILE pragma: module IdAgda where idAgda : {A : Set} → A → A idAgda x = x {-# COMPILE GHC idAgda as idAgdaFromHs #-} This tells the compiler that the Agda function idAgda should be compiled to a Haskell function called idAgdaFromHs. Without this pragma, functions are compiled to Haskell functions with unpredictable names and, as a result, cannot be invoked from Haskell. The type of idAgdaFromHs will be the translated type of idAgda. The compiled and exported function idAgdaFromHs can then be imported and invoked from Haskell like this: -- file UseIdAgda.hs module UseIdAgda where import MAlonzo.Code.IdAgda (idAgdaFromHs) -- idAgdaFromHs :: () -> a -> a idAgdaApplied :: a -> a idAgdaApplied = idAgdaFromHs () Polymorphic functions Agda is a monomorphic language, so polymorphic functions are modeled as functions taking types as arguments. These arguments will be present in the compiled code as well, so when calling polymorphic Haskell functions they have to be discarded explicitly. For instance, postulate ioReturn : {A : Set} → A → IO A {-# COMPILE GHC ioReturn = \ _ x -> return x #-} In this case compiled calls to ioReturn will still have A as an argument, so the compiled definition ignores its first argument and then calls the polymorphic Haskell return function. 34 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Level-polymorphic types Level-polymorphic types face a similar problem to polymorphic functions. Since Haskell does not have universe levels the Agda type will have more arguments than the corresponding type. This can be solved by defining a Haskell type synonym with the appropriate number of phantom arguments. For instance data Either {a b} (A : Set a) (B : Set b) : Set (a left : A → Either A B right : B → Either A B b) where {-# FOREIGN GHC type AgdaEither a b = Either #-} {-# COMPILE GHC Either = data AgdaEither (Left | Right) #-} Handling typeclass constraints There is (currently) no way to map a Haskell type with type class constraints to an Agda type. This means that functions with class constraints cannot be used from Agda. However, this can be worked around by wrapping class constraints in Haskell data types, and providing Haskell functions using explicit dictionary passing. For instance, suppose we have a simple GUI library in Haskell: module GUILib where class Widget w setVisible :: Widget w => w -> Bool -> IO () data Window instance Widget Window newWindow :: IO Window To use this library from Agda we first define a Haskell type for widget dictionaries and map this to an Agda type Widget: {-# FOREIGN GHC import GUILib #-} {-# FOREIGN GHC data WidgetDict w = Widget w => WidgetDict #-} postulate Widget : Set → Set {-# COMPILE GHC Widget = type WidgetDict #-} We can then expose setVisible as an Agda function taking a Widget instance argument: postulate setVisible : {w : Set} {{_ : Widget w}} → w → Bool → IO {-# COMPILE GHC setVisible = \ _ WidgetDict -> setVisible #-} Note that the Agda Widget argument corresponds to a WidgetDict argument on the Haskell side. When we match on the WidgetDict constructor in the Haskell code, the packed up dictionary will become available for the call to setVisible. The window type and functions are mapped as expected and we also add an Agda instance packing up the Widget Window Haskell instance into a WidgetDict: postulate Window : Set newWindow : IO Window instance WidgetWindow : Widget Window (continues on next page) 3.8. Foreign Function Interface 35 Agda User Manual, Release 2.6.0 (continued from previous page) {-# COMPILE GHC Window = type Window #-} {-# COMPILE GHC newWindow = newWindow #-} {-# COMPILE GHC WidgetWindow = WidgetDict #-} We can then write code like this: openWindow : IO Window openWindow = newWindow >>= 𝜆 w → setVisible w true >>= 𝜆 _ → return w 3.8.3 JavaScript FFI The JavaScript backend recognizes COMPILE pragmas of the following form: {-# COMPILE JS = #-} where is a postulate, constructor, or data type. The code for a data type is used to compile pattern matching and should be a function taking a value of the data type and a table of functions (corresponding to case branches) indexed by the constructor names. For instance, this is the compiled code for the List type, compiling lists to JavaScript arrays: data List {a} (A : Set a) : Set a where [] : List A __ : (x : A) (xs : List A) → List A {-# COMPILE JS List = function(x,v) { if (x.length < 1) { return v["[]"](); } else { return v["__"](x[0], x.slice(1)); } } #-} {-# COMPILE JS [] = Array() #-} {-# COMPILE JS __ = function (x) { return function(y) { return Array(x).concat(y); }; ˓→} #-} 3.9 Function Definitions 3.9.1 Introduction A function is defined by first declaring its type followed by a number of equations called clauses. Each clause consists of the function being defined applied to a number of patterns, followed by = and a term called the right-hand side. For example: not : Bool → Bool not true = false not false = true Functions are allowed to call themselves recursively, for example: 36 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 twice : Nat → Nat twice zero = zero twice (suc n) = suc (suc (twice n)) 3.9.2 General form The general form for defining a function is f : (x1 : A1 ) → ... → (x : A) → B f p1 ... p = d ... f q1 ... q = e where f is a new identifier, p and q are patterns of type A, and d and e are expressions. The declaration above gives the identifier f the type (x1 : A1 ) → ... → (x : A) → B and f is defined by the defining equations. Patterns are matched from top to bottom, i.e., the first pattern that matches the actual parameters is the one that is used. By default, Agda checks the following properties of a function definition: • The patterns in the left-hand side of each clause should consist only of constructors and variables. • No variable should occur more than once on the left-hand side of a single clause. • The patterns of all clauses should together cover all possible inputs of the function. • The function should be terminating on all possible inputs, see Termination Checking. 3.9.3 Special patterns In addition to constructors consisting of constructors and variables, Agda supports two special kinds of patterns: dot patterns and absurd patterns. Dot patterns A dot pattern (also called inaccessible pattern) can be used when the only type-correct value of the argument is determined by the patterns given for the other arguments. The syntax for a dot pattern is .t. As an example, consider the datatype Square defined as follows data Square : Nat → Set where sq : (m : Nat) → Square (m * m) Suppose we want to define a function root : (n : Nat) → Square n → Nat that takes as its arguments a number n and a proof that it is a square, and returns the square root of that number. We can do so as follows: root : (n : Nat) → Square n → Nat root .(m * m) (sq m) = m Notice that by matching on the argument of type Square n with the constructor sq : Square (m * m), n is forced to be equal to m * m. (m : Nat) → In general, when matching on an argument of type D i1 ... i with a constructor c : (x1 : A1 ) → ... → (x : A) → D j1 ... j, Agda will attempt to unify i1 ... i with j1 ... j. When the unification algorithm instantiates a variable x with value t, the corresponding argument of the function can be replaced by 3.9. Function Definitions 37 Agda User Manual, Release 2.6.0 a dot pattern .t. Using a dot pattern is optional, but can help readability. The following are also legal definitions of root: Since Agda 2.4.2.4: root1 : (n : Nat) → Square n → Nat root1 _ (sq m) = m Since Agda 2.5.2: root2 : (n : Nat) → Square n → Nat root2 n (sq m) = m In the case of root2 , n evaluates to m * m in the body of the function and is thus equivalent to root3 : (n : Nat) → Square n → Nat root3 _ (sq m) = let n = m * m in m Absurd patterns Absurd patterns can be used when none of the constructors for a particular argument would be valid. The syntax for an absurd pattern is (). As an example, if we have a datatype Even defined as follows data Even : Nat → Set where even-zero : Even zero even-plus2 : {n : Nat} → Even n → Even (suc (suc n)) then we can define a function one-not-even : Even 1 → by using an absurd pattern: one-not-even : Even 1 → one-not-even () Note that if the left-hand side of a clause contains an absurd pattern, its right-hand side must be omitted. In general, when matching on an argument of type D i1 ... i with an absurd pattern, Agda will attempt for each constructor c : (x1 : A1 ) → ... → (x : A) → D j1 ... j of the datatype D to unify i1 ... i with j1 ... j. The absurd pattern will only be accepted if all of these unifications end in a conflict. As-patterns As-patterns (or @-patterns) can be used to name a pattern. The name has the same scope as normal pattern variables (i.e. the right-hand side, where clause, and dot patterns). The name reduces to the value of the named pattern. For example: module _ {A : Set} (_<_ : A → A → Bool) where merge : List A → List A → List A merge xs [] = xs merge [] ys = ys merge xs@(x xs1 ) ys@(y ys1 ) = if x < y then x merge xs1 ys else y merge xs ys1 As-patterns are properly supported since Agda 2.5.2. 38 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 3.9.4 Case trees Internally, Agda represents function definitions as case trees. For example, a function definition max max max max : Nat → Nat → Nat zero n = n m zero = m (suc m) (suc n) = suc (max m n) will be represented internally as a case tree that looks like this: max m n = case m of zero → n suc m' → case n of zero → suc m' suc n' → suc (max m' n') Note that because Agda uses this representation of the function max, the clause max m zero = m does not hold definitionally (i.e. as a reduction rule). If you would try to prove that this equation holds, you would not be able to write refl: data __ {A : Set} (x : A) : A → Set where refl : x x -- Does not work! lemma : (m : Nat) → max m zero lemma = refl m Clauses which do not hold definitionally are usually (but not always) the result of writing clauses by hand instead of using Agda’s case split tactic. These clauses are highlighted by Emacs. The --exact-split command-line and pragma option causes Agda to raise an error whenever a clause in a definition by pattern matching cannot be made to hold definitionally. Specific clauses can be excluded from this check by means of the {-# CATCHALL #-} pragma. For instance, the above definition of max will be rejected when using the --exact-split flag because its second clause does not to hold definitionally. When using the --exact-split flag, catch-all clauses have to be marked as such, for instance: eq : Nat → Nat → Bool eq zero zero = true eq (suc m) (suc n) = eq m n {-# CATCHALL #-} eq _ _ = false The --no-exact-split command-line and pragma option can be used to override a global --exact-split in a file, by adding a pragma {-# OPTIONS --no-exact-split #-}. This option is enabled by default. 3.10 Function Types Function types are written (x : A) → B, or in the case of non-dependent functions simply A → B. For instance, the type of the addition function for natural numbers is: Nat → Nat → Nat and the type of the addition function for vectors is: 3.10. Function Types 39 Agda User Manual, Release 2.6.0 (A : Set) → (n : Nat) → (u : Vec A n) → (v : Vec A n) → Vec A n where Set is the type of sets and Vec A n is the type of vectors with n elements of type A. Arrows between consecutive hypotheses of the form (x : A) may also be omitted, and (x : A) (y : A) may be shortened to (x y : A): (A : Set) (n : Nat)(u v : Vec A n) → Vec A n Functions are constructed by lambda abstractions, which can be either typed or untyped. For instance, both expressions below have type (A : Set) → A → A (the second expression checks against other types as well): example1 = \ (A : Set)(x : A) → x example2 = \ A x → x You can also use the Unicode symbol 𝜆 (type “\lambda” in the Emacs Agda mode) instead of \\. The application of a function f : is B[x := a]. (x : A) → B to an argument a : A is written f a and the type of this 3.10.1 Notational conventions Function types: prop1 prop2 prop3 prop4 prop5 : : : : : ((x : A) (y : B) → C) ((x y : A) → C) (forall (x : A) → C) (forall x → C) (forall x y → C) is-the-same-as is-the-same-as is-the-same-as is-the-same-as is-the-same-as ((x : A) → (y : B) → C) ((x : A)(y : A) → C) ((x : A) → C) ((x : _) → C) (forall x → forall y → C) You can also use the Unicode symbol (type “\all” in the Emacs Agda mode) instead of forall. Functional abstraction: (\x y → e) is-the-same-as (\x → (\y → e)) Functional application: (f a b) is-the-same-as ((f a) b) 3.11 Implicit Arguments It is possible to omit terms that the type checker can figure out for itself, replacing them by _. If the type checker cannot infer the value of an _ it will report an error. For instance, for the polymorphic identity function id : (A : Set) → A → A the first argument can be inferred from the type of the second argument, so we might write id _ zero for the application of the identity function to zero. We can even write this function application without the first argument. In that case we declare an implicit function space: id : {A : Set} → A → A 40 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 and then we can use the notation id zero. Another example: _==_ : {A : Set} → A → A → Set subst : {A : Set} (C : A → Set) {x y : A} → x == y → C x → C y Note how the first argument to _==_ is left implicit. Similarly, we may leave out the implicit arguments A, x, and y in an application of subst. To give an implicit argument explicitly, enclose in curly braces. The following two expressions are equivalent: x1 = subst C eq cx x2 = subst {_} C {_} {_} eq cx It is worth noting that implicit arguments are also inserted at the end of an application, if it is required by the type. For example, in the following, y1 and y2 are equivalent. y1 : a == b → C a → C b y1 = subst C y2 : a == b → C a → C b y2 = subst C {_} {_} Implicit arguments are inserted eagerly in left-hand sides so y3 and y4 are equivalent. An exception is when no type signature is given, in which case no implicit argument insertion takes place. Thus in the definition of y5 there only implicit is the A argument of subst. y3 : {x y : A} → x == y → C x → C y y3 = subst C y4 : {x y : A} → x == y → C x → C y y4 {x} {y} = subst C {_} {_} y5 = subst C It is also possible to write lambda abstractions with implicit arguments. For example, given id : A → A, we can define the identity function with implicit type argument as (A : Set) → id’ = 𝜆 {A} → id A Implicit arguments can also be referred to by name, so if we want to give the expression e explicitly for y without giving a value for x we can write subst C {y = e} eq cx When constructing implicit function spaces the implicit argument can be omitted, so both expressions below are valid expressions of type {A : Set} → A → A: z1 = 𝜆 {A} x → x z2 = 𝜆 x → x The (or forall) syntax for function types also has implicit variants: : ( {x : A} → B) : ( {x} → B) : ( {x y} → B) is-the-same-as is-the-same-as is-the-same-as 3.11. Implicit Arguments ({x : A} → B) ({x : _} → B) ( {x} → {y} → B) 41 Agda User Manual, Release 2.6.0 There are no restrictions on when a function space can be implicit. Internally, explicit and implicit function spaces are treated in the same way. This means that there are no guarantees that implicit arguments will be solved. When there are unsolved implicit arguments the type checker will give an error message indicating which application contains the unsolved arguments. The reason for this liberal approach to implicit arguments is that limiting the use of implicit argument to the cases where we guarantee that they are solved rules out many useful cases in practice. 3.11.1 Metavariables 3.11.2 Unification 3.12 Instance Arguments • Usage – Defining type classes – Declaring instances – Examples • Instance resolution Instance arguments are the Agda equivalent of Haskell type class constraints and can be used for many of the same purposes. In Agda terms, they are implicit arguments that get solved by a special instance resolution algorithm, rather than by the unification algorithm used for normal implicit arguments. In principle, an instance argument is resolved, if a unique instance of the required type can be built from declared instances and the current context. 3.12.1 Usage Instance arguments are enclosed in double curly braces {{ }}, e.g. {{x : T}}. Alternatively they can be enclosed, with proper spacing, e.g. x : T , in the unicode braces (U+2983 and U+2984, which can be typed as \{{ and \}} in the Emacs mode). For instance, given a function _==_ _==_ : {A : Set} {{eqA : Eq A}} → A → A → Bool for some suitable type Eq, you might define elem : {A : Set} {{eqA : Eq A}} → A → List A → Bool elem x (y xs) = x == y || elem x xs elem x [] = false Here the instance argument to _==_ is solved by the corresponding argument to elem. Just like ordinary implicit arguments, instance arguments can be given explicitly. The above definition is equivalent to elem : {A : Set} {{eqA : Eq A}} → A → List A → Bool elem {{eqA}} x (y xs) = _==_ {{eqA}} x y || elem {{eqA}} x xs elem x [] = false A very useful function that exploits this is the function it which lets you apply instance resolution to solve an arbitrary goal: 42 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 it : {a} {A : Set a} {{_ : A}} → A it {{x}} = x Note that instance arguments in types are always named, but the name can be _: _==_ : {A : Set} → {{Eq A}} → A → A → Bool _==_ : {A : Set} {{_ : Eq A}} → A → A → Bool -- INVALID -- VALID Defining type classes The type of an instance argument should have the form {Γ} → C vs, where C is a postulated name, a bound variable, or the name of a data or record type, and {Γ} denotes an arbitrary number of (ordinary) implicit arguments (see Dependent instances below for an example where Γ is non-empty). Instance arguments that do not have this form are currently accepted, but instance resolution may or may not work as described below for such arguments. Other than that there are no requirements on the type of an instance argument. In particular, there is no special declaration to say that a type is a “type class”. Instead, Haskell-style type classes are usually defined as record types. For instance, record Monoid {a} (A : Set a) : Set a where field mempty : A _<>_ : A → A → A In order to make the fields of the record available as functions taking instance arguments you can use the special module application open Monoid {{...}} public This will bring into scope mempty : _<>_ : {a} {A : Set a} {{_ : Monoid A}} → A {a} {A : Set a} {{_ : Monoid A}} → A → A → A Superclass dependencies can be implemented using Instance fields. See Module application and Record modules for details about how the module application is desugared. If defined by hand, mempty would be mempty : {a} {A : Set a} {{_ : Monoid A}} → A mempty {{mon}} = Monoid.mempty mon Although record types are a natural fit for Haskell-style type classes, you can use instance arguments with data types to good effect. See the Examples below. Declaring instances As seen above, instance arguments in the context are available when solving instance arguments, but you also need to be able to define top-level instances for concrete types. This is done using the instance keyword, which starts a block in which each definition is marked as an instance available for instance resolution. For example, an instance Monoid (List A) can be defined as 3.12. Instance Arguments 43 Agda User Manual, Release 2.6.0 instance ListMonoid : {a} {A : Set a} → Monoid (List A) ListMonoid = record { mempty = []; _<>_ = _++_ } Or equivalently, using copatterns: instance ListMonoid : {a} {A : Set a} → Monoid (List A) mempty {{ListMonoid}} = [] _<>_ {{ListMonoid}} xs ys = xs ++ ys Top-level instances must target a named type (Monoid in this case), and cannot be declared for types in the context. You can define local instances in let-expressions in the same way as a top-level instance. For example: mconcat : {a} {A : Set a} {{_ : Monoid A}} → List A → A mconcat [] = mempty mconcat (x xs) = x <> mconcat xs sum : List Nat → Nat sum xs = let instance NatMonoid : Monoid Nat NatMonoid = record { mempty = 0; _<>_ = _+_ } in mconcat xs Instances can have instance arguments themselves, which will be filled in recursively during instance resolution. For instance, record Eq {a} (A : Set a) : Set a where field _==_ : A → A → Bool open Eq {{...}} public instance eqList : {a} {A : Set a} {{_ : Eq A}} → Eq (List A) _==_ {{eqList}} [] [] = true _==_ {{eqList}} (x xs) (y ys) = x == y && xs == ys _==_ {{eqList}} _ _ = false eqNat : Eq Nat _==_ {{eqNat}} = natEquals ex : Bool ex = (1 2 3 []) == (1 2 []) -- false Note the two calls to _==_ in the right-hand side of the second clause. The first uses the Eq A instance and the second uses a recursive call to eqList. In the example ex, instance resolution, needing a value of type Eq (List Nat), will try to use the eqList instance and find that it needs an instance argument of type Eq Nat, it will then solve that with eqNat and return the solution eqList {{eqNat}}. Note: At the moment there is no termination check on instances, so it is possible to construct non-sensical instances like loop : {a} {A : Set a} {{_ : Eq A}} → Eq A. To prevent looping in cases like this, the search depth of instance search is limited, and once the maximum depth is reached, a type error will be thrown. You can set the maximum depth using the --instance-search-depth flag. 44 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Constructor instances Although instance arguments are most commonly used for record types, mimicking Haskell-style type classes, they can also be used with data types. In this case you often want the constructors to be instances, which is achieved by declaring them inside an instance block. Typically arguments to constructors are not instance arguments, so during instance resolution explicit arguments are treated as instance arguments. See Instance resolution below for the details. A simple example of a constructor that can be made an instance is the reflexivity constructor of the equality type: data __ {a} {A : Set a} (x : A) : A → Set a where instance refl : x x This allows trivial equality proofs to be inferred by instance resolution, which can make working with functions that have preconditions less of a burden. As an example, here is how one could use this to define a function that takes a natural number and gives back a Fin n (the type of naturals smaller than n): data Fin : Nat → Set where zero : {n} → Fin (suc n) suc : {n} → Fin n → Fin (suc n) mkFin mkFin mkFin mkFin : {n} (m : Nat) {{_ : suc m - n {zero} m {{}} {suc n} zero = zero {suc n} (suc m) = suc (mkFin m) 0}} → Fin n five : Fin 6 five = mkFin 5 -- OK In the first clause of mkFin we use an absurd pattern to discharge the impossible assumption suc m next section for another example of constructor instances. 0. See the Record fields can also be declared instances, with the effect that the corresponding projection function is considered a top-level instance. Examples Proof search Instance arguments are useful not only for Haskell-style type classes, but they can also be used to get some limited form of proof search (which, to be fair, is also true for Haskell type classes). Consider the following type, which models a proof that a particular element is present in a list as the index at which the element appears: infix 4 __ data __ {A : Set} (x : A) : List A → Set where instance zero : {xs} → x x xs suc : {y xs} → x xs → x y xs Here we have declared the constructors of __ to be instances, which allows instance resolution to find proofs for concrete cases. For example, ex1 : 1 + 2 1 2 3 4 [] ex1 = it -- computes to suc (suc zero) ex2 : {A : Set} (x y : A) (xs : List A) → x y y x xs (continues on next page) 3.12. Instance Arguments 45 Agda User Manual, Release 2.6.0 (continued from previous page) ex2 x y xs = it -- suc (suc zero) ex3 : {A : Set} (x y : A) (xs : List A) {{i : x ex3 x y xs = it -- suc (suc i) xs}} → x y y xs It will fail, however, if there are more than one solution, since instance arguments must be unique. For example, fail1 : 1 1 2 1 [] fail1 = it -- ambiguous: zero or suc (suc zero) fail2 : {A : Set} (x y : A) (xs : List A) {{i : x fail2 x y xs = it -- suc zero or suc (suc i) xs}} → x y x xs Dependent instances Consider a variant on the Eq class where the equality function produces a proof in the case the arguments are equal: record Eq {a} (A : Set a) : Set a where field _==_ : (x y : A) → Maybe (x y) open Eq {{...}} public A simple boolean-valued equality function is problematic for types with dependencies, like the Σ-type data Σ {a b} (A : Set a) (B : A → Set b) : Set (a _,_ : (x : A) → B x → Σ A B b) where since given two pairs x , y and x1 , y1 , the types of the second components y and y1 can be completely different and not admit an equality test. Only when x and x1 are really equal can we hope to compare y and y1 . Having the equality function return a proof means that we are guaranteed that when x and x1 compare equal, they really are equal, and comparing y and y1 makes sense. An Eq instance for Σ can be defined as follows: instance eqΣ : {a b} ˓→Eq (Σ A B) _==_ {{eqΣ}} _==_ {{eqΣ}} _==_ {{eqΣ}} _==_ {{eqΣ}} _==_ {{eqΣ}} {A : Set a} {B : A → Set b} {{_ : Eq A}} {{_ : (x (x (x (x (x , , , , , y) y) y) y) y) (x1 (x1 (.x (.x (.x , , , , , {x} → Eq (B x)}} → y1 ) with x == x1 y1 ) | nothing = nothing y1 ) | just refl with y == y1 y1 ) | just refl | nothing = nothing .y) | just refl | just refl = just refl Note that the instance argument for B states that there should be an Eq instance for B x, for any x : A. The argument x must be implicit, indicating that it needs to be inferred by unification whenever the B instance is used. See Instance resolution below for more details. 3.12.2 Instance resolution Given a goal that should be solved using instance resolution we proceed in the following four stages: Verify the goal First we check that the goal is not already solved. This can happen if there are unification constraints determining the value, or if it is of singleton record type and thus solved by eta-expansion. 46 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Next we check that the goal type has the right shape to be solved by instance resolution. It should be of the form {Γ} → C vs, where the target type C is a variable from the context or the name of a data or record type, and {Γ} denotes a telescope of implicit arguments. If this is not the case instance resolution fails with an error message1 . Finally we have to check that there are no unconstrained metavariables in vs. A metavariable 𝛼 is considered constrained if it appears in an argument that is determined by the type of some later argument, or if there is an existing constraint of the form 𝛼 us = C vs, where C inert (i.e. a data or type constructor). For example, 𝛼 is constrained in T 𝛼 xs if T : (n : Nat) → Vec A n → Set, since the type of the second argument of T determines the value of the first argument. The reason for this restriction is that instance resolution risks looping in the presence of unconstrained metavariables. For example, suppose the goal is Eq 𝛼 for some metavariable 𝛼. Instance resolution would decide that the eqList instance was applicable if setting 𝛼 := List 𝛽 for a fresh metavariable 𝛽, and then proceed to search for an instance of Eq 𝛽. Find candidates In the second stage we compute a set of candidates. Let-bound variables and top-level definitions in scope are candidates if they are defined in an instance block. Lambda-bound variables, i.e. variables bound in lambdas, function types, left-hand sides, or module parameters, are candidates if they are bound as instance arguments using {{ }}. Only candidates that compute something of type C us, where C is the target type computed in the previous stage, are considered. Check the candidates We attempt to use each candidate in turn to build an instance of the goal type {Γ} → C vs. First we extend the current context by Γ. Then, given a candidate c : ∆ → A we generate fresh metavariables 𝛼s : ∆ for the arguments of c, with ordinary metavariables for implicit arguments, and instance metavariables, solved by a recursive call to instance resolution, for explicit arguments and instance arguments. Next we unify A[∆ := 𝛼s] with C vs and apply instance resolution to the instance metavariables in 𝛼s. Both unification and instance resolution have three possible outcomes: yes, no, or maybe. In case we get a no answer from any of them, the current candidate is discarded, otherwise we return the potential solution 𝜆 {Γ} → c 𝛼s. Compute the result From the previous stage we get a list of potential solutions. If the list is empty we fail with an error saying that no instance for C vs could be found (no). If there is a single solution we use it to solve the goal (yes), and if there are multiple solutions we check if they are all equal. If they are, we solve the goal with one of them (yes), but if they are not, we postpone instance resolution (maybe), hoping that some of the maybes will turn into nos once we know more about the involved metavariables. If there are left-over instance problems at the end of type checking, the corresponding metavariables are printed in the Emacs status buffer together with their types and source location. The candidates that gave rise to potential solutions can be printed with the show constraints command (C-c C-=). 3.13 Irrelevance Since version 2.2.8 Agda supports irrelevancy annotations. The general rule is that anything prepended by a dot (.) is marked irrelevant, which means that it will only be typechecked but never evaluated. 3.13.1 Motivating example One intended use case of irrelevance is data structures with embedded proofs, like sorted lists. data __ : Nat → Nat → Set where zero : {n : Nat} → zero n sucsuc : {m n : Nat} → m n → suc m suc n (continues on next page) 1 Instance goal verification is buggy at the moment. See issue #1322. 3.13. Irrelevance 47 Agda User Manual, Release 2.6.0 (continued from previous page) postulate p1 : 0 p2 : 0 1 1 module No-Irrelevance where data SList (bound : Nat) : Set where [] : SList bound scons : (head : Nat) → (head bound) → (tail : SList head) → SList bound Usually, when we define datatypes with embedded proofs we are forced to reason about the values of these proofs. For example, suppose we have two lists l1 and l2 with the same elements but different proofs: l1 : SList 1 l1 = scons 0 p1 [] l2 : SList 1 l2 = scons 0 p2 [] Now suppose we want to prove that l1 and l2 are equal: l1 l2 : l1 l2 l1 l2 = refl It’s not so easy! Agda gives us an error: p1 != p2 of type 0 1 when checking that the expression refl has type l1 We can’t show that l1 l2 by refl when p1 and p2 are relevant. Instead, we need to reason about proofs of 0 postulate proof-equality : p1 Now we can prove l1 l2 1. p2 l2 by rewriting with this equality: l1 l2 : l1 l2 l1 l2 rewrite proof-equality = refl Reasoning about equality of proofs becomes annoying quickly. We would like to avoid this kind of reasoning about proofs here - in this case we only care that a proof of head bound exists, i.e. any proof suffices. We can use irrelevance annotations to tell Agda we don’t care about the values of the proofs: data SList (bound : Nat) : Set where [] : SList bound scons : (head : Nat) → .(head bound) -- note the dot! → (tail : SList head) → SList bound The effect of the irrelevant type in the signature of scons is that scons’s second argument is never inspected after Agda has ensured that it has the right type. The type-checker ignores irrelevant arguments when checking equality, so two lists can be equal even if they contain different proofs: 48 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 l1 : SList 1 l1 = scons 0 p1 [] l2 : SList 1 l2 = scons 0 p2 [] l1 l2 : l1 l2 l1 l2 = refl 3.13.2 Irrelevant function types For starters, consider irrelevant non-dependent function types: f : .A → B This type implies that f does not depend computationally on its argument. What can be done to irrelevant arguments Example 1. We can prove that two applications of an unknown irrelevant function to two different arguments are equal. -- an unknown function that does not use its second argument postulate f : {A B : Set} -> A -> .B -> A -- the second argument is irrelevant for equality proofIrr : {A : Set}{x y z : A} -> f x y f x z proofIrr = refl Example 2. We can use irrelevant arguments as arguments to other irrelevant functions. id : {A B : Set} -> (.A -> B) -> .A -> B id g x = g x Example 3. We can match on an irrelevant argument of an empty type with an absurd pattern (). data : Set where zero-not-one : .(0 zero-not-one () 1) → Example 4. We can match on an irrelevant record (see Record Types) as long as we only use the fields irrelevantly. record _×_ (A B : Set) : Set where constructor _,_ field fst : A snd : B irrElim : {A B C : Set} → .(A × B) → (.A → .B → C) → C irrElim (a , b) f = f a b lemma : {A B C : Set} {a a' : A} {b b' : B} (continues on next page) 3.13. Irrelevance 49 Agda User Manual, Release 2.6.0 (continued from previous page) → (f : .A -> .B -> C) -> irrElim (a , b) f lemma f = refl f a' b' What can’t be done to irrelevant arguments Example 1. You can’t use an irrelevant value in a non-irrelevant context. bad-plus : Nat → .Nat → Nat bad-plus n m = m + n Variable m is declared irrelevant, so it cannot be used here when checking that the expression m has type Nat Example 2. You can’t declare the function’s return type as irrelevant. bad : Nat → .Nat bad n = 1 Invalid dotted expression when checking that the expression .Nat has type Set _47 Example 3. You can’t pattern match on an irrelevant value. badMatching : Nat → .Nat → Nat badMatching n zero = n badMatching n (suc m) = n Cannot pattern match against irrelevant argument of type Nat when checking that the pattern zero has type Nat 3.13.3 Irrelevant declarations Postulates and functions can be marked as irrelevant by prefixing the name with a dot when the name is declared. Irrelevant definitions can only be used as arguments of functions of an irrelevant function type .A → B. Examples: .irrFunction : Nat → Nat irrFunction zero = zero irrFunction (suc n) = suc (suc (irrFunction n)) postulate .assume-false : (A : Set) → A An important example is the irrelevance axiom irrAx: postulate .irrAx : {} {A : Set } -> .A -> A This axiom is not provable inside Agda, but it is often very useful when working with irrelevance. 50 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 3.13.4 Irrelevant record fields Record fields (see Record Types) can be marked as irrelevant by prefixing their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record type containing pairs of numbers satisfying certain properties. record InterestingNumbers : Set where field n : Nat m : Nat .prop1 : n + m n * m + 2 .prop2 : suc m n Example 2. For any type A, we can define a squashed’ version ‘‘Squash A‘ where all elements are equal. record Squash (A : Set) : Set where constructor squash field .proof : A open Squash .unsquash : {A} → Squash A → A unsquash x = proof x Example 3. We can define the subset of x : A satisfying P x with irrelevant membership certificates. record Subset (A : Set) (P : A -> Set) : Set where constructor _#_ field elem : A .certificate : P elem .certificate : {A : Set}{P : A -> Set} -> (x : Subset A P) -> P (Subset.elem x) certificate (a # p) = irrAx p 3.13.5 Dependent irrelevant function types Just like non-dependent functions, we can also make dependent functions irrelevant. The basic syntax is as in the following examples: f f f f f f : : : : : : .(x y : A) → B .{x y z : A} → B .(xs {ys zs} : A) → B x .y → B x .{y} {z} .v → B .{{x : A}} → B The declaration f : .(x : A) → B[x] f x = t[x] requires that x is irrelevant both in t[x] and in B[x]. This is possible if, for instance, B[x] = C x, with C : .A → Set. 3.13. Irrelevance 51 Agda User Manual, Release 2.6.0 Dependent irrelevance allows us to define the eliminator for the Squash type: elim-Squash : {A : Set} (P : Squash A → Set) (ih : .(a : A) → P (squash a)) → (a : Squash A) → P a elim-Squash P ih (squash a) = ih a Note that this would not type-check with (ih : (a : A) → P (squash a)). 3.13.6 Irrelevant instance arguments Contrary to normal instance arguments, irrelevant instance arguments (see Instance Arguments) are not required to have a unique solution. record : Set where instance constructor tt NonZero : Nat → Set NonZero zero = NonZero (suc _) = pred : (n : Nat) .{{_ : NonZero n}} → Nat pred zero {{}} pred (suc n) = n find-nonzero : (n : Nat) {{x y : NonZero n}} → Nat find-nonzero n = pred n 3.14 Lambda Abstraction 3.14.1 Pattern matching lambda Anonymous pattern matching functions can be defined using the syntax: \ { p11 .. p1n -> e1 ; ... ; pm1 .. pmn -> em } (where, as usual, \ and -> can be replaced by 𝜆 and →). Internally this is translated into a function definition of the following form: .extlam p11 .. p1n = e1 ... .extlam pm1 .. pmn = em This means that anonymous pattern matching functions are generative. For instance, refl will not be accepted as an inhabitant of the type (𝜆 { true → true ; false → false }) == (𝜆 { true → true ; false → false }) because this is equivalent to extlam1 extlam2 for some distinct fresh names extlam1 and extlam2. Currently the where and with constructions are not allowed in (the top-level clauses of) anonymous pattern matching functions. Examples: 52 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 and : Bool → Bool → Bool and = 𝜆 { true x → x ; false _ → false } xor : Bool → Bool → Bool xor = 𝜆 { true true → false ; false false → false ; _ _ → true } fst : {A : Set} {B : A → Set} → Σ A B → A fst = 𝜆 { (a , b) → a } snd : {A : Set} {B : A → Set} (p : Σ A B) → B (fst p) snd = 𝜆 { (a , b) → b } 3.15 Local Definitions: let and where There are two ways of declaring local definitions in Agda: • let-expressions • where-blocks 3.15.1 let-expressions A let-expression defines an abbreviation. In other words, the expression that we define in a let-expression can neither be recursive nor defined by pattern matching. Example: f : Nat f = let h : Nat → Nat h m = suc (suc m) in h zero + h (suc zero) let-expressions have the general form let f1 : A11 → ... → A1 → A1 f1 x1 ... x = e1 ... f : A1 → ... → A → A f x1 ... x = e in e’ where previous definitions are in scope in later definitions. The type signatures can be left out if Agda can infer them. After type-checking, the meaning of this is simply the substitution e’[f1 := 𝜆 x1 ... x → e; ...; f := 𝜆 x1 ... x → e]. Since Agda substitutes away let-bindings, they do not show up in terms Agda prints, nor in the goal display in interactive mode. 3.15.2 where-blocks where-blocks are much more powerful than let-expressions, as they support arbitrary local definitions. A where can be attached to any function clause. 3.15. Local Definitions: let and where 53 Agda User Manual, Release 2.6.0 where-blocks have the general form clause where decls or clause module M where decls A simple instance is g ps = e where f : A1 → ... → A → A f p11 ... p1 = e1 ... ... f p1 ... p= e Here, the p are patterns of the corresponding types and e is an expression that can contain occurrences of f. Functions defined with a where-expression must follow the rules for general definitions by pattern matching. Example: reverse : {A : Set} → List A → List A reverse {A} xs = rev-append xs [] where rev-append : List A → List A → List A rev-append [] ys = ys rev-append (x xs) ys = rev-append xs (x ys) Variable scope The pattern variables of the parent clause of the where-block are in scope; in the previous example, these are A and xs. The variables bound by the type signature of the parent clause are not in scope. This is why we added the hidden binder {A}. Scope of the local declarations The where-definitions are not visible outside of the clause that owns these definitions (the parent clause). If the where-block is given a name (form module M where), then the definitions are available as qualified by M, since module M is visible even outside of the parent clause. The special form of an anonymous module (module _ where) makes the definitions visible outside of the parent clause without qualification. If the parent function of a named where-block (form module M where) is private, then module M is also private. However, the declarations inside M are not private unless declared so explicitly. Thus, the following example scope checks fine: module Parent1 where private parent = local module Private where (continues on next page) 54 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) local = Set module Public = Private test1 = Parent1 .Public.local Likewise, a private declaration for a parent function does not affect the privacy of local functions defined under a module _ where-block: module Parent2 where private parent = local module _ where local = Set test2 = Parent2 .local They can be declared private explicitly, though: module Parent3 where parent = local module _ where private local = Set Now, Parent3 .local is not in scope. A private declaration for the parent of an ordinary where-block has no effect on the local definitions, of course. They are not even in scope. 3.15.3 Proving properties Sometimes one needs to refer to local definitions in proofs about the parent function. In this case, the module where variant is preferable. reverse : {A : Set} → List A → List A reverse {A} xs = rev-append xs [] module Rev where rev-append : List A → List A → List A rev-append [] ys = ys rev-append (x :: xs) ys = rev-append xs (x :: ys) This gives us access to the local function as Rev.rev-append : {A : Set} (xs : List A) → List A → List A → List A Alternatively, we can define local functions as private to the module we are working in; hence, they will not be visible in any module that imports this module but it will allow us to prove some properties about them. private rev-append : {A : Set} → List A → List A → List A rev-append [] ys = ys rev-append (x xs) ys = rev-append xs (x ys) reverse' : {A : Set} → List A → List A reverse' xs = rev-append xs [] 3.15. Local Definitions: let and where 55 Agda User Manual, Release 2.6.0 3.15.4 More Examples (for Beginners) Using a let-expression: tw-map : {A : Set} → List A → List (List A) tw-map {A} xs = let twice : List A → List A twice xs = xs ++ xs in map (\ x → twice [ x ]) xs Same definition but with less type information: tw-map' : {A : Set} → List A → List (List A) tw-map' {A} xs = let twice : _ twice xs = xs ++ xs in map (\ x → twice [ x ]) xs Same definition but with a where-expression tw-map'' : {A : Set} → List A → List (List A) tw-map'' {A} xs = map (\ x → twice [ x ]) xs where twice : List A → List A twice xs = xs ++ xs Even less type information using let: g : Nat → List Nat g zero = [ zero ] g (suc n) = let sing = [ suc n ] in sing ++ g n Same definition using where: g' : Nat → List Nat g' zero = [ zero ] g' (suc n) = sing ++ g' n where sing = [ suc n ] More than one definition in a let: h : Nat → Nat h n = let add2 : Nat add2 = suc (suc n) twice : Nat → Nat twice m = m * m in twice add2 More than one definition in a where: fibfact : Nat → Nat fibfact n = fib n + fact n where fib : Nat → Nat fib zero = suc zero fib (suc zero) = suc zero fib (suc (suc n)) = fib (suc n) + fib n (continues on next page) 56 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) fact : Nat → Nat fact zero = suc zero fact (suc n) = suc n * fact n Combining let and where: k : Nat → Nat k n = let aux : Nat → Nat aux m = pred (h m) + fibfact m in aux (pred n) where pred : Nat → Nat pred zero = zero pred (suc m) = m 3.16 Lexical Structure Agda code is written in UTF-8 encoded plain text files with the extension .agda. Most unicode characters can be used in identifiers and whitespace is important, see Names and Layout below. 3.16.1 Tokens Keywords and special symbols Most non-whitespace unicode can be used as part of an Agda name, but there are two kinds of exceptions: special symbols Characters with special meaning that cannot appear at all in a name. These are .;{}()@". keywords Reserved words that cannot appear as a name part, but can appear in a name together with other characters. = | -> → : ? \ 𝜆 .. ... abstract codata coinductive constructor data do eta-equality field forall hiding import in inductive infix infixl infixr instance let macro module mutual no-eta-equality open overlap pattern postulate primitive private public quote quoteContext quoteGoal quoteTerm record renaming rewrite Set syntax tactic unquote unquoteDecl unquoteDef using where with The Set keyword can appear with a number suffix, optionally subscripted (see Universe Levels). For instance Set42 and Set42 are both keywords. Names A qualified name is a non-empty sequence of names separated by dots (.). A name is an alternating sequence of name parts and underscores (_), containing at least one name part. A name part is a non-empty sequence of unicode characters, excluding whitespace, _, and special symbols. A name part cannot be one of the keywords above, and cannot start with a single quote, ' (which are used for character literals, see Literals below). Examples • Valid: data?, ::, if_then_else_, 0b, ___, x=y • Invalid: data_?, foo__bar, _, a;b, [_.._] The underscores in a name indicate where the arguments go when the name is used as an operator. For instance, the application _+_ 1 2 can be written as 1 + 2. See Mixfix Operators for more information. Since most sequences of 3.16. Lexical Structure 57 Agda User Manual, Release 2.6.0 characters are valid names, whitespace is more important than in other languages. In the example above the whitespace around + is required, since 1+2 is a valid name. Qualified names are used to refer to entities defined in other modules. For instance Prelude.Bool.true refers to the name true defined in the module Prelude.Bool. See Module System for more information. Literals There are four types of literal values: integers, floats, characters, and strings. See Built-ins for the corresponding types, and Literal Overloading for how to support literals for user-defined types. Integers Integer values in decimal or hexadecimal (prefixed by 0x) notation. Non-negative numbers map by default to built-in natural numbers, but can be overloaded. Negative numbers have no default interpretation and can only be used through overloading. Examples: 123, 0xF0F080, -42, -0xF Floats Floating point numbers in the standard notation (with square brackets denoting optional parts): float ::= [-] decimal . decimal [exponent] | [-] decimal exponent exponent ::= (e | E) [+ | -] decimal These map to built-in floats and cannot be overloaded. Examples: 1.0, -5.0e+12, 1.01e-16, 4.2E9, 50e3. Characters Character literals are enclosed in single quotes ('). They can be a single (unicode) character, other than ' or \, or an escaped character. Escaped characters start with a backslash \ followed by an escape code. Escape codes are natural numbers in decimal or hexadecimal (prefixed by x) between 0 and 0x10ffff (1114111), or one of the following special escape codes: Code a v " ETX BEL VT SI DC3 ETB ESC US ASCII 7 11 " 3 7 11 15 19 23 27 31 Code b f NUL EOT BS FF DLE DC4 CAN FS SP ASCII 8 12 0 4 8 12 16 20 24 28 32 Code t \ SOH ENQ HT CR DC1 NAK EM GS DEL ASCII 9 \ 1 5 9 13 17 21 25 29 127 Code n ' STX ACK LF SO DC2 SYN SUB RS ASCII 10 ' 2 6 10 14 18 22 26 30 Character literals map to the built-in character type and cannot be overloaded. Examples: 'A', '', '\x2200', '\ESC', '\32', '\n', '\'', '"'. Strings String literals are sequences of, possibly escaped, characters enclosed in double quotes ". They follow the same rules as character literals except that double quotes " need to be escaped rather than single quotes '. String literals map to the built-in string type by default, but can be overloaded. Example: " \"\"\n". 58 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Holes Holes are an integral part of the interactive development supported by the Emacs mode. Any text enclosed in {! and !} is a hole and may contain nested holes. A hole with no contents can be written ?. There are a number of Emacs commands that operate on the contents of a hole. The type checker ignores the contents of a hole and treats it as an unknown (see Implicit Arguments). Example: {! f {!x!} 5 !} Comments Single-line comments are written with a double dash -- followed by arbitrary text. Multi-line comments are enclosed in {- and -} and can be nested. Comments cannot appear in string literals. Example: {- Here is a {- nested -} comment -} s : String --line comment {s = "{- not a comment -}" Pragmas Pragmas are special comments enclosed in {-# and #-} that have special meaning to the system. See Pragmas for a full list of pragmas. 3.16.2 Layout Agda is layout sensitive using similar rules as Haskell, with the exception that layout is mandatory: you cannot use explicit {, } and ; to avoid it. A layout block contains a sequence of statements and is started by one of the layout keywords: abstract do field instance let macro mutual postulate primitive private where The first token after the layout keyword decides the indentation of the block. Any token indented more than this is part of the previous statement, a token at the same level starts a new statement, and a token indented less lies outside the block. data Nat : Set where -- starts a layout block -- comments are not tokens zero : Nat -- statement 1 suc : Nat → -- statement 2 Nat -- also statement 2 one : Nat -- outside the layout block one = suc zero Note that the indentation of the layout keyword does not matter. An Agda file contains one top-level layout block, with the special rule that the contents of the top-level module need not be indented. 3.16. Lexical Structure 59 Agda User Manual, Release 2.6.0 module Example where NotIndented : Set1 NotIndented = Set 3.16.3 Literate Agda Agda supports literate programming where everything in a file is a comment unless enclosed in \begin{code}, \end{code}. Literate Agda files have the extension .lagda instead of .agda. The main use case for literate Agda is to generate LaTeX documents from Agda code. See Generating LaTeX for more information. \documentclass{article} % some preamble stuff \begin{document} Introduction usually goes here \begin{code} module MyPaper where open import Prelude five : Nat five = 2 + 3 \end{code} Now, conclusions! \end{document} 3.17 Literal Overloading 3.17.1 Natural numbers By default natural number literals are mapped to the built-in natural number type. This can be changed with the FROMNAT built-in, which binds to a function accepting a natural number: {-# BUILTIN FROMNAT fromNat #-} This causes natural number literals n to be desugared to fromNat n. Note that the desugaring happens before implicit argument are inserted so fromNat can have any number of implicit or instance arguments. This can be exploited to support overloaded literals by defining a type class containing fromNat: module number-simple where record Number {a} (A : Set a) : Set a where field fromNat : Nat → A open Number {{...}} public {-# BUILTIN FROMNAT fromNat #-} This definition requires that any natural number can be mapped into the given type, so it won’t work for types like Fin n. This can be solved by refining the Number class with an additional constraint: record Number {a} (A : Set a) : Set (lsuc a) where field Constraint : Nat → Set a (continues on next page) 60 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) fromNat : (n : Nat) {{_ : Constraint n}} → A open Number {{...}} public using (fromNat) {-# BUILTIN FROMNAT fromNat #-} This is the definition used in Agda.Builtin.FromNat. A Number instance for Nat is simply this: instance NumNat : Number Nat NumNat .Number.Constraint _ = NumNat .Number.fromNat m = m A Number instance for Fin n can be defined as follows: __ : (m n : Nat) → Set zero n = suc m zero = suc m suc n = m n fromN fromN fromN fromN : m n → m n → Fin (suc n) zero _ _ = zero (suc _) zero () (suc m) (suc n) p = suc (fromN m n p) instance NumFin : {n} → Number (Fin (suc n)) NumFin {n} .Number.Constraint m = m n NumFin {n} .Number.fromNat m {{mn}} = fromN m n mn test : Fin 5 test = 3 It is important that the constraint for literals is trivial. Here, 3 tion. 5 evaluates to whose inhabitant is found by unifica- Using predefined function from the standard library and instance NumNat, the NumFin instance can be simply: open import Data.Fin using (Fin; #_) open import Data.Nat using (suc; _?_) open import Relation.Nullary.Decidable using (True) instance NumFin : {n} → Number (Fin n) NumFin {n} .Number.Constraint m = True (suc m ? n) NumFin {n} .Number.fromNat m {{m #-}, and can be given wherever fixity declarations can be given. The listed polarities apply to the given postulate’s arguments (explicit/implicit/instance), from left to right. Polarities currently cannot be given for module parameters. If the postulate takes n arguments (excluding module parameters), then the number of polarities given must be between 0 and n (inclusive). Polarity pragmas make it possible to use postulated type formers in recursive types in the following way: postulate _ : Set → Set (continues on next page) 72 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) {-# POLARITY _ ++ #-} data D : Set where c : D → D Note that one can use postulates that may seem benign, together with polarity pragmas, to prove that the empty type is inhabited: postulate __ : Set → Set → Set lambda : {A B : Set} → (A → B) → A B apply : {A B : Set} → A B → A → B {-# POLARITY __ ++ #-} data : Set where data D : Set where c : D → D not-inhabited : D → not-inhabited (c f) = apply f (c f) d : D d = c (lambda not-inhabited) bad : bad = not-inhabited d Polarity pragmas are not allowed in safe mode. 3.23 Postulates A postulate is a declaration of an element of some type without an accompanying definition. With postulates we can introduce elements in a type without actually giving the definition of the element itself. The general form of a postulate declaration is as follows: postulate c11 ... c1i : ... cn1 ... cnj : Example: postulate A B : a : b : _=AB=_ : a==b : Set A B A -> B -> Set a =AB= b Introducing postulates is in general not recommended. Once postulates are introduced the consistency of the whole development is at risk, because there is nothing that prevents us from introducing an element in the empty set. 3.23. Postulates 73 Agda User Manual, Release 2.6.0 data False : Set where postulate bottom : False A preferable way to work is to define a module parametrised by the elements we need module Absurd (bt : False) where -- ... module M (A B : Set) (a : A) (b : B) (_=AB=_ : A -> B -> Set) (a==b : a =AB= b) where -- ... 3.23.1 Postulated built-ins Some Built-ins such as Float and Char are introduced as a postulate and then given a meaning by the corresponding {-# BUILTIN ... #-} pragma. 3.24 Pragmas Pragmas are comments that are not ignored by Agda but have some special meaning. The general format is: {-# #-} 3.24.1 Index of pragmas • BUILTIN • CATCHALL • COMPILE • FOREIGN • NO_POSITIVITY_CHECK • NO_TERMINATION_CHECK • NON_TERMINATING • POLARITY • STATIC • TERMINATING • INLINE • NOINLINE • WARNING_ON_USAGE See also Command-line and pragma options. 74 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 The INLINE and NOINLINE pragmas A definition marked with an INLINE pragma is inlined during compilation. If it is a simple definition that does no pattern matching, it is also inlined in function bodies at type-checking time. Definitions are automatically marked INLINE if they satisfy the following criteria: • No pattern matching. • Uses each argument at most once. • Does not use all its arguments. Automatic inlining can be prevented using the NOINLINE pragma. Example: -- Would be auto-inlined since it doesn't use the type arguments. __ : {A B C : Set} → (B → C) → (A → B) → A → C (f g) x = f (g x) {-# NOINLINE __ #-} -- prevents auto-inlining -- Would not be auto-inlined since it's using all its arguments _o_ : (Set → Set) → (Set → Set) → Set → Set (F o G) X = F (G X) {-# INLINE _o_ #-} -- force inlining The WARNING_ON_USAGE pragma A library author can use a WARNING_ON_USAGE pragma to attach to a defined name a warning to be raised whenever this name is used. This would typically be used to declare a name ‘DEPRECATED’ and advise the end-user to port their code before the feature is dropped. Example: -- The new name for the identity id : {A : Set} → A → A id x = x -- The deprecated name 𝜆x→x = id -- The warning {-# WARNING_ON_USAGE 𝜆x→x "DEPRECATED: Use `id` instead of `𝜆x→x`" #-} 3.25 Record Types • Declaring, constructing and decomposing records – Declarating record types 3.25. Record Types 75 Agda User Manual, Release 2.6.0 – Constructing record values – Decomposing record values – Record update • Record modules • Eta-expansion • Recursive records • Instance fields Records are types for grouping values together. They generalise the dependent product type by providing named fields and (optional) further components. Record types can be declared using the record keyword record Pair (A B : Set) : Set where field fst : A snd : B This defines a new type Pair : Set → Set → Set and two projection functions Pair.fst : {A B : Set} → Pair A B → A Pair.snd : {A B : Set} → Pair A B → B Elements of record types can be defined using a record expression p23 : Pair Nat Nat p23 = record { fst = 2; snd = 3 } or using copatterns p34 : Pair Nat Nat Pair.fst p34 = 3 Pair.snd p34 = 4 If you use the constructor keyword, you can also use the named constructor to define elements of the record type: record Pair (A B : Set) : Set where constructor _,_ field fst : A snd : B p45 : Pair Nat Nat p45 = 4 , 5 In this sense, record types behave much like single constructor datatypes (but see Eta-expansion below). 3.25.1 Declaring, constructing and decomposing records Declarating record types The general form of a record declaration is as follows: 76 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 record : Set where constructor field : : -- ... All the components are optional, and can be given in any order. In particular, fields can be given in more than one block, interspersed with other declarations. Each field is a component of the record. Types of later fields can depend on earlier fields. The directives available are eta-equality, no-eta-equality (see Eta-expansion), inductive and co-inductive (see Recursive records). Constructing record values Record values are constructed by giving a value for each record field: record { = ; = ; ... } where the types of the terms matches the types of the fields. If a constructor has been declared for the record, this can also be written ... For named definitions, this can also be expressed using copatterns: : . = . = ... Records can also be constructed by updating other records. Building records from modules The record { } syntax also accept module names. Fields are defined using the corresponding definitions from the given module. For instance assuming this record type R and module M: record R : Set where field x : X y : Y z : Z module M where x = ... y = ... r : R r = record { M; z = ... } 3.25. Record Types 77 Agda User Manual, Release 2.6.0 This construction supports any combination of explicit field definitions and applied modules. If a field is both given explicitly and available in one of the modules, then the explicit one takes precedence. If a field is available in more than one module then this is ambiguous and therefore rejected. As a consequence the order of assignments does not matter. The modules can be both applied to arguments and have import directives such as hiding, using, and renaming. Here is a contrived example building on the example above: module M2 (a : A) where w = ... z = ... r2 : A → R r2 a = record { M hiding (y); M2 a renaming (w to y) } Decomposing record values With the field name, we can project the corresponding component out of a record value. It is also possible to pattern match against inductive records: sum : Pair Nat Nat → Nat sum (x , y) = x + y Internally, this is translated to sum' : Pair Nat Nat → Nat sum' p = (Pair.fst p) + (Pair.snd p) Note: Naming the constructor is not required to enable pattern matching against record values. Record expressions can appear as patterns. Record update Assume that we have a record type and a corresponding value: record MyRecord : Set where field a b c : Nat old : MyRecord old = record { a = 1; b = 2; c = 3 } Then we can update (some of) the record value’s fields in the following way: new : MyRecord new = record old { a = 0; c = 5 } Here new normalises to record { a = 0; b = 2; c = 5 }. Any expression yielding a value of type MyRecord can be used instead of old. Using that records can be built from module names, together with the fact that all records define a module, this can also be written as new' : MyRecord new' = record { MyRecord old; a = 0; c = 5} 78 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Record updating is not allowed to change types: the resulting value must have the same type as the original one, including the record parameters. Thus, the type of a record update can be inferred if the type of the original record can be inferred. The record update syntax is expanded before type checking. When the expression record old { upd-fields } is checked against a record type R, it is expanded to let r = old in record { new-fields } where old is required to have type R and new-fields is defined as follows: for each field x in R, • if x = e is contained in upd-fields then x = e is included in new-fields, and otherwise • if x is an explicit field then x = R.x r is included in new-fields, and • if x is an implicit or instance field, then it is omitted from new-fields. The reason for treating implicit and instance fields specially is to allow code like the following: data Vec (A : Set) : Nat → Set where [] : Vec A zero __ : {n} → A → Vec A n → Vec A (suc n) record R : Set where field {length} : Nat vec : Vec Nat length -- More fields ... xs : R xs = record { vec = 0 1 ys = record xs { vec = 0 2 [] } [] } Without the special treatment the last expression would need to include a new binding for length (for instance length = _). 3.25.2 Record modules Along with a new type, a record declaration also defines a module with the same name, parameterised over an element of the record type containing the projection functions. This allows records to be “opened”, bringing the fields into scope. For instance swap : {A B : Set} → Pair A B → Pair B A swap p = snd , fst where open Pair p In the example, the record module Pair has the shape module Pair {A B : Set} (p : Pair A B) where fst : A snd : B It’s possible to add arbitrary definitions to the record module, by defining them inside the record declaration 3.25. Record Types 79 Agda User Manual, Release 2.6.0 record Functor (F : Set → Set) : Set1 where field fmap : {A B} → (A → B) → F A → F B _<$_ : {A B} → A → F B → F A x <$ fb = fmap (𝜆 _ → x) fb Note: In general new definitions need to appear after the field declarations, but simple non-recursive function definitions without pattern matching can be interleaved with the fields. The reason for this restriction is that the type of the record constructor needs to be expressible using let-expressions. In the example below D1 can only contain declarations for which the generated type of mkR is well-formed. record R Γ : Set where constructor mkR field f1 : A1 D1 field f2 : A2 mkR : {Γ} (f1 : A1 ) (let D1 ) (f2 : A2 ) → R Γ 3.25.3 Eta-expansion The eta rule for a record type record R : Set where field a : A b : B c : C states that every x : R is definitionally equal to record { a = R.a x ; b = R.b x ; c = R.c x }. By default, all (inductive) record types enjoy eta-equality if the positivity checker has confirmed it is safe to have it. The keywords eta-equality/no-eta-equality enable/disable eta rules for the record type being declared. 3.25.4 Recursive records Recursive records need to be declared as either inductive or coinductive. record Tree (A : Set) : Set where inductive constructor tree field elem : A subtrees : List (Tree A) record Stream (A : Set) : Set where coinductive constructor _::_ field head : A tail : Stream A 80 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Inductive records have eta-equality on by default, while no-eta-equality is the default for coinductive records. In fact, the eta-equality and coinductive directives are not allowed together, since this can easily make Agda loop. This can be overridden at your own risk by using the pragma ETA instead. It is possible to pattern match on inductive records, but not on coinductive ones. 3.25.5 Instance fields Instance fields, that is record fields marked with {{ }} can be used to model “superclass” dependencies. For example: record Eq (A : Set) : Set where field _==_ : A → A → Bool open Eq {{...}} record Ord (A : Set) : Set where field _<_ : A → A → Bool {{eqA}} : Eq A open Ord {{...}} hiding (eqA) Now anytime you have a function taking an Ord A argument the Eq A instance is also available by virtue of 𝜂expansion. So this works as you would expect: __ : {A : Set} {{OrdA : Ord A}} → A → A → Bool x y = (x == y) || (x < y) There is a problem however if you have multiple record arguments with conflicting instance fields. For instance, suppose we also have a Num record with an Eq field record Num (A : Set) : Set where field fromNat : Nat → A {{eqA}} : Eq A open Num {{...}} hiding (eqA) _3 : {A : Set} {{OrdA : Ord A}} {{NumA : Num A}} → A → Bool x 3 = (x == fromNat 3) || (x < fromNat 3) Here the Eq A argument to _==_ is not resolved since there are two conflicting candidates: Ord.eqA OrdA and Num.eqA NumA. To solve this problem you can declare instance fields as overlappable using the overlap keyword: record Ord (A : Set) : Set where field _<_ : A → A → Bool overlap {{eqA}} : Eq A open Ord {{...}} hiding (eqA) record Num (A : Set) : Set where field fromNat : Nat → A overlap {{eqA}} : Eq A (continues on next page) 3.25. Record Types 81 Agda User Manual, Release 2.6.0 (continued from previous page) open Num {{...}} hiding (eqA) _3 : {A : Set} {{OrdA : Ord A}} {{NumA : Num A}} → A → Bool x 3 = (x == fromNat 3) || (x < fromNat 3) Whenever there are multiple valid candidates for an instance goal, if all candidates are overlappable, the goal is solved by the left-most candidate. In the example above that means that the Eq A goal is solved by the instance from the Ord argument. Clauses for instance fields can be omitted when defining values of record types. For instance we can define Nat instances for Eq, Ord and Num as follows, leaving out cases for the eqA fields: instance EqNat : Eq Nat _==_ {{EqNat}} = Agda.Builtin.Nat._==_ OrdNat : Ord Nat _<_ {{OrdNat}} = Agda.Builtin.Nat._<_ NumNat : Num Nat fromNat {{NumNat}} n = n 3.26 Reflection 3.26.1 Builtin types Names The built-in QNAME type represents quoted names and comes equipped with equality, ordering and a show function. postulate Name : Set {-# BUILTIN QNAME Name #-} primitive primQNameEquality : Name → Name → Bool primQNameLess : Name → Name → Bool primShowQName : Name → String Name literals are created using the quote keyword and can appear both in terms and in patterns nameOfNat : Name nameOfNat = quote Nat isNat : Name → Bool isNat (quote Nat) = true isNat _ = false Note that the name being quoted must be in scope. Metavariables Metavariables are represented by the built-in AGDAMETA type. They have primitive equality, ordering and show: 82 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 postulate Meta : Set {-# BUILTIN AGDAMETA Meta #-} primitive primMetaEquality : Meta → Meta → Bool primMetaLess : Meta → Meta → Bool primShowMeta : Meta → String Builtin metavariables show up in reflected terms. Literals Literals are mapped to the built-in AGDALITERAL datatype. Given the appropriate built-in binding for the types Nat, Float, etc, the AGDALITERAL datatype has the following shape: data Literal : Set where nat : (n : Nat) → word64 : (n : Word64) → float : (x : Float) → char : (c : Char) → string : (s : String) → name : (x : Name) → meta : (x : Meta) → {-# {-# {-# {-# {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN AGDALITERAL AGDALITNAT AGDALITWORD64 AGDALITFLOAT AGDALITCHAR AGDALITSTRING AGDALITQNAME AGDALITMETA Literal Literal Literal Literal Literal Literal Literal Literal nat word64 float char string name meta #-} #-} #-} #-} #-} #-} #-} #-} Arguments Arguments can be (visible), {hidden}, or {{instance}}: data Visibility : Set where visible hidden instance : Visibility {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN HIDING VISIBLE HIDDEN INSTANCE Visibility #-} visible #-} hidden #-} instance #-} Arguments can be relevant or irrelevant: data Relevance : Set where relevant irrelevant : Relevance {-# BUILTIN RELEVANCE Relevance #-} {-# BUILTIN RELEVANT relevant #-} {-# BUILTIN IRRELEVANT irrelevant #-} Visibility and relevance characterise the behaviour of an argument: 3.26. Reflection 83 Agda User Manual, Release 2.6.0 data ArgInfo : Set where arg-info : (v : Visibility) (r : Relevance) → ArgInfo data Arg (A : Set) : Set where arg : (i : ArgInfo) (x : A) → Arg A {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN ARGINFO ARGARGINFO ARG ARGARG ArgInfo arg-info Arg arg #-} #-} #-} #-} Patterns Reflected patterns are bound to the AGDAPATTERN built-in using the following data type. data Pattern : Set where con : (c : Name) (ps : dot : Pattern var : (s : String) → lit : (l : Literal) → proj : (f : Name) → absurd : Pattern {-# {-# {-# {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN AGDAPATTERN AGDAPATCON AGDAPATDOT AGDAPATVAR AGDAPATLIT AGDAPATPROJ AGDAPATABSURD List (Arg Pattern)) → Pattern Pattern Pattern Pattern Pattern con dot var lit proj absurd #-} #-} #-} #-} #-} #-} #-} Name abstraction data Abs (A : Set) : Set where abs : (s : String) (x : A) → Abs A {-# BUILTIN ABS Abs #-} {-# BUILTIN ABSABS abs #-} Terms Terms, sorts and clauses are mutually recursive and mapped to the AGDATERM, AGDASORT and AGDACLAUSE builtins respectively. Types are simply terms. Terms use de Bruijn indices to represent variables. data data data Type Term : Set Sort : Set Clause : Set = Term data Term where var : (x : Nat) (args : List (Arg Term)) → Term con : (c : Name) (args : List (Arg Term)) → Term (continues on next page) 84 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) def lam pat-lam pi agda-sort lit meta unknown data Sort set lit unknown : : : : : : : : (f : Name) (args : List (Arg Term)) → Term (v : Visibility) (t : Abs Term) → Term (cs : List Clause) (args : List (Arg Term)) → Term (a : Arg Type) (b : Abs Type) → Term (s : Sort) → Term (l : Literal) → Term (x : Meta) → List (Arg Term) → Term Term -- Treated as '_' when unquoting. where : (t : Term) → Sort -- A Set of a given (possibly neutral) level. : (n : Nat) → Sort -- A Set of a given concrete level. : Sort data Clause where clause : (ps : List (Arg Pattern)) (t : Term) → Clause absurd-clause : (ps : List (Arg Pattern)) → Clause {-# BUILTIN AGDASORT {-# BUILTIN AGDATERM {-# BUILTIN AGDACLAUSE {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN Sort #-} Term #-} Clause #-} AGDATERMVAR AGDATERMCON AGDATERMDEF AGDATERMMETA AGDATERMLAM AGDATERMEXTLAM AGDATERMPI AGDATERMSORT AGDATERMLIT AGDATERMUNSUPPORTED var con def meta lam pat-lam pi agda-sort lit unknown #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} {-# BUILTIN AGDASORTSET set #-} {-# BUILTIN AGDASORTLIT lit #-} {-# BUILTIN AGDASORTUNSUPPORTED unknown #-} {-# BUILTIN AGDACLAUSECLAUSE clause #-} {-# BUILTIN AGDACLAUSEABSURD absurd-clause #-} Absurd lambdas 𝜆 () are quoted to extended lambdas with an absurd clause. The built-in constructors AGDATERMUNSUPPORTED and AGDASORTUNSUPPORTED are translated to meta variables when unquoting. Declarations There is a built-in type AGDADEFINITION representing definitions. AGDATCMGETDEFINITION built-in described below. data Definition function : data-type : ˓→constructors record-type : ˓→constructor Values of this type is returned by the : Set where (cs : List Clause) → Definition (pars : Nat) (cs : List Name) → Definition -- parameters and (c : Name) (fs : List (Arg Name)) → -- c: name of record (continues on next page) 3.26. Reflection 85 Agda User Manual, Release 2.6.0 (continued from previous page) data-cons axiom prim-fun {-# {-# {-# {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN Definition : (d : Name) → Definition : Definition : Definition AGDADEFINITION AGDADEFINITIONFUNDEF AGDADEFINITIONDATADEF AGDADEFINITIONRECORDDEF AGDADEFINITIONDATACONSTRUCTOR AGDADEFINITIONPOSTULATE AGDADEFINITIONPRIMITIVE -- fs: fields -- d: name of data type Definition function data-type record-type data-cons axiom prim-fun #-} #-} #-} #-} #-} #-} #-} Type errors Type checking computations (see below) can fail with an error, which is a list of ErrorParts. This allows metaprograms to generate nice errors without having to implement pretty printing for reflected terms. -- Error messages can contain embedded names and terms. data ErrorPart : Set where strErr : String → ErrorPart termErr : Term → ErrorPart nameErr : Name → ErrorPart {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN AGDAERRORPART AGDAERRORPARTSTRING AGDAERRORPARTTERM AGDAERRORPARTNAME ErrorPart strErr termErr nameErr #-} #-} #-} #-} Type checking computations Metaprograms, i.e. programs that create other programs, run in a built-in type checking monad TC: postulate TC : returnTC : bindTC : {a} → Set a → Set a {a} {A : Set a} → A → TC A {a b} {A : Set a} {B : Set b} → TC A → (A → TC B) → TC B {-# BUILTIN AGDATCM TC #-} {-# BUILTIN AGDATCMRETURN returnTC #-} {-# BUILTIN AGDATCMBIND bindTC #-} The TC monad provides an interface to the Agda type checker using the following primitive operations: postulate -- Unify two terms, potentially solving metavariables in the process. unify : Term → Term → TC -- Throw a type error. Can be caught by catchTC. typeError : {a} {A : Set a} → List ErrorPart → TC A -- Block a type checking computation on a metavariable. This will abort -- the computation and restart it (from the beginning) when the (continues on next page) 86 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 (continued from previous page) -- metavariable is solved. blockOnMeta : {a} {A : Set a} → Meta → TC A -- Prevent current solutions of metavariables from being rolled back in -- case 'blockOnMeta' is called. commitTC : TC -- Backtrack and try the second argument if the first argument throws a -- type error. catchTC : {a} {A : Set a} → TC A → TC A → TC A -- Infer the type of a given term inferType : Term → TC Type -- Check a term against a given type. This may resolve implicit arguments -- in the term, so a new refined term is returned. Can be used to create -- new metavariables: newMeta t = checkType unknown t checkType : Term → Type → TC Term -- Compute the normal form of a term. normalise : Term → TC Term -- Compute the weak head normal form of a term. reduce : Term → TC Term -- Get the current context. Returns the context in reverse order, so that -- it is indexable by deBruijn index. Note that the types in the context are -- valid in the rest of the context. To use in the current context they need -- to be weakened by 1 + their position in the list. getContext : TC (List (Arg Type)) -- Extend the current context with a variable of the given type. extendContext : {a} {A : Set a} → Arg Type → TC A → TC A -- Set the current context. Takes a context telescope with the outer-most -- entry first, in contrast to 'getContext'. Each type should be valid in the -- context formed by the preceding elements in the list. inContext : {a} {A : Set a} → List (Arg Type) → TC A → TC A -- Quote a value, returning the corresponding Term. quoteTC : {a} {A : Set a} → A → TC Term -- Unquote a Term, returning the corresponding value. unquoteTC : {a} {A : Set a} → Term → TC A -- Create a fresh name. freshName : String → TC Name -- Declare a new function of the given type. The function must be defined -- later using 'defineFun'. Takes an Arg Name to allow declaring instances -- and irrelevant functions. The Visibility of the Arg must not be hidden. declareDef : Arg Name → Type → TC -- Declare a new postulate of the given type. The Visibility of the Arg -- must not be hidden. It fails when executed from command-line with --safe -- option. declarePostulate : Arg Name → Type → TC (continues on next page) 3.26. Reflection 87 Agda User Manual, Release 2.6.0 (continued from previous page) -- Define a declared function. The function may have been declared using -- 'declareDef' or with an explicit type signature in the program. defineFun : Name → List Clause → TC -- Get the type of a defined name. Replaces 'primNameType'. getType : Name → TC Type -- Get the definition of a defined name. Replaces 'primNameDefinition'. getDefinition : Name → TC Definition -- Check if a name refers to a macro isMacro : Name → TC Bool -- Change the behaviour of inferType, checkType, quoteTC, getContext -- to normalise (or not) their results. The default behaviour is no -- normalisation. withNormalisation : {a} {A : Set a} → Bool → TC A → TC A -- Prints the third argument if the corresponding verbosity level is turned -- on (with the -v flag to Agda). debugPrint : String → Nat → List ErrorPart → TC {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# {-# BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN BUILTIN AGDATCMUNIFY AGDATCMTYPEERROR AGDATCMBLOCKONMETA AGDATCMCATCHERROR AGDATCMINFERTYPE AGDATCMCHECKTYPE AGDATCMNORMALISE AGDATCMREDUCE AGDATCMGETCONTEXT AGDATCMEXTENDCONTEXT AGDATCMINCONTEXT AGDATCMQUOTETERM AGDATCMUNQUOTETERM AGDATCMFRESHNAME AGDATCMDECLAREDEF AGDATCMDECLAREPOSTULATE AGDATCMDEFINEFUN AGDATCMGETTYPE AGDATCMGETDEFINITION AGDATCMCOMMIT AGDATCMISMACRO AGDATCMWITHNORMALISATION AGDATCMDEBUGPRINT unify typeError blockOnMeta catchTC inferType checkType normalise reduce getContext extendContext inContext quoteTC unquoteTC freshName declareDef declarePostulate defineFun getType getDefinition commitTC isMacro withNormalisation debugPrint #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} #-} 3.26.2 Metaprogramming There are three ways to run a metaprogram (TC computation). To run a metaprogram in a term position you use a macro. To run metaprograms to create top-level definitions you can use the unquoteDecl and unquoteDef primitives (see Unquoting Declarations). 88 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Macros Macros are functions of type t1 → t2 → .. → Term → TC that are defined in a macro block. The last argument is supplied by the type checker and will be the representation of a metavariable that should be instantiated with the result of the macro. Macro application is guided by the type of the macro, where Term and Name arguments are quoted before passed to the macro. Arguments of any other type are preserved as-is. For example, the macro application f u v w where f : desugars into: Term → Name → Bool → Term → TC unquote (f (quoteTerm u) (quote v) w) where quoteTerm u takes a u of arbitrary type and returns its representation in the Term data type, and unquote m runs a computation in the TC monad. Specifically, when checking unquote m : A for some type A the type checker proceeds as follows: • Check m : Term → TC . • Create a fresh metavariable hole : • Let qhole : A. Term be the quoted representation of hole. • Execute m qhole. • Return (the now hopefully instantiated) hole. Reflected macro calls are constructed using the def constructor, so given a macro g : def (quote g) [] unquotes to a macro call to g. Term → TC the term Note: The quoteTerm and unquote primitives are available in the language, but it is recommended to avoid using them in favour of macros. Limitations: • Macros cannot be recursive. This can be worked around by defining the recursive function outside the macro block and have the macro call the recursive function. Silly example: macro plus-to-times : Term → Term → TC plus-to-times (def (quote _+_) (a b ˓→b [])) plus-to-times v hole = unify hole v thm : (a b : Nat) → plus-to-times (a + b) thm a b = refl [])) hole = unify hole (def (quote _*_) (a a * b Macros lets you write tactics that can be applied without any syntactic overhead. For instance, suppose you have a solver: magic : Type → Term that takes a reflected goal and outputs a proof (when successful). You can then define the following macro: 3.26. Reflection 89 Agda User Manual, Release 2.6.0 macro by-magic : Term → TC by-magic hole = bindTC (inferType hole) 𝜆 goal → unify hole (magic goal) This lets you apply the magic tactic as a normal function: thm : ¬ P NP thm = by-magic Unquoting Declarations While macros let you write metaprograms to create terms, it is also useful to be able to create top-level definitions. You can do this from a macro using the declareDef and defineFun primitives, but there is no way to bring such definitions into scope. For this purpose there are two top-level primitives unquoteDecl and unquoteDef that runs a TC computation in a declaration position. They both have the same form: unquoteDecl x1 .. x = m unquoteDef x1 .. x = m except that the list of names can be empty for unquoteDecl, but not for unquoteDef. In both cases m should have type TC . The main difference between the two is that unquoteDecl requires m to both declare (with declareDef) and define (with defineFun) the x whereas unquoteDef expects the x to be already declared. In other words, unquoteDecl brings the x into scope, but unquoteDef requires them to already be in scope. In m the x stand for the names of the functions being defined (i.e. x : Name) rather than the actual functions. One advantage of unquoteDef over unquoteDecl is that unquoteDef is allowed in mutual blocks, allowing mutually recursion between generated definitions and hand-written definitions. 3.27 Rewriting Note: This is a stub. 3.28 Safe Agda By using the command-line option --safe, a user can specify that Agda should ensure that features leading to possible inconsistencies should be disabled. Here is a list of the features --safe is incompatible with: • postulate can be used to assume any axiom. • --allow-unsolved-metas forces Agda to accept unfinished proofs. • --no-positivity-check makes it possible to write non-terminating programs by structural “induction” on non strictly positive datatypes. • --no-termination-check gives loopy programs any type. • --type-in-type and --omega-in-omega allow the user to encode the Girard-Hurken paradox. 90 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 • --injective-type-constructors together with excluded middle leads to an inconsistency via ChnugKil Hur’s construction. • guardedness-preserving-type-constructors is based on a rather operational understanding of ∞/_; it’s not yet clear if this extension is consistent. • --experimental-irrelevance enables potentially unsound irrelevance features (irrelevant levels, irrelevant data matching). • --rewriting turns any equation into one that holds definitionally. It can at the very least break convergence. 3.28.1 Known Issues Pragma Option It is possible to specify {-# OPTIONS --safe #-} at the top of a file. Unfortunately a known bug (see #2487) means that the option choice is not repercuted in the imported file. Therefore only the command-line option can be trusted. Standard Library The standard library uses a lot of unsafe features (e.g. postulate in the Foreign Function Interface) and these are not isolated in separate modules. As a consequence virtually any project relying on the standard library will not be successfully typechecked with the --safe option. There is work in progress to fix this issue. 3.29 Sized Types Note: This is a stub. Sizes help the termination checker by tracking the depth of data structures across definition boundaries. The built-in combinators for sizes are described in Sized types. 3.29.1 Example for coinduction: finite languages See Abel 2017 and Traytel 2017. Decidable languages can be represented as infinite trees. Each node has as many children as the number of characters in the alphabet A. Each path from the root of the tree to a node determines a possible word in the language. Each node has a boolean label, which is true if and only if the word corresponding to that node is in the language. In particular, the root node of the tree is labelled true if and only if the word 𝜖 belongs to the language. These infinite trees can be represented as the following coinductive data-type: record Lang (i : Size) (A : Set) : Set where coinductive field 𝜈 : Bool 𝛿 : {j : Size< i} → A → Lang j A open Lang 3.29. Sized Types 91 Agda User Manual, Release 2.6.0 As we said before, given a language a : Lang A, 𝜈 a true iff 𝜖 a. On the other hand, the language 𝛿 a x : Lang A is the Brzozowski derivative of a with respect to the character x, that is, w 𝛿 a x iff xw a. With this data type, we can define some regular languages. The first one, the empty language, contains no words; so all the nodes are labelled false: : 𝜈 𝛿 {i A} → Lang i A = false _ = The second one is the language containing a single word; the empty word. The root node is labelled true, and all the others are labelled false: 𝜖 : {i A} → Lang i A 𝜈 𝜖 = true 𝛿 𝜖 _ = To compute the union (or sum) of two languages, we do a point-wise or operation on the labels of their nodes: _+_ : {i A} → Lang i A → Lang i A → Lang i A 𝜈 (a + b) = 𝜈 a 𝜈 b 𝛿 (a + b) x = 𝛿 a x + 𝛿 b x infixl 10 _+_ Now, lets define concatenation. The base case (𝜈) is straightforward: 𝜖 For the derivative (𝛿), assume that we have a word w, w and 𝛽 b. a · b iff 𝜖 a and 𝜖 b. 𝛿 (a · b) x. This means that xw = 𝛼𝛽, with 𝛼 a We have to consider two cases: 1. 𝜖 a. Then, either: • 𝛼 = 𝜖, and 𝛽 = xw, where w • 𝛼 = x𝛼’, with 𝛼’ 2. 𝜖 𝛿 b x. 𝛿 a x, and w = 𝛼’𝛽 𝛿 a x · b. a. Then, only the second case above is possible: • 𝛼 = x𝛼’, with 𝛼’ 𝛿 a x, and w = 𝛼’𝛽 𝛿 a x · b. _·_ : {i A} → Lang i A → Lang i A → Lang i A 𝜈 (a · b) = 𝜈 a 𝜈 b 𝛿 (a · b) x = if 𝜈 a then 𝛿 a x · b + 𝛿 b x else 𝛿 a x · b infixl 20 _·_ Here is where sized types really shine. Without sized types, the termination checker would not be able to recognize that _+_ or if_then_else are not inspecting the tree, which could render the definition non-productive. By contrast, with sized types, we know that the a + b is defined to the same depth as a and b are. In a similar spirit, we can define the Kleene star: _* : {i A} → Lang i A → Lang i A 𝜈 (a *) = true 𝛿 (a *) x = 𝛿 a x · a * infixl 30 _* 92 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Again, because the types tell us that _·_ preserves the size of its inputs, we can have the recursive call to a * under a function call to _·_. Testing First, we want to give a precise notion of membership in a language. We consider a word as a List of characters. __ : {i} {A} → List i A → Lang i A → Bool [] a = 𝜈 a (x w) a = w 𝛿 a x Note how the size of the word we test for membership cannot be larger than the depth to which the language tree is defined. If we want to use regular, non-sized lists, we need to ask for the language to have size ∞. __ : {A} → List A → Lang ∞ A → Bool [] a = 𝜈 a (x w) a = w 𝛿 a x Intuitively, ∞ is a Size larger than the size of any term than one could possibly define in Agda. Now, let’s consider binary strings as words. First, we define the languages x containing the single word “x” of length 1, for alphabet A = Bool: _ : {i} → Bool → Lang i Bool 𝜈 _ = false 𝛿 𝛿 𝛿 𝛿 false true false true false true true false = 𝜖 = 𝜖 = = Now we can define the bip-bop language, consisting of strings of even length alternating letters “true” and “false”. bip-bop = ( true · false )* Let’s test a few words for membership in the language bip-bop! test1 : (true test1 = refl false true false true false test2 : (true test2 = refl false true false true []) test3 : (true test3 = refl true false []) bip-bop []) bip-bop bip-bop true false false 3.29.2 References Equational Reasoning about Formal Languages in Coalgebraic Style, Andreas Abel. Formal Languages, Formally and Coinductively, Dmitriy Traytel, LMCS Vol. 13(3:28)2017, pp. 1–22 (2017). 3.29. Sized Types 93 Agda User Manual, Release 2.6.0 3.30 Syntactic Sugar • Do-notation – Desugaring – Example • Idiom brackets 3.30.1 Do-notation A do-block consists of the layout keyword do followed by a sequence of do-statements, where do-stmt ::= | | lam-clause ::= pat ← expr [where lam-clauses] let decls expr pat → expr The where clause of a bind is used to handle the cases not matched by the pattern left of the arrow. See details below. Note: Arrows can use either unicode (←/→) or ASCII (<-/->) variants. For example: filter : {A : Set} → (A → Bool) → List A → List A filter p xs = do x ← xs true ← p x [] where false → [] x [] Do-notation is desugared before scope checking and is translated into calls to _>>=_ and _>>_, whatever those happen to be bound in the context of the do-block. This means that do-blocks are not tied to any particular notion of monad. In fact if there are no monadic statements in the do block it can be used as sugar for a let: pure-do : Nat → Nat pure-do n = do let p2 m = m * m p4 m = p2 (p2 m) p4 n check-pure-do : pure-do 5 check-pure-do = refl 94 625 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Desugaring Statement Simple bind Sugar Desugars to do x ← m m' m >>= 𝜆 x → m' do p ← m where p → m m' m >>= 𝜆 where p → m' p → m do m m' m >> m' do let ds m' let ds in m' Pattern bind Non-binding statement Let If the pattern in the bind is exhaustive, the where-clause can be omitted. Example Do-notation becomes quite powerful together with pattern matching on indexed data. As an example, let us write a correct-by-construction type checker for simply typed 𝜆-calculus. First we define the raw terms, using de Bruijn indices for variables and explicit type annotations on the lambda: infixr 6 _=>_ data Type : Set where nat : Type _=>_ : (A B : Type) → Type data Raw : Set where var : (x : Nat) → Raw lit : (n : Nat) → Raw suc : Raw app : (s t : Raw) → Raw lam : (A : Type) (t : Raw) → Raw Next up, well-typed terms: Context = List Type -- A proof of x xs is the index into xs where x is located. infix 2 __ data __ {A : Set} (x : A) : List A → Set where zero : {xs} → x x xs suc : {y xs} → x xs → x y xs data Term (Γ var : {A} lit : (n : suc : Term : Context) : Type → Set where (x : A Γ) → Term Γ A Nat) → Term Γ nat Γ (nat => nat) (continues on next page) 3.30. Syntactic Sugar 95 Agda User Manual, Release 2.6.0 (continued from previous page) app : lam : {A B} (s : Term Γ (A => B)) (t : Term Γ A) → Term Γ B A {B} (t : Term (A Γ) B) → Term Γ (A => B) Given a well-typed term we can mechaincally erase all the type information (except the annotation on the lambda) to get the corresponding raw term: rawIndex : {A} {x : A} {xs} → x xs → Nat rawIndex zero = zero rawIndex (suc i) = suc (rawIndex i) eraseTypes eraseTypes eraseTypes eraseTypes eraseTypes eraseTypes : {Γ A} → (var x) = (lit n) = suc = (app s t) = (lam A t) = Term Γ A → Raw var (rawIndex x) lit n suc app (eraseTypes s) (eraseTypes t) lam A (eraseTypes t) Now we’re ready to write the type checker. The goal is to have a function that takes a raw term and either fails with a type error, or returns a well-typed term that erases to the raw term it started with. First, lets define the return type. It’s parameterised by a context and the raw term to be checked: data WellTyped Γ e : Set where ok : (A : Type) (t : Term Γ A) → eraseTypes t e → WellTyped Γ e We’re going to need a corresponding type for variables: data InScope Γ n : Set where ok : (A : Type) (i : A Γ) → rawIndex i n → InScope Γ n Lets also have a type synonym for the case when the erasure proof is refl: infix 2 _ofType_ pattern _ofType_ x A = ok A x refl Since this is a do-notation example we had better have a monad. Lets use the either monad with string errors: TC : Set → Set TC A = Either String A typeError : {A} → String → TC A typeError = left For the monad operations, we are using instance arguments to infer which monad is being used. We are going to need to compare types for equality. This is our first opportunity to take advantage of pattern matching binds: _=?=_ : (A B : Type) → nat =?= nat = nat =?= (_ => _) = (_ => _) =?= nat = (A => B) =?= (A1 => B1 ) refl ← A =?= A1 refl ← B =?= B1 pure refl TC (A B) pure refl typeError "type mismatch: nat _ => _" typeError "type mismatch: _ => _ nat" = do We will also need to look up variables in the context: 96 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 lookupVar : Γ n → TC (InScope Γ n) lookupVar [] n = typeError "variable out of scope" lookupVar (A Γ) zero = pure (zero ofType A) lookupVar (A Γ) (suc n) = do i ofType B ← lookupVar Γ n pure (suc i ofType B) Note how the proof obligation that the well-typed deBruijn index erases to the given raw index is taken care of completely under the hood (in this case by the refl pattern in the ofType synonym). Finally we are ready to implement the actual type checker: infer : Γ e → TC (WellTyped Γ e) infer Γ (var x) = do i ofType A ← lookupVar Γ x pure (var i ofType A) infer Γ (lit n) = pure (lit n ofType nat) infer Γ suc = pure (suc ofType nat => nat) infer Γ (app e e1 ) = do s ofType A => B ← infer Γ e where _ ofType nat → typeError "numbers cannot be applied to arguments" t ofType A1 ← infer Γ e1 refl ← A =?= A1 pure (app s t ofType B) infer Γ (lam A e) = do t ofType B ← infer (A Γ) e pure (lam A t ofType A => B) In the app case we use a where-clause to handle the error case when the function to be applied is well-typed, but does not have a function type. 3.30.2 Idiom brackets Idiom brackets is a notation used to make it more convenient to work with applicative functors, i.e. functors F equipped with two operations pure : _<*>_ : {A} → A → F A {A B} → F (A → B) → F A → F B As do-notation, idiom brackets desugar before scope checking, so whatever the names pure and _<*>_ are bound to gets used when desugaring the idiom brackets. The syntax for idiom brackets is (| e a1 .. a |) or using unicode lens brackets (U+2987) and (U+2988): e a1 .. a This expands to (assuming left associative _<*>_) pure e <*> a1 <*> .. <*> a Idiom brackets work well with operators, for instance 3.30. Syntactic Sugar 97 Agda User Manual, Release 2.6.0 (| if a then b else c |) desugars to pure if_then_else_ <*> a <*> b <*> c Limitations: • Binding syntax and operator sections cannot appear immediately inside idiom brackets. • The top-level application inside idiom brackets cannot include implicit applications, so (| foo {x = e} a b |) is illegal. In case the e is pure you can write (| (foo {x = e}) a b |) which desugars to pure (foo {x = e}) <*> a <*> b 3.31 Syntax Declarations Note: This is a stub It is now possible to declare user-defined syntax that binds identifiers. Example: postulate State : Set → Set → Set put : {S} → S → State S get : {S} → State S S return : {A S} → A → State S A bind : {A B S} → State S B → (B → State S A) → State S A syntax bind e1 (𝜆 x → e2 ) = x ← e1 , e2 increment : State increment = x ← get , put (suc x) The syntax declaration for bind implies that x is in scope in e2 , but not in e1 . You can give fixity declarations along with syntax declarations: infixr 40 bind syntax bind e1 (𝜆 x → e2 ) = x ← e1 , e2 The fixity applies to the syntax, not the name; syntax declarations are also restricted to ordinary, non-operator names. The following declaration is disallowed: syntax _==_ x y = x === y Syntax declarations must also be linear; the following declaration is disallowed: 98 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 syntax wrong x = x + x Syntax declarations can have implicit arguments. For example: id : {a}{A : Set a} -> A -> A id x = x syntax id {A} x = x A 3.32 Telescopes Note: This is a stub. 3.33 Termination Checking Note: This is a stub. 3.33.1 With-functions 3.33.2 Pragmas and Options • The NON_TERMINATING pragma This is a safer version of TERMINATING which doesn’t treat the affected functions as terminating. This means that NON_TERMINATING functions do not reduce during type checking. They do reduce at run-time and when invoking C-c C-n at top-level (but not in a hole). The pragma was added in Agda 2.4.2. • The TERMINATING pragma Switches off termination checker for individual function definitions and mutual blocks and marks them as terminating. Since Agda 2.4.2.1 replaced the NO_TERMINATION_CHECK pragma. The pragma must precede a function definition or a mutual block. The pragma cannot be used in --safe mode. Examples: – Skipping a single definition: before type signature: {-# TERMINATING #-} a : A a = a – Skipping a single definition: before first clause: b : A {-# TERMINATING #-} b = b – Skipping an old-style mutual block: Before mutual keyword: 3.32. Telescopes 99 Agda User Manual, Release 2.6.0 {-# TERMINATING #-} mutual c : A c = d d : A d = c – Skipping an old-style mutual block: Somewhere within mutual block before a type signature or first function clause: mutual {-# TERMINATING #-} e : A e = f f : A f = e – Skipping a new-style mutual block: Anywhere before a type signature or first function clause in the block: g : A h : A g = h {-# TERMINATING #-} h = g 3.34 Universe Levels Note: This is a stub. 3.35 With-Abstraction • Usage – Generalisation – Nested with-abstractions – Simultaneous abstraction – Using underscores and variables in pattern repetition – Rewrite – The inspect idiom – Alternatives to with-abstraction – Performance considerations 100 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 • Technical details – Examples – Ill-typed with-abstractions With-abstraction was first introduced by Conor McBride [McBride2004] and lets you pattern match on the result of an intermediate computation by effectively adding an extra argument to the left-hand side of your function. 3.35.1 Usage In the simplest case the with construct can be used just to discriminate on the result of an intermediate computation. For instance filter filter filter filter filter : p p p p {A : Set} → (A → Bool) → List A → List A [] = [] (x xs) with p x (x xs) | true = x filter p xs (x xs) | false = filter p xs The clause containing the with-abstraction has no right-hand side. Instead it is followed by a number of clauses with an extra argument on the left, separated from the original arguments by a vertical bar (|). When the original arguments are the same in the new clauses you can use the ... syntax: filter : {A : Set} → (A → Bool) → List A → List A filter p [] = [] filter p (x xs) with p x ... | true = x filter p xs ... | false = filter p xs In this case ... expands to filter p (x side: xs). There are three cases where you have to spell out the left-hand • If you want to do further pattern matching on the original arguments. • When the pattern matching on the intermediate result refines some of the other arguments (see Dot patterns). • To disambiguate the clauses of nested with-abstractions (see Nested with-abstractions below). Generalisation The power of with-abstraction comes from the fact that the goal type and the type of the original arguments are generalised over the value of the scrutinee. See Technical details below for the details. This generalisation is important when you have to prove properties about functions defined using with. For instance, suppose we want to prove that the filter function above satisfies some property P. Starting out by pattern matching of the list we get the following (with the goal types shown in the holes) postulate postulate postulate postulate P : {A} → List A → Set p-nil : P [] Q : Set q-nil : Q proof : {A : Set} (p : A → Bool) (xs : List A) → P (filter p xs) proof p [] = {! P [] !} proof p (x xs) = {! P (filter p xs | p x) !} 3.35. With-Abstraction 101 Agda User Manual, Release 2.6.0 In the cons case we have to prove that P holds for filter p xs | p x. This is the syntax for a stuck withabstraction—filter cannot reduce since we don’t know the value of p x. This syntax is used for printing, but is not accepted as valid Agda code. Now if we with-abstract over p x, but don’t pattern match on the result we get: proof : {A : Set} (p : A → Bool) (xs : List A) → P (filter p xs) proof p [] = p-nil proof p (x xs) with p x ... | r = {! P (filter p xs | r) !} Here the p x in the goal type has been replaced by the variable r introduced for the result of p x. If we pattern match on r the with-clauses can reduce, giving us: proof : {A : Set} (p : A → Bool) (xs : List A) → P (filter p xs) proof p [] = p-nil proof p (x xs) with p x ... | true = {! P (x filter p xs) !} ... | false = {! P (filter p xs) !} Both the goal type and the types of the other arguments are generalised, so it works just as well if we have an argument whose type contains filter p xs. proof2 : {A : Set} (p : A → Bool) (xs : List A) → P (filter p xs) → Q proof2 p [] _ = q-nil proof2 p (x xs) H with p x ... | true = {! H : P (filter p xs) !} ... | false = {! H : P (x filter p xs) !} The generalisation is not limited to scrutinees in other with-abstractions. All occurrences of the term in the goal type and argument types will be generalised. Note that this generalisation is not always type correct and may result in a (sometimes cryptic) type error. See Ill-typed with-abstractions below for more details. Nested with-abstractions With-abstractions can be nested arbitrarily. The only thing to keep in mind in this case is that the ... syntax applies to the closest with-abstraction. For example, suppose you want to use ... in the definition below. compare compare compare compare compare compare : x x x x x Nat → y with y | y | y | y | Nat → Comparison x < y false with y < x false | false = equal false | true = greater true = less You might be tempted to replace compare x y with ... in all the with-clauses as follows. compare : Nat → compare x y with ... | ... ... ... | Nat → Comparison x < y false with y < x | false = equal | true = greater true = less -- WRONG This, however, would be wrong. In the last clause the ... is interpreted as belonging to the inner with-abstraction (the whitespace is not taken into account) and thus expands to compare x y | false | true. In this case you have to spell out the left-hand side and write 102 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 compare : Nat → compare x y with ... | ... ... compare x y | Nat → Comparison x < y false with y < x | false = equal | true = greater true = less Simultaneous abstraction You can abstract over multiple terms in a single with-abstraction. To do this you separate the terms with vertical bars (|). compare : Nat → compare x y with ... | ... | ... | Nat → x < y true _ false Comparison | y < x | _ = less | true = greater | false = equal In this example the order of abstracted terms does not matter, but in general it does. Specifically, the types of later terms are generalised over the values of earlier terms. For instance postulate plus-commute : (a b : Nat) → a + b postulate P : Nat → Set b + a thm : (a b : Nat) → P (a + b) → P (b + a) thm a b t with a + b | plus-commute a b thm a b t | ab | eq = {! t : P ab, eq : ab b + a !} Note that both the type of t and the type of the result eq of plus-commute a b have been generalised over a + b. If the terms in the with-abstraction were flipped around, this would not be the case. If we now pattern match on eq we get thm : (a b : Nat) → P (a + b) → P (b + a) thm a b t with a + b | plus-commute a b thm a b t | .(b + a) | refl = {! t : P (b + a) !} and can thus fill the hole with t. In effect we used the commutativity proof to rewrite a + b to b + a in the type of t. This is such a useful thing to do that there is special syntax for it. See Rewrite below. A limitation of generalisation is that only occurrences of the term that are visible at the time of the abstraction are generalised over, but more instances of the term may appear once you start filling in the right-hand side or do further matching on the left. For instance, consider the following contrived example where we need to match on the value of f n for the type of q to reduce, but we then want to apply q to a lemma that talks about f n: postulate R : Set P : Nat → Set f : Nat → Nat lemma : n → P (f n) → R Q : Nat → Set Q zero = Q (suc n) = P (suc n) 3.35. With-Abstraction 103 Agda User Manual, Release 2.6.0 proof proof proof proof : n n n (n : Nat) → Q (f n) → R q with f n () | zero q | suc fn = {! q : P (suc fn) !} Once we have generalised over f n we can no longer apply the lemma, which needs an argument of type P (f n). To solve this problem we can add the lemma to the with-abstraction: proof proof proof proof : n n n (n : Nat) → Q (f n) → R q with f n | lemma n () | zero | _ q | suc fn | lem = lem q In this case the type of lemma n (P (f n) → R) is generalised over f n so in the right-hand side of the last clause we have q : P (suc fn) and lem : P (suc fn) → R. See The inspect idiom below for an alternative approach. Using underscores and variables in pattern repetition If an ellipsis . . . cannot be used, the with-clause has to repeat (or refine) the patterns of the parent clause. Since Agda 2.5.3, such patterns can be replaced by underscores _ if the variables they bind are not needed. Here is a (slightly contrived) example: record R : Set where coinductive -- disallows matching field f : Bool n : Nat data P (r : R) : Nat → Set where fTrue : R.f r true → P r zero nSuc : P r (suc (R.n r)) data Q : (b : Bool) (n : Nat) → Set where true! : Q true zero suc! : {b n} → Q b (suc n) test : (r : R) {n : Nat} (p : P r n) → Q (R.f r) n test r nSuc = suc! test r (fTrue p) with R.f r test _ (fTrue ()) | false test _ _ | true = true! -- underscore instead of (isTrue _) Since Agda 2.5.4, patterns can also be replaced by a variable: f f f f : List Nat → List Nat [] = [] (x xs) with f xs xs0 | r = ? The variable xs0 is treated as a let-bound variable with value .x .xs (where .x : Nat and .xs : List Nat are out of scope). Since with-abstraction may change the type of variables, the instantiation of such let-bound variables are type checked again after with-abstraction. 104 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Rewrite Remember example of simultaneous abstraction from above. postulate plus-commute : (a b : Nat) → a + b b + a thm : (a b : Nat) → P (a + b) → P (b + a) thm a b t with a + b | plus-commute a b thm a b t | .(b + a) | refl = t This pattern of rewriting by an equation by with-abstracting over it and its left-hand side is common enough that there is special syntax for it: thm : (a b : Nat) → P (a + b) → P (b + a) thm a b t rewrite plus-commute a b = t The rewrite construction takes a term eq of type lhs rhs, where __ is the built-in equality type, and expands to a with-abstraction of lhs and eq followed by a match of the result of eq against refl: f ps rewrite eq = v --> f ps with lhs | eq ... | .rhs | refl = v One limitation of the rewrite construction is that you cannot do further pattern matching on the arguments after the rewrite, since everything happens in a single clause. You can however do with-abstractions after the rewrite. For instance, postulate T : Nat → Set isEven isEven isEven isEven : Nat → Bool zero = true (suc zero) = false (suc (suc n)) = isEven n thm1 thm1 thm1 thm1 (a b : Nat) b t rewrite b t | true b t | false : a a a → T (a + b) → T (b + a) plus-commute a b with isEven a = t = t Note that the with-abstracted arguments introduced by the rewrite (lhs and eq) are not visible in the code. The inspect idiom When you with-abstract a term t you lose the connection between t and the new argument representing its value. That’s fine as long as all instances of t that you care about get generalised by the abstraction, but as we saw above this is not always the case. In that example we used simultaneous abstraction to make sure that we did capture all the instances we needed. An alternative to that is to use the inspect idiom, which retains a proof that the original term is equal to its abstraction. In the simplest form, the inspect idiom uses a singleton type: data Singleton {a} {A : Set a} (x : A) : Set a where _with_ : (y : A) → x y → Singleton x (continues on next page) 3.35. With-Abstraction 105 Agda User Manual, Release 2.6.0 (continued from previous page) inspect : {a} {A : Set a} (x : A) → Singleton x inspect x = x with refl Now instead of with-abstracting t, you can abstract over inspect t. For instance, filter : {A : Set} → (A → Bool) → List A → List A filter p [] = [] filter p (x xs) with inspect (p x) ... | true with eq = {! eq : p x true !} ... | false with eq = {! eq : p x false !} Here we get proofs that p x true and p x false in the respective branches that we can use on the right. Note that since the with-abstraction is over inspect (p x) rather than p x, the goal and argument types are no longer generalised over p x. To fix that we can replace the singleton type by a function graph type as follows (see Anonymous modules to learn about the use of a module to bind the type arguments to Graph and inspect): module _ {a b} {A : Set a} {B : A → Set b} where data Graph (f : ingraph : f x x → B x) (x : A) (y : B x) : Set b where y → Graph f x y inspect : (f : x → B x) (x : A) → Graph f x (f x) inspect _ _ = ingraph refl To use this on a term g v you with-abstract over both g v and inspect g v. For instance, applying this to the example from above we get postulate R : Set P : Nat → Set f : Nat → Nat lemma : n → P (f n) → R Q : Nat → Set Q zero = Q (suc n) = P (suc n) proof proof proof proof : n n n (n : Nat) → Q (f n) → R q with f n | inspect f n () | zero | _ q | suc fn | ingraph eq = {! q : P (suc fn), eq : f n We could then use the proof that f n suc fn !} suc fn to apply lemma to q. This version of the inspect idiom is defined (using slightly different names) in the standard library in the module Relation.Binary.PropositionalEquality and in the agda-prelude in Prelude.Equality. Inspect (reexported by Prelude). Alternatives to with-abstraction Although with-abstraction is very powerful there are cases where you cannot or don’t want to use it. For instance, you cannot use with-abstraction if you are inside an expression in a right-hand side. In that case there are a couple of alternatives. 106 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 Pattern lambdas Agda does not have a primitive case construct, but one can be emulated using pattern matching lambdas. First you define a function case_of_ as follows: case_of_ : {a b} {A : Set a} {B : Set b} → A → (A → B) → B case x of f = f x You can then use this function with a pattern matching lambda as the second argument to get a Haskell-style case expression: filter : {A : Set} → (A → Bool) → List A → List A filter p [] = [] filter p (x xs) = case p x of 𝜆 { true → x filter p xs ; false → filter p xs } This version of case_of_ only works for non-dependent functions. For dependent functions the target type will in most cases not be inferrable, but you can use a variant with an explicit B for this case: case_return_of_ : {a b} {A : Set a} (x : A) (B : A → Set b) → ( x → B x) → B x case x return B of f = f x The dependent version will let you generalise over the scrutinee, just like a with-abstraction, but you have to do it manually. Two things that it will not let you do is • further pattern matching on arguments on the left-hand side, and • refine arguments on the left by the patterns in the case expression. For instance if you matched on a Vec A n the n would be refined by the nil and cons patterns. Helper functions Internally with-abstractions are translated to auxiliary functions (see Technical details below) and you can always1 write these functions manually. The downside is that the type signature for the helper function needs to be written out explicitly, but fortunately the Emacs Mode has a command (C-c C-h) to generate it using the same algorithm that generates the type of a with-function. Performance considerations The generalisation step of a with-abstraction needs to normalise the scrutinee and the goal and argument types to make sure that all instances of the scrutinee are generalised. The generalisation also needs to be type checked to make sure that it’s not ill-typed. This makes it expensive to type check a with-abstraction if • the normalisation is expensive, • the normalised form of the goal and argument types are big, making finding the instances of the scrutinee expensive, • type checking the generalisation is expensive, because the types are big, or because checking them involves heavy computation. In these cases it is worth looking at the alternatives to with-abstraction from above. 1 The termination checker has special treatment for with-functions, so replacing a with by the equivalent helper function might fail termination. 3.35. With-Abstraction 107 Agda User Manual, Release 2.6.0 3.35.2 Technical details Internally with-abstractions are translated to auxiliary functions—there are no with-abstractions in the Core language. This translation proceeds as follows. Given a with-abstraction 𝑓 :Γ→𝐵 𝑓 𝑝𝑠 with 𝑡1 | . . . | 𝑡𝑚 𝑓 𝑝𝑠1 | 𝑞11 | . . . | 𝑞1𝑚 = 𝑣1 .. . 𝑓 𝑝𝑠𝑛 | 𝑞𝑛1 | . . . | 𝑞𝑛𝑚 = 𝑣𝑛 where ∆ ⊢ 𝑝𝑠 : Γ (i.e. ∆ types the variables bound in 𝑝𝑠), we • Infer the types of the scrutinees 𝑡1 : 𝐴1 , . . . , 𝑡𝑚 : 𝐴𝑚 . • Partition the context ∆ into ∆1 and ∆2 such that ∆1 is the smallest context where ∆1 ⊢ 𝑡𝑖 : 𝐴𝑖 for all 𝑖, i.e., where the scrutinees are well-typed. Note that the partitioning is not required to be a split, ∆1 ∆2 can be a (well-formed) reordering of ∆. • Generalise over the 𝑡𝑖 s, by computing 𝐶 = (𝑤1 : 𝐴1 )(𝑤1 : 𝐴′2 ) . . . (𝑤𝑚 : 𝐴′𝑚 ) → ∆′2 → 𝐵 ′ such that the normal form of 𝐶 does not contain any 𝑡𝑖 and 𝐴′𝑖 [𝑤1 := 𝑡1 . . . 𝑤𝑖−1 := 𝑡𝑖−1 ] ≃ 𝐴𝑖 (∆′2 → 𝐵 ′ )[𝑤1 := 𝑡1 . . . 𝑤𝑚 := 𝑡𝑚 ] ≃ ∆2 → 𝐵 where 𝑋 ≃ 𝑌 is equality of the normal forms of 𝑋 and 𝑌 . The type of the auxiliary function is then ∆1 → 𝐶. • Check that ∆1 → 𝐶 is type correct, which is not guaranteed (see below). • Add a function 𝑓𝑎𝑢𝑥 , mutually recursive with 𝑓 , with the definition 𝑓𝑎𝑢𝑥 : ∆1 → 𝐶 𝑓𝑎𝑢𝑥 𝑝𝑠11 qs 1 𝑝𝑠21 = 𝑣1 .. . 𝑓𝑎𝑢𝑥 𝑝𝑠1𝑛 qs 𝑛 𝑝𝑠2𝑛 = 𝑣𝑛 where qs 𝑖 = 𝑞𝑖1 . . . 𝑞𝑖𝑚 , and 𝑝𝑠1𝑖 : ∆1 and 𝑝𝑠2𝑖 : ∆2 are the patterns from 𝑝𝑠𝑖 corresponding to the variables of 𝑝𝑠. Note that due to the possible reordering of the partitioning of ∆ into ∆1 and ∆2 , the patterns 𝑝𝑠1𝑖 and 𝑝𝑠2𝑖 can be in a different order from how they appear 𝑝𝑠𝑖 . • Replace the with-abstraction by a call to 𝑓𝑎𝑢𝑥 resulting in the final definition 𝑓 :Γ→𝐵 𝑓 𝑝𝑠 = 𝑓𝑎𝑢𝑥 xs 1 𝑡𝑠 xs 2 where 𝑡𝑠 = 𝑡1 . . . 𝑡𝑚 and xs 1 and xs 2 are the variables from ∆ corresponding to ∆1 and ∆2 respectively. Examples Below are some examples of with-abstractions and their translations. 108 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 postulate A : Set _+_ : A → A → A T : A → Set mkT : x → T x P : x → T x → Set --f1 f1 f1 the type A of the with argument has no free variables, so the with argument will come first : (x y : A) (t : T (x + y)) → T (x + y) x y t with x + y x y t | w = {!!} -- Generated with function f-aux1 : (w : A) (x y : A) (t : T w) → T w f-aux1 w x y t = {!!} --f2 f2 f2 x and p are not needed to type the with argument, so the context is reordered with only y before the with argument : (x y : A) (p : P y (mkT y)) → P y (mkT y) x y p with mkT y x y p | w = {!!} f-aux2 : (y : A) (w : T y) (x : A) (p : P y w) → P y w f-aux2 y w x p = {!!} postulate H : x y → T (x + y) → Set ---f3 f3 f3 Multiple with arguments are always inserted together, so in this case t ends up on the left since it’s needed to type h and thus x + y isn’t abstracted from the type of t : (x y : A) (t : T (x + y)) (h : H x y t) → T (x + y) x y t h with x + y | h x y t h | w1 | w2 = {! t : T (x + y), goal : T w1 !} f-aux3 : (x y : A) (t : T (x + y)) (h : H x y t) (w1 : A) (w2 : H x y t) → T w1 f-aux3 x y t h w1 w2 = {!!} -f4 f4 f4 But earlier with : (x y : A) (t : x y t with x + y x y t | w1 arguments are abstracted from the types of later ones T (x + y)) → T (x + y) | t | w2 = {! t : T (x + y), w2 : T w1 , goal : T w1 !} f-aux4 : (x y : A) (t : T (x + y)) (w1 : A) (w2 : T w1 ) → T w1 f-aux4 x y t w1 w2 = {!!} Ill-typed with-abstractions As mentioned above, generalisation does not always produce well-typed results. This happens when you abstract over a term that appears in the type of a subterm of the goal or argument types. The simplest example is abstracting over the first component of a dependent pair. For instance, postulate A : Set (continues on next page) 3.35. With-Abstraction 109 Agda User Manual, Release 2.6.0 (continued from previous page) B : A → Set H : (x : A) → B x → Set bad-with : (p : Σ A B) → H (fst p) (snd p) bad-with p with fst p ... | _ = {!!} Here, generalising over fst p results in an ill-typed application H w (snd p) and you get the following type error: fst p != w of type A when checking that the type (p : Σ A B) (w : A) → H w (snd p) of the generated with function is well-formed This message can be a little difficult to interpret since it only prints the immediate problem (fst p != w) and the full type of the with-function. To get a more informative error, pointing to the location in the type where the error is, you can copy and paste the with-function type from the error message and try to type check it separately. 3.36 Without K The option --without-K makes pattern matching more restricted. If the option is activated, then Agda only accepts certain case splits. This option was added in Agda 2.2.10. Since Agda 2.4.0 when the option --without-K is enabled, then the unification algorithm for checking case splits cannot make use of the deletion rule to solve equations of the form x = x. For example, the obvious implementation of the J rule is accepted: J : {A : Set} (P : (x y : A) → x y → Set) → ((x : A) → P x x refl) → (x y : A) (xy : x J P p x .x refl = p x y) → P x y xy Pattern matching with the constructor refl on the argument xy causes x to be unified with y. The same applies to Christine Paulin-Mohring’s version of the J rule: J : {A : Set} {x : A} (P : (y : A) → x y → Set) → P x refl → (y : A) (xy : x y) → P y xy J P p ._ refl = p On the other hand, the obvious implementation of the K rule is not accepted: K : {A : Set} {x : A} (P : x x → Set) → P refl → (xx : x x) → P xx K P p refl = p Pattern matching with the constructor refl on the argument xx causes x to be unified with x, which fails because the deletion rule cannot be used when --without-K is enabled. For more details, see the paper Eliminating dependent pattern matching without K [Cockx, Devriese, and Piessens (2016)]. The option --with-K can be used to override a global --without-K in a file, by adding a pragma {-# OPTIONS --with-K #-}. This option was added in Agda 2.4.2 and it is on by default. Since Agda 2.4.2 termination checking --without-K restricts structural descent to arguments ending in data types or Size. Likewise, guardedness is only tracked when result type is data or record type: 110 Chapter 3. Language Reference Agda User Manual, Release 2.6.0 data : Set where mutual data WOne : Set where wrap : FOne → WOne FOne = → WOne postulate iso : WOne FOne noo : (X : Set) → (WOne X) → X → noo .WOne refl (wrap f) = noo FOne iso f noo is rejected since at type X the structural descent f < wrap f is discounted --without-K: data Pandora : Set where C : ∞ → Pandora postulate foo : Pandora loop : (A : Set) → A Pandora → A loop .Pandora refl = C ( (loop foo)) loop is rejected since guardedness is not tracked at type A --without-K. See issues #1023, #1264, #1292. 3.36. Without K 111 Agda User Manual, Release 2.6.0 112 Chapter 3. Language Reference CHAPTER 4 Tools 4.1 Automatic Proof Search (Auto) Agda supports (since version 2.2.6) the command Auto, that searches for type inhabitants and fills a hole when one is found. The type inhabitant found is not necessarily unique. Auto can be used as an aid when interactively constructing terms in Agda. In a system with dependent types it can be meaningful to use such a tool for finding fragments of, not only proofs, but also programs. For instance, giving the type signature of the map function over vectors, you will get the desired function as the first solution. The tool is based on a term search implementation independent of Agda called Agsy. Agsy is a general purpose search algorithm for a dependently typed language similar to Agda. One should not expect it to handle large problems of any particular kind, but small enough problems of almost any kind. Any solution coming from Auto is checked by Agda. Also, the main search algorithm has a timeout mechanism. Therefore, there is little harm in trying Auto and it might save you key presses. 4.1.1 Usage The tool is invoked by placing the cursor on a hole and choosing Auto in the goal menu or pressing C-c C-a. Agsy’s behaviour can be changed by using various options which are passed directly in the hole. Option -t N -c -d ID -m -r -l -s N Meaning Set timeout to N seconds Allow Agsy to use case-split Attempt to disprove the goal Use definition ID as a hint Use the definitions in the current module as hints Use the unqualified definitions in scope as hints List up to ten solutions, does not commit to any Commit to the N th solution 113 Agda User Manual, Release 2.6.0 Giving no arguments is fine and results in a search with default parameters. The search carries on until either a (not necessarily unique) solution is found, the search space is fully (and unsuccessfully) explored or it times out (one second by default). Here follows a list of the different modes and parameters. Case split Auto normally only tries to find a term that could replace the current hole. However, if the hole constitutes the entire RHS of the clause (same as for the make-case command), you can instruct Auto to try case splitting by writing (since version 2.2.8) -c. Note that if a solution is found the whole file will be reloaded (as for make-case) resulting in a delay. Case splitting cannot yet be combined with -l or -s . Equality reasoning If the constants __ begin_ ___ _ sym cong from the standard library are in scope, then Auto will do equality reasoning using these constructs. However, it will not do anything more clever than things like not nesting several sym or cong. Hence long chains of equalities should not be expected and required arithmetical rules have to be given as hints. Hints Auto does not by default try using constants in scope. If there is a lemma around that might help in constructing the term you can include it in the search by giving hints. There are two ways of doing this. One way is to provide the exact list of constants to include. Such a list is given by writing a number of constant names separated by space: .... The other way is to write -m. This includes all constants in scope which are defined or postulated in the innermost module surrounding the current hole. It is also possible to combine -m with a list of named constants (not included by -m). There are a few exceptions to what you have to specify as hints: • Datatypes and constants that can be deduced by unifying the two sides of an equality constraint can be omitted. E.g., if the constraint ? = List A occurs during the search, then refining ? to List ... will happen without having to provide List as a hint. The constants that you can leave out overlap more or less with the ones appearing in hidden arguments, i.e. you wouldn’t have written them when giving the term by hand either. • Constructors and projection functions are automatically tried, so should never be given as hints. • Recursive calls, although currently only the function itself, not all functions in the same mutual block. Timeout The timeout is one second by default but can be changed by adding -t to the parameters, where is the number of seconds. Listing and choosing among several solutions Normally, Auto replaces the hole with the first solution found. If you are not happy with that particular solution, you can list the ten (at most) first solutions encountered by including the flag -l. 114 Chapter 4. Tools Agda User Manual, Release 2.6.0 You can then pick a particular solution by writing -s where is the number of solutions to skip (as well as the number appearing before the solution in the list). The options -l and -s can be combined to list solutions other than the ten first ones. Disproving If you are uncertain about the validity of what you are trying to prove, you can use Auto to try to find a counterproof. The flag -d makes Auto negate the current goal and search for a term disproving it. If such a term is found, it will be displayed in the info buffer. The flag -d can be combined with -l and -l -s . Auto refine / suggest There is a special mode for searching (part of) the scope of constants for possible refinement candidates. The flag -r chooses this mode. By default all constants which are in scope unqualified are included. The constants that matches the current goal are sorted in order of how many constructs their result type contains. This means that the constants which in some sense match the goal most specifically will appear first and the most general ones last. By default, Auto will present a list of candidates, rather than refining using the topmost constant. To select one of them for refinement, combine -r with -s . In order to list constants other than the ten first ones, write -r -l -s . The auto refine feature has little to do with the rest of the Auto tool. If it turns out to be useful it could be improved and made into a separate Emacs mode command. Dependencies between meta variables If the goal type or type of local variables contain meta variables, then the constraints for these are also included in the search. If a solution is found it means that Auto has also found solutions for the occurring meta variables. Those solutions will be inserted into your file along with that of the hole from where you called Auto. Also, any unsolved equality constraints that contain any of the involved meta variables are respected in the search. 4.1.2 Limitations • Irrelevance is not yet respected. Agsy will happily use a parameter marked as irrelevant and does not disregard irrelevant arguments when comparing terms. • Records that lack a constructor name are still deconstructed when case splitting, but the name of the record type is used instead of a constructor name in the resulting pattern. • Literals representing natural numbers are supported (but any generated natural number will be given in constructor form). Apart from this, literals are not supported. • Primitive functions are not supported. • Copatterns are not supported. • Termination checking for recursive calls is done locally, so a non-terminating set of clauses might be sent back to Agda. • Agsy currently does not automatically pick a datatype when instantiating types. A frequently occurring situation is when you try to disprove a generic property. Then Agsy must come up with a particular type as part of the disproof. You can either fix your generic type to e.g. Nat or Fin n (for an arbitrary n if you wish), or you give Nat or Fin as a hint to the search. • Case split mode currently does not do case on expressions (with). 4.1. Automatic Proof Search (Auto) 115 Agda User Manual, Release 2.6.0 • Case split mode sometimes gives a unnecessarily complex RHS for some clause when the solution consists of several clauses. • The special constraints that apply to codata are not respected by Agsy. Agsy treats codata just like data. • Agsy has universe subtyping, so sometimes suggests solutions not accepted by Agda. • Universe polymorphism is only partially supported. Agsy may fail when trying to construct universe polymorphic definitions, but will probably succeed (with respect to this) when constructing terms which refer to, or whose type is defined in terms of, universe polymorphic definitions. • In case split and disproving modes, the current goal may not depend on any other meta variables. For disproving mode this means that there may be implicitly universally quantified but not existentially quantified stuff. • Searching for simultaneous solutions of several holes does not combine well with parameterised modules and recursive calls. 4.1.3 User feedback When sending bug reports, please use Agda’s bug tracker. Apart from that, receiving nice examples (via the bug tracker) would be much appreciated. Both such examples which Auto does not solve, but you have a feeling it’s not larger than for that to be possible. And examples that Auto only solves by increasing timeout. The examples sent in will be used for tuning the heuristics and hopefully improving the performance. 4.2 Command-line options 4.2.1 Command-line options Agda accepts the following options. General options --version -V Show version number --help[=TOPIC] -?[TOPIC] Show basically this help, or more help about TOPIC. Current topics available: warning. --interactive -I Start in interactive mode (no longer supported) --interaction For use with the Emacs mode (no need to invoke yourself) --interaction-json For use with other editors such as Atom (no need to invoke yourself) Compilation See Compilers for backend-specific options. --no-main Do not treat the requested module as the main module of a program when compiling --compile-dir=DIR Set DIR as directory for compiler output (default: the project root) --no-forcing Disable the forcing optimisation 116 Chapter 4. Tools Agda User Manual, Release 2.6.0 Generating highlighted source code --vim Generate Vim highlighting files --latex Generate LaTeX with highlighted source code (see Generating LaTeX) --latex-dir=DIR Set directory in which LaTeX files are placed to DIR (default: latex) --count-clusters Count extended grapheme clusters when generating LaTeX code (see Counting Extended Grapheme Clusters) --html Generate HTML files with highlighted source code (see Generating HTML) --html-dir=DIR Set directory in which HTML files are placed to DIR (default: html) --css=URL Set URL of the CSS file used by the HTML files to URL (can be relative) --dependency-graph=FILE Generate a Dot file FILE with a module dependency graph Imports and libraries (see Library Management) --ignore-interfaces Ignore interface files (re-type check everything) --include-path=DIR -i=DIR Look for imports in DIR --library=DIR -l=LIB Use library LIB --library-file=FILE Use FILE instead of the standard libraries file --no-libraries Don’t use any library files --no-default-libraries Don’t use default library files Sharing and caching --sharing Enable sharing and call-by-need evaluation (experimental) (default: OFF) --no-sharing Disable sharing and call-by-need evaluation --caching Enable caching of typechecking (experimental) (default: OFF) --no-caching Disable caching of typechecking --only-scope-checking Only scope-check the top-level module, do not type-check it 4.2.2 Command-line and pragma options The following options can also be given in .agda files in the {-# OPTIONS --{opt1 } --{opt2 } ... #-} form at the top of the file. Printing and debugging --show-implicit Show implicit arguments when printing --show-irrelevant Show irrelevant arguments when printing --no-unicode Don’t use unicode characters to print terms --verbose=N -v=N Set verbosity level to N 4.2. Command-line options 117 Agda User Manual, Release 2.6.0 Copatterns and projections --copatterns Enable definitions by copattern matching (default; see Copatterns) --no-copatterns Disable definitions by copattern matching --postfix-projections Make postfix projection notation the default Experimental features --injective-type-constructors Enable injective type constructors (makes Agda anti-classical and possibly inconsistent) --guardedness-preserving-type-constructors Treat type constructors as inductive constructors when checking productivity --experimental-irrelevance Enable potentially unsound irrelevance features (irrelevant levels, irrelevant data matching) (see Irrelevance) --rewriting Enable declaration and use of REWRITE rules (see Rewriting) --cubical Enable cubical features (see Cubical Type Theory in Agda) Errors and warnings --allow-unsolved-metas Succeed and create interface file regardless of unsolved meta variables (see Metavariables) --no-positivity-check Do not warn about not strictly positive data types (see Positivity Checking) --no-termination-check Do not warn about possibly nonterminating code (see Termination Checking) --warning=GROUP|FLAG -W=GROUP|FLAG Set warning group or flag (see Warnings) Pattern matching and equality --without-K Disables definitions using Streicher’s K axiom (see Without K) --with-K Overrides a global --without-K in a file (see Without K) --no-pattern-matching Disable pattern matching completely --exact-split Require all clauses in a definition to hold as definitional equalities unless marked CATCHALL (see Case trees) --no-exact-split Do not require all clauses in a definition to hold as definitional equalities (default) --no-eta-equality Default records to no-eta-equality (see Eta-expansion) Search depth --termination-depth=N Allow termination checker to count decrease/increase upto N (default: 1; see Termination Checking) --instance-search-depth=N Set instance search depth to N (default: 500; see Instance Arguments) --inversion-max-depth=N Set maximum depth for pattern match inversion to N (default: 50). Should only be needed in pathological cases. 118 Chapter 4. Tools Agda User Manual, Release 2.6.0 Other features --safe Disable postulates, unsafe OPTION pragmas and primTrustMe (see Safe Agda) --type-in-type Ignore universe levels (this makes Agda inconsistent; see Universe Levels) --omega-in-omega Enable typing rule Set𝜔 : Set𝜔 (this makes Agda inconsistent). --sized-types Use sized types (default, inconsistent with “musical” coinduction; see Sized Types) --no-sized-types Disable sized types (see Sized Types) --universe-polymorphism Enable universe polymorphism (default; see Universe Levels) --no-universe-polymorphism Disable universe polymorphism (see Universe Levels) --no-irrelevant-projections Disable projection of irrelevant record fields (see Irrelevance) --no-auto-inline Disable automatic compile-time inlining. Only definitions marked INLINE will be inlined. --no-print-pattern-synonyms Always expand Pattern Synonyms during printing. With this option enabled you can use pattern synonyms freely, but Agda will not use any pattern synonyms when printing goal types or error messages, or when generating patterns for case splits. Warnings The -W or --warning option can be used to disable or enable different warnings. The flag -W error (or --warning=error) can be used to turn all warnings into errors, while -W noerror turns this off again. A group of warnings can be enabled by -W {group}, where group is one of the following: all All of the existing warnings warn Default warning level ignore Ignore all warnings Individual warnings can be turned on and off by -W {Name} and -W {noName} respectively. The flags available are: AbsurdPatternRequiresNoRHS RHS given despite an absurd pattern in the LHS. CoverageIssue Failed coverage checks. CoverageNoExactSplit Failed exact split checks. DeprecationWarning Feature deprecation. EmptyAbstract Empty abstract blocks. EmptyInstance Empty instance blocks. EmptyMacro Empty macro blocks. EmptyMutual Empty mutual blocks. EmptyPostulate Empty postulate blocks. EmptyPrivate Empty private blocks. EmptyRewritePragma Empty REWRITE pragmas. InvalidCatchallPragma CATCHALL pragmas before a non-function clause. InvalidNoPositivityCheckPragma No positivity checking pragmas before non-data‘, record or mutual blocks. 4.2. Command-line options 119 Agda User Manual, Release 2.6.0 InvalidTerminationCheckPragma Termination checking pragmas before non-function or mutual blocks. InversionDepthReached Inversions of pattern-matching failed due to exhausted inversion depth. MissingDefinitions Names declared without an accompanying definition. ModuleDoesntExport Names mentioned in an import statement which are not exported by the module in question. NotAllowedInMutual Declarations not allowed in a mutual block. NotStrictlyPositive Failed strict positivity checks. OldBuiltin Deprecated BUILTIN pragmas. OverlappingTokensWarning Multi-line comments spanning one or more literate text blocks. PolarityPragmasButNotPostulates Polarity pragmas for non-postulates. SafeFlagNoPositivityCheck NO_POSITIVITY_CHECK pragmas with the safe flag. SafeFlagNonTerminating NON_TERMINATING pragmas with the safe flag. SafeFlagPolarity POLARITY pragmas with the safe flag. SafeFlagPostulate postulate blocks with the safe flag SafeFlagPragma Unsafe OPTIONS pragmas with the safe flag. SafeFlagPrimTrustMe primTrustMe usages with the safe flag. SafeFlagTerminating TERMINATING pragmas with the safe flag. TerminationIssue Failed termination checks. UnknownFixityInMixfixDecl Mixfix names without an associated fixity declaration. UnknownNamesInFixityDecl Names not declared in the same scope as their syntax or fixity declaration. UnknownNamesInPolarityPragmas Names not declared in the same scope as their polarity pragmas. UnreachableClauses Unreachable function clauses. UnsolvedConstraints Unsolved constraints. UnsolvedInteractionMetas Unsolved interaction meta variables. UnsolvedMetaVariables Unsolved meta variables. UselessAbstract abstract blocks where they have no effect. UselessInline INLINE pragmas where they have no effect. UselessInstance instance blocks where they have no effect. UselessPrivate private blocks where they have no effect. UselessPublic public blocks where they have no effect. For example, the following command runs Agda with all warnings enabled, except for warnings about empty abstract blocks: agda -W all --warning noEmptyAbstract file.agda 120 Chapter 4. Tools Agda User Manual, Release 2.6.0 4.3 Compilers • Backends – GHC Backend – JavaScript Backend • Optimizations – Builtin natural numbers – Erasable types See also Foreign Function Interface. 4.3.1 Backends GHC Backend The GHC backend translates Agda programs into GHC Haskell programs. Usage The backend can be invoked from the command line using the flag --compile: agda --compile [--compile-dir= ] [--ghc-flag= ] .agda Pragmas Example The following “Hello, World!” example requires some Built-ins and uses the Foreign Function Interface: module HelloWorld where open import Agda.Builtin.IO open import Agda.Builtin.Unit open import Agda.Builtin.String postulate putStrLn : String → IO {-# FOREIGN GHC import qualified Data.Text.IO as Text #-} {-# COMPILE GHC putStrLn = Text.putStrLn #-} main : IO main = putStrLn "Hello, World!" After compiling the example 4.3. Compilers 121 Agda User Manual, Release 2.6.0 agda --compile HelloWorld.agda you can run the HelloWorld program which prints Hello, World!. Required libraries for the Built-ins • primFloatEquality requires the ieee754 library. JavaScript Backend The JavaScript backend translates Agda code to JavaScript code. Usage The backend can be invoked from the command line using the flag --js: agda --js [--compile-dir= ] .agda 4.3.2 Optimizations Builtin natural numbers Builtin natural numbers are represented as arbitrary-precision integers. The builtin functions on natural numbers are compiled to the corresponding arbitrary-precision integer functions. Note that pattern matching on an Integer is slower than on an unary natural number. Code that does a lot of unary manipulations and doesn’t use builtin arithmetic likely becomes slower due to this optimization. If you find that this is the case, it is recommended to use a different, but isomorphic type to the builtin natural numbers. Erasable types A data type is considered erasable if it has a single constructor whose arguments are all erasable types, or functions into erasable types. The compilers will erase • calls to functions into erasable types • pattern matches on values of erasable type At the moment the compilers only have enough type information to erase calls of top-level functions that can be seen to return a value of erasable type without looking at the arguments of the call. In other words, a function call will not be erased if it calls a lambda bound variable, or the result is erasable for the given arguments, but not for others. Typical examples of erasable types are the equality type and the accessibility predicate used for well-founded recursion: data __ {a} {A : Set a} (x : A) : A → Set a where refl : x x data Acc {a} {A : Set a} (_<_ : A → A → Set a) (x : A) : Set a where acc : ( y → y < x → Acc _<_ y) → Acc _<_ x 122 Chapter 4. Tools Agda User Manual, Release 2.6.0 The erasure means that equality proofs will (mostly) be erased, and never looked at, and functions defined by wellfounded recursion will ignore the accessibility proof. 4.4 Emacs Mode 4.4.1 Introduction 4.4.2 Configuration If you want to you can customise the Emacs mode. Just start Emacs and type the following: M-x load-library RET agda2-mode RET M-x customize-group RET agda2 RET If you want some specific settings for the Emacs mode you can add them to agda2-mode-hook. For instance, if you do not want to use the Agda input method (for writing various symbols like →𝜋) you can add the following to your .emacs: (add-hook 'agda2-mode-hook '(lambda () ; If you do not want to use any input method: (deactivate-input-method) ; (In some versions of Emacs you should use ; inactivate-input-method instead of ; deactivate-input-method.) Note that, on some systems, the Emacs mode changes the default font of the current frame in order to enable many Unicode symbols to be displayed. This only works if the right fonts are available, though. If you want to turn off this feature, then you should customise the agda2-fontset-name variable. 4.4.3 Keybindings Notation for key combinations The following notation is used when describing key combinations: C-c means hitting the c key while pressing the Ctrl key. M-x means hitting the x key while pressing the Meta key, which is called Alt on many systems. Alternatively one can type Escape followed by x (in separate key strokes). RET is the Enter, Return or key. SPC is the space bar. Commands working with types can be prefixed with C-u to compute type without further normalisation and with C-u C-u to compute normalised types. Global commands C-c C-l Load file C-c C-x C-c Compile file C-c C-x C-q Quit, kill the Agda process 4.4. Emacs Mode 123 Agda User Manual, Release 2.6.0 C-c C-x C-r Kill and restart the Agda process C-c C-x C-a Abort a command C-c C-x C-d Remove goals and highlighting (deactivate) C-c C-x C-h Toggle display of hidden arguments C-c C-= Show constraints C-c C-s Solve constraints C-c C-? Show all goals C-c C-f Move to next goal (forward) C-c C-b Move to previous goal (backwards) C-c C-d Infer (deduce) type C-c C-o Module contents C-c C-z Search through definitions in scope C-c C-n Compute normal form C-u C-c C-n Compute normal form, ignoring abstract C-u C-u C-c C-n Compute and print normal form of show C-c C-x M-; Comment/uncomment rest of buffer C-c C-x C-s Switch to a different Agda version Commands in context of a goal Commands expecting input (for example which variable to case split) will either use the text inside the goal or ask the user for input. C-c C-SPC Give (fill goal) C-c C-r Refine. Partial give: makes new holes for missing arguments C-c C-a Automatic Proof Search (Auto) C-c C-c Case split C-c C-h Compute type of helper function and add type signature to kill ring (clipboard) C-c C-t Goal type C-c C-e Context (environment) C-c C-d Infer (deduce) type C-c C-, Goal type and context C-c C-. Goal type, context and inferred type C-c C-; Goal type, context and checked term C-c C-o Module contents C-c C-n Compute normal form C-u C-c C-n Compute normal form, ignoring abstract C-u C-u C-c C-n Compute and print normal form of show 124 Chapter 4. Tools Agda User Manual, Release 2.6.0 Other commands TAB Indent current line, cycles between points S-TAB Indent current line, cycles in opposite direction M-. Go to definition of identifier under point Middle mouse button Go to definition of identifier clicked on M-* Go back (Emacs < 25.1) M-, Go back (Emacs 25.1) 4.4.4 Unicode input How can I write Unicode characters using Emacs? The Agda Emacs mode comes with an input method for easily writing Unicode characters. Most Unicode character can be input by typing their corresponding TeX/LaTeX commands, eg. typing \lambda will input 𝜆. Some characters have key bindings which have not been taken from TeX/LaTeX (typing \bN results in being inserted, for instance), but all bindings start with \. To see all characters you can input using the Agda input method type M-x describe-input-method RET Agda or type M-x agda-input-show-translations RET RET (with some exceptions in certain versions of Emacs). If you know the Unicode name of a character you can input it using M-x ucs-insert RET (which supports tabcompletion) or C-x 8 RET. Example: Type C-x 8 RET not SPACE a SPACE sub TAB RET to insert the character “NOT A SUBSET OF” (). (The Agda input method has one drawback: if you make a mistake while typing the name of a character, then you need to start all over again. If you find this terribly annoying, then you can use Abbrev mode instead. However, note that Abbrev mode cannot be used in the minibuffer, which is used to give input to many Agda and Emacs commands.) The Agda input method can be customised via M-x customize-group RET agda-input. OK, but how can I find out what to type to get the . . . character? To find out how to input a specific character, eg from the standard library, position the cursor over the character and type M-x describe-char or C-u C-x =. For instance, for I get the following: character: (displayed as ) (codepoint 8759, #o21067, #x2237) preferred charset: unicode (Unicode (ISO10646)) code point in charset: 0x2237 script: symbol syntax: w which means: word category: .:Base, c:Chinese to input: type "\::" with Agda input method buffer code: #xE2 #x88 #xB7 file code: #xE2 #x88 #xB7 (encoded by coding system utf-8-unix) display: by this font (glyph code) x:-misc-fixed-medium-r-normal--20-200-75-75-c-100-iso10646-1 (#x2237) Character code properties: customize what to show name: PROPORTION (continues on next page) 4.4. Emacs Mode 125 Agda User Manual, Release 2.6.0 (continued from previous page) general-category: Sm (Symbol, Math) decomposition: (8759) ('') There are text properties here: fontified t Here it says that I can type \:: to get a . If there is no “to input” line, then you can add a key binding to the Agda input method by using M-x customize-variable RET agda-input-user-translations. Show me some commonly used characters Many common characters have a shorter input sequence than the corresponding TeX command: • Arrows: \r- for →. You can replace r with another direction: u, d, l. Eg. \d- for ↓. Replace - with = or == to get a double and triple arrows. • Greek letters can be input by \G followed by the first character of the letters Latin name. Eg. \Gl will input 𝜆 while \GL will input Λ. • Negation: you can get the negated form of many characters by appending n to the name. Eg. while \ni inputs , \nin will input . • Subscript and superscript: you can input subscript or superscript forms by prepending the character with \_ (subscript) or \^ (superscript). Note that not all characters have a subscript or superscript counterpart in Unicode. Some characters which were used in this documentation or which are commonly used in the standard library (sorted by hexadecimal code): Hex code 00ac 00d7 02e2 03bb 041f 0432 0435 0438 043c 0440 0442 1d62 2032 207f 2081 2082 2083 2084 2096 2098 2099 126 Character ¬ × 𝜆 1 2 3 4 Short key-binding \x \^s \Gl \_i \'1 \^n \_1 \_2 \_3 \_4 \_k \_m \_n TeX command \neg \times \lambda \prime Chapter 4. Tools Agda User Manual, Release 2.6.0 Hex code 2113 Hex code 2115 2191 2192 21a6 2200 2208 220b 220c 2218 2237 223c 2248 2261 2264 2284 228e 2294 22a2 22a4 22a5 266d 266f 27e8 27e9 Hex code 2983 2984 2985 2986 Hex code 2c7c Character (PDF TODO) Character ↑ → Short key-binding Short key-binding \bN \u \r\r-| \all \nin \o \:: \~ \~~ \== \<= \subn \u+ \lub \|- TeX command \ell TeX command \Bbb{N} \uparrow \to \mapsto \forall \in \ni \circ \sim \approx \equiv \le \uplus \vdash \top \bot \b \# \< \> Character (PDF TODO) (PDF TODO) (PDF TODO) (PDF TODO) Character Short key-binding \{{ \}} \(( \)) Short key-binding \_j TeX command TeX command 4.4.5 Highlight Clauses which do not hold definitionally (see Case trees) are highlighted in white smoke. 4.5 Literate Programming Agda supports a limited form of literate programming, i.e. code interspersed with prose, if the corresponding filename extension is used. 4.5. Literate Programming 127 Agda User Manual, Release 2.6.0 4.5.1 Literate TeX Files ending in .lagda or .lagda.tex are interpreted as literate TeX files. All code has to appear in code blocks: Ignored by Agda. \begin{code}[ignored by Agda] module Whatever where -- Agda code goes here \end{code} Text outside of code blocks is ignored, as well as text right after begin{code}, on the same line. If you provide a suitable definition for the code environment, then literate Agda files can double as LaTeX document sources. Example definition: \usepackage{fancyvrb} \DefineVerbatimEnvironment {code}{Verbatim} {} % Add fancy options here if you like. The LaTeX backend or the preprocessor lhs2TeX can also be used to produce tex code from literate Agda files. See Unicode and LaTeX for how to make LaTeX accept Agda files using the UTF-8 character encoding. 4.5.2 Literate reStructuredText Files ending in .lagda.rst are interpreted as literate reStructuredText files. Agda will parse code following a line ending in ::, as long as that line does not start with ..: This line is ordinary text, which is ignored by Agda. :: module Whatever where -- Agda code goes here Another non-code line. :: .. This line is also ignored reStructuredText source files can be turned into other formats such as HTML or LaTeX using Sphinx. • Code blocks inside an rST comment block will be type-checked by Agda, but not rendered. • Code blocks delimited by .. code-block:: but not type-checked by Agda. agda or .. code-block:: lagda will be rendered, • All lines inside a codeblock must be further indented than the first line of the code block. • Indentation must be consistent between code blocks. In other words, the file as a whole must be a valid Agda file if all the literate text is replaced by white space. 4.5.3 Literate Markdown Files ending in .lagda.md are interpreted as literate Markdown files. Code blocks start with ``` or ```agda on its own line, and end with ```, also on its own line: 128 Chapter 4. Tools Agda User Manual, Release 2.6.0 This line is ordinary text, which is ignored by Agda. ``` module Whatever where -- Agda code goes here ``` Here is another code block: ```agda data : Set where zero : suc : → ``` Markdown source files can be turned into many other formats such as HTML or LaTeX using PanDoc. • Code blocks which should be type-checked by Agda but should not be visible when the Markdown is rendered may be enclosed in HTML comment delimiters (). • Code blocks which should be ignored by Agda, but rendered in the final document may be indented by four spaces. • Note that inline code fragments are not supported due to the difficulty of interpreting their indentation level with respect to the rest of the file. 4.6 Generating HTML To generate highlighted, hyperlinked web pages from source code, run the following command in a shell: $ agda --html --html-dir={output directory} {root module} You can change the way in which the code is highlighted by providing your own CSS file instead of the default, included one (use the --css option). 4.6.1 Options --html-dir directory Changes the directory where the output is placed to directory. Default: html. --css URL The CSS file used by the HTML files (URL can be relative). 4.7 Generating LaTeX An experimental LaTeX-backend was added in Agda 2.3.2. It can be used as follows: $ agda --latex {file}.lagda $ cd latex $ {latex-compiler} {file}.tex where latex-compiler could be pdflatex, xelatex or lualatex, and file.lagda is a literate Agda TeX file. The source file must import the agda latex package by including the line \usepackage{agda}. Only the top-most module is processed, like with lhs2tex but unlike with the HTML-backend. If you want to process imported modules you have to call agda --latex manually on each of those modules. 4.6. Generating HTML 129 Agda User Manual, Release 2.6.0 The latex-backend checks if agda.sty is found by the latex environment. If it isn’t, a default agda.sty is copied from Agda’s data directory into the working directory (and thus made available to the latex environment). Colors, fonts, spacing etc can be modified by editing agda.sty and putting it somewhere where the latex environment can find it. 4.7.1 Unicode and LaTeX LaTeX does not by default understand the UTF-8 character encoding. You can tell LaTeX to treat the input as UTF-8 using the ucs package by inserting the following code in the preamble of your source file: \usepackage{ucs} \usepackage[utf8x]{inputenc} Unicode characters are translated to LaTeX commands, so e.g. the following packages might be needed. You may need more packages if you use more unicode characters: \usepackage{amssymb} \usepackage{bbm} \usepackage[greek,english]{babel} The ucs package does not contain definitions for all Unicode characters. If LaTeX complains about a missing definition for some character U+xxxx, then you can define it yourself: \DeclareUnicodeCharacter{"xxxx}{ } It may also be necessary to instruct LaTeX to use a specific font encoding. The autofe package (from the ucs bundle) tries to select the font encoding automatically: \usepackage{autofe} A complete LaTeX template can be found below. Note: LaTeX was never written with unicode in mind. Hacks like the ucs package makes it possible to use them, but for the best possible output consider using xelatex or lualatex instead. If you do, agda.sty is using the more complete XITS font by default. 4.7.2 Features Hiding code Code that you do not want to show up in the output can be hidden by giving the argument hide to the code block: \begin{code}[hide] -- the code here will not be part of the final document \end{code} Alignment Two or more spaces can be used to make the backend align code, as in the following example: 130 Chapter 4. Tools Agda User Manual, Release 2.6.0 \begin{code} data : Set where zero : suc : → _+_ : → → zero + n = n suc m + n = suc (m + n) \end{code} In more detail, the constraint on the indentation of the first token t on a line is determined as follows: • Let T be the set containing every previous token (in any code block) that is either the initial token on its line or preceded by at least one whitespace character. • Let S be the set containing all tokens in T that are not shadowed by other tokens in T. A token t1 is shadowed by t2 if t2 is further down than t1 and does not start to the right of t1 . • Let L be the set containing all tokens in S that start to the left of t, and E be the set containing all tokens in S that start in the same column as t. • The constraint is that t must be indented further than every token in L, and aligned with every token in E. Note that if any token in L or E belongs to a previous code block, then the constraint may not be satisfied unless (say) the AgdaAlign environment is used in an appropriate way. If custom settings are used, for instance if \AgdaIndent is redefined, then the constraint discussed above may not be satisfied. Examples: • Here C is indented further than B: postulate A B C : Set • Here C is not (necessarily) indented further than B, because X shadows B: postulate A B : Set X C : Set These rules are inspired by, but not identical to, the one used by lhs2TeX’s poly mode (see Section 8.4 of the manual for lhs2TeX version 1.17). Vertical space Code blocks are by default surrounded by vertical space. Use \AgdaNoSpaceAroundCode{} to avoid this vertical space, and \AgdaSpaceAroundCode{} to reenable it. Note that, if \AgdaNoSpaceAroundCode{} is used, then empty lines before or after a code block will not necessarily lead to empty lines in the generated document. However, empty lines inside the code block do (by default) lead to empty lines in the output. The height of such empty lines can be controlled by the length \AgdaEmptySkip, which by default is \baselineskip. 4.7. Generating LaTeX 131 Agda User Manual, Release 2.6.0 Breaking up code blocks Sometimes one might want to break up a code block into multiple pieces, but keep code in different blocks aligned with respect to each other. Then one can use the AgdaAlign environment. Example usage: \begin{AgdaAlign} \begin{code} code code (more code) \end{code} Explanation... \begin{code} aligned with "code" code (aligned with (more code)) \end{code} \end{AgdaAlign} Note that AgdaAlign environments should not be nested. Sometimes one might also want to hide code in the middle of a code block. This can be accomplished in the following way: \begin{AgdaAlign} \begin{code} visible \end{code} \begin{code}[hide] hidden \end{code} \begin{code} visible \end{code} \end{AgdaAlign} However, the result may be ugly: extra space is perhaps inserted around the code blocks. The AgdaSuppressSpace environment ensures that extra space is only inserted before the first code block, and after the last one (but not if \AgdaNoSpaceAroundCode{} is used). Example usage: \begin{AgdaAlign} \begin{code} code more code \end{code} Explanation... \begin{AgdaSuppressSpace} \begin{code} aligned with "code" aligned with "more code" \end{code} \begin{code}[hide] hidden code \end{code} \begin{code} also aligned with "more code" \end{code} \end{AgdaSuppressSpace} \end{AgdaAlign} 132 Chapter 4. Tools Agda User Manual, Release 2.6.0 Note that AgdaSuppressSpace environments should not be nested. There is also a combined environment, AgdaMultiCode, that combines the effects of AgdaAlign and AgdaSuppressSpace. Controlling the typesetting of individual tokens The typesetting of (certain) individual tokens can be controlled by redefining the \AgdaFormat command. Example: \usepackage{ifthen} % Insert extra space before some tokens. \DeclareRobustCommand{\AgdaFormat}[2]{% \ifthenelse{ \equal{#1}{} \OR \equal{#1}{} \OR \equal{#1}{} }{\ }{}#2} Note the use of \DeclareRobustCommand. The first argument to \AgdaFormat is the token, and the second argument the thing to be typeset. Hyperlinks (experimental) If the hyperref latex package is loaded before the agda package and the links option is passed to the agda package, then the agda package provides a function called \AgdaTarget. Identifiers which have been declared targets, by the user, will become clickable hyperlinks in the rest of the document. Here is a small example: \documentclass{article} \usepackage{hyperref} \usepackage[links]{agda} \begin{document} \AgdaTarget{} \AgdaTarget{zero} \begin{code} data : Set where zero : suc : → \end{code} See next page for how to define \AgdaFunction{two} (doesn't turn into a link because the target hasn't been defined yet). We could do it manually though; \hyperlink{two}{\AgdaDatatype{two}}. \newpage \AgdaTarget{two} \hypertarget{two}{} \begin{code} two : two = suc (suc zero) \end{code} \AgdaInductiveConstructor{zero} is of type \AgdaDatatype{}. \AgdaInductiveConstructor{suc} has not been defined to be a target so it doesn't turn into a link. (continues on next page) 4.7. Generating LaTeX 133 Agda User Manual, Release 2.6.0 (continued from previous page) \newpage Now that the target for \AgdaFunction{two} has been defined the link works automatically. \begin{code} data Bool : Set where true false : Bool \end{code} The AgdaTarget command takes a list as input, enabling several targets to be specified as follows: \AgdaTarget{if, then, else, if\_then\_else\_} \begin{code} if_then_else_ : {A : Set} → Bool → A → A → A if true then t else f = t if false then t else f = f \end{code} \newpage Mixfix identifier need their underscores escaped: \AgdaFunction{if\_then\_else\_}. \end{document} The borders around the links can be suppressed using hyperref’s hidelinks option: \usepackage[hidelinks]{hyperref} Warning: The current approach to links does not keep track of scoping or types, and hence overloaded names might create links which point to the wrong place. Therefore it is recommended to not overload names when using the links option at the moment. This might get fixed in the future. Typesetting inline code The backend only typesets code inside code blocks; inline code has to be typeset manually, e.g.: Below we postulate a set called \AgdaDatatype{apa}. \begin{code} postulate apa : Set \end{code} You can find all the commands used by the backend (and which you can use manually) in the latex/agda.sty file. If you are doing a lot of manual typesetting, then you might want to introduce shorter command names, e.g.: \newcommand{\D}{\AgdaDatatype} \newcommand{\F}{\AgdaFunction} etc. Long names were chosen by default to avoid name clashes. 134 Chapter 4. Tools Agda User Manual, Release 2.6.0 Semi-automatically typesetting inline code (experimental) Since Agda version 2.4.2 there is experimental support for semi-automatically typesetting code inside text, using the references option. After loading the agda package with this option, inline Agda snippets will be typeset in the same way as code blocks — after post-processing — if referenced using the \AgdaRef command. Only the current module is used; should you need to reference identifiers in other modules, then you need to specify which other module manually by using \AgdaRef[module]{identifier}. In order for the snippets to be typeset correctly, they need to be post-processed by the postprocess-latex.pl script from the Agda data directory. You can copy it into the current directory by issuing the command $ cp $(dirname $(dirname $(agda-mode locate)))/postprocess-latex.pl . In order to generate a PDF, you can then do the following: $ $ $ $ $ agda --latex {file}.lagda cd latex/ perl ../postprocess-latex.pl {file}.tex > {file}.processed mv {file}.processed {file}.tex xelatex {file}.tex Here is a full example, consisting of a Literate Agda file Example.lagda and a makefile Makefile. Listing 1: Example.lagda \documentclass{article} \usepackage[references]{agda} \begin{document} Here we postulate \AgdaRef{apa}. \begin{code} postulate apa : Set \end{code} \end{document} Listing 2: Makefile AGDA=agda AFLAGS=-i. --latex SOURCE=Example POSTPROCESS=postprocess-latex.pl LATEX=latexmk -pdf -use-make -xelatex all: $(AGDA) $(AFLAGS) $(SOURCE).lagda cd latex/ && \ perl ../$(POSTPROCESS) $(SOURCE).tex > $(SOURCE).processed && \ mv $(SOURCE).processed $(SOURCE).tex && \ $(LATEX) $(SOURCE).tex && \ mv $(SOURCE).pdf .. See Issue #1054 on the bug tracker for implementation details. 4.7. Generating LaTeX 135 Agda User Manual, Release 2.6.0 Warning: Overloading identifiers should be avoided. If multiple identifiers with the same name exist, then AgdaRef will typeset according to the first one it finds. Emulating %format rules The LaTeX-backend has no feature analogue to lhs2TeX’s %format rules, however most systems come with sed which can be used to achieve similar goals. Given a file called, for example, replace.sed, containing: # Super- and subscripts. #s/\\textasciicircum\([^\}]*\)\([^\}]*\)/\$\^\\AgdaFontStyle\{\\scriptscriptstyle \1\} ˓→\_\\AgdaFontStyle\{\\scriptscriptstyle \2\}\$/g s/\\textasciicircum\([^\}]*\)/\{\^\\AgdaFontStyle\{\\scriptscriptstyle \1\}\}/g #s/\([^\}]*\)/\$\_\\AgdaFontStyle\{\\scriptscriptstyle \1\}\$/g # Σ[ x X ] into (x : X) × s/\\AgdaRecord{Σ\[} \(.*\) \\AgdaRecord{} \(.*\) \\AgdaRecord{]}/\\AgdaSymbol\{(\}\1 ˓→\\AgdaSymbol\{:\} \2\\AgdaSymbol\{)\} \\AgdaFunction\{×\}/g # Bind, Kleisli extension and fmap. #s/>>=/\$\\mathbin\{>\\!\\!\\!>\\mkern-6.7mu=\}\$/g s/>>=/\\mathbin\{>\\!\\!\\!>\\mkern-6.7mu=\}/g #s/>>/\$\\mathbin\{>\\!\\!\\!>}\$/g #s/=</\$\\mathop\{<\\!\\!\\!\\$\\!\\!\\!>\}\$/g s/<\\$>/\\mathop\{<\\!\\!\\!\\$\\!\\!\\!>\}/g # Append. s/++/+\\!+/g # Comments. #s/AgdaComment{\-\-/AgdaComment\{\-\\!\-/g We can postprocess the tex output as follows: $ sed -f replace.sed {file}.tex > {file}.sedded $ mv {file}.sedded {file}.tex Note that the above sed file assumes the use of {xe|lua}latex where code is in math mode rather than text mode (as is the case when using the pdflatex compiler). The commented out variants of the substitution rules are their pdflatex equivalents. The substitution rules dealing with super- and subscripts lets us write apa^bepa in the code for things we want superscripted in the output (\undertie does the same for subscripts). Including Agda code into a larger LaTeX document Sometimes you might want to include a bit of code without necessarily making the whole document a literate Agda file. Here is how to do this in the context of a beamer presentation, but the method should work similarly also for other documents. Given two files Presentation.tex and Code.lagda as follows: 136 Chapter 4. Tools Agda User Manual, Release 2.6.0 Listing 3: Presentation.tex \documentclass{beamer} % When using XeLaTeX, the following should be used instead: % \documentclass[xetex]{beamer} % \usefonttheme[onlymath]{serif} \usepackage{latex/agda} \usepackage{catchfilebetweentags} \begin{document} \begin{frame}\frametitle{Some Agda code} \begin{itemize} \item The natural numbers \end{itemize} \ExecuteMetaData[latex/Code.tex]{nat} \begin{itemize} \item Addition (\AgdaFunction{\_+\_}) \end{itemize} \ExecuteMetaData[latex/Code.tex]{plus} \end{frame} \end{document} Listing 4: Code.lagda %<*nat> \begin{code} data : Set where zero : suc : (n : ) → \end{code} % %<*plus> \begin{code} _+_ : → → zero + n = n suc m + n = suc (m + n) \end{code} % we can use pdflatex to compile a presentation as follows: $ agda --latex Code.lagda $ latexmk -pdf -use-make Presentation.tex If you are using a lot of unicode it might be more convenient to use xelatex instead. See comments about xelatex in Presentation.tex and compile as follows: $ agda --latex Code.lagda $ latexmk -xelatex -pdf -use-make Presentation.tex 4.7. Generating LaTeX 137 Agda User Manual, Release 2.6.0 The \ExecuteMetaData command is part of the catchfilebetweentags package. Also see the following thread on the mailing list for other methods of including Agda code into a presentation. 4.7.3 Options The following command-line options change the behaviour of the LaTeX backend: --latex-dir=directory Changes the output directory where agda.sty and the output .tex are placed to directory. Default: latex. --only-scope-checking Generates highlighting without typechecking the file. See Quicker generation without typechecking. --count-clusters Count extended grapheme clusters when generating LaTeX code. This option can be given in OPTIONS pragmas. See Counting Extended Grapheme Clusters. The following options can be given when loading agda.sty by using usepackage[options]agda: bw Colour scheme which highlights in black and white. conor Colour scheme similar to the colours used in Epigram1. nofontsetup Instructs the package to not select any fonts, and to not change the font encoding. noinputencodingsetup Instructs the package to not change the input encoding, and to not load the ucs package. references Enables inline typesetting of referenced code. links Enables hyperlink support. 4.7.4 Counting Extended Grapheme Clusters The alignment feature regards the string +, containing + and a combining character, as having length two. However, it seems more reasonable to treat it as having length one, as it occupies a single column, if displayed “properly” using a monospace font. The flag --count-clusters is an attempt at fixing this. When this flag is enabled the backend counts “extended grapheme clusters” rather than code points. Note that this fix is not perfect: a single extended grapheme cluster might be displayed in different ways by different programs, and might, in some cases, occupy more than one column. Here are some examples of extended grapheme clusters, all of which are treated as a single character by the alignment algorithm: + O Note also that the layout machinery does not count extended grapheme clusters, but code points. The following code is syntactically correct, but if --count-clusters is used, then the LaTeX backend does not align the two field keywords: record + : Set1 where field A : Set field B : Set The --count-clusters flag is not enabled in all builds of Agda, because the implementation depends on the ICU library, the installation of which could cause extra trouble for some users. The presence of this flag is controlled by the Cabal flag enable-cluster-counting. 138 Chapter 4. Tools Agda User Manual, Release 2.6.0 4.7.5 Quicker generation without typechecking A faster variant of the backend is available by invoking QuickLaTeX from the Emacs mode, or using agda --latex --only-scope-checking. When this variant of the backend is used the top-level module is not type-checked, only scope-checked. Note that this can affect the generated document. For instance, scope-checking does not resolve overloaded constructors. If the module has already been type-checked successfully, then this information is reused; in this case QuickLaTeX behaves like the regular LaTeX backend. 4.7.6 Known issues The unicode-math package and older versions of the polytable package (still in the Debian stable) are incompatible, which can result in errors in generated latex code. The workaround is to download a more up to date version of polytable and either put it with the generated files, or install it globally. 4.7.7 Complete LaTeX Template for Literate Agda with Unicode This template has been tested under Debian and Ubuntu with TexLive, but should also work in other distributions. For xelatex or lualatex, only \usepackage{agda} should be needed. \documentclass{article} \usepackage{agda} % % % % The following packages are needed because unicode is translated (using the next set of packages) to latex commands. You may need more packages if you use more unicode characters: \usepackage{amssymb} \usepackage{bbm} \usepackage[greek,english]{babel} % This handles the translation of unicode to latex: \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{autofe} % Some characters that are not automatically defined % (you figure out by the latex compilation errors you get), % and you need to define: \DeclareUnicodeCharacter{8988}{\ensuremath{\ulcorner}} \DeclareUnicodeCharacter{8989}{\ensuremath{\urcorner}} \DeclareUnicodeCharacter{8803}{\ensuremath{\overline{\equiv}}} % Add more as you need them (shouldn't happen often). \begin{document} \begin{code} -- your Agda code goes here \end{code} (continues on next page) 4.7. Generating LaTeX 139 Agda User Manual, Release 2.6.0 (continued from previous page) \end{document} 4.8 Library Management Agda has a simple package management system to support working with multiple libraries in different locations. The central concept is that of a library. 4.8.1 Example: Using the standard library Before we go into details, here is some quick information for the impatient on how to tell Agda about the location of the standard library, using the library management system. Let’s assume you have downloaded the standard library into a directory which we will refer to by AGDA_STDLIB (as an absolute path). A library file standard-library.agda-lib should exist in this directory, with the following content: name: standard-library include: src To use the standard library by default in your Agda projects, you have to do two things: 1. Create a file AGDA_DIR/libraries with the following content: AGDA_STDLIB/standard-library.agda-lib (Of course, replace AGDA_STDLIB by the actual path.) The AGDA_DIR defaults to ~/.agda on unix-like systems and C:\Users\USERNAME\AppData\Roaming\agda or similar on Windows. (More on AGDA_DIR below.) Remark: The libraries file informs Agda about the libraries you want it to know about. 2. Create a file AGDA_DIR/defaults with the following content: standard-library Remark: The defaults file informs Agda which of the libraries pointed to by libraries should be used by default (i.e. in the default include path). That’s the short version, if you want to know more, read on! 4.8.2 Library files A library consists of • a name • a set of dependencies 140 Chapter 4. Tools Agda User Manual, Release 2.6.0 • a set of include paths Libraries are defined in .agda-lib files with the following syntax: name: LIBRARY-NAME depend: LIB1 LIB2 LIB3 LIB4 include: PATH1 PATH2 PATH3 -- Comment Dependencies are library names, not paths to .agda-lib files, and include paths are relative to the location of the library-file. Each of the three fields is optional. Naturally, unnamed libraries cannot be depended upon. But dropping the name is possible if the library file only serves to list include pathes and/or dependencies of the current project. 4.8.3 Installing libraries To be found by Agda a library file has to be listed (with its full path) in a libraries file • AGDA_DIR/libraries-VERSION, or if that doesn’t exist • AGDA_DIR/libraries where VERSION is the Agda version (for instance 2.5.1). The AGDA_DIR defaults to ~/.agda on unix-like systems and C:\Users\USERNAME\AppData\Roaming\agda or similar on Windows, and can be overridden by setting the AGDA_DIR environment variable. Each line of the libraries file shall be the absolute file system path to the root of a library. Environment variables in the paths (of the form $VAR or ${VAR}) are expanded. The location of the libraries file used can be overridden using the --library-file=FILE command line option. You can find out the precise location of the libraries file by calling agda -l fjdsk Dummy.agda at the command line and looking at the error message (assuming you don’t have a library called fjdsk installed). Note that if you want to install a library so that it is used by default, it must also be listed in the defaults file (details below). 4.8.4 Using a library There are three ways a library gets used: • You supply the --library=LIB (or -l LIB) option to Agda. This is equivalent to adding a -iPATH for each of the include paths of LIB and its (transitive) dependencies. • No explicit --library flag is given, and the current project root (of the Agda file that is being loaded) or one of its parent directories contains an .agda-lib file defining a library LIB. This library is used as if a --library=LIB option had been given, except that it is not necessary for the library to be listed in the AGDA_DIR/libraries file. • No explicit --library flag, and no .agda-lib file in the project root. In this case the file AGDA_DIR/ defaults is read and all libraries listed are added to the path. The defaults file should contain a list of library names, each on a separate line. In this case the current directory is also added to the path. To disable default libraries, you can give the flag --no-default-libraries. To disable using libraries altogether, use the --no-libraries flag. 4.8. Library Management 141 Agda User Manual, Release 2.6.0 4.8.5 Default libraries If you want to usually use a variety of libraries, it is simplest to list them all in the AGDA_DIR/defaults file. Each line of the defaults file shall be the name of a library resolvable using the paths listed in the libraries file. For example, standard-library library2 library3 where of course library2 and library3 are the libraries you commonly use. While it is safe to list all your libraries in library, be aware that listing libraries with name clashes in defaults can lead to difficulties, and should be done with care (i.e. avoid it unless you really must). 4.8.6 Version numbers Library names can end with a version number (for instance, mylib-1.2.3). When resolving a library name (given in a --library flag, or listed as a default library or library dependency) the following rules are followed: • If you don’t give a version number, any version will do. • If you give a version number an exact match is required. • When there are multiple matches an exact match is preferred, and otherwise the latest matching version is chosen. For example, suppose you have the following libraries installed: mylib, mylib-1.0, otherlib-2.1, and otherlib-2.3. In this case, aside from the exact matches you can also say --library=otherlib to get otherlib-2.3. 4.8.7 Upgrading If you are upgrading from a pre 2.5 version of Agda, be aware that you may have remnants of the previous library management system in your preferences. In particular, if you get warnings about agda2-include-dirs, you will need to find where this is defined. This may be buried deep in .el files, whose location is both operating system and emacs version dependant. 142 Chapter 4. Tools CHAPTER 5 Contribute See also the HACKING file in the root of the agda repo. 5.1 Documentation Documentation is written in reStructuredText format. The Agda documentation is shipped together with the main Agda repository in the doc/user-manual subdirectory. The content of this directory is automatically published to https://agda.readthedocs.io. 5.1.1 Rendering documentation locally • To build the user manual locally, you need to install the following dependencies: – Python 3.3 – Sphinx and sphinx-rtd-theme pip install –user -r doc/user-manual/requirements.txt Note that the --user option puts the Sphinx binaries in $HOME/.local/bin. – LaTeX – dvipng To see the list of available targets, execute make help in doc/user-manual. E.g., call make html to build the documentation in html format. 5.1.2 Type-checking code examples You can include code examples in your documentation. 143 Agda User Manual, Release 2.6.0 If your give the documentation file the extension .lagda.rst, Agda will recognise it as an Agda file and type-check it. Tip: If you edit .lagda.rst documentation files in Emacs, you can use Agda’s interactive mode to write your code examples. Run M-x agda2-mode to switch to Agda mode, and M-x rst-mode to switch back to rST mode. You can check that all the examples in the manual are type-correct by running make user-manual-test from the root directory. This check will be run as part of the continuous integration build. Warning: Remember to run fix-agda-whitespace to remove trailing whitespace before submitting the documentation to the repository. 5.1.3 Syntax for code examples The syntax for embedding code examples depends on: 1. Whether the code example should be visible to the reader of the documentation. 2. Whether the code example contains valid Agda code (which should be type-checked). Visible, checked code examples This is code that the user will see, and that will be also checked for correctness by Agda. Ideally, all code in the documentation should be of this form: both visible and valid. It can appear stand-alone: :: data Bool : Set where true false : Bool Or at the end of a paragraph:: data Bool : Set where true false : Bool Here ends the code fragment. Result: It can appear stand-alone: data Bool : Set where true false : Bool Or at the end of a paragraph: data Bool : Set where true false : Bool Here ends the code fragment. 144 Chapter 5. Contribute Agda User Manual, Release 2.6.0 Warning: Remember to always leave a blank like after the ::. Otherwise, the code will be checked by Agda, but it will appear as regular paragraph text in the documentation. Visible, unchecked code examples This is code that the reader will see, but will not be checked by Agda. It is useful for examples of incorrect code, program output, or code in languages different from Agda. .. code-block:: agda -- This is not a valid definition 𝜔 : a → a 𝜔 x = x .. code-block:: haskell -- This is haskell code data Bool = True | False Result: -- This is not a valid definition 𝜔 : a → a 𝜔 x = x -- This is haskell code data Bool = True | False Invisible, checked code examples This is code that is not shown to the reader, but which is used to typecheck the code that is actually displayed. This might be definitions that are well known enough that do not need to be shown again. .. :: data Nat : Set where zero : Nat suc : Nat → Nat :: add : Nat → Nat → Nat add zero y = y add (suc x) y = suc (add x y) Result: 5.1. Documentation 145 Agda User Manual, Release 2.6.0 add : Nat → Nat → Nat add zero y = y add (suc x) y = suc (add x y) File structure Documentation literate files (.lagda.*) are type-checked as whole Agda files, as if all literate text was replaced by whitespace. Thus, indentation is interpreted globally. Namespacing In the documentation, files are typechecked starting from the doc/user-manual/ root. For example, the file doc/usermanual/language/data-types.lagda.rst should start with a hidden code block declaring the name of the module as language.data-types: .. :: module language.data-types where Scoping Sometimes you will want to use the same name in different places in the same documentation file. You can do this by using hidden module declarations to isolate the definitions from the rest of the file. .. :: module scoped-1 where :: foo : Nat foo = 42 .. :: module scoped-2 where :: foo : Nat foo = 66 Result: foo : Nat foo = 42 146 Chapter 5. Contribute CHAPTER 6 The Agda License Copyright (c) 2005-2017 remains with the authors. Agda 2 was originally written by Ulf Norell, partially based on code from Agda 1 by Catarina Coquand and Makoto Takeyama, and from Agdalight by Ulf Norell and Andreas Abel. Agda 2 is currently actively developed mainly by Andreas Abel, Guillaume Allais, Jesper Cockx, Nils Anders Danielsson, Philipp Hausmann, Fredrik Nordvall Forsberg, Ulf Norell, Víctor López Juan, Andrés Sicard-Ramírez, and Andrea Vezzosi. Further, Agda 2 has received contributions by, amongst others, Stevan Andjelkovic, Marcin Benke, Jean-Philippe Bernardy, Guillaume Brunerie, James Chapman, Dominique Devriese, Péter Diviánszki, Olle Fredriksson, Adam Gundry, Daniel Gustafsson, Kuen-Bang Hou (favonia), Patrik Jansson, Alan Jeffrey, Wolfram Kahl, Wen Kokke, Fredrik Lindblad, Francesco Mazzoli, Stefan Monnier, Darin Morrison, Guilhem Moulin, Nicolas Pouillard, Nobuo Yamashita, Christian Sattler, and Makoto Takeyama. The full list of contributors is available at https://github.com/ agda/agda/graphs/contributors Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. The file src/full/Agda/Utils/Parser/ReadP.hs is Copyright (c) The University of Glasgow 2002 and is licensed under a BSD-like license as follows: Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 147 Agda User Manual, Release 2.6.0 • Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. • Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. • Neither name of the University nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE UNIVERSITY COURT OF THE UNIVERSITY OF GLASGOW AND THE CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE UNIVERSITY COURT OF THE UNIVERSITY OF GLASGOW OR THE CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. The file src/full/Agda/Utils/Maybe/Strict.hs (and the following license text?) uses the following license: Copyright (c) Roman Leshchinskiy 2006-2007 Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. Neither the name of the author nor the names of his contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHORS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. 148 Chapter 6. The Agda License CHAPTER 7 The Agda Team Authors: • Ulf Norell • Andreas Abel • Nils Anders Danielsson • Makoto Takeyama • Catarina Coquand Contributors (alphabetically sorted): • Stevan Andjelkovic • Marcin Benke • Jean-Philippe Bernardy • James Chapman • Jesper Cockx • Dominique Devriese • Peter Divanski • Fredrik Nordvall Forsberg • Olle Fredriksson • Daniel Gustafsson • Philipp Hausmann • Patrik Jansson • Alan Jeffrey • Wolfram Kahl • Fredrik Lindblad 149 Agda User Manual, Release 2.6.0 • Francesco Mazzoli • Stefan Monnier • Darin Morrison • Guilhem Moulin • Nicolas Pouillard • Andrés Sicard-Ramírez • Andrea Vezzosi • and many more 150 Chapter 7. The Agda Team CHAPTER 8 Indices and tables • genindex • search 151 Agda User Manual, Release 2.6.0 152 Chapter 8. Indices and tables Bibliography [McBride2004] C. McBride and J. McKinna. The view from the left. Journal of Functional Programming, 2004. http://strictlypositive.org/vfl.pdf. 153

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