0736219 · Idris-dev · git.iko.soy

docs/_static/theme_overrides.cssnew file+13

compare with master

unifiedsplit

@@ -0,0 +1,13 @@

1+/* override table width restrictions */

2+@media screen and (min-width: 767px) {

3+

4+ .wy-table-responsive table td {

5+ /* !important prevents the common CSS stylesheets from overriding

6+ this as on RTD they are loaded after this stylesheet */

7+ white-space: normal !important;

8+ }

9+

10+ .wy-table-responsive {

11+ overflow: visible !important;

12+ }

13+}

docs/conf.py+7−1

@@ -51,7 +51,7 @@ master_doc = 'index'

51

52# General information about the project.

53project = u'Idris'

54−copyright = u'2017, The Idris Community'

54+copyright = u'2017-2019, The Idris Community'

55author = u'The Idris Community'

56

57# The version info for the project you're documenting, acts as replacement for

@@ -151,6 +151,12 @@ if not on_rtd: # only import and set the theme if we're building docs locally

151# so a file named "default.css" will overwrite the builtin "default.css".

152html_static_path = ['_static']

153

154+html_context = {

155+ 'css_files': [

156+ '_static/theme_overrides.css', # override wide tables in RTD theme

157+ ],

158+ }

159+

154# Add any extra paths that contain custom files (such as robots.txt or

160# Add any extra paths that contain custom files (such as robots.txt or

155# .htaccess) here, relative to this directory. These files are copied

161# .htaccess) here, relative to this directory. These files are copied

156# directly to the root of the documentation.

162# directly to the root of the documentation.

docs/image/README.mdnew file+16

@@ -0,0 +1,16 @@

1+Many of the images in this directory have two files with the same name but a different extension:

2+

3+* The SVG file, being vector graphics, is best for drawing and modifying.

4+* The PNG file, being raster graphics, gives more consistent rendering.

5+

6+The image has therefore been generated as an SVG file and then exported to the PNG file.

7+

8+So, if you need to modify an image, I suggest the following sequence:

9+

10+* Edit the SVG file using a suitable editor (I use Inkscape).

11+* Export the changes to the PNG file, overwriting the old version.

12+* upload both files.

13+

14+Note, when exporting in Inkscape:

15+* Select everything first, this stops an enormous boundary being exported.

16+* After selecting the file to export to and clicking OK, don't forget to click on export button as well.

docs/image/attack.pngnew file

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docs/proofs/elabReflection.rstnew file+60

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1+Extending Idris using Elaborator Reflection

2+===========================================

3+

4+Idris provides a mechanism to modify the language without having to recompile Idris itself. We can think of this in terms of metaprogramming or domain specific languages or just building in new capabilities.

5+

6+In order to extend the language we need to know something about how Idris is compiled. This page explains only what is needed to customise the elaboration. For more information about the compiler's implementation see `Edwin Brady's 2013 paper`_ and for customising the elaboration process see `Elaborator reflection: extending Idris in Idris`_ and `David Christiansen's PhD thesis`_.

7+

8+Compilation of Idris proceeds through a number of stages.

9+

10+- First, Idris is desugared by inserting placeholders for terms to be guessed by the compiler and replacing certain syntactic forms, such as do-notation, with the functions that implement them.

11+- Then, this desugared Idris is translated into a much simpler core language, called TT. This translation process is called elaboration.

12+- Finally, TT is type checked a second time to rule out errors, and then compiled into the target language.

13+

14+.. image:: ../image/idrisTopLevel.png

15+ :width: 484px

16+ :height: 147px

17+ :alt: diagram illustrating these stages of Idris compilation

18+

19+TT is a core language which is syntactically very simple. This makes it easy for computers to process but very verbose and hard for humans to read. The Idris elaborator is written in Haskell using an elaboration library that was inspired by the tactics in interactive proof assistants such as Coq.

20+

21+.. list-table::

22+

23+ * - There are some similarities with a proof assistant but in Idris the elaborator is an interpreter of Idris source in the elaboration monad, where each syntactic construct of Idris is interpreted as a sequence of tactics.

24+ - .. image:: ../image/compareToProofAssist.png

25+ :width: 206px

26+ :height: 112px

27+ :alt: diagram comparing elaboration with proof assistant

28+

29+The primitives in the elaboration library are not just useful for the implementors of Idris itself. They can also be used by authors of extensions to the compiler, using a mechanism known as elaborator reflection.

30+During elaboration TT (Raw) structure contains:

31+

32+- holes - placeholders for terms that have not yet been filled in.

33+- guesses - similar to let bindings, except with no reduction rule, so that elaboration programs can control the precise shape of terms that are under construction.

34+

35+For more information about holes and guesses see `Dependently Typed Functional Programs and their Proofs by McBride 1999`_.

36+

37+The following diagram is intended to illustrate a high level view of the tactics and how this eventually results in the TT language being generated. It is not necessary to understand the details at this stage. The intention is to help build up some intuition so that, when we get into the details, we can recognise how this fits into the big picture.

38+

39+.. image:: ../image/elabOverview.png

40+ :width: 410px

41+ :height: 282px

42+ :alt: diagram illustrating overview of TT language being generated from tactics.

43+

44+As already mentioned the TT core language is kept syntactically very simple, for instance, here are the binders in TT with corresponding code and logic type validity rules:

45+

46+.. list-table::

47+

48+ * - This diagram illustrates the basis of the compilation process in logic (in this case for binders). It is not necessary to be an expert logician to understand elaborator reflection. However, when learning about tactics, they may appear arbitrary without knowing some theory. For more information about this see `Edwin Brady's 2013 paper`_.

49+ - .. image:: ../image/binders.png

50+ :width: 310px

51+ :height: 203px

52+ :alt: diagram illustrating basis of code in logic

53+

54+.. target-notes::

55+.. _`Edwin Brady's 2013 paper`: https://eb.host.cs.st-andrews.ac.uk/drafts/impldtp.pdf

56+.. _`Elaborator reflection: extending Idris in Idris`: https://dl.acm.org/citation.cfm?doid=2951913.2951932

57+.. _`David Christiansen's PhD thesis`: https://davidchristiansen.dk/david-christiansen-phd.pdf

58+.. _`Dependently Typed Functional Programs and their Proofs by McBride 1999`: https://www.era.lib.ed.ac.uk/handle/1842/374

59+

60+

docs/proofs/example1.rstnew file+192

@@ -0,0 +1,192 @@

1+Elaborator Reflection - Identity Example

2+========================================

3+

4+.. list-table::

5+

6+ * - This example of elaborator reflection steps through this metaprogram that generates the identity function:

7+ - .. code-block:: idris

8+

9+ %language ElabReflection

10+

11+ idNat : Nat -> Nat

12+ idNat = %runElab (do intro `{{x}}

13+ fill (Var `{{x}})

14+ solve)

15+

16+.. list-table::

17+ :widths: 200 100

18+

19+ * - At the beginning of executing the elaboration script, the initial state consists of a single hole of type Nat -> Nat.

20+

21+ As a first approximation, the state consists of a term with holes in it, an indicator of which hole is focused, a queue of the next holes to focus on, and miscellaneous information like a source of fresh names. The intro tactic modifies this state, replacing the focused hole with a lambda and focusing on the lambda's body.

22+ - .. image:: ../image/tree.png

23+ :width: 119px

24+ :height: 109px

25+

26+The following is a walkthough looking at the state after each tactic:

27+

28+.. list-table::

29+

30+ * - Start with the type signature like this:

31+ - .. code-block:: idris

32+

33+ %language ElabReflection

34+

35+ idNat : Nat -> Nat

36+ idNat = %runElab (do

37+

38+ * - In order to investigate how the program works this table shows the proof state at each stage as the tactics are applied. So here is the proof state at the start:

39+ - .. image:: ../image/elabProofStateEx1_1.png

40+ :width: 310px

41+ :height: 115px

42+

43+ * - This table shows the hole types and what they depend on. The aim is to illustrate the types by analogy with proofs, as a line with the premises above it and the conclusion below it.

44+ - .. image:: ../image/elabLogicEx1_1.png

45+ :width: 277px

46+ :height: 15px

47+

48+ * - The term is:

49+ - ?{hole_0} ≈ ? {hole_2} . {hole_2} . {hole_0}

50+

51+ * - It is possible to read the state from the script by calling getEnv, getGoal and getHoles.

52+

53+ - The output of these calls contain structures with TT code. To show the results I hacked this: `my code`_. TT code is not really designed to be readable by humans, all the names are fully expanded, everything has a type down to universes (type of types). This is shown here to illustrate the information available.

54+

55+ .. code-block:: idris

56+

57+ getEnv=[]

58+

59+ getGoal=(hole_2, __pi_arg:(Nat.["Nat", "Prelude"]:{

60+ type constructor tag=8 number=0}.Type:U=(20:./Prelude/Nat.idr)->.

61+ {name ref{type constructor tag=8 number=0}Nat.["Nat","Prelude"]:

62+ Type:U=(20:./Prelude/Nat.idr)

63+ })

64+ })

65+

66+ getHoles=[{hole_2},{hole_0}]

67+

68+ * - getGuess

69+ - error no guess

70+

71+ * - Introduce a lambda binding around the current hole and focus on the body.

72+ - intro \`{{x}}

73+

74+ * - The state now looks like this:

75+ - .. image:: ../image/elabProofStateEx1_2.png

76+ :width: 312px

77+ :height: 84px

78+

79+ * - The hole types now looks like this:

80+ - .. image:: ../image/elabLogicEx1_2.png

81+ :width: 279px

82+ :height: 26px

83+

84+ * - The term now looks like this:

85+ - ?{hole_0} ≈ λ x . ? {hole_2} . {hole_2} . {hole_0}

86+

87+ * - Again we can check the state by calling getEnv, getGoal and getHoles: see `my code`_

88+

89+ - .. code-block:: idris

90+

91+ getEnv=[(x, {λ (Nat.["Nat", "Prelude"]:{

92+ type constructor tag=8 number=0}).

93+ Type:U=(20:./Prelude/Nat.idr)

94+ })]

95+

96+ getGoal=(hole_2, {name ref{type constructor tag=8 number=0}

97+ Nat.["Nat","Prelude"]:Type:U=(20:./Prelude/Nat.idr)

98+ })

99+

100+ getHoles=[{hole_2},{hole_0}]

101+

102+ * - getGuess

103+ - error no guess

104+

105+ * - Place a term into a hole, unifying its type

106+ - fill (Var \`{{x}})

107+

108+ * - The state still looks like this:

109+ - .. image:: ../image/elabProofStateEx1_3.png

110+ :width: 312px

111+ :height: 57px

112+

113+ * - The hole types now looks like this:

114+ - .. image:: ../image/elabLogicEx1_3.png

115+ :width: 290px

116+ :height: 26px

117+

118+ * - The term now looks like this:

119+ - ?{hole_0} ≈ λ x . ?{hole_2} ≈ x . {hole_2} . {hole_0}

120+

121+ * - Again we can check the state by calling getEnv, getGoal and getHoles: see `my code`_

122+

123+ - .. code-block:: idris

124+

125+ getEnv=[(x, {λ (Nat.["Nat", "Prelude"]:

126+ {type constructor tag=8 number=0}).

127+ Type:U=(20:./Prelude/Nat.idr)

128+ })]

129+

130+ getGoal=(hole_2, {name ref{type constructor tag=8 number=0}

131+ Nat.["Nat","Prelude"]:Type:U=(20:./Prelude/Nat.idr)

132+ })

133+

134+ getHoles=[{hole_2}, {hole_0}]

135+

136+ * - getGuess

137+ - .. code-block:: idris

138+

139+ {name ref bound x:

140+ {name ref{type constructor tag=8 number=0}

141+ Nat.["Nat","Prelude"]:Type:U=(20:./Prelude/Nat.idr)

142+ }

143+ }

144+

145+ * - Substitute a guess into a hole.

146+ - solve

147+

148+ * - The hole types now looks like this:

149+ - .. image:: ../image/elabLogicEx1_4.png

150+ :width: 131px

151+ :height: 14px

152+

153+ * - The term now looks like this:

154+ - ?{hole_0} ≈ λ x . x . {hole_0}

155+

156+ * - getEnv

157+

158+ getGoal

159+

160+ getHoles

161+

162+ - .. code-block:: idris

163+

164+ getEnv=[]

165+

166+ getGoal=(hole_0, __pi_arg:(Nat.["Nat", "Prelude"]:{

167+ type constructor tag=8 number=0}.

168+ Type:U=(20:./Prelude/Nat.idr)

169+ ->.{name ref

170+ {type constructor tag=8 number=0}

171+ Nat.["Nat","Prelude"]:Type:U=(20:./Prelude/Nat.idr)

172+ })

173+ })

174+

175+ getHoles=[{hole_0}]

176+

177+ * - getGuess

178+ - .. code-block:: idris

179+

180+ x:({λ (Nat.["Nat", "Prelude"]:{

181+ type constructor tag=8 number=0}).

182+ Type:U=(20:./Prelude/Nat.idr)

183+ }.{

184+ name ref bound

185+ x:{name ref {type constructor tag=8 number=0}

186+ Nat.["Nat","Prelude"]:Type:U=(20:./Prelude/Nat.idr)

187+ }

188+ })

189+ }

190+

191+.. target-notes::

192+.. _`my code`: https://github.com/martinbaker/Idris-dev/blob/uglyTTPrinter/libs/prelude/Language/Reflection/TTPrinter.idr

docs/proofs/generatingData.rstnew file+385

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1+Generating Datatypes and Functions at Compile Time

2+==================================================

3+

4+Program elements, such as datatypes and functions can be constructed at compile-time in the Elab monad.

5+This can allow proofs to be generated for user defined types or it could allow types to be automatically generated to support user defined types.

6+An example is the code, from `Elaborator reflection: extending Idris in Idris`_, that automatically generates accessibility predicates using the Bove-Capretta method.

7+

8+Generating Datatypes

9+--------------------

10+

11+There are two main 'tactics' associated with generating datatypes:

12+

13+- declareDatatype

14+- defineDatatype

15+

16+Which declare and define the datatype as the names suggest.

17+

18+.. list-table::

19+

20+ * - These 'tactics' and the data structures associated with them are listed in the tables later on this page, for now, here is a summary:

21+ - .. image:: ../image/generateDatatype.png

22+ :width: 332px

23+ :height: 246px

24+ :alt: diagram illustrating data structures associated with declareDatatype defineDatatype.

25+

26+.. list-table::

27+

28+ * - As a first example, the following boolean-like type can be constructed. When the compiler has run it will be available to us as if we had compiled it in the usual way:

29+ - .. code-block:: idris

30+

31+ λΠ> :printdef Two

32+ data Two : Type where

33+ F : Two

34+ T : Two

35+

36+This was generated by the following code:

37+

38+.. code-block:: idris

39+

40+ module TwoDef

41+ %language ElabReflection

42+

43+ addTwo : Elab ()

44+ addTwo = do let twoname : TTName = `{{TwoDef.Two}}

45+ let F2 = `{{TwoDef.F}}

46+ let T2 = `{{TwoDef.T}}

47+ declareDatatype $ Declare twoname [] `(Type)

48+ defineDatatype $ DefineDatatype twoname [

49+ Constructor `{{F}} [] (Var `{{TwoDef.Two}}),

50+ Constructor `{{T}} [] (Var `{{TwoDef.Two}})

51+ ]

52+

53+ %runElab addTwo

54+

55+.. list-table::

56+

57+ * - The constructors T and F can be called as would be expected:

58+ - .. code-block:: idris

59+

60+ λΠ> F

61+ F : Two

62+ λΠ> T

63+ T : Two

64+

65+Generating Functions

66+--------------------

67+

68+There are two main 'tactics' associated with generating functions:

69+

70+- declareType

71+- defineFunction

72+

73+Which declare and define the function as the names suggest.

74+

75+.. list-table::

76+

77+ * - These 'tactics' and the data structures associated with them are listed in the tables later on this page, for now, here is a summary:

78+ - .. image:: ../image/generateFunction.png

79+ :width: 332px

80+ :height: 246px

81+ :alt: diagram illustrating data structures associated with function declare and define.

82+

83+Note: The left hand side (lhs) and right hand side (rhs) of FunClause typically is of type 'Raw'.

84+

85+.. list-table::

86+

87+ * - Bound pattern variables are represented by 'PVar' binders:

88+ This diagram shows an example of a possible Raw structure that might be used in a function definition.

89+ - .. image:: ../image/generateFunction2.png

90+ :width: 264px

91+ :height: 239px

92+ :alt: diagram illustrating data structures associated with functions.

93+

94+.. list-table::

95+

96+ * - Some function definitions can now be added to the above datatype. This is what they will look like:

97+ - .. code-block:: idris

98+

99+ λΠ> :printdef perm1

100+ perm1 : Two -> Two

101+ perm1 F = F

102+ perm1 T = T

103+ λΠ> :printdef perm2

104+ perm2 : Two -> Two

105+ perm1 F = T

106+ perm1 T = F

107+

108+This was generated with the following code:

109+

110+.. code-block:: idris

111+

112+ let perm1 = `{{TwoDef.perm1}}

113+ declareType (Declare perm1 [MkFunArg `{{code}} (Var twoname) Explicit NotErased] (Var twoname))

114+ defineFunction $ DefineFun perm1 [

115+ MkFunClause (RApp (Var perm1) (Var `{{TwoDef.F}})) (Var F2),

116+ MkFunClause (RApp (Var perm1) (Var `{{TwoDef.T}})) (Var T2)

117+ ]

118+

119+ let perm2 = `{{TwoDef.perm2}}

120+ declareType (Declare perm2 [MkFunArg `{{code}} (Var twoname) Explicit NotErased] (Var twoname))

121+ defineFunction $ DefineFun perm2 [

122+ MkFunClause (RApp (Var perm1) (Var `{{TwoDef.F}})) (Var T2),

123+ MkFunClause (RApp (Var perm1) (Var `{{TwoDef.T}})) (Var F2)

124+ ]

125+

126+.. list-table::

127+

128+ * - This is what happens when we call the functions:

129+ - .. code-block:: idris

130+

131+ λΠ> perm1 F

132+ F : Two

133+ λΠ> perm1 T

134+ T : Two

135+ λΠ> perm2 F

136+ T : Two

137+ λΠ> perm2 T

138+ F : Two

139+

140+So far these datatypes and functions could have been written, statically, in the usual way. However, it is possible to imagine situations where we may need a lot of functions to be generated automatically at compile time. For example, if we extend this Boolean datatype to a datatype with more simple constructors (a finite set), we could generate a function for every possible permutation of that datatype back to itself.

141+

142+A Different Example which has Type Parameters

143+---------------------------------------------

144+

145+.. list-table::

146+

147+ * - Here is an example of a datatype with type parameters:

148+ - .. code-block:: idris

149+

150+ data N : Nat -> Type where

151+ MkN : N x

152+ MkN' : (x : Nat) -> N (S x)

153+

154+This was produced by the following code:

155+

156+.. code-block:: idris

157+

158+ module DataDef

159+ %language ElabReflection

160+

161+ addData : Elab ()

162+ addData = do

163+ let dataname : TTName = `{{DataDef.N}}

164+ declareDatatype $ Declare dataname [MkFunArg `{{n}} `(Nat) Explicit NotErased] `(Type)

165+ defineDatatype $ DefineDatatype dataname [

166+ Constructor `{{MkN}} [MkFunArg `{{x}} `(Nat) Implicit NotErased]

167+ (RApp (Var dataname) (Var `{{x}})),

168+ Constructor `{{MkN'}} [MkFunArg `{{x}} `(Nat) Explicit NotErased]

169+ (RApp (Var dataname) (RApp (Var `{S}) (Var `{{x}})))

170+ ]

171+

172+ %runElab addData

173+

174+So this declares and defines the following data structure 'N' with a constructor 'MkN' which can have an implicit or an explicit Nat argument. Which can be used like this:

175+

176+.. code-block:: idris

177+

178+ λΠ> :t N

179+ N : Nat -> Type

180+ λΠ> N 2

181+ N 2 : Type

182+ λΠ> N 0

183+ N 0 : Type

184+ λΠ> :t MkN

185+ MkN : N x

186+

187+Table of 'tactics' for Generating Data and Functions

188+----------------------------------------------------

189+

190+These are the functions that we can use to create data and functions in the Elab monad:

191+

192+.. list-table::

193+ :widths: 10 30

194+ :stub-columns: 1

195+

196+ * - declareType

197+ - Add a type declaration to the global context.

198+

199+ Signature:

200+

201+ declareType : TyDecl -> Elab ()

202+ * - defineFunction

203+ - Define a function in the global context. The function must have already been declared, either in ordinary Idris code or using `declareType`.

204+

205+ Signature:

206+

207+ defineFunction : FunDefn Raw -> Elab ()

208+

209+ * - declareDatatype

210+ - Declare a datatype in the global context. This step only establishes the type constructor; use `defineDatatype` to give it constructors.

211+

212+ Signature:

213+

214+ declareDatatype : TyDecl -> Elab ()

215+

216+ * - defineDatatype

217+ - Signature:

218+

219+ defineDatatype : DataDefn -> Elab ()

220+

221+ * - addImplementation

222+ - Register a new implementation for interface resolution.

223+

224+ Arguments:

225+

226+ - ifaceName the name of the interface for which an implementation is being registered

227+ - implName the name of the definition to use in implementation search

228+

229+ Signature:

230+

231+ addImplementation : (ifaceName, implName : TTName) -> Elab ()

232+

233+ * - isTCName

234+ - Determine whether a name denotes an interface.

235+

236+ Arguments:

237+

238+ - name - a name that might denote an interface.

239+

240+ Signature:

241+

242+ isTCName : (name : TTName) -> Elab Bool

243+

244+Table of Datatypes Associated with Generating Data and Functions

245+----------------------------------------------------------------

246+

247+The above functions use the following data/records:

248+

249+.. list-table::

250+ :widths: 10 30

251+ :stub-columns: 1

252+

253+ * - Plicity

254+ - How an argument is provided in high-level Idris

255+

256+ .. code-block:: idris

257+

258+ data Plicity=

259+ ||| The argument is directly provided at the application site

260+ Explicit |

261+ ||| The argument is found by Idris at the application site

262+ Implicit |

263+ ||| The argument is solved using interface resolution

264+ Constraint

265+

266+ * - FunArg

267+ - Function arguments

268+

269+ These are the simplest representation of argument lists, and are used for functions. Additionally, because a FunArg provides enough

270+ information to build an application, a generic type lookup of a top-level identifier will return its FunArgs, if applicable.

271+

272+ .. code-block:: idris

273+

274+ record FunArg where

275+ constructor MkFunArg

276+ name : TTName

277+ type : Raw

278+ plicity : Plicity

279+ erasure : Erasure

280+

281+ * - TyConArg

282+ - Type constructor arguments

283+

284+ Each argument is identified as being either a parameter that is

285+

286+ consistent in all constructors, or an index that varies based on

287+

288+ which constructor is selected.

289+

290+ .. code-block:: idris

291+

292+ data TyConArg =

293+ ||| Parameters are uniform across the constructors

294+ TyConParameter FunArg |

295+ ||| Indices are not uniform

296+ TyConIndex FunArg

297+

298+ * - TyDecl

299+ - A type declaration for a function or datatype

300+

301+ .. code-block:: idris

302+

303+ record TyDecl where

304+ constructor Declare

305+ ||| The fully-qualified name of the function or datatype being declared.

306+ name : TTName

307+ ||| Each argument is in the scope of the names of previous arguments.

308+ arguments : List FunArg

309+ ||| The return type is in the scope of all the argument names.

310+ returnType : Raw

311+

312+ * - FunClause

313+ - A single pattern-matching clause

314+

315+ .. code-block:: idris

316+

317+ data FunClause : Type -> Type where

318+ MkFunClause : (lhs, rhs : a) -> FunClause a

319+ MkImpossibleClause : (lhs : a) -> FunClause a

320+

321+ * - FunDefn

322+ - A reflected function definition.

323+

324+ .. code-block:: idris

325+

326+ record FunDefn a where

327+ constructor DefineFun

328+ name : TTName

329+ clauses : List (FunClause a)

330+

331+ * - ConstructorDefn

332+ - A constructor to be associated with a new datatype.

333+

334+ .. code-block:: idris

335+

336+ record ConstructorDefn where

337+ constructor Constructor

338+ ||| The name of the constructor. The name must _not_ be qualified -

339+ ||| that is, it should begin with the `UN` or `MN` constructors.

340+ name : TTName

341+ ||| The constructor arguments. Idris will infer which arguments are

342+ ||| datatype parameters.

343+ arguments : List FunArg

344+ ||| The specific type constructed by the constructor.

345+ returnType : Raw

346+

347+ * - DataDefn

348+ - A definition of a datatype to be added during an elaboration script.

349+

350+ .. code-block:: idris

351+

352+ record DataDefn where

353+ constructor DefineDatatype

354+ ||| The name of the datatype being defined. It must be

355+ ||| fully-qualified, and it must have been previously declared as a

356+ ||| datatype.

357+ name : TTName

358+ ||| A list of constructors for the datatype.

359+ constructors : List ConstructorDefn

360+

361+ * - CtorArg

362+ - CtorParameter

363+

364+ .. code-block:: idris

365+

366+ data CtorArg = CtorParameter FunArg | CtorField FunArg

367+

368+ * - Datatype

369+ - A reflected datatype definition

370+

371+ .. code-block:: idris

372+

373+ record Datatype where

374+ constructor MkDatatype

375+ ||| The name of the type constructor

376+ name : TTName

377+ ||| The arguments to the type constructor

378+ tyConArgs : List TyConArg

379+ ||| The result of the type constructor

380+ tyConRes : Raw

381+ ||| The constructors for the family

382+ constructors : List (TTName, List CtorArg, Raw)</td>

383+

384+.. target-notes::

385+.. _`Elaborator reflection: extending Idris in Idris`: https://dl.acm.org/citation.cfm?doid=2951913.2951932

docs/proofs/holes.rstnew file+196

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1+Elaborator Reflection - Holes

2+=============================

3+

4+The process of doing proofs and elaborator reflection tends to involve stating with a desired conclusion and working back to known premises. This often needs intermediate sub-goals which may only be partially solved, these are encoded using 'holes' and 'guesses'.

5+

6+* A hole is a term (an expression - a chunk of code) which is yet to be determined. We do have information about its type (this process tends to be type driven).

7+* A guess is like a hole that is not yet bound.

8+

9+The theory around this was developed in `Dependently Typed Functional Programs and their Proofs by McBride 1999`_.

10+

11+Notation for Holes and Guesses

12+------------------------------

13+

14+There is a notation used in the McBride 1999 thesis which is adapted for the TT language. When working on elaborator reflection it is useful to know this notation, for instance when reading the output of the 'debug' tactic.

15+

16+* A focused hole is notated like this ?x:t . t

17+* A guess is notated like this ?x ≈ t:t . t

18+

19+The following example shows how this is used:

20+

21+Simple Example

22+--------------

23+

24+Start with a code file that just contains:

25+

26+.. code-block:: idris

27+

28+ %language ElabReflection

29+

30+ testFn : Nat

31+ testFn = %runElab (do debug {a = ()})

32+

33+when this is loaded the following is displayed:

34+

35+.. code-block:: idris

36+

37+ Holes:

38+ ----------------------------------

39+ {hole_2} : Prelude.Nat.Nat

40+

41+ ----------------------------------

42+ {hole_0} : Prelude.Nat.Nat

43+

44+ Term:

45+ ?{hole_0} ≈ ? {hole_2} . {hole_2} . {hole_0}

46+

47+This shows information about the state when debug is encountered, during tactic execution, which allows us to investigate what is happening at each stage.

48+

49+* The "Holes" part shows the types of the holes and the local context of each

50+* The "Term" part shows where these holes are in the expression being constructed.

51+

52+So starting with the "Term" part we have.

53+

54+.. code-block:: idris

55+

56+ ?{hole_0} ≈ ? {hole_2} . {hole_2} . {hole_0}

57+

58+.. sidebar:: attack tactic

59+

60+ This kind of thing tends to arise from "attack", which helps keep binding forms in order.

61+

62+The meaning of this is not immediately apparent so it helps to add some parentheses to make the structure clearer:

63+

64+.. code-block:: idris

65+

66+ (?{hole_0} ≈ (? {hole_2} . {hole_2}) . {hole_0})

67+

68+First lets look at the inner part:

69+

70+.. code-block:: idris

71+

72+ ? {hole_2} . {hole_2}

73+

74+We can substitute in the type from the "Holes" part:

75+

76+.. code-block:: idris

77+

78+ ? {hole_2}:Nat . {hole_2}:Nat

79+

80+So we are looking for a hole of type Nat and all we know is it has type Nat.

81+

82+Going back to the full term, the above is wrapped in a guess, so it means: 'a guess that is itself a hole'.

83+

84+Since the elaborator does not have any further information it has to be given a value:

85+

86+.. code-block:: idris

87+

88+ %language ElabReflection

89+

90+ testFn : Nat

91+ testFn = %runElab (do fill `(Z)

92+ debug {a = ()}

93+ solve)

94+

95+Now we have a guess for hole_2 which is itself a guess for hole_0:

96+

97+.. code-block:: idris

98+

99+ ?{hole_0} ≈ ?{hole_2} ≈ Prelude.Nat.Z . {hole_2} . {hole_0}

100+

101+The guesses can be accepted by calling the 'solve' tactic.

102+

103+Example Showing Patterns

104+------------------------

105+

106+In this next example a parameter 'n' has been added to the function. This allows us to see how patterns are used. Starting with this file:

107+

108+.. code-block:: idris

109+

110+ %language ElabReflection

111+

112+ testFn : Nat -> Nat

113+ testFn n = %runElab (do debug {a = ()})

114+

115+when loaded gives:

116+

117+.. code-block:: idris

118+

119+ Holes:

120+ n : Prelude.Nat.Nat

121+ ----------------------------------

122+ {hole_3} : Prelude.Nat.Nat

123+

124+ n : Prelude.Nat.Nat

125+ ----------------------------------

126+ {hole_1} : Prelude.Nat.Nat

127+

128+ ----------------------------------

129+ {hole_0} : patTy n : Prelude.Nat.Nat . Prelude.Nat.Nat

130+

131+ Term:

132+ ?{hole_0} ≈ pat n : Prelude.Nat.Nat .

133+ ?{hole_1} ≈ ? {hole_3} . {hole_3} . {hole_1} .

134+ {hole_0}

135+

136+The ns above the lines show the context of the holes on the right hand side - they include the variable n that is an argument!

137+

138+patTy is a binding form in Idris's core language that introduces a pattern variable. The idea is that the left-hand side and right-hand side of each pattern should have the same type. Because pattern variables may occur multiple times on either side of the equation, we can achieve this by documenting their types with a binding form that wraps each side. This new binding form is why an "attack" was necessary prior to elaborating the RHS.

139+

140+patTy is a type former, and pat is the corresponding introduction form. So you can think of patTy as being a bit like a dependent function type, and pat as being a bit like lambda, except they don't introduce functions (they instead are used for pattern-matching definitions).

141+

142+More Complicated Example

143+------------------------

144+

145+This example does not introduce any new notation but the extra complexity gives a more realistic idea of how it is used. Here we start with this file:

146+

147+.. code-block:: idris

148+

149+ %language ElabReflection

150+

151+ testFn : (n : Nat) -> (n = plus n Z) -> (S n = S (plus n Z))

152+ testFn k ih = %runElab (do debug {a = ()})

153+

154+when loaded gives:

155+

156+.. code-block:: idris

157+

158+ Holes:

159+ k : Prelude.Nat.Nat

160+ ih : = Prelude.Nat.Nat Prelude.Nat.Nat k

161+ (Prelude.Nat.plus k Prelude.Nat.Z)

162+ ----------------------------------

163+ {hole_4} : = Prelude.Nat.Nat Prelude.Nat.Nat (Prelude.Nat.S k)

164+ (Prelude.Nat.S (Prelude.Nat.plus k Prelude.Nat.Z))

165+

166+ k : Prelude.Nat.Nat

167+ ih : = Prelude.Nat.Nat Prelude.Nat.Nat k

168+ (Prelude.Nat.plus k Prelude.Nat.Z)

169+ ----------------------------------

170+ {hole_2} : = Prelude.Nat.Nat Prelude.Nat.Nat (Prelude.Nat.S k)

171+ (Prelude.Nat.S (Prelude.Nat.plus k Prelude.Nat.Z))

172+

173+ k : Prelude.Nat.Nat

174+ ----------------------------------

175+ {hole_1} : patTy ih : = Prelude.Nat.Nat Prelude.Nat.Nat k

176+ (Prelude.Nat.plus k Prelude.Nat.Z) .

177+ = Prelude.Nat.Nat Prelude.Nat.Nat (Prelude.Nat.S k)

178+ (Prelude.Nat.S (Prelude.Nat.plus k Prelude.Nat.Z))

179+

180+ ----------------------------------

181+ {hole_0} : patTy k : Prelude.Nat.Nat .

182+ patTy ih : = Prelude.Nat.Nat Prelude.Nat.Nat k

183+ (Prelude.Nat.plus k Prelude.Nat.Z) .

184+ = Prelude.Nat.Nat Prelude.Nat.Nat (Prelude.Nat.S k)

185+ (Prelude.Nat.S (Prelude.Nat.plus k Prelude.Nat.Z))

186+

187+ Term:

188+ ?{hole_0} ≈ pat k : Prelude.Nat.Nat .

189+ ?{hole_1} ≈ pat ih : = Prelude.Nat.Nat Prelude.Nat.Nat k

190+ (Prelude.Nat.plus k Prelude.Nat.Z) .

191+ ?{hole_2} ≈ ? {hole_4} . {hole_4} . {hole_2} .

192+ {hole_1} .

193+ {hole_0}

194+

195+.. target-notes::

196+.. _`Dependently Typed Functional Programs and their Proofs by McBride 1999`: https://www.era.lib.ed.ac.uk/handle/1842/374

docs/proofs/index.rst+10−5

@@ -1,10 +1,10 @@

1.. _proofs-index:

2

3−###############

3+#########################################

4−Theorem Proving

4+Theorem Proving and Elaborator Reflection

5−###############

5+#########################################

6

7−A tutorial on theorem proving in Idris.

7+A tutorial on theorem proving in Idris.

8

9.. note::

10

@@ -25,4 +25,9 @@ A tutorial on theorem proving in Idris.

25 propositional

26 interactive

27 interactiveOld

28−

28+ elabReflection

29+ tactics

30+ holes

31+ example1

32+ primitive

33+ generatingData

docs/proofs/primitive.rstnew file+518

@@ -0,0 +1,518 @@

1+Primitive Operators

2+===================

3+

4+.. list-table::

5+ :widths: 10 30

6+ :stub-columns: 1

7+

8+ * - gensym

9+ - Generate a unique name based on some hint.

10+

11+ Useful when establishing a new binder.

12+

13+ **NB**: the generated name is unique **for this run of the elaborator**.

14+

15+ Do not assume that they are globally unique.

16+

17+ Signature:

18+

19+ gensym : (hint : String) -> Elab TTName

20+

21+ * - solve

22+ - Substitute a guess into a hole.

23+

24+

25+ .. image:: ../image/solve.png

26+ :width: 141px

27+ :height: 74px

28+ :alt: diagram illustrating solve tactic

29+

30+

31+ Substitute the focused guess throughout its scope, eliminating it and moving focus to the next element of the hole queue. Fail if the focus is not a guess.

32+

33+ Signature:

34+

35+ solve : Elab ()

36+

37+ Example:

38+

39+ .. code-block:: idris

40+

41+ %language ElabReflection

42+

43+ testFn : Nat

44+ testFn = %runElab (do fill `(Z)

45+ solve)

46+

47+ * - fill

48+ - Place a term into a hole, unifying its type. Fails if the focus is not a hole.

49+

50+

51+ .. image:: ../image/fill.png

52+ :width: 140px

53+ :height: 74px

54+ :alt: diagram illustrating fill tactic

55+

56+

57+ Signature:

58+

59+ fill : (e : Raw) -> Elab ()

60+

61+ Place a term e with type tc into the focused hole:

62+

63+ ?h : th.e'

64+

65+ and convert it to a guess:

66+

67+ ?h ≈ e:t.e'

68+

69+ and fail if the current focus is not a hole. The type t of the guess is constructed by unifying tc and th, which may instantiate holes that they refer to. Fail if the current focus is not a hole or if unification fails.

70+

71+ This unification can result in the solution of further holes or the establishment of additional unsolved unification constraints.

72+

73+ Example:

74+

75+ .. code-block:: idris

76+

77+ %language ElabReflection

78+

79+ testFn : Nat

80+ testFn = %runElab (do fill `(Z)

81+ solve)

82+

83+ * - apply

84+ - Attempt to apply an operator to fill the current hole, potentially solving arguments by unification.

85+

86+ A hole is established for each argument.

87+

88+ The return value is the list of holes established for the arguments to the function.

89+

90+ Note that not all of the returned hole names still exist, as they may have been solved.

91+

92+ - @ op the term to apply

93+ - @ argSpec - A list of booleans, one for each argument that the operator will be applied to. If true then attempt to solve the argument by unification.

94+

95+ Signature:

96+

97+ apply : (op : Raw) -> (argSpec : List Bool) -> Elab (List TTName)

98+

99+ Example: elaborating an application of a function f that takes one implicit and two explicit arguments might invoke:

100+

101+ apply \`(f) [False, True, True]

102+

103+ Here is an example of an elab script that uses apply to insert the term plus Z (S Z) into a goal of type Nat.

104+

105+ .. code-block:: idris

106+

107+ do [x,y] <- apply `(plus) [False, False]

108+ solve

109+ focus x; fill `(Z); solve

110+ focus y; fill `(S Z); solve

111+

112+ The names of the established holes are returned.

113+

114+ Note: This was added to the original tactic language to allow elaborator reflection.

115+

116+ * - matchApply

117+ - Attempt to apply an operator to fill the current hole, potentially solving arguments by matching.

118+

119+ The return value is the list of holes established for the arguments to the function.

120+

121+ Note that not all of the returned hole names still exist, as they may have been solved.

122+

123+ - @ op the term to apply

124+ - @ argSpec instructions for finding the arguments to the term, where the Boolean states whether or not to attempt to solve the argument by matching.

125+

126+ Signature:

127+

128+ matchApply : (op : Raw) -> (argSpec : List Bool) -> Elab (List TTName)

129+

130+ * - focus

131+ - Move the focus to the specified hole, bringing it to the front of the hole queue. Fails if the hole does not exist.

132+

133+ @ hole the hole to focus on

134+

135+ Signature:

136+

137+ focus : (hole : TTName) -> Elab ()

138+

139+ * - unfocus

140+ - Send the currently-focused hole to the end of the hole queue and focus on the next hole.

141+

142+ Signature:

143+

144+ unfocus : TTName -> Elab ()

145+

146+ * - attack

147+ - Convert a hole to make it suitable for bindings.

148+

149+

150+ .. image:: ../image/attack.png

151+ :width: 152px

152+ :height: 70px

153+ :alt: diagram illustrating attack tactic

154+

155+

156+ The binding tactics require that a hole be directly under its binding, or else the scopes of the generated terms won't make sense. This tactic creates a new hole of the proper form, and points the old hole at it.

157+

158+ Signature:

159+

160+ attack : Elab ()

161+

162+ * - claim

163+ - Establish a new hole binding named n with type t, surrounding the current focus.

164+

165+ Introduce a new hole with a specified name and type.

166+

167+ The new hole will be focused, and the previously-focused hole will be immediately after it in the hole queue. Because this tactic introduces a new binding, you may need to 'attack' first.

168+

169+ Signature:

170+

171+ claim : TTName -> Raw -> Elab ()

172+

173+ * - patvar

174+ - Convert a hole into a pattern variable.

175+

176+ Signature:

177+

178+ patvar : TTName -> Elab ()

179+

180+ * - compute

181+ - Normalise the goal.

182+

183+ Often this is not necessary because normanisation is applied during other tactics.

184+

185+ Signature:

186+

187+ compute : Elab ()

188+

189+ * - normalise

190+ - Normalise a term in some lexical environment

191+

192+ - @ env the environment in which to compute (get one of these from `getEnv`)

193+ - @ term the term to normalise

194+

195+ Signature:

196+

197+ normalise : (env : List (TTName, Binder TT)) -> (term : TT) -> Elab TT

198+

199+ * - whnf

200+ - Reduce a closed term to weak-head normal form

201+

202+ @ term the term to reduce

203+

204+ Signature:

205+

206+ whnf : (term : TT) -> Elab TT

207+

208+ * - convertsInEnv

209+ - Check that two terms are convertible in the current context and in some environment.

210+

211+ - @ env a lexical environment to compare the terms in (see `getEnv`)

212+ - @ term1 the first term to convert

213+ - @ term2 the second term to convert

214+

215+ Signature:

216+

217+ convertsInEnv : (env : List (TTName, Binder TT)) -> (term1, term2 : TT) -> Elab ()

218+

219+ * - converts

220+ - Check that two terms are convertable in the current context and environment

221+

222+ - @ term1 the first term to convert

223+ - @ term2 the second term to convertconverts : (term1, term2 : TT) -> Elab ()

224+

225+ converts term1 term2 = convertsInEnv !getEnv term1 term2

226+

227+ * - getSourceLocation

228+ - Find the source context for the elaboration script

229+

230+ Signature:

231+

232+ getSourceLocation : Elab SourceLocation

233+

234+ * - sourceLocation

235+ - Attempt to solve the current goal with the source code

236+

237+ locationsourceLocation : Elab ()

238+

239+ .. code-block:: idris

240+

241+ sourceLocation = do loc <- getSourceLocation

242+ fill (quote loc)

243+ solve

244+

245+ * - currentNamespace

246+ - Get the current namespace at the point of tactic execution. This allows scripts to define top-level names conveniently.

247+

248+ The namespace is represented as a reverse-order list of strings, just as in the representation of names.

249+

250+ Signature:

251+

252+ currentNamespace : Elab (List String)

253+

254+ * - rewriteWith

255+ - Attempt to rewrite the goal using an equality.

256+

257+ The tactic searches the goal for applicable subterms, and constructs a context for `replace` using them. In some cases, this is not possible, and `replace` must be called manually with an appropriate context.

258+

259+ Because this tactic internally introduces a `let` binding, it requires that the hole be immediately under its binder (use 'attack' if it might not be).

260+

261+ Signature:

262+

263+ rewriteWith : Raw -> Elab ()

264+

265+ * - resolveTC

266+ - Attempt to solve the current goal with an interface dictionary

267+

268+ @ fn the name of the definition being elaborated (to prevent Idris from looping)

269+

270+ Signature:

271+

272+ resolveTC : (fn : TTName) -> Elab ()

273+

274+ * - search

275+ - Use Idris's internal proof search.

276+

277+ Signature:

278+

279+ search : Elab ()

280+

281+ * - search'

282+ - Use Idris's internal proof search, with more control.

283+

284+ - @ depth the search depth

285+ - @ hints additional names to try

286+

287+ Signature:

288+

289+ search' : (depth : Int) -> (hints : List TTName) -> Elab ()

290+

291+ * - operatorFixity

292+ - Look up the declared fixity for an operator.

293+

294+ The lookup fails if the operator does not yet have a fixity or if the string is not a valid operator.

295+

296+ @ operator the operator string to look up

297+

298+ Signature:

299+

300+ operatorFixity : (operator : String) -> Elab Fixity

301+

302+ * - debug

303+ - Halt elaboration, dumping the internal state for inspection.

304+

305+ This is intended for elaboration script developers, not for end-users. Use `fail` for final scripts.

306+

307+ Signature:

308+

309+ debug : Elab a

310+

311+ If 'debug' is not the last tactic then make sure its type is sufficiently constrained. In particular, its type is Elab a, but there's no way for Idris to find out which type was meant for a. This can be fixed by either writing an explicit type (e.g. debug {a = ()}) or by using a helper that constrains the type (such as simple in Pruviloj, e.g. simple debug as a line).

312+

313+ .. code-block:: idris

314+

315+ %language ElabReflection

316+

317+ idNat : Nat -> Nat

318+ idNat = %runElab (do intro `{{x}}

319+ debug {a = ()}

320+ fill (Var `{{x}})

321+ solve)

322+

323+ * - debugMessage

324+ - Halt elaboration, dumping the internal state and displaying a message.

325+

326+ This is intended for elaboration script developers, not for end-users. Use `fail` for final scripts.

327+

328+ @ msg the message to display

329+

330+ Signature:

331+

332+ debugMessage : (msg : List ErrorReportPart) -> Elab a

333+

334+ If 'debugMessage' is not the last tactic then make sure its type is sufficiently constrained. In particular, its type is Elab a, but there's no way for Idris to find out which type was meant for a. This can be fixed by either writing an explicit type (e.g. debugMessage [TextPart "message"] {a = ()}) or by using a helper that constrains the type (such as simple in Pruviloj, e.g. simple debug as a line).

335+

336+ .. code-block:: idris

337+

338+ %language ElabReflection

339+ idNat : Nat -> Nat

340+ idNat = %runElab (do intro `{{x}}

341+ debugMessage [TextPart "error message"] {a = ()}

342+ fill (Var `{{x}})

343+ solve)

344+

345+ * - metavar

346+ - Create a new top-level metavariable to solve the current hole.

347+

348+ @ name the name for the top-level variable

349+

350+ Signature:

351+

352+ metavar : (name : TTName) -> Elab ()

353+

354+ * - runElab

355+ - Recursively invoke the reflected elaborator with some goal.

356+

357+ The result is the final term and its type.

358+

359+ Signature:

360+

361+ runElab : Raw -> Elab () -> Elab (TT, TT)

362+

363+

364+Read and Write State

365+--------------------

366+

367+.. list-table::

368+ :widths: 10 30

369+ :stub-columns: 1

370+

371+ * - getEnv

372+ - Look up the lexical binding at the focused hole. Fails if no holes are present.

373+

374+ Signature:

375+

376+ getEnv : Elab (List (TTName, Binder TT))

377+

378+ * - getGoal

379+ - Get the name and type of the focused hole. Fails if not holes are present.

380+

381+ Signature:

382+

383+ getGoal : Elab (TTName, TT)

384+

385+ * - getHoles

386+ - Get the hole queue, in order.

387+

388+ Signature:

389+

390+ getHoles : Elab (List TTName)

391+

392+ * - getGuess

393+ - If the current hole contains a guess, return it. Otherwise, fail.

394+

395+ Signature:

396+

397+ getGuess : Elab TT

398+

399+ * - lookupTy

400+ - Look up the types of every overloading of a name.

401+

402+ Signature:

403+

404+ lookupTy : TTName -> Elab (List (TTName, NameType, TT))

405+

406+ * - lookupTyExact

407+ - Get the type of a fully-qualified name. Fail if it doesn't resolve uniquely.

408+

409+ Signature:

410+

411+ lookupTyExact : TTName -> Elab (TTName, NameType, TT)

412+

413+ * - lookupDatatype

414+ - Find the reflected representation of all datatypes whose names are overloadings of some name.

415+

416+ Signature:

417+

418+ lookupDatatype : TTName -> Elab (List Datatype)

419+

420+ * - lookupDatatypeExact

421+ - Find the reflected representation of a datatype, given its fully-qualified name. Fail if the name does not uniquely resolve to a datatype.

422+

423+ Signature:

424+

425+ lookupDatatypeExact : TTName -> Elab Datatype

426+

427+ * - lookupFunDefn

428+ - Find the reflected function definition of all functions whose names are overloadings of some name.

429+

430+ Signature:

431+

432+ lookupFunDefn : TTName -> Elab (List (FunDefn TT))

433+

434+ * - lookupFunDefnExact

435+ - Find the reflected function definition of a function, given its fully-qualified name. Fail if the name does not uniquely resolve to a function.

436+

437+ Signature:

438+

439+ lookupFunDefnExact : TTName -> Elab (FunDefn TT)

440+

441+ * - lookupArgs

442+ - Get the argument specification for each overloading of a name.

443+

444+ Signature:

445+

446+ lookupArgs : TTName -> Elab (List (TTName, List FunArg, Raw))

447+

448+ * - lookupArgsExact

449+ - Get the argument specification for a name. Fail if the name does not uniquely resolve.

450+

451+ Signature:

452+

453+ lookupArgsExact : TTName -> Elab (TTName, List FunArg, Raw)

454+

455+ * - check

456+ - Attempt to type-check a term, getting back itself and its type.

457+

458+ - @ env the environment within which to check the type

459+ - @ tm the term to check

460+

461+ Signature:

462+

463+ check : (env : List (TTName, Binder TT)) -> (tm : Raw) -> Elab (TT, TT)

464+

465+Error Handling

466+--------------

467+

468+.. list-table::

469+ :widths: 10 30

470+ :stub-columns: 1

471+

472+ * - tryCatch

473+ - `tryCatch t (\err => t')` will run `t`, and if it fails, roll back the elaboration state and run `t'`,

474+ but with access to the knowledge of why `t` failed.

475+

476+ Signature:

477+

478+ tryCatch : Elab a -> (Err -> Elab a) -> Elab a

479+

480+ .. code-block:: idris

481+

482+ %language ElabReflection

483+

484+ f : Err -> Elab ()

485+ f (Msg _) = fill `("message error")

486+ f (CantUnify _ _ _ _ _ _) = fill `("unification error")

487+ f _ = fill `("other")

488+

489+ s2 : String

490+ s2 = %runElab (do tryCatch (fill `(True)) f ; solve)

491+

492+ * - fail

493+ - Halt elaboration with an error

494+

495+ Signature:

496+

497+ fail : List ErrorReportPart -> Elab a

498+

499+ Note: we may need to make sure the return type 'a' is sufficiently constrained. If required add an explicit type {a = ()}.

500+

501+ Below is some code which includes 'fail'. This will always fail but we could replace 'True' with some more useful condition.

502+

503+ .. code-block:: idris

504+

505+ %language ElabReflection

506+

507+ id1 : Elab ()

508+ id1 = do

509+ intro `{{x}}

510+ fill (Var `{{x}})

511+ if True

512+ then

513+ fail [TextPart "put error message here"]

514+ else

515+ solve

516+

517+ idNat : Nat -> Nat

518+ idNat = %runElab id1

docs/proofs/tactics.rstnew file+430

@@ -0,0 +1,430 @@

1+Elaborator Reflection - Tactics

2+===============================

3+

4+The Idris part of the code for elaborator reflection is in

5+Elab.idr `<https://github.com/idris-lang/Idris-dev/blob/master/libs/prelude/Language/Reflection/Elab.idr>`_

6+Before looking at the Elab monad we need to know how to construct terms.

7+

8+

9+Proof State

10+-----------

11+

12+The terminology 'Proof State' is used by analogy to proof assistants but, as used here, it's really more of a metaprogramming state.

13+

14+Tactics operate on the proof state. The proof state contains various pieces of information, at this stage, the important ones for us are:

15+

16+- A hole queue - This contains a list of the names of the holes used in the proof term. The solution to each hole may depend on the other holes. For more information about holes and guesses see `Dependently Typed Functional Programs and their Proofs by McBride 1999`_.

17+- A goal type - The type of the term that is under construction for the current goal.

18+- A possibly incomplete proof term, which is being calculated and should be the goal type at the end of elaboration.

19+- Context - A collection of open unification problems, representing recoverable failures of unification that may yet unify once more variables are solved.

20+

21+There are many other pieces of information in the proof state such as the names used , such as the bodies of case blocks that need to be elaborated.but, for now, we will leave the system to handle these.

22+

23+.. image:: ../image/elabProofState.png

24+ :width: 283px

25+ :height: 232px

26+

27+Names TTName

28+------------

29+

30+Names in an Idris program are evaluated at runtime but sometimes a 'variable name' is needed, which can be referred to as an unevaluated symbol.

31+The names used in terms have different constructors depending on their type:

32+

33++---------------------------+-----------------------------------------------+

34+| UN String | A user-provided name |

35++---------------------------+-----------------------------------------------+

36+| NS (UN "foo") ["B", "A"]) | A name in some namespace |

37++---------------------------+-----------------------------------------------+

38+| MN Int String | Machine-chosen names |

39++---------------------------+-----------------------------------------------+

40+| SN SpecialName | Special names, to make conflicts impossible |

41+| | and language features predictable |

42++---------------------------+-----------------------------------------------+

43+

44+

45+A user defined name can be constructed like this:

46+

47+.. code-block:: idris

48+

49+ Idris> UN "x"

50+ UN "x" : TTName

51+

52+.. code-block:: idris

53+

54+ Idris> NS (UN "x") ["b","a"]

55+ NS (UN "x") ["b", "a"] : TTName

56+

57+.. list-table:: machine-chosen names

58+

59+ * - A machine-chosen name needs to be generated within an Elab monad (see below for details) and is unique within that monad. This produced: {abc_140}{abc_141} so although gensym "abc" was called twice each one had a different integer:

60+ - example of unique names:

61+

62+ .. code-block:: idris

63+

64+ implementation Show TTName where

65+ show (UN s) = "`{{"++s++"}}"

66+ show (NS t a) = "{"++(show t)++

67+ "."++(show a)++"}"

68+ show (MN i s) = "{"++s++"_"++(show i)++"}"

69+ show (SN sn) = "TTName special name"

70+

71+ showGensym : String

72+ showGensym = %runElab (do

73+ debugMessage [TextPart (

74+ show !(gensym "abc") ++

75+ show !(gensym "abc")

76+ )]

77+ )

78+

79+Quasiquotation

80+--------------

81+

82+Since names are used frequently in elaborator reflection there is a shortcut for constructing them:

83+

84+.. list-table::

85+

86+ * - An unresolved variable "x" is wrapped in backtick and double braces:

87+ - .. code-block:: idris

88+

89+ Idris> `{{x}}

90+ UN "x" : TTName

91+

92+

93+ * - Single braces are used for existing variables:

94+ - .. code-block:: idris

95+

96+ Idris> `{x}

97+ No such variable x

98+ Idris> :let x=2

99+ Idris> `{x}

100+ UN "x" : TTName

101+

102+

103+ * - brackets are used for an expression:

104+

105+ here type is inferable

106+ - .. code-block:: idris

107+

108+ Idris> :let a=2

109+ Idris> `(a)

110+ P Ref (UN "a") (TConst (AType (ATInt ITBig))) : TT

111+

112+

113+ * - Expression with explicit type:

114+ - .. code-block:: idris

115+

116+ Idris> `(a:Integer)

117+ P Ref (UN "a") (TConst (AType (ATInt ITBig))) : TT

118+

119+

120+ * - If we want the value we can escape from quasiquotation by using anti-quotation (tilde)

121+ - .. code-block:: idris

122+

123+ Idris> `(~a)

124+ 2 : Integer

125+

126+quasiquotation summary:

127+

128++------------+-----------+----------------------------------------------------------+

129+| |Reification| |

130++============+===========+==========================================================+

131+| \`{{n}} | TTName | Use for new names. Unresolved quotation of the name n. |

132++------------+-----------+----------------------------------------------------------+

133+| \`{n} | TTName | Use for existing names. Resolved quotation of the name |

134+| | | n. n is a reference to a unique name in scope. |

135++------------+-----------+----------------------------------------------------------+

136+| \`(e) | | expression e for which a type is inferable. |

137++------------+-----------+----------------------------------------------------------+

138+| \`(e:t) | | expression e with a given type e. |

139++------------+-----------+----------------------------------------------------------+

140+| ~a | | anti-quotation - sub region which can be evaluated rather|

141+| | | than quoted. |

142++------------+-----------+----------------------------------------------------------+

143+|(Var\`{{x}})| Raw | |

144++------------+-----------+----------------------------------------------------------+

145+

146+TT

147+--

148+

149+There is a notation for a term in TT as it is being constructed (based on a BNF-like grammar), this is used for example in the debug output, it is a compact way to see the state of the term so it is used here.

150+So internally the program is stored as a tree structure using the following syntax:

151+

152++------------+-------+-------------+---------------------------------------------+

154++============+=======+=============+=============================================+

155+| term | | t | |

156++------------+-------+-------------+---------------------------------------------+

157+| binding | | b | |

158++------------+-------+-------------+---------------------------------------------+

159+| constant | t ::= | c | |

160++------------+-------+-------------+---------------------------------------------+

161+| variable | t ::= | x | |

162++------------+-------+-------------+---------------------------------------------+

163+| variable | t ::= | b.t | so a dot '.' tells us this is some sort of |

165++------------+-------+-------------+---------------------------------------------+

166+| application| t ::= | t t | As with Idris, juxtaposition indicates |

167+| | | | function application. Note: the same symbol |

168+| | | | 't' is used for both terms, this does not |

169+| | | | imply they are the same term. |

170++------------+-------+-------------+---------------------------------------------+

171+| Type | t ::= | T | |

172+| constructor| | | |

173++------------+-------+-------------+---------------------------------------------+

174+| Data | t ::= | C | |

175+| constructor| | | |

176++------------+-------+-------------+---------------------------------------------+

177+| function | b::= | λ c:t | colon ':' separates parameters from body of |

178+| | | | binding. |

179++------------+-------+-------------+---------------------------------------------+

180+| let binding| b::= | let\|-> t:t | |

181++------------+-------+-------------+---------------------------------------------+

182+| function | b::= | ∀ x:t | |

183++------------+-------+-------------+---------------------------------------------+

184+| Type | c::= | \*i | The universe hierarchy is usually handled |

186+| | | | type of types. |

187++------------+-------+-------------+---------------------------------------------+

188+| integer | c:== | i | |

189+| literal | | | |

190++------------+-------+-------------+---------------------------------------------+

191+|integer type| c:== | Integer | |

192++------------+-------+-------------+---------------------------------------------+

193+| string | c:== | s | |

194+| literal | | | |

195++------------+-------+-------------+---------------------------------------------+

196+| string type| c:== | String | |

197++------------+-------+-------------+---------------------------------------------+

198+|focused hole| | ?x : t | Conor McBride 1999 thesis. |

199++------------+-------+-------------+---------------------------------------------+

200+| guess | | ?x ≈ t : t | Conor McBride 1999 thesis. |

201++------------+-------+-------------+---------------------------------------------+

202+

203+Sometimes the part of the term in focus is underlined.

204+

205+Reflection of the well typed core language

206+

207+.. code-block:: idris

208+

209+ data TT =

210+ ||| A reference to some name (P for Parameter)

211+ P NameType TTName TT |

212+ ||| de Bruijn variables

213+ V Int |

214+ ||| Bind a variable

215+ Bind TTName (Binder TT) TT |

216+ ||| Apply one term to another

217+ App TT TT |

218+ ||| Embed a constant

219+ TConst Const |

220+ ||| Erased terms

221+ Erased |

222+ ||| The type of types along (with universe constraints)

223+ TType TTUExp |

224+ ||| Alternative universes for dealing with uniqueness

225+ UType Universe |

226+

227+TT stores local bound variables using De Bruijn index, when working in Idris this does not concern the user because string names are used for variables. Converting bound variables internally to index values means that the same variable name can be used, in different lambda terms, without ambiguity and without the need for α-substitution.

228+De Bruijn index which is a integer where:

229+

230+- 0=inside current (inner) lambda term

231+- 1= next outer lambda term

232+- 2= next outer and so on

233+

234+Raw

235+---

236+

237+Raw is similar to TT except it is used before types are known. The types should be resolved by the type checker.

238+

239+.. code-block:: idris

240+

241+ data Raw =

242+ ||| Variables, global or local

243+ Var TTName |

244+ ||| Bind a variable

245+ RBind TTName (Binder Raw) Raw |

246+ ||| Application

247+ RApp Raw Raw |

248+ ||| The type of types

249+ RType |

250+ ||| Alternative universes for dealing with uniqueness

251+ RUType Universe |

252+ ||| Embed a constant

253+ RConstant Const |

254+

255+Expression Syntax

256+-----------------

257+

258+There is a way of notating expressions such as those used in the proof state (example: goal type and proof terms) which is reasonably standard in the papers written about this subject.

259+

260+This notation is not entered directly by metaprogrammers but it is seen, for example in debug output. So this notation is explained here because it is useful to be familiar with it.

261+

262+The notation assumes right-associativity, in the absence of brackets, the term to the right binds more tightly than the one on the left.

263+So, for nested lambda terms:

264+

265++---------------+--------+-------------------+

266+| λ a . λ b . f | means | λ a .( λ b . f) |

267++---------------+--------+-------------------+

268+

269+and the same for function application:

270+

271++---------------+--------+-------------------+

272+| f g x | means | f (g x) |

273++---------------+--------+-------------------+

274+

275+In contrast, in lambda calculus, function application is usually regarded as left-associative,

276+Here are some typical examples of the notation used for expressions:

277+

278++-----------------------+

279+| ? {hole_0} . {hole_0} |

280++-----------------------+

281+

282+The term, to be derived, may start off in this state following something like this:

283+

284+.. code-block:: idris

285+

286+ myScript : Elab ()

287+ myScript= do

288+

289+The dot '.' tells us this is some sort of binding.

290+

291++----------------------------------------------+

292+| ?{hole_0} ≈ ? {hole_2} . {hole_2} . {hole_0} |

293++----------------------------------------------+

294+

295+This is a slightly more complicated example arising from:

296+

297+.. code-block:: idris

298+

299+ idNat : Nat -> Nat

300+ idNat = %runElab (do

301+

302+This introduces a guess that hole_0 = hole_2 .

303+

304++----------------------------------------------------+

305+| ?{hole_0} ≈ λ x . ? {hole_2} . {hole_2} . {hole_0} |

306++----------------------------------------------------+

307+

308+Following on from the previous example a lambda function is introduced like this:

309+

310+.. code-block:: idris

311+

312+ intro `{{x}}

313+

314+So now the expression is wrapped in a lambda binding.

315+

316++-------------------------------------------------------+

317+| ?{hole_0} ≈ λ x . ?{hole_2} ≈ x . {hole_2} . {hole_0} |

318++-------------------------------------------------------+

319+

320+Following on, we can use the fill tactic like this:

321+

322+.. code-block:: idris

323+

324+ fill (Var `{{x}})

325+

326+which introduces another guess.

327+

328++--------------------------------+

329+| ?{hole_0} ≈ λ x . x . {hole_0} |

330++--------------------------------+

331+

332+The solve tactic completes the proof

333+

334+Binders

335+-------

336+

337+Here we look at each tactic in turn to see how they affect the proof state.

338+

339+Introduction tactics for binders. The binder types are:

340+

341+- lambda function (intro)

342+- dependent function (forall)

343+- let (letBind)

344+- pattern (patbind)

345+

346+A precondition of these tactics is that the focused hole is of the form:

347+

348++-----------+

349+| ?h : t.h |

350++-----------+

351+

352+that is, that the body of its scope consists directly of a reference to the hole-bound variable. If a hole binder were of the form:

353+

354++--------------------+

355+| ?h : t1 -> t2.f h |

356++--------------------+

357+

358+and a tactic such as

359+

360++------------------+

361+| intro \`{{x}} |

362++------------------+

363+

364+were applied, the result would be the term

365+

366++---------------------------+

367+| ?h : t2 . λ x:t1. f h |

368++---------------------------+

369+

370+However this would cause the application of f to be ill-typed, as it expects an argument of type t1->t2, not an argument of type t2. Additionally, some binding tactics require that t, the type of the hole h, have a specific form, because the binder to be established may have a typing rule associated with it.

371+

372+.. list-table:: Binders

373+ :widths: 10 30

374+ :stub-columns: 1

375+

376+ * - intro

377+ - Introduce a lambda binding around the current hole and focus on the body.

378+

379+ Requires that the hole be in binding form (use 'attack' if it might not be).

380+

381+ @ n the name to use for the argument.

382+

383+ Signature:

384+

385+ intro : (n : TTName) -> Elab ()

386+

387+ Result

388+

389+ λn:t1.?h:[n/x]t2.h

390+

391+ * - intro'

392+ - Introduce a lambda binding around the current hole and focus on the body, using the name provided by the type of the hole.

393+

394+ Requires that the hole be immediately under its binder (use 'attack' if it might not be).

395+

396+ Signature:

397+

398+ intro' : Elab ()

399+

400+ * - forall

401+ - Introduce a dependent function type binding into the current hole, and focus on the body. Requires that the hole be immediately under its binder

402+

403+ (use 'attack' if it might not be).

404+

405+ Signature:

406+

407+ forall : TTName -> Raw -> Elab ()

408+

409+ * - patbind

410+ - Introduce a new pattern binding. Requires that the hole be immediately under its binder (use 'attack' if it might not be).

411+

412+ Signature:

413+

414+ patbind : TTName -> Elab ()

415+

416+ * - letbind

417+ - Introduce a new let binding.

418+

419+ Requires that the hole be immediately under its binder (use 'attack' if it might not be).

420+

421+ - @ n the name to let bind

422+ - @ ty the type of the term to be let-bound

423+ - @ tm the term to be bound

424+

425+ Signature:

426+

427+ letbind : (n : TTName) -> (ty, tm : Raw) -> Elab ()

428+

429+.. target-notes::

430+.. _`Dependently Typed Functional Programs and their Proofs by McBride 1999`: https://www.era.lib.ed.ac.uk/handle/1842/374