This function generates simple .dot files for visualizing automata transition functions using Graphviz. It's primary purpose is debugging large sets of automatically generated transitions (e.g., the inflections of Latin verbs).
prepGraph :: ( ... ) => NFA c b a -> [String]
prepGraph nfa = "digraph finite_state_machine {"
: wrapSp "rankdir = LR"
: wrapSp ("node [shape = circle]" ++ (mapSp (states nfa \\ terminal nfa)))
: wrapSp ("node [shape = doublecircle]" ++ (mapSp $ terminal nfa))
: formatGraph nfa ++ ["}"]
formatGraph :: ( ... ) => NFA c b a -> [String]
formatGraph = map formatDelta . deltaTuples
where formatDelta (a, a', bc) = wrapSp (mkArrow a a' ++ " " ++ mkLabel bc)
mkArrow x y = show x ++ " -> " ++ show y
mkLabel (y, z) = case z of
(Just t) -> "[ label = \"(" ++ show y ++ ", " ++ show t ++ ")\" ]"
Nothing -> "[ label = \"(" ++ show y ++ ", " ++ "Null" ++ ")\" ]"
where wrap, wrapSp and mapSp are formatting functions, as is deltaTuples.
The problem is that formatGraph retains double quotes around Strings, which causes errors in Graphviz. E.g., when I print unlines $ prepGraph to a file, I get things like:
0 -> 1 [ label = "('a', "N. SF")" ];
instead of
0 -> 1 [ label = "('a', N. SF)" ];
(However, "Null" seems to work fine, and outputs perfectly well). Now of course the string "N. SF" isn't the actual form I use to store inflections, but that form does include a String or two. So how can I tell Haskell: when you show a String values, don't double-quote it?
Check out how Martin Erwig handled the same problem in Data.Graph.Inductive.Graphviz:
http://hackage.haskell.org/packages/archive/fgl/5.4.2.3/doc/html/src/Data-Graph-Inductive-Graphviz.html
The function you're looking for is "sq" at the bottom:
sq :: String -> String
sq s#[c] = s
sq ('"':s) | last s == '"' = init s
| otherwise = s
sq ('\'':s) | last s == '\'' = init s
| otherwise = s
sq s = s
(check out the context and adapt for your own code, of course)
Use dotgen package - it has special safeguards in place to prevent forbidden chars from sneaking into attribute values.
You could define your own typeClass like this:
class GShow a where
gShow :: a -> String
gShow = show
instance GShow String where
show = id
instance GShow Integer
instance GShow Char
-- And so on for all the types you need.
The default implementation for "gShow" is "show", so you don't need a "where" clause for every instance. But you do need all the instances, which is a bit of a drag.
Alternatively you could use overlapping instances. I think (although I haven't tried it) that this will let you replace the list of instances using the default "gShow" by a single line:
instance (Show a) => GShow a
The idea is that with overlapping instances the compiler will chose the most specific instance available. So for strings it will pick the string instance over the more general one, and for everything else the general one is the only one that matches.
It seems a little ugly, but you could apply a filter to show t
filter (/='"') (show t)
Related
I'm really new in Haskell. perhaps this is a simple question for the master coder. I want to remove the quotation mark outside the string. such as "A1" to A1.
I tried my best to solve this problem. but it doesn't work. I already used read :: Read a => String -> a, id function, and regular expression.
initialGuess :: ([Pitch],GameState)
initialGuess = (startGuess, GameState (chords))
where startGuess = ["A1", "B1", "C1"]
standard = [[n, o] | n <- ['A'..'G'], o <- ['1'..'3']] <br/>
chords = [[a,b,c] | a <- standard, b <- standard, c <-
standard, a /= b, b /= c, a /= c]
initialGuess aim to takes no input arguments.And returns a pair of an initial guess and a game state. Running this code I can get
["A1","B1","C1"],GameState [["A1","A2","A3"],["A1","A2","B1"],["A1","A2","B2"],["A1","A2","B3"],["A1","A2","C1"]................["A1","A2","C2"],["A1","A2","C3"],["A1","A2","D1"],["A1","A2","D2"],["A1","A2","D3"]]
however, I want to remove these quotation marks such as
[A1,B1,C1],GameState [[A1,A2,A3],[A1,A2,B1],[A1,A2,B2],[A1,A2,B3],[A1,A2,C1]................[A1,A2,C2],[A1,A2,C3],[A1,A2,D1],[A1,A2,D2],[A1,A2,D3]]
If all you are trying to do is make the states prettier, you can provide your own display function:
-- |Display a list of strings without quotation marks
showLStr :: [String] -> String
showLStr p = "[" ++ intercalate "," p ++ "]"
The intercalate function puts the "," between each element of the list of strings.
showLStr ["A1", "A2", "A3"]
should display as
[A1,A2,A3]
Edit: Now you can use showLStr to display the game state:
showGameState :: GameState -> String
showGameState (GameState chords) =
"GameState [" ++ (intercalate "," $ map showLStr chords) ++ "]"
-- |Make GameState "Show"-able
instance Show GameState where
show = showGameState
showGuess :: ([Pitch],GameState) -> String
showGuess (pitch, gameState) = showLStr pitch ++ ", " ++ show gameState
I am trying to build a string representation for the show function of a typeclass representing a polynomial. I keep getting type errors of a mismatch from 'Char' to '[Char]', but from my understanding haskell's "append" function should be able to concatenate a Char to a string/[Char]. I don't understand where the problem lies, or where to look for a solution based on the errors I receive. here is the faulty code:
newtype Poly a = P [a]
instance (Num a, Show a) => Show (Poly a) where
show p = ["" : form (p !! i) i | i <- [l,(l-1)..0]]
where
l = length p
form e i
| i == 0 = elem
| i == 1 = elem ++ "x + "
| otherwise = elem ++ "x^" ++ (show i) ++ " + "
where elem = show e
any help would be greatly appreciated, thanks in advance.
You write
from my understanding haskell's "append" function should be able to concatenate a Char to a string/[Char].
I have no idea where you got this idea. It's wrong. I'm guessing you've defined
type Poly a = [a]
and I'll go with that assumption.
instance (Num a, Show a) => Show (Poly a) where
This is wrong. Poly is a type synonym. You can only declare instances for proper first-class types (the application of a type constructor to zero or more type variables). You can fix this by using, instead,
newtype Poly a = Poly {getPoly :: [a]}
but then you need to wrap/unwrap the Poly data constructor as required. Once you've gotten this right, you'll probably see that the Num constraint you've given is unnecessary.
show p = ["" ++ form (p !! i) i | i <- [(length p)..0]]
There are a few problems. The big one is that this does not define a string (list of characters) but rather a list of strings. You can fix this, generally, by applying concat to the result. The second one is that "" ++ anything is just anything, because concatenating the empty list to another list doesn't do anything. The third problem is that you're trying to count down, but you've done it wrong. That notation only counts up. To count down, you have to show that you want to count down:
let lp = length p in [lp, (lp-1) .. 0]
The last thing I see immediately (some of these mistakes are repeated in the preceding two lines):
| otherwise = e ++ "x^" ++ i ++ " + "
Now i is an Int, and ++ only works for lists. So that will not work. You need to first convert i to a string using show. e is of type a, and needs to be converted to a string using show as well.
I'm working on a way of representing memory in Haskell that looks like this...
data MemVal = Stored Value | Unbound
deriving Show
type Memory = ([Ide],Ide -> MemVal)
As an Identifier is called its added to the list of Identifiers. If an error occurs in the program I want to be able to recall the identifiers used up to date. So far I have this...
display :: Memory -> String
display m = "Memory = " ++ show (map (snd m) (fst m)) ++ " "
But was wondering if there were a way to map the name of the identifier to (fst m) as well as the function (snd m) so the output will be similar to...
Memory = [sum = stored Numeric 1, x = stored Boolean true]
Thank you.
You probably want something like this
display :: Memory -> String
display (ides, mem) =
"Memory = [" ++ unwords (map (\x -> x ++ "=" ++ mem x) ides) ++ "]"
I'm guessing this is what you are after:
import Data.List (intercalate)
display (Memory ids f) = "Memory = [" ++ (intercalates ", " assigns) ++ "]"
where assigns = [ show i ++ " = " ++ show (f i) | i <- ids ]
Here assigns is a list like:
[ "sum = stored Numeric 1", "x = stored Boolean true", ...]
and intercalate ", " assigns joins the strings together.
I've used destructuring to avoid having to refer to fst ... and snd ...
I have a "public safe" that may fail with a (potentially informative) errors:
data EnigmaError = BadRotors
| BadWindows
| MiscError String
instance Show EnigmaError where
show BadRotors = "Bad rotors"
show BadWindows = "Bad windows"
show (MiscError str) = str
configEnigma :: String -> String -> String -> String -> Except EnigmaError EnigmaConfig
configEnigma rots winds plug rngs = do
unless (and $ [(>=1),(<=26)] <*> rngs') (throwError BadRotors)
unless (and $ (`elem` letters) <$> winds') (throwError BadWindows)
-- ...
return EnigmaConfig {
components = components',
positions = zipWith (\w r -> (mod (numA0 w - r + 1) 26) + 1) winds' rngs',
rings = rngs'
}
where
rngs' = reverse $ (read <$> (splitOn "." $ "01." ++ rngs ++ ".01") :: [Int])
winds' = "A" ++ reverse winds ++ "A"
components' = reverse $ splitOn "-" $ rots ++ "-" ++ plug
but it is unclear how I should call this, particularly (and specifically) in implementing Read and Arbitrary (for QuickCheck).
For the former, I can get as far as
instance Read EnigmaConfig where
readsPrec _ i = case runExcept (configEnigma c w s r) of
Right cfg -> [(cfg, "")]
Left err -> undefined
where [c, w, s, r] = words i
but this seems to end up hiding error information available in err; while for the latter, I'm stuck at
instance Arbitrary EnigmaConfig where
arbitrary = do
nc <- choose (3,4) -- This could cover a wider range
ws <- replicateM nc capitals
cs <- replicateM nc (elements rotors)
uk <- elements reflectors
rs <- replicateM nc (choose (1,26))
return $ configEnigma (intercalate "-" (uk:cs))
ws
"UX.MO.KZ.AY.EF.PL" -- TBD - Generate plugboard and test <<<
(intercalate "." $ (printf "%02d") <$> (rs :: [Int]))
which fails with a mismatch between the expected and actual types:
Expected type: Gen EnigmaConfig
Actual type: Gen (transformers-0.4.2.0:Control.Monad.Trans.Except.Except Crypto.Enigma.EnigmaError EnigmaConfig)
How do I call a ("public safe") constructor when it may fail, particularly when using it in implementing Read and Arbitrary for my class?
The Read typeclass represents parses as lists of successes (with failures being the same as no successes); so rather than undefined you should return []. As for losing information about what went wrong: that's true, and the type of readsPrec means you can't do much about that. If you really, really wanted to [note: I don't think you should want this] you could define a newtype wrapper around Except EnigmaError EnigmaConfig and give that a Read instance that had successful parses of configuration errors.
For Arbitrary you have a couple choices. One choice is so-called rejection sampling; e.g.
arbitrary = do
-- ...
case configEnigma ... of
Left err -> arbitrary -- try again
Right v -> return v
You might also consider an Arbitrary instance to be part of your internal API, and use unsafe, internal calls rather than using the safe, public API for constructing your configuration. Other options include calling error or fail. (I consider these four options to be in roughly preference order -- rejection sampling, then unsafe internal calls, then error, then fail -- though your judgement may differ.)
How can one create LaTeX source for natural deduction proof trees (like those shown here) via Haskell eg using HaTeX? I'd like to emulate LaTeX .stys like bussproofs.sty or proof.sty.
I'm using your question as an excuse to improve and demo a Haskell
call-tracing library I'm working
on. In the context of
tracing, an obvious way to create a proof tree is to trace a type
checker and then format the trace as a natural-deduction proof. To
keep things simple my example logic is the simply-typed lambda
calculus
(STLC),
which corresponds to the implicational fragment of propositional
intuitionistic
logic.
I am using proofs.sty, but not via HaTeX or any other Haskell
Latex library. The Latex for proof trees is very simple and using a
Haskell Latex library would just complicate things.
I've written the proof-tree generation code twice:
in a self-contained way, by writing a type checker that also
returns a proof tree;
using my tracing library, by call-tracing a type checker and then
post processing the trace into a proof tree.
Since you didn't ask about call-tracing libraries, you may be less
interested in the call-trace based version, but I think it's
interesting to compare both versions.
Examples
Let's start with some output examples first, to see what all this gets us.
The first three examples are motivated
by an axiom system for implicational propositional
calculus;
the first two also happen to correspond to S and
K:
The first axiom, K, with proof terms:
The second axiom, S, with proof terms, but with premises in the
context, not lambda bound:
The fourth axiom, modus ponens, without proof terms:
The third axiom in that Wikipedia article (Peirce's law) is
non-constructive and so we can't prove it here.
For a different kind of example, here's a failed type check of the Y
combinator:
The arrows are meant to lead you to the error, which is marked with a
bang (!).
Code
Now I'll describe the code which generated those examples. The code is
from this
file
unless otherwise noted. I'm not including every line of code here;
see that link if you want something you can actually build using GHC
7.6.3.
Most of the code -- the grammar, parser, and pretty printer -- is the
same for both versions; only the type checkers and proof-tree
generators differ. All of the common code is in the file just
referenced.
STLC grammar
The STLC grammar in ASCII:
-- Terms
e ::= x | \x:T.e | e e
-- Types
T ::= A | T -> T
-- Contexts
C ::= . | C,x:T
And the corresponding Haskell:
type TmVar = String
type TyVar = String
data Tm = Lam TmVar Ty Tm
| TmVar TmVar
| Tm :#: Tm
deriving Show
data Ty = TyVar TyVar
| Ty :->: Ty
deriving (Eq , Show)
type Ctx = [(TmVar,Ty)]
Type checking + proof tree generation
Both versions implement the same abstract STLC type checker. In ASCII:
(x:T) in C
---------- Axiom
C |- x : T
C,x:T1 |- e : T2
--------------------- -> Introduction
C |- \x:T1.e : T1->T2
C |- e : T1 -> T2 C |- e1 : T1
--------------------------------- -> Elimination
C |- e e1 : T2
Version 1: self-contained with inline proof-tree generation
The full code for this version is
here.
The proof-tree generation happens in the type checker, but the actual
proof-tree generation code is factored out into addProof and
conclusion.
Type checking
-- The mode is 'True' if proof terms should be included.
data R = R { _ctx :: Ctx , _mode :: Bool }
type M a = Reader R a
extendCtx :: TmVar -> Ty -> M a -> M a
extendCtx x t = local extend where
extend r = r { _ctx = _ctx r ++ [(x,t)] }
-- These take the place of the inferred type when there is a type
-- error.
here , there :: String
here = "\\,!"
there = "\\,\\uparrow"
-- Return the inferred type---or error string if type inference
-- fails---and the latex proof-tree presentation of the inference.
--
-- This produces different output than 'infer' in the error case: here
-- all premises are always computed, whereas 'infer' stops at the
-- first failing premise.
inferProof :: Tm -> M (Either String Ty , String)
inferProof tm#(Lam x t e) = do
(et' , p) <- extendCtx x t . inferProof $ e
let et'' = (t :->:) <$> et'
addProof et'' [p] tm
inferProof tm#(TmVar x) = do
mt <- lookup x <$> asks _ctx
let et = maybe (Left here) Right mt
addProof et [] tm
inferProof tm#(e :#: e1) = do
(et , p) <- inferProof e
(et1 , p1) <- inferProof e1
case (et , et1) of
(Right t , Right t1) ->
case t of
t1' :->: t2 | t1' == t1 -> addProof (Right t2) [p , p1] tm
_ -> addProof (Left here) [p , p1] tm
_ -> addProof (Left there) [p , p1] tm
Proof tree generation
The addProof corresponds to proofTree in the other version:
-- Given the inferred type, the proof-trees for all premise inferences
-- (subcalls), and the input term, annotate the inferred type with a
-- result proof tree.
addProof :: Either String Ty -> [String] -> Tm -> M (Either String Ty , String)
addProof et premises tm = do
R { _mode , _ctx } <- ask
let (judgment , rule) = conclusion _mode _ctx tm et
let tex = "\\infer[ " ++ rule ++ " ]{ " ++
judgment ++ " }{ " ++
intercalate " & " premises ++ " }"
return (et , tex)
The code for conclusion is common to both versions:
conclusion :: Mode -> Ctx -> Tm -> Either String Ty -> (String , String)
conclusion mode ctx tm e = (judgment mode , rule tm)
where
rule (TmVar _) = "\\textsc{Axiom}"
rule (Lam {}) = "\\to \\text{I}"
rule (_ :#: _) = "\\to \\text{E}"
tyOrError = either id pp e
judgment True = pp ctx ++ " \\vdash " ++ pp tm ++ " : " ++ tyOrError
judgment False = ppCtxOnlyTypes ctx ++ " \\vdash " ++ tyOrError
Version 2: via call-tracing, with proof-tree generation as post processing
Here the type checker is not even aware of proof-tree generation, and
adding call-tracing is just one line.
Type checking
type Mode = Bool
type Stream = LogStream (ProofTree Mode)
type M a = ErrorT String (ReaderT Ctx (Writer Stream)) a
type InferTy = Tm -> M Ty
infer , infer' :: InferTy
infer = simpleLogger (Proxy::Proxy "infer") ask (return ()) infer'
infer' (TmVar x) = maybe err pure . lookup x =<< ask where
err = throwError $ "Variable " ++ x ++ " not in context!"
infer' (Lam x t e) = (t :->:) <$> (local (++ [(x,t)]) . infer $ e)
infer' (e :#: e1) = do
t <- infer e
t1 <- infer e1
case t of
t1' :->: t2 | t1' == t1 -> pure t2
_ -> throwError $ "Can't apply " ++ show t ++ " to " ++ show t1 ++ "!"
The LogStream
type
and ProofTree
class
are from the library. The LogStream is the type of log events that
the "magic"
simpleLogger
logs. Note the line
infer = simpleLogger (Proxy::Proxy "infer") ask (return ()) infer'
which defines infer to be a logged version of infer', the actual
type checker. That's all you have to do to trace a monadic function!
I won't get into how simpleLogger actually works here, but the
result is that each call to infer gets logged, including the
context, arguments, and return value, and these data get grouped
together with all logged subcalls (here only to infer). It would be
easy to manually write such logging code for infer, but it's nice
that with the library you don't have to.
Proof-tree generation
To generate the Latex proof trees, we implement ProofTree to post
process infer's call trace. The library provides a proofTree
function that calls the ProofTree methods and assembles the proof
trees; we just need to specify how the conclusions of the typing
judgments will be formatted:
instance ProofTree Mode (Proxy (SimpleCall "infer" Ctx InferTy ())) where
callAndReturn mode t = conclusion mode ctx tm (Right ty)
where
(tm , ()) = _arg t
ty = _ret t
ctx = _before t
callAndError mode t = conclusion mode ctx tm (Left error)
where
(tm , ()) = _arg' t
how = _how t
ctx = _before' t
error = maybe "\\,!" (const "\\,\\uparrow") how
The pp calls are to a user defined pretty printer; obviously, the
library can't know how to pretty print your data types.
Because calls can be erroneous -- the library detects errors
-- we have to say how to format
successful and failing calls. Refer to the Y-combinator example above
for an example a failing type check, corresponding to the
callAndError case here.
The library's proofTree
function
is quite simple: it builds a proofs.sty proof tree with the current
call as conclusion, and the subcalls as premises:
proofTree :: mode -> Ex2T (LogTree (ProofTree mode)) -> String
proofTree mode (Ex2T t#(CallAndReturn {})) =
"\\infer[ " ++ rule ++ " ]{ " ++ conclusion ++ " }{ " ++ intercalate " & " premises ++ " }"
where
(conclusion , rule) = callAndReturn mode t
premises = map (proofTree mode) (_children t)
proofTree mode (Ex2T t#(CallAndError {})) =
"\\infer[ " ++ rule ++ " ]{ " ++ conclusion ++ " }{ " ++ intercalate " & " premises ++ " }"
where
(conclusion , rule) = callAndError mode t
premises = map (proofTree mode)
(_children' t ++ maybe [] (:[]) (_how t))
I use proofs.sty in the library because it allows arbitrarily many
premises, although bussproofs.sty would work for this STLC example
since no rule has more than five premises (the limit for
bussproofs). Both Latex packages are described
here.
Pretty printing
Now we return to code that is common between both versions.
The pretty printer that defines the pp used above is rather long --
it handles precedence and associativity, and is written in a way that
should be extensible if more terms, e.g. products, are added to the
calculus -- but mostly straightforward. First, we set up a precedence
table and a precedence-and-associativity-aware parenthesizer:
- Precedence: higher value means tighter binding.
type Prec = Double
between :: Prec -> Prec -> Prec
between x y = (x + y) / 2
lowest , highest , precLam , precApp , precArr :: Prec
highest = 1
lowest = 0
precLam = lowest
precApp = between precLam highest
precArr = lowest
-- Associativity: left, none, or right.
data Assoc = L | N | R deriving Eq
-- Wrap a pretty print when the context dictates.
wrap :: Pretty a => Assoc -> a -> a -> String
wrap side ctx x = if prec x `comp` prec ctx
then pp x
else parens . pp $ x
where
comp = if side == assoc x || assoc x == N
then (>=)
else (>)
parens s = "(" ++ s ++ ")"
And then we define the individual pretty printers:
class Pretty t where
pp :: t -> String
prec :: t -> Prec
prec _ = highest
assoc :: t -> Assoc
assoc _ = N
instance Pretty Ty where
pp (TyVar v) = v
pp t#(t1 :->: t2) = wrap L t t1 ++ " {\\to} " ++ wrap R t t2
prec (_ :->: _) = precArr
prec _ = highest
assoc (_ :->: _) = R
assoc _ = N
instance Pretty Tm where
pp (TmVar v) = v
pp (Lam x t e) = "\\lambda " ++ x ++ " {:} " ++ pp t ++ " . " ++ pp e
pp e#(e1 :#: e2) = wrap L e e1 ++ " " ++ wrap R e e2
prec (Lam {}) = precLam
prec (_ :#: _) = precApp
prec _ = highest
assoc (_ :#: _) = L
assoc _ = N
instance Pretty Ctx where
pp [] = "\\cdot"
pp ctx#(_:_) =
intercalate " , " [ x ++ " {:} " ++ pp t | (x,t) <- ctx ]
By adding a "mode" argument, it would be easy to use the same pretty
printer to print plain ASCII, which would be useful with other
call-trace post processors, such as the (unfinished) UnixTree
processor.
Parsing
A parser is not essential to the example, but of course I did not enter the
example input terms directly as Haskell Tms.
Recall the STLC grammar in ASCII:
-- Terms
e ::= x | \x:T.e | e e
-- Types
T ::= A | T -> T
-- Contexts
C ::= . | C,x:T
This grammar is ambiguous: both the term application e e
and function type T -> T have no associativity given by the
grammar. But in STLC term application is left associative and function
types are right associative, and so the corresponding disambiguated
grammar we actually parse is
-- Terms
e ::= e' | \x:T.e | e e'
e' ::= x | ( e )
-- Types
T ::= T' | T' -> T
T' ::= A | ( T )
-- Contexts
C ::= . | C,x:T
The parser is maybe too simple -- I'm not using a languageDef and
it's whitespace sensitive -- but it gets the job done:
type P a = Parsec String () a
parens :: P a -> P a
parens = Text.Parsec.between (char '(') (char ')')
tmVar , tyVar :: P String
tmVar = (:[]) <$> lower
tyVar = (:[]) <$> upper
tyAtom , arrs , ty :: P Ty
tyAtom = parens ty
<|> TyVar <$> tyVar
arrs = chainr1 tyAtom arrOp where
arrOp = string "->" *> pure (:->:)
ty = arrs
tmAtom , apps , lam , tm :: P Tm
tmAtom = parens tm
<|> TmVar <$> tmVar
apps = chainl1 tmAtom appOp where
appOp = pure (:#:)
lam = uncurry Lam <$> (char '\\' *> typing)
<*> (char '.' *> tm)
tm = apps <|> lam
typing :: P (TmVar , Ty)
typing = (,) <$> tmVar
<*> (char ':' *> ty)
ctx :: P Ctx
ctx = typing `sepBy` (char ',')
To clarify what the input terms look like, here are the examples from
the Makefile:
# OUTFILE CONTEXT TERM
./tm2latex.sh S.ctx 'x:P->Q->R,y:P->Q,z:P' 'xz(yz)'
./tm2latex.sh S.lam '' '\x:P->Q->R.\y:P->Q.\z:P.xz(yz)'
./tm2latex.sh S.err '' '\x:P->Q->R.\y:P->Q.\z:P.xzyz'
./tm2latex.sh K.ctx 'x:P,y:Q' 'x'
./tm2latex.sh K.lam '' '\x:P.\y:Q.x'
./tm2latex.sh I.ctx 'x:P' 'x'
./tm2latex.sh I.lam '' '\x:P.x'
./tm2latex.sh MP.ctx 'x:P,y:P->Q' 'yx'
./tm2latex.sh MP.lam '' '\x:P.\y:P->Q.yx'
./tm2latex.sh ZERO '' '\s:A->A.\z:A.z'
./tm2latex.sh SUCC '' '\n:(A->A)->(A->A).\s:A->A.\z:A.s(nsz)'
./tm2latex.sh ADD '' '\m:(A->A)->(A->A).\n:(A->A)->(A->A).\s:A->A.\z:A.ms(nsz)'
./tm2latex.sh MULT '' '\m:(A->A)->(A->A).\n:(A->A)->(A->A).\s:A->A.\z:A.m(ns)z'
./tm2latex.sh Y.err '' '\f:A->A.(\x:A.f(xx))(\x:A.f(xx))'
./tm2latex.sh Y.ctx 'a:A->(A->A),y:(A->A)->A' '\f:A->A.(\x:A.f(axx))(y(\x:A.f(axx)))'
Latex document generation
The ./tm2latex.sh script just calls pdflatex on the output of the
Haskell programs described above. The Haskell programs produce the proof
tree and then wrap it in a minimal Latex document:
unlines
[ "\\documentclass[10pt]{article}"
, "\\usepackage{proof}"
, "\\usepackage{amsmath}"
, "\\usepackage[landscape]{geometry}"
, "\\usepackage[cm]{fullpage}"
-- The most slender font I could find:
-- http://www.tug.dk/FontCatalogue/iwonalc/
, "\\usepackage[light,condensed,math]{iwona}"
, "\\usepackage[T1]{fontenc}"
, "\\begin{document}"
, "\\tiny"
, "\\[" ++ tex ++ "\\]"
, "\\end{document}"
]
As you can see, most of the Latex is devoted to making the proof trees
as small as possible; I plan to also write an ASCII proof tree post
processor, which may be more useful in practice when the examples are
larger.
Conclusion
As always, it takes a bit of code to write a parser, type checker, and
pretty printer. On top of that, adding proof-tree generation is
pretty simple in both versions. This is a fun toy example, but I
expect to do something similar in the context of a "real"
unification-based type checker for a dependently-typed language; there
I expect call tracing and proof-tree generation (in ASCII) to provide
significant help in debugging the type checker.