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Merge pull request #4700 from martinbaker/mydoc

Add Documentation for Elaborator Reflection

commit0736219e780e28305724f962f1a748bbaf6538b7 authorNiklas Larsson <metaniklas@gmail.com> dateMon 2 Dec 2019, 01:08 (7 years ago) parent21434e0cf3071d
50 files changed+7552−6
compare with master
unifiedsplit
docs/_static/theme_overrides.css+13 docs/conf.py+7−1 docs/image/README.md+16 docs/image/attack.png docs/image/attack.svg+199 docs/image/binders.png docs/image/binders.svg+326 docs/image/compareToProofAssist.png docs/image/compareToProofAssist.svg+260 docs/image/elab.png docs/image/elab.svg+139 docs/image/elabLogicEx1_1.png docs/image/elabLogicEx1_1.svg+99 docs/image/elabLogicEx1_2.png docs/image/elabLogicEx1_2.svg+112 docs/image/elabLogicEx1_3.png docs/image/elabLogicEx1_3.svg+112 docs/image/elabLogicEx1_4.png docs/image/elabLogicEx1_4.svg+80 docs/image/elabOverview.png docs/image/elabOverview.svg+585 docs/image/elabProofState.png docs/image/elabProofState.svg+356 docs/image/elabProofStateEx1_1.png docs/image/elabProofStateEx1_1.svg+312 docs/image/elabProofStateEx1_2.png docs/image/elabProofStateEx1_2.svg+281 docs/image/elabProofStateEx1_3.png docs/image/elabProofStateEx1_3.svg+244 docs/image/fill.png docs/image/fill.svg+163 docs/image/generateDatatype.png docs/image/generateDatatype.svg+540 docs/image/generateFunction.png docs/image/generateFunction.svg+588 docs/image/generateFunction2.png docs/image/generateFunction2.svg+430 docs/image/idrisTopLevel.png docs/image/idrisTopLevel.svg+524 docs/image/solve.png docs/image/solve.svg+163 docs/image/tree.png docs/image/tree.svg+212 docs/proofs/elabReflection.rst+60 docs/proofs/example1.rst+192 docs/proofs/generatingData.rst+385 docs/proofs/holes.rst+196 docs/proofs/index.rst+10−5 docs/proofs/primitive.rst+518 docs/proofs/tactics.rst+430
docs/_static/theme_overrides.cssnew file+13
@@ -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'
5151
5252# General information about the project.
5353project = u'Idris'
54−copyright = u'2017, The Idris Community'
54+copyright = u'2017-2019, The Idris Community'
5555author = u'The Idris Community'
5656
5757# 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
151151# so a file named "default.css" will overwrite the builtin "default.css".
152152html_static_path = ['_static']
153153
154+html_context = {
155+ 'css_files': [
156+ '_static/theme_overrides.css', # override wide tables in RTD theme
157+ ],
158+ }
159+
154160# Add any extra paths that contain custom files (such as robots.txt or
155161# .htaccess) here, relative to this directory. These files are copied
156162# directly to the root of the documentation.
docs/image/README.mdnew file+16
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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.
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155+ id="tspan4184">binding</tspan></text>
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162+ xml:space="preserve"
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165+ style="font-style:normal;font-variant:normal;font-weight:bold;font-stretch:normal;font-size:20px;font-family:sans-serif;-inkscape-font-specification:'sans-serif Bold'"
166+ y="307.36218"
167+ x="19"
168+ id="tspan4164"
169+ sodipodi:role="line">function</tspan><tspan
170+ style="font-style:normal;font-variant:normal;font-weight:bold;font-stretch:normal;font-size:20px;font-family:sans-serif;-inkscape-font-specification:'sans-serif Bold'"
171+ y="332.36218"
172+ x="19"
173+ sodipodi:role="line"
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175+ <text
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177+ style="font-style:normal;font-weight:normal;font-size:40px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#ff0000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;"
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187+ y="235.3622"
188+ style="font-style:normal;font-variant:normal;font-weight:bold;font-stretch:normal;font-size:20px;font-family:sans-serif;-inkscape-font-specification:'sans-serif Bold';fill:#ff0000;">λ x:t</tspan></text>
189+ <text
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192+ y="222.3622"
193+ x="259"
194+ style="font-style:normal;font-weight:normal;font-size:40px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#0000ff;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;"
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200+ x="259"
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202+ sodipodi:role="line">Γ ⊢ S : Type</tspan></text>
203+ <text
204+ sodipodi:linespacing="125%"
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206+ y="417.36218"
207+ x="124"
208+ style="font-style:normal;font-weight:normal;font-size:40px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#ff0000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;"
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239+ <text
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294+ style="font-style:normal;font-weight:normal;font-size:40px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#0000ff;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;"
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302+ sodipodi:role="line">Γ ⊢ S : Type Γ ⊢ s:S </tspan></text>
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324+ style="font-style:normal;font-variant:normal;font-weight:bold;font-stretch:normal;font-size:20px;font-family:sans-serif;-inkscape-font-specification:'sans-serif Bold';fill:#0000ff;">Γ ; let x ↦ s : S ⊢ valid</tspan></text>
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155+ style="font-style:normal;font-variant:normal;font-weight:bold;font-stretch:normal;font-size:15px;font-family:sans-serif;-inkscape-font-specification:'sans-serif Bold'">fill holes</tspan></text>
156+ <text
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160+ x="130"
161+ style="font-style:normal;font-weight:normal;font-size:40px;line-height:125%;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#ff0000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;"
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212+</svg>
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
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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
@@ -0,0 +1,196 @@
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 @@
11.. _proofs-index:
22
3−###############
4−Theorem Proving
5−###############
3+#########################################
4+Theorem Proving and Elaborator Reflection
5+#########################################
66
7−A tutorial on theorem proving in Idris.
7+A tutorial on theorem proving in Idris.
88
99.. note::
1010
@@ -25,4 +25,9 @@ A tutorial on theorem proving in Idris.
2525 propositional
2626 interactive
2727 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++------------+-------+-------------+---------------------------------------------+
153+| | | Syntax | More Information |
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 |
164+| binding | | | binding. |
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 |
185+| universe | | | automatically so we can just use\* for the |
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