Much ado about nothing

23 files changed, 1495 insertions, 263 deletions

diff --git a/.gitignore b/.gitignore
index dbb9822..8bf5f01 100644
--- a/.gitignore
+++ b/.gitignore
@@ -5,3 +5,6 @@ literature.pdf
 /thesis.meta.yml
 /thesis.pdf
 **/*.cabal
+**.log
+**.prof
+**.prof.*
diff --git a/edit-lens/package.yaml b/edit-lens/package.yaml
index b506ec1..88a35ca 100644
--- a/edit-lens/package.yaml
+++ b/edit-lens/package.yaml
@@ -9,32 +9,38 @@ extra-source-files:
   - ChangeLog.md
 git: https://git.yggdrasil.li/gkleen/pub/bachelor-thesis
 
+default-extensions:
+  - TypeFamilies
+  - FlexibleContexts
+  - FlexibleInstances
+  - MultiParamTypeClasses
+  - FunctionalDependencies
+  - AllowAmbiguousTypes
+  - TypeApplications
+  - GADTs
+  - RecordWildCards
+  - NamedFieldPuns
+  - PatternGuards
+  - TupleSections
+  - RankNTypes
+  - ViewPatterns
+  - DerivingStrategies
+
+dependencies:
+  - base
+  - lens
+  - containers
+  - composition-tree
+  - monad-memo
+  - Diff
+  - mtl
+  - wl-pprint
+  - intervals
+
+# ghc-options: [ -O2 ]
+
 library:
-  default-extensions:
-    - TypeFamilies
-    - FlexibleContexts
-    - FlexibleInstances
-    - MultiParamTypeClasses
-    - FunctionalDependencies
-    - AllowAmbiguousTypes
-    - TypeApplications
-    - GADTs
-    - RecordWildCards
-    - NamedFieldPuns
-    - PatternGuards
-    - TupleSections
-    - RankNTypes
-    - ViewPatterns
   source-dirs: src
-  dependencies:
-    - base
-    - lens
-    - containers
-    - composition-tree
-    - Diff
-    - mtl
-    - wl-pprint
-    - intervals
   exposed-modules:
     - Control.Edit
     - Control.Lens.Edit
@@ -42,4 +48,17 @@ library:
     - Control.FST
     - Control.DFST.Lens
     
-  
+tests:
+  test:
+    source-dirs:
+      - test
+      - src
+    main: Driver.hs
+    dependencies:
+      - tasty
+      - tasty-discover
+      - tasty-hedgehog
+      - tasty-hunit
+      - HUnit
+      - hedgehog
+      - deepseq
diff --git a/edit-lens/src/Control/DFST.lhs b/edit-lens/src/Control/DFST.lhs
index eae2e66..6e16c74 100644
--- a/edit-lens/src/Control/DFST.lhs
+++ b/edit-lens/src/Control/DFST.lhs
@@ -1,6 +1,7 @@
+\begin{comment}
 \begin{code}
 {-# LANGUAGE ScopedTypeVariables
-#-}
+  #-}
 
 {-|
 Description: Deterministic finite state transducers
@@ -11,8 +12,8 @@ module Control.DFST
   , toFST
   ) where
 
-import Data.Map.Strict (Map, (!?))
-import qualified Data.Map.Strict as Map
+import Data.Map.Lazy (Map, (!?))
+import qualified Data.Map.Lazy as Map
 
 import Data.Set (Set)
 import qualified Data.Set as Set
@@ -29,18 +30,34 @@ import Control.Monad.State
 
 import Control.FST (FST(FST))
 import qualified Control.FST as FST
+\end{code}
+\end{comment}
 
+\begin{defn}[deterministic finite state transducer]
+  Wir nennen einen FST \emph{deterministic}, wenn jedes Paar aus Ausgabezustand und Eingabesymbol maximal eine Transition zulässt, $\epsilon$-Transitionen keine Schleifen bilden und die Menge von initialen Zustände einelementig ist.
 
+  Zusätzlich ändern wir die Darstellung indem wir $\epsilon$-Transitionen kontrahieren.
+  Wir erweitern hierfür die Ausgabe pro Transition von einem einzelnen Zeichen zu einem Wort beliebiger Länge und fügen, bei jeder Kontraktion einer $\epsilon$-Transition $A \rightarrow B$, die Ausgabe der Transition vorne an die Ausgabe aller Transitionen $B \rightarrow \ast$ von $B$ an.
+\end{defn}
+  
+\begin{code}
 data DFST state input output = DFST
   { stInitial :: state
   , stTransition :: Map (state, input) (state, Seq output)
-    -- ^ All @(s, c)@-combinations not mapped are assumed to map to @(s, Nothing)@
   , stAccept :: Set state
   }
+\end{code}
 
+Die in der Definition von DFSTs beschriebene Umwandlung lässt sich umkehren, sich also jeder DFST auch als FST auffassen.
 
+Hierfür trennen wir Transitionen $A \overset{(\sigma, \delta^\ast)}{\rightarrow} B$ mit Eingabe $\sigma$ und Ausgabe-Wort $\delta^\ast = \delta_1 \delta_2 \ldots \delta_n$ in eine Serie von Transitionen $A \overset{(\sigma, \delta_1)}{\rightarrow} A_1 \overset{(\epsilon, \delta_2)}{\rightarrow} \ldots \overset{(\epsilon, \delta_n)}{\rightarrow} B$ auf.
+
+\begin{code}
 toFST :: forall state input output. (Ord state, Ord input, Ord output) => DFST state input output -> FST (state, Maybe (input, Natural)) input output
--- ^ Split apart non-singleton outputs into a series of epsilon-transitions
+-- ^ View a DFST as a FST splitting apart non-singleton outputs into a series of epsilon-transitions
+\end{code}
+\begin{comment}
+\begin{code}
 toFST DFST{..} = flip execState initialFST $ forM_ (Map.toList stTransition) handleTransition
   where
     initialFST = FST
@@ -62,21 +79,31 @@ toFST DFST{..} = flip execState initialFST $ forM_ (Map.toList stTransition) han
               -- Both calls to `handleTransition'` (one in `handleTransition`, the other below) satisfy one of the above cases
       addTransition (from, inS) (next, Just outS)
       handleTransition' next Nothing oo to
-  
+\end{code}
+\end{comment}
+
+Das Ausführen eines DFST auf eine gegebene Eingabe ist komplett analog zum Ausführen eines FST.
+Unsere Implementierung nutzt die restriktivere Struktur aus unserer Definition von DFSTs um den Determinismus bereits im Typsystem zu kodieren.
+
+\begin{code}
 runDFST :: forall state input output. (Ord state, Ord input) => DFST state input output -> Seq input -> Maybe (Seq output)
-runDFST dfst@DFST{..} str = let (finalState, str') = runDFST' dfst stInitial str Seq.empty
-                            in str' <$ guard (finalState `Set.member` stAccept)
+\end{code}
+\begin{comment}
+\begin{code}
+runDFST dfst@DFST{..} str = do
+  let (str', finalState') = runDFST' dfst stInitial str Seq.empty
+  finalState <- finalState'
+  str' <$ guard (finalState `Set.member` stAccept)
 
 runDFST' :: forall state input output. (Ord state, Ord input)
          => DFST state input output
          -> state               -- ^ Current state
          -> Seq input           -- ^ Remaining input
          -> Seq output          -- ^ Accumulator containing previous output
-         -> (state, Seq output) -- ^ Next state, altered output
-runDFST' _             st  Empty     acc = (st, acc)
-runDFST' dfst@DFST{..} st (c :<| cs) acc
-  | Just (st', mc') <- stTransition !? (st, c)
-  = runDFST' dfst st' cs $ acc <> mc'
-  | otherwise
-  = runDFST' dfst st cs acc
+         -> (Seq output, Maybe state) -- ^ Altered output, Next state
+runDFST' _             st  Empty     acc = (acc, Just st)
+runDFST' dfst@DFST{..} st (c :<| cs) acc = case stTransition !? (st, c) of
+  Just (st', mc') -> runDFST' dfst st' cs $ acc <> mc'
+  Nothing -> (acc, Nothing)
 \end{code}
+\end{comment}
diff --git a/edit-lens/src/Control/DFST/Lens.lhs b/edit-lens/src/Control/DFST/Lens.lhs
index 95be34e..fe33bd6 100644
--- a/edit-lens/src/Control/DFST/Lens.lhs
+++ b/edit-lens/src/Control/DFST/Lens.lhs
@@ -1,12 +1,14 @@
+\begin{comment}
 \begin{code}
 {-# LANGUAGE ScopedTypeVariables
            , TemplateHaskell
            , ConstraintKinds
+           , GeneralizedNewtypeDeriving
 #-}
 
 module Control.DFST.Lens
-  ( StringEdit(..)
-  , StringEdits(..)
+  ( StringEdit(..), sePos, seInsertion
+  , StringEdits(..), _StringEdits, _SEFail, stringEdits
   , insert, delete, replace
   , DFSTAction(..), DFSTComplement
   , dfstLens
@@ -16,7 +18,7 @@ module Control.DFST.Lens
 
 import Control.DFST
 import Control.FST hiding (stInitial, stTransition, stAccept)
-import qualified Control.FST as FST (stInitial, stTransition, stAccept)
+import qualified Control.FST as FST (stInitial, stTransition, stAccept, step)
 import Control.Lens.Edit
 import Control.Lens
 import Control.Lens.TH
@@ -32,11 +34,11 @@ import Data.Sequence (Seq((:<|), (:|>)))
 import qualified Data.Sequence as Seq
 import Data.Set (Set)
 import qualified Data.Set as Set
-import Data.Map.Strict (Map)
-import qualified Data.Map.Strict as Map
+import Data.Map.Lazy (Map)
+import qualified Data.Map.Lazy as Map
 
-import Data.Compositions.Snoc (Compositions)
-import qualified Data.Compositions.Snoc as Comp
+import Data.Compositions (Compositions)
+import qualified Data.Compositions as Comp
 
 import Data.Algorithm.Diff (Diff, getDiff)
 import qualified Data.Algorithm.Diff as Diff
@@ -48,69 +50,72 @@ import Data.Function (on)
 import Data.Foldable (toList)
 import Data.List (partition)
 
-import Debug.Trace
+import Control.Exception (assert)
 
+import System.IO.Unsafe
+import Text.PrettyPrint.Leijen (Pretty(..))
 
-data StringEdit char = Insert { _sePos :: Natural, _seInsertion :: char }
-                     | Delete { _sePos :: Natural }
+\end{code}
+\end{comment}
+
+
+Wir betrachten, zur Einfachheit, ein minimiales Set von Edits auf Strings\footnote{Wie in der Konstruktion zum Longest Common Subsequence Problem}:
+
+\begin{defn}[Atomare edits of strings]
+\begin{code}
+data StringEdit pos char = Insert { _sePos :: pos, _seInsertion :: char }
+                         | Delete { _sePos :: pos }
   deriving (Eq, Ord, Show, Read)
 
+-- Automatically derive van-leerhoven-lenses:
+--
+-- @sePos :: Lens' (StringEdits pos char) pos@
+-- @seInsertion :: Traversal' (StringEdits pos char) char@
 makeLenses ''StringEdit
+\end{code}
+\end{defn}
 
-data StringEdits char = StringEdits (Seq (StringEdit char))
-                      | SEFail
+Atomare edits werden, als Liste, zu edits komponiert.
+Wir führen einen speziellen edit ein, der nicht-Anwendbarkeit der edits repräsentiert:
+\begin{code}
+data StringEdits pos char = StringEdits (Seq (StringEdit pos char))
+                          | SEFail
   deriving (Eq, Ord, Show, Read)
 
 makePrisms ''StringEdits
 
-stringEdits :: Traversal' (StringEdits char) (StringEdit char)
+stringEdits :: Traversal (StringEdits pos char) (StringEdits pos' char') (StringEdit pos char) (StringEdit pos' char')
+\end{code}
+\begin{comment}
+\begin{code}
 stringEdits = _StringEdits . traverse
-
-affected :: forall char. StringEdits char -> Maybe (Interval Natural)
--- ^ For a given set of edits @es@ return the interval @i = a ... b@ such that for any given string @str@ of sufficient length the following holds:
---
---   - For all @n :: Natural@: @n < a ==> str ! n == (str `apply` es) ! n@
---   - There exists a @k :: Integer@ such that for all @n :: Integer@: @n > b ==> str ! (n + k) == (str `apply` es) ! n@
---
--- Intuitively: for any character @c@ of the new string @str `apply` es@ there exists a corresponding character in @str@ (offset by either 0 or a constant shift @k@) if the index of @c@ is /not/ contained in @affected es@.
-affected SEFail = Nothing
-affected (StringEdits es) = Just . toInterval $ go es Map.empty
-  where
-    toInterval :: Map Natural Integer -> Interval Natural
-    toInterval map
-      | Just (((minK, _), _), ((maxK, _), _)) <- (,) <$> Map.minViewWithKey map <*> Map.maxViewWithKey map
-      = let
-          maxV' = maximum . (0 :) $ do
-            offset <- [0..maxK]
-            v <- maybeToList $ Map.lookup (maxK - offset) map
-            v' <- maybeToList . fmap fromInteger $ negate v <$ guard (v <= 0)
-            guard $ v' >= succ offset
-            return $ v' - offset
-        in (minK Int.... maxK + maxV')
-      | otherwise
-      = Int.empty
-    go :: Seq (StringEdit char) -> Map Natural Integer -> Map Natural Integer
-    go Seq.Empty offsets = offsets
-    go (es :> e) offsets = go es offsets'
-      where
-        p = e ^. sePos
-        p' = fromIntegral $ Map.foldrWithKey (\k o p -> bool (fromIntegral p) (o + p) $ k < fromIntegral p) (fromIntegral p) offsets
-        offsets' = Map.alter (Just . myOffset . fromMaybe 0) p offsets
-        myOffset :: Integer -> Integer
-        myOffset
-          | Insert _ _ <- e = pred
-          | Delete _   <- e = succ
-
-insert :: Natural -> char -> StringEdits char
+\end{code}
+\end{comment}
+\begin{code}
+insert :: pos -> char -> StringEdits pos char
+\end{code}
+\begin{comment}
+\begin{code}
 insert n c = StringEdits .  Seq.singleton $ Insert n c
-  
-delete :: Natural -> StringEdits char
+\end{code}
+\end{comment}
+\begin{code}
+delete :: pos -> StringEdits pos char
+\end{code}
+\begin{comment}
+\begin{code}
 delete n = StringEdits .  Seq.singleton $ Delete n
-
-replace :: Natural -> char -> StringEdits char
+\end{code}
+\end{comment}
+\begin{code}
+replace :: Eq pos => pos -> char -> StringEdits pos char
+\end{code}
+\begin{comment}
+\begin{code}
 replace n c = insert n c <> delete n
 
-instance Monoid (StringEdits char) where
+-- | Rudimentarily optimize edit composition
+instance Eq pos => Monoid (StringEdits pos char) where
   mempty = StringEdits Seq.empty
   SEFail `mappend` _ = SEFail
   _ `mappend` SEFail = SEFail
@@ -122,12 +127,16 @@ instance Monoid (StringEdits char) where
     , n == n'
     = StringEdits bs `mappend` StringEdits as
     | otherwise = StringEdits $ x `mappend` y
+\end{code}
+\end{comment}
 
-instance Module (StringEdits char) where
-  type Domain (StringEdits char) = Seq char
+Da wir ein minimales set an atomaren edits gewählt haben, ist die Definiton der Modulnstruktur über Strings des passenden Alphabets recht einfach:
+\begin{code}
+instance Module (StringEdits Natural char) where
+  type Domain (StringEdits Natural char) = Seq char
   apply str SEFail = Nothing
   apply str (StringEdits Seq.Empty) = Just str
-  apply str (StringEdits (es :|> Insert n c)) = (flip apply) (StringEdits es) =<< go str n c
+  apply str (StringEdits (es :|> Insert n c)) = flip apply (StringEdits es) =<< go str n c
     where
       go Seq.Empty n c
         | n == 0 = Just $ Seq.singleton c
@@ -135,7 +144,7 @@ instance Module (StringEdits char) where
       go str@(x :<| xs) n c
         | n == 0 = Just $ c <| str
         | otherwise = (x <|) <$> go xs (pred n) c
-  apply str (StringEdits (es :|> Delete n)) = (flip apply) (StringEdits es) =<< go str n
+  apply str (StringEdits (es :|> Delete n)) = flip apply (StringEdits es) =<< go str n
     where
       go Seq.Empty _ = Nothing
       go (x :<| xs) n
@@ -146,99 +155,128 @@ instance Module (StringEdits char) where
   divInit = StringEdits . Seq.unfoldl go . (0,)
     where
       go (_, Seq.Empty) = Nothing
-      go (n, (c :<| cs)) = Just ((succ n, cs), Insert n c)
+      go (n, c :<| cs ) = Just ((succ n, cs), Insert n c)
 
 \end{code}
 
 % TODO Make notation mathy
 
-Um zunächst eine asymmetrische edit-lens `StringEdits -> StringEdits` mit akzeptabler Komplexität für einen bestimmten `DFST s` (entlang der \emph{Richtung} des DFSTs) zu konstruieren möchten wir folgendes Verfahren anwenden:
+Um zunächst eine asymmetrische edit-lens \texttt{StringEdits -> StringEdits} mit akzeptabler Komplexität für einen bestimmten DFST (entlang der \emph{Richtung} des DFSTs) zu konstruieren möchten wir folgendes Verfahren anwenden:
 
-Gegeben eine Sequenz (`StringEdits`) von zu übersetzenden Änderungen genügt es die Übersetzung eines einzelnen `StringEdit`s in eine womöglich längere Sequenz von `StringEdits` anzugeben, alle `StringEdits` aus der Sequenz zu übersetzen (hierbei muss auf die korrekte Handhabung des Komplements geachtet werden) und jene Übersetzungen dann zu concatenieren.
+Gegeben eine Sequenz von zu übersetzenden Änderungen genügt es die Übersetzung eines einzelnen \texttt{StringEdit}s in eine womöglich längere Sequenz von \texttt{StringEdits} anzugeben, alle \texttt{StringEdits} aus der Sequenz derart zu übersetzen (hierbei muss auf die korrekte Handhabung des Komplements geachtet werden) und jene Übersetzungen dann zu concatenieren.
 
-Wir definieren zunächst die \emph{Wirkung} eines DFST auf einen festen String als eine Abbildung `state -> (state, String)`, die den aktuellen Zustand vorm Parsen des Strings auf den Zustand danach und die (womöglich leere) Ausgabe schickt.
+Wir definieren zunächst die \emph{Wirkung} eines DFST auf einen festen String als eine Abbildung \texttt{state -> (Seq output, Maybe state)}, die den aktuellen Zustand vor dem Parsen des Strings auf den Zustand danach und die (womöglich leere) Ausgabe schickt.
+Wir annotieren Wirkungen zudem mit dem konsumierten String.
 Diese Wirkungen bilden einen Monoiden analog zu Endomorphismen, wobei die Resultat-Strings concateniert werden.
 
-Die Unterliegende Idee ist nun im Komplement der edit-lens eine Liste von Wirkungen (eine für jedes Zeichen der Eingabe des DFSTs) und einen Cache der monoidalen Summen aller kontinuirlichen Teillisten zu halten.
-Da wir wissen welche Stelle im input-String von einem gegebenen edit betroffen ist können wir, anhand der Wirkung des Teilstücks bis zu jener Stelle, den output-String in einen durch den edit unveränderten Prefix und einen womöglich betroffenen Suffix unterteilen.
-Die Wirkung ab der betroffenen Stelle im input-String können wir also Komposition der Wirkung der durch den edit betroffenen Stelle und derer aller Zeichen danach bestimmen.
-Nun gilt es nur noch die Differenz (als `StringEdits`) des vorherigen Suffixes im output-String und des aus der gerade berechneten Wirkung Bestimmten zu bestimmen.
-
-
-% Für die Rückrichtung bietet es sich an eine Art primitive Invertierung des DFSTs zu berechnen.
-% Gegeben den aktuellen DFST $A$ möchten wir einen anderen $A^{-1}$ finden, sodass gilt:
-
-% \begin{itemize}
-%   \item $A^{-1}$ akzeptiert einen String $s^{-1}$ (endet seinen Lauf in einem finalen Zustand) gdw. es einen String $s$ gibt, der unter $A$ die Ausgabe $s^{-1}$ produziert.
-%   \item Wenn $A^{-1}$ einen String $s^{-1}$ akzeptiert so produziert die resultierende Ausgabe $s$ unter $A$ die Ausgabe $s^{-1}$.
-% \end{itemize}
-
-% Kann nicht funktionieren, denn $A^{-1}$ ist notwendigerweise nondeterministisch. Wird $A^{-1}$ dann zu einem DFST forciert (durch arbiträre Wahl einer Transition pro Zustand) gehen Informationen verloren—$A^{-1}$ produziert nicht den minimale edit auf dem input string (in der Tat beliebig schlecht) für einen gegeben edit auf dem output string.
-  
-% Stelle im bisherigen Lauf isolieren, an der edit im output-string passieren soll, breitensuche auf pfaden, die sich von dieser stelle aus unterscheiden?
-% Gegeben einen Pfad und eine markierte Transition, finde Liste aller Pfade aufsteigend sortiert nach Unterschied zu gegebenem Pfad, mit Unterschieden "nahe" der markierten Transition zuerst — zudem jeweils edit auf dem Eingabestring
-% Einfacher ist Breitensuche ab `stInitial` und zunächst diff auf eingabe-strings.
-  
 \begin{code}
-
 data DFSTAction state input output = DFSTAction
-  { runDFSTAction :: state -> (state, Seq output)
+  { runDFSTAction :: state -> (Seq output, Maybe state)
   , dfstaConsumes :: Seq input
   }
 
 instance Monoid (DFSTAction state input output) where
-  mempty = DFSTAction (\x -> (x, Seq.empty)) Seq.empty
+\end{code}
+\begin{comment}
+\begin{code}
+  mempty = DFSTAction (\x -> (Seq.empty, Just x)) Seq.empty
   DFSTAction f cf `mappend` DFSTAction g cg = DFSTAction
-    { runDFSTAction = \s -> let ((f -> (s', out')), out) = g s in (s', out <> out')
+    { runDFSTAction = \x ->
+        let (outG, x') = g x
+            (outF, x'') = maybe (mempty, Nothing) f x'
+         in (outG <> outF, x'')
     , dfstaConsumes = cg <> cf
     }
+\end{code}
+\end{comment}
+\begin{code}
 
 type DFSTComplement state input output = Compositions (DFSTAction state input output)
 
-runDFSTAction' :: DFSTComplement state input output -> state -> (state, Seq output)
+runDFSTAction' :: DFSTComplement state input output -> state -> (Seq output, Maybe state)
 runDFSTAction' = runDFSTAction . Comp.composed
 
 dfstaConsumes' :: DFSTComplement state input output -> Seq input
 dfstaConsumes' = dfstaConsumes . Comp.composed
 
-dfstaProduces :: DFST state input output -> DFSTComplement state input output -> Seq output
-dfstaProduces DFST{..} = snd . flip runDFSTAction' stInitial
+dfstaProduces :: DFSTComplement state input output -> state -> Seq output
+dfstaProduces = fmap fst . runDFSTAction'
+\end{code}
 
-type Debug state input output = (Show state, Show input, Show output)
+Die Unterliegende Idee von $\Rrightarrow$ ist nun im Komplement der edit-lens eine Liste von Wirkungen (eine für jedes Zeichen der Eingabe des DFSTs) und einen Cache der monoidalen Summen aller kontinuirlichen Teillisten zu halten.
 
-type LState state input output = (Natural, (state, Maybe (input, Natural)))
+Wir können die alte DFST-Wirkung zunächst anhand des Intervalls indem der input-String von allen gegebenen edits betroffen ist in einen unveränderten Prefix und einen womöglich betroffenen Suffix unterteilen.
+
+Da wir wissen welche Stelle im input-String vom ersten gegebenen edit betroffen ist können wir, anhand der Wirkung des Teilstücks bis zu jener Stelle, den betroffenen Suffix wiederum teilen.
+Die Wirkung ab der betroffenen Stelle im input-String können wir als Komposition der Wirkung der durch den edit betroffenen Stelle und derer aller Zeichen danach bestimmen.
+Nun gilt es nur noch die Differenz (als `StringEdits`) des vorherigen Suffixes im output-String und des aus der gerade berechneten Wirkung zu bestimmen, wir bedienen uns hierzu dem Unix Standard-Diff-Algorithmus zwischen der ursprünglichen Ausgabe und dem Ergebnis der Iteration des Verfahrens auf alle gegebenen edits.
+
+Für die asymmetrische edit-lens entgegen der DFST-Richtung $\Lleftarrow$ verwenden wir Breitensuche über die Zustände des DFST innerhalb eines iterative vergrößerten Intervalls:
+
+Wir bestimmen zunächst (`affected`) eine obere Schranke an das Intervall in dem der Ausgabe-String vom edit betroffen ist und generieren eine von dort quadratisch wachsende Serie von Intervallen.
+
+Für jedes Intervall ("lokalere" Änderungen werden präferiert) schränken wir zunächst den DFST (zur einfachereren Implementierung in seiner Darstellung als FST) vermöge \texttt{restrictOutput} derart ein, dass nur die gewünschte Ausgabe produziert werden kann.
 
-dfstLens :: forall state input output. (Ord state, Ord input, Ord output, Debug state input output) => DFST state input output -> EditLens (DFSTComplement state input output) (StringEdits input) (StringEdits output)
+Wir betrachten dann in jedem Schritt (beginnend mit dem initialen Zustand des DFST) alle ausgehenden Transitionen und ziehen hierbei jene vor, die im vorherigen Lauf (gespeichert im Komplement der edit-lens), ebenfalls genommen wurden.
+Abweichungen vom im Komplement gespeicherten Lauf lassen wir nur innerhalb des betrachteten Intervalls zu und wählen in diesem Fall einen Edit auf der Eingabe, der die gewählte Abweichung produziert.
+Es wird zudem, wie für Breitensuche üblich, jeder besuchte Zustand markiert und ausgehende Transitionen nicht ein zweites mal betrachtet.
+
+Erreichen wir einen finalen Zustand (wegen der Einschränkung des DFSTs wurde dann auch genau die gewünschte Ausgabe produziert), so fügen wir an die gesammelten Eingabe-edits eine Serie von deletions an, die den noch nicht konsumierten suffix der Eingabe verwerfen und brechen die Suche unter Rückgabe der Eingabe-edits und des neuen Laufs ab.
+
+In Haskell formulieren wir das vorzeitige Abbrechen der Suche indem wir eine vollständige Liste von Rückgabe-Kandidaten konstruieren und dann immer ihr erstes Element zurück geben.
+Wegen der verzögerten Auswertungsstrategie von Haskell wird auch tatsächlich nur der erste Rückgabe-Kandidat konstruiert.
+
+\begin{comment}
+\begin{code}
+type LState state input output = (Natural, (state, Maybe (input, Natural)))
+\end{code}
+\end{comment}
+\begin{code}
+dfstLens :: forall state input output. (Ord state, Ord input, Ord output, Show state, Show input, Show output) => DFST state input output -> EditLens (DFSTComplement state input output) (StringEdits Natural input) (StringEdits Natural output)
+\end{code}
+\begin{comment}
+\begin{code}
 dfstLens dfst@DFST{..} = EditLens ground propR propL
   where
     ground :: DFSTComplement state input output
-    ground = Comp.fromList []
+    ground = mempty
 
-    propR :: (DFSTComplement state input output, StringEdits input) -> (DFSTComplement state input output, StringEdits output)
+    propR :: (DFSTComplement state input output, StringEdits Natural input) -> (DFSTComplement state input output, StringEdits Natural output)
     propR (c, SEFail) = (c, SEFail)
     propR (c, StringEdits Seq.Empty) = (c, mempty)
-    propR (c, StringEdits (es :> e))
-      | fst (runDFSTAction' c' stInitial) `Set.member` stAccept = (c', es' <> es'')
-      | otherwise                                               = (c', SEFail)
+    propR (c, es'@(StringEdits (es :> e)))
+      | (_, Just final) <- runDFSTAction' c' stInitial
+      , final `Set.member` stAccept
+      = (c', rEs)
+      | otherwise
+      = (c, SEFail)
       where
+        Just int = affected es'
+        (cAffSuffix, cAffPrefix) = Comp.splitAt (Comp.length c - fromIntegral (Int.inf int)) c
         (cSuffix, cPrefix) = Comp.splitAt (Comp.length c - (e ^. sePos . from enum)) c
         cSuffix'
-          | Delete _       <- e = Comp.take (pred $ Comp.length cSuffix) cSuffix -- TODO unsafe
+          | Delete _       <- e
+          , Comp.length cSuffix > 0 = Comp.take (pred $ Comp.length cSuffix) cSuffix
           | Insert _ nChar <- e = cSuffix <> Comp.singleton (DFSTAction (\x -> runDFST' dfst x (pure nChar) Seq.empty) (Seq.singleton nChar))
-        (pState, pOutput)  = runDFSTAction' cPrefix stInitial
-        (_, sOutput ) = runDFSTAction' cSuffix  pState
-        (_, sOutput') = runDFSTAction' cSuffix' pState
-        (c', es') = propR (cSuffix' <> cPrefix, StringEdits es)
-        es'' = strDiff sOutput sOutput' & stringEdits . sePos . from enum +~ Seq.length pOutput
+          | otherwise = Comp.singleton $ DFSTAction (\_ -> (Seq.empty, Nothing)) Seq.empty
+        (c', _) = propR (cSuffix' <> cPrefix, StringEdits es)
+        (cAffSuffix', _) = Comp.splitAt (Comp.length c' - Comp.length cAffPrefix) c'
+        (_, Just pFinal) = runDFSTAction' cPrefix stInitial
+        rEs = strDiff (fst $ runDFSTAction' cAffSuffix pFinal) (fst $ runDFSTAction' cAffSuffix' pFinal) & stringEdits . sePos . from enum +~ length (dfstaProduces cAffPrefix stInitial)
         
 
-    propL :: (DFSTComplement state input output, StringEdits output) -> (DFSTComplement state input output, StringEdits input)
+    propL :: (DFSTComplement state input output, StringEdits Natural output) -> (DFSTComplement state input output, StringEdits Natural input)
     propL (c, StringEdits Seq.Empty) = (c, mempty)
     propL (c, es) = fromMaybe (c, SEFail) $ do
+      let prevOut = dfstaProduces c stInitial
       newOut <- prevOut `apply` es
       affected' <- affected es
       let outFST :: FST (LState state input output) input output
-          outFST = wordFST newOut `productFST` toFST dfst
+          -- outFST = wordFST newOut `productFST` toFST dfst
+          outFST = restrictOutput newOut $ toFST dfst
+      
+          trace x y = flip seq y . unsafePerformIO $ appendFile "lens.log" (x <> "\n\n")
+
           inflate by int
             | Int.null int = Int.empty
             | inf >= by = inf - by Int.... sup + by
@@ -251,53 +289,90 @@ dfstLens dfst@DFST{..} = EditLens ground propR propL
               max = fromIntegral $ Seq.length newOut
               all = 0 Int.... max
           runCandidates :: Interval Natural -- ^ Departure from complement-run only permitted within interval (to guarantee locality)
-                       -> [ ( Seq (LState state input output, Maybe output) -- ^ Computed run
-                            , StringEdits input
-                            , DFSTComplement state input output
-                            )
-                          ]
-          runCandidates focus = continueRun (Seq.empty, mempty) (c, mempty) 0
+                        -> [ ( Seq (LState state input output, Maybe output) -- ^ Computed run
+                             , StringEdits Natural input
+                             , DFSTComplement state input output
+                             )
+                           ]
+          runCandidates focus = map ((,,) <$> view _1 <*> view _2 <*> view (_3 . _2)) $ go Set.empty [(Seq.empty, mempty, (c, mempty), 0)]
             where
-              continueRun :: (Seq (LState state input output, Maybe output), StringEdits input)
+              go _ [] = []
+              go visited (args@(run, edits, compZipper, inP) : alts) = 
+                  [ (run', finalizeEdits remC inP' edits', compZipper', inP') | (run', edits', compZipper'@(remC, _), inP') <- args : conts, isFinal run' ]
+                  ++ go visited' (alts ++ conts)
+                where
+                  conts
+                    | lastSt <- view _1 <$> Seq.lookup (pred $ Seq.length run) run
+                    , lastSt `Set.member` visited = []
+                    | otherwise = continueRun edits compZipper inP run
+                  visited' = Set.insert (view _1 <$> Seq.lookup (pred $ Seq.length run) run) visited
+
+              isFinal :: Seq (LState state input output, Maybe output) -> Bool
+              -- ^ Is the final state of the run a final state of the DFST?
+              isFinal Seq.Empty = (0, (stInitial, Nothing)) `Set.member` FST.stAccept outFST
+                               && (0 Int.... fromIntegral (Seq.length newOut)) `Int.isSubsetOf` focus
+              isFinal (_ :> (lastSt, _)) = lastSt `Set.member` FST.stAccept outFST
+
+              finalizeEdits :: DFSTComplement state input output -- ^ Remaining complement
+                            -> Natural -- ^ Input position
+                            -> StringEdits Natural input -> StringEdits Natural input
+              finalizeEdits remC inP = mappend . mconcat . replicate (Seq.length $ dfstaConsumes' remC) $ delete inP
+              
+              continueRun :: StringEdits Natural input
                           -> (DFSTComplement state input output, DFSTComplement state input output) -- ^ Zipper into complement
                           -> Natural -- ^ Input position
-                          -> [(Seq (LState state input output, Maybe output), StringEdits input, DFSTComplement state input output)]
-              continueRun (run, inEdits) (c', remC) inP = do
+                          -> Seq (LState state input output, Maybe output)
+                          -> [ ( Seq (LState state input output, Maybe output)
+                               , StringEdits Natural input
+                               , (DFSTComplement state input output, DFSTComplement state input output)
+                               , Natural
+                               )
+                             ]
+              -- ^ Nondeterministically make a single further step, continueing a given run
+              continueRun inEdits (c', remC) inP run = do
                 let
                   pos :: Natural
-                  pos = fromIntegral $ Comp.length c - Comp.length c'
+                  -- pos = fromIntegral $ Comp.length c - Comp.length c' -- FIXME: should use length of dfstaProduces
+                  pos = fromIntegral . Seq.length $ dfstaProduces remC stInitial
                   (c'', step) = Comp.splitAt (pred $ Comp.length c') c' -- TODO: unsafe?
                   current :: LState state input output
                   current
                     | Seq.Empty <- run      = (0, (stInitial, Nothing))
                     | (_ :> (st, _)) <- run = st
                   current' :: state
-                  current' = let (_, (st, _)) = current
-                              in st
-                  next' :: state
-                  next' = fst . runDFSTAction' step $ current'
                   oldIn :: Maybe input
-                  oldIn = Seq.lookup 0 $ dfstaConsumes' step
+                  (current', oldIn)
+                    | (_ :> ((_, (st, _)), _)) <- rest
+                    , (_ :> ((_, (_, Just (partialIn, _))), _)) <- partial = (st, Just partialIn)
+                    | (_ :> ((_, (_, Just (partialIn, _))), _)) <- partial = (stInitial, Just partialIn)
+                    | Seq.Empty <- rest = (stInitial, Seq.lookup 0 $ dfstaConsumes' step)
+                    | (_ :> ((_, (st, _)), _)) <- rest = (st, Seq.lookup 0 $ dfstaConsumes' step)
+                    where
+                      (partial, rest) = Seq.spanr (\((_, (_, inp)), _) -> isJust inp) run
+                next' <- trace (show ("next'", pos, focus, run, (current', oldIn), current, dfstaConsumes' step, runDFST' dfst current' (maybe Seq.empty Seq.singleton oldIn) Seq.empty)) . maybeToList . snd $ runDFST' dfst current' (maybe Seq.empty Seq.singleton oldIn) Seq.empty
+                let
                   outgoing :: LState state input output -> [(LState state input output, Maybe input, Maybe output)]
-                  outgoing current = let go (st, minS) os acc
-                                           | st == current = ($ acc) $ Set.fold (\(st', moutS) -> (. ((st', minS, moutS) :))) id os
+                  outgoing current = let go (st, minS) outs acc
+                                           | st == current = Set.foldr (\(st', moutS) -> ((st', minS, moutS) :)) acc outs
                                            | otherwise     = acc
                                       in Map.foldrWithKey go [] $ FST.stTransition outFST 
                   isPreferred :: (LState state input output, Maybe input, Maybe output) -> Bool
-                  isPreferred ((_, (st, Nothing)), inS, _) = st == next' && (fromMaybe True $ (==) <$> oldIn <*> inS)
-                  isPreferred (st, _, _) = any isPreferred $ outgoing st -- By construction of `outFST`, `outgoing st` is a singleton
+                  isPreferred ((_, (st, Nothing)), _, _) = st == next'
+                  isPreferred (st@(_, (_, Just (inS , _))), _, _) = maybe True (== inS) oldIn && any isPreferred (outgoing st) -- By construction of `outFST`, `outgoing st` is a singleton in this case
                   (preferred, alternate) = partition isPreferred $ outgoing current
                   assocEdit :: (LState state input output, Maybe input, Maybe output) -- ^ Transition
                             -> [ ( (DFSTComplement state input output, DFSTComplement state input output) -- ^ new `(c', remC)`, i.e. complement-zipper `(c', remC)` but with edit applied
-                                 , StringEdits input
+                                 , StringEdits Natural input
                                  , Natural
                                  )
                                ]
                   assocEdit (_, Just inS, _)
-                    | oldIn == Just inS = [((c'', step <> remC), mempty, succ inP)]
-                    | isJust oldIn      = [((c'', altStep inS <> remC), replace inP inS, succ inP), ((c', altStep inS <> remC), insert inP inS, succ inP)]
-                    | otherwise         = [((c', altStep inS <> remC), insert inP inS, succ inP)]
-                  assocEdit (_, Nothing, _) = [((c', remC), mempty, inP)] -- TODO: is this correct?
+                    | oldIn == Just inS = [ ((c'', step <> remC), mempty, succ inP) ]
+                    | isJust oldIn      = [ ((c', altStep inS <> remC), insert inP inS, succ inP)
+                                          , ((c'', altStep inS <> remC), replace inP inS, succ inP)
+                                          ]
+                    | otherwise         = [ ((c', altStep inS <> remC), insert inP inS, succ inP) ]
+                  assocEdit (_, Nothing, _) = [((c', remC), mempty, inP)]
                   altStep :: input -> DFSTComplement state input output
                   altStep inS = Comp.singleton DFSTAction{..}
                     where
@@ -306,7 +381,7 @@ dfstLens dfst@DFST{..} = EditLens ground propR propL
                   options
                     | pos `Int.member` focus = preferred ++ alternate
                     | otherwise              = preferred
-                choice@(next, inS, outS) <- options
+                choice@(next, inS, outS) <- trace (unlines $ show (pretty outFST) : map show options) options
                 ((c3, remC'), inEdits', inP') <- assocEdit choice
                 -- let
                 --   -- | Replace prefix of old complement to reflect current candidate
@@ -317,27 +392,70 @@ dfstLens dfst@DFST{..} = EditLens ground propR propL
                 --   fin
                 --     | (trans, inEs, newComplement) <- acc = (trans, dropSuffix <> inEs, newComplement)
                 let
-                  acc = (run :> (next, outS), inEdits' <> inEdits)
-                  dropSuffix = mconcat (replicate (Seq.length $ dfstaConsumes' c3) $ delete inP')
-                  fin
-                    | (trans, inEs) <- acc = (trans, dropSuffix <> inEs, remC')
-                bool id (fin :) (next `Set.member` FST.stAccept outFST) $ continueRun acc (c3, remC') inP'
+                  trans = run :> (next, outS)
+                  inEs = inEdits' <> inEdits
+                --   dropSuffix = mconcat (replicate (Seq.length $ dfstaConsumes' c3) $ delete inP')
+                --   fin
+                --     | (trans, inEs) <- acc = (trans, dropSuffix <> inEs, remC')
+                -- bool id (over _BFS $ cons fin) (next `Set.member` FST.stAccept outFST) $ continueRun acc (c3, remC') inP'
+                return (trans, inEs, (c3, remC'), inP')
               
       -- Properties of the edits computed are determined mostly by the order candidates are generated below
       -- (_, inEs, c') <- (\xs -> foldr (\x f -> x `seq` f) listToMaybe xs $ xs) $ traceShowId fragmentIntervals >>= (\x -> (\y@(y1, y2, _) -> traceShow (y1, y2) y) <$> runCandidates x)
 
-      (_, inEs, c') <- listToMaybe $ runCandidates =<< fragmentIntervals
-      
-      return (c', inEs)
-      where
-        (_, prevOut) = runDFSTAction' c stInitial
+      fmap ((,) <$> view _3 <*> view _2) .  listToMaybe $ runCandidates =<< fragmentIntervals
 
-strDiff :: forall sym. Eq sym => Seq sym -> Seq sym -> StringEdits sym
+strDiff :: forall sym pos. (Eq sym, Integral pos) => Seq sym -> Seq sym -> StringEdits pos sym
 -- ^ @strDiff a b@ calculates a set of edits, which, when applied to @a@, produce @b@
-strDiff a b = snd . foldr toEdit (0, mempty) $ (getDiff `on` toList) a b
+strDiff a b = snd . foldl toEdit (0, mempty) $ (getDiff `on` toList) a b
+  where
+    toEdit :: (pos, StringEdits pos sym) -> Diff sym -> (pos, StringEdits pos sym)
+    toEdit (n, es) (Diff.Both _ _) = (succ n, es)
+    toEdit (n, es) (Diff.First _ ) = (n, delete n <> es)
+    toEdit (n, es) (Diff.Second c) = (succ n, insert n c <> es)
+\end{code}
+\end{comment}
+
+Um eine obere Schranke an das von einer Serie von edits betroffene Intervall zu bestimmen ordnen wir zunächst jeder von mindestens einem atomaren edit betroffenen Position $n$ im Eingabe-Wort einen $\text{offset}_n = \text{\# deletions} - \text{\# inserts}$ zu.
+Das gesuchte Intervall ist nun $(\text{minK}, \text{maxK})$, mit $\text{minK}$ der Position im Eingabe-Wort mit niedrigstem $\text{offset}$ und $\text{maxK}$ die Position im Eingabe-Wort mit höchstem $\text{offset}$, $\text{maxK}^\prime$, modifiziert um das Maximum aus $\{ 0 \} \cup \{ \text{maxK}_n \colon n \in \{ 0 \ldots \text{maxK}^\prime \} \}$ wobei $\text{maxK}_n = -1 \cdot (n + \text{offset}_n)$ an Position $n$ ist.
+
+\begin{code}
+affected :: forall char. StringEdits Natural char -> Maybe (Interval Natural)
+-- ^ For a given set of edits @es@ return the interval @i = a ... b@ such that for any given string @str@ of sufficient length the following holds:
+--
+--   - For all @n :: Natural@: @n < a ==> str ! n == (str `apply` es) ! n@
+--   - There exists a @k :: Integer@ such that for all @n :: Integer@: @n > b ==> str ! (n + k) == (str `apply` es) ! n@
+--
+-- Intuitively: for any character @c@ of the new string @str `apply` es@ there exists a corresponding character in @str@ (offset by either 0 or a constant shift @k@) if the index of @c@ is /not/ contained in @affected es@.
+\end{code}
+\begin{comment}
+\begin{code}
+affected SEFail = Nothing
+affected (StringEdits es) = Just . toInterval $ go es Map.empty
   where
-    toEdit :: Diff sym -> (Natural, StringEdits sym) -> (Natural, StringEdits sym)
-    toEdit (Diff.Both _ _) (n, es) = (succ n, es)
-    toEdit (Diff.First _ ) (n, es) = (n, delete n <> es)
-    toEdit (Diff.Second c) (n, es) = (succ n, insert n c <> es)
+    toInterval :: Map Natural Integer -> Interval Natural
+    toInterval map
+      | Just (((minK, _), _), ((maxK, _), _)) <- (,) <$> Map.minViewWithKey map <*> Map.maxViewWithKey map
+      = let
+          maxV' = maximum . (0 :) $ do
+            offset <- [0..maxK]
+            v <- maybeToList $ Map.lookup (maxK - offset) map
+            v' <- maybeToList . fmap fromInteger $ negate v <$ guard (v <= 0)
+            guard $ v' >= succ offset
+            return $ v' - offset
+        in (minK Int.... maxK + maxV')
+      | otherwise
+      = Int.empty
+    go :: Seq (StringEdit Natural char) -> Map Natural Integer -> Map Natural Integer
+    go Seq.Empty offsets = offsets
+    go (es :> e) offsets = go es offsets'
+      where
+        p = e ^. sePos
+        -- p' = fromIntegral $ Map.foldrWithKey (\k o p -> bool (fromIntegral p) (o + p) $ k < fromIntegral p) (fromIntegral p) offsets
+        offsets' = Map.alter (Just . myOffset . fromMaybe 0) p offsets
+        myOffset :: Integer -> Integer
+        myOffset
+          | Insert _ _ <- e = pred
+          | Delete _   <- e = succ
 \end{code}
+\end{comment}
diff --git a/edit-lens/src/Control/Edit.lhs b/edit-lens/src/Control/Edit.lhs
index 19fe336..8c4f045 100644
--- a/edit-lens/src/Control/Edit.lhs
+++ b/edit-lens/src/Control/Edit.lhs
@@ -16,7 +16,7 @@ Ein Modul $M$ ist eine \emph{partielle Monoidwirkung} zusammen mit einem schwach
 
 und einem Element $\init_M \in \Dom M$, sodass gilt:
 
-$$ \forall m \in \Dom M \ \exists \partial m \in \partial M \colon m = \init_M \cdot \partial m$$
+$$\forall m \in \Dom M \ \exists \partial m \in \partial M \colon m = \init_M \cdot \partial m$$
 
 Wir führen außerdem eine Abbildung $(\init_M \cdot)^{-1} \colon \Dom M \to \partial m$ ein, die ein $m$ auf ein arbiträr gewähltes $\partial m$ abbildet für das $\init_M \cdot \partial m = m$ gilt.
 
@@ -24,6 +24,7 @@ In Haskell charakterisieren wir Moduln über ihren Monoid, d.h. die Wahl des Mon
 Eine Repräsentierung als Typklasse bietet sich an:
 
 \begin{code}
+-- `apply` binds one level weaker than monoid composition `(<>)`
 infix 5 `apply`
 
 class Monoid m => Module m where
@@ -42,11 +43,12 @@ class Monoid m => Module m where
 
 infixl 5 `apply'`
 apply' :: Module m => Maybe (Domain m) -> m -> Maybe (Domain m)
+-- ^ `apply` under `Maybe`s monad-structure
 apply' md e = flip apply e =<< md
 \end{code}
 \end{defn}
 
-Wir weichen von der originalen Definition von Moduln aus \cite{hofmann2012edit} darin ab, dass wir für das ausgezeichnete Element $\init_X$ des Trägers explizit (und konstruktiv) fordern, dass es ein schwach-initiales Element bzgl. der Monoidwirkung sei.
+Wir weichen von der originalen Definition von Moduln aus \cite{hofmann2012edit} darin ab, dass wir für das ausgezeichnete Element $\init_X$ des Trägers explizit (und konstruktiv\footnote{$(\init_M \cdot)^{-1}$}) fordern, dass es ein schwach-initiales Element bzgl. der Monoidwirkung sei.
 
 \begin{comment}
 \begin{defn}[Modulhomomorphismen]
diff --git a/edit-lens/src/Control/FST.lhs b/edit-lens/src/Control/FST.lhs
index 9298e11..9aa5341 100644
--- a/edit-lens/src/Control/FST.lhs
+++ b/edit-lens/src/Control/FST.lhs
@@ -1,3 +1,4 @@
+\begin{comment}
 \begin{code}
 {-# LANGUAGE ScopedTypeVariables
 #-}
@@ -7,16 +8,18 @@ Description: Finite state transducers with epsilon-transitions
 -}
 module Control.FST
   ( FST(..)
+  -- * Using FSTs
+  , runFST, runFST', step
   -- * Constructing FSTs
   , wordFST
   -- * Operations on FSTs
-  , productFST, restrictFST
+  , productFST, restrictOutput, restrictFST
   -- * Debugging Utilities
   , liveFST
   ) where
 
-import Data.Map.Strict (Map, (!?))
-import qualified Data.Map.Strict as Map
+import Data.Map.Lazy (Map, (!?), (!))
+import qualified Data.Map.Lazy as Map
 
 import Data.Set (Set)
 import qualified Data.Set as Set
@@ -24,7 +27,7 @@ import qualified Data.Set as Set
 import Data.Sequence (Seq)
 import qualified Data.Sequence as Seq
 
-import Data.Maybe (mapMaybe, fromMaybe, isJust, fromJust)
+import Data.Maybe (mapMaybe, fromMaybe, isJust, fromJust, isNothing)
 
 import Numeric.Natural
 
@@ -35,12 +38,28 @@ import Control.Monad.State.Strict
 import Text.PrettyPrint.Leijen (Pretty(..))
 import qualified Text.PrettyPrint.Leijen as PP
 
-data FST state input output = FST
+\end{code}
+\end{comment}
+
+\begin{defn}[Finite state transducers]
+Unter einem finite state transducer verstehen wir ein 6-Tupel $(\Sigma, \Delta, Q, I, F, E)$ mit $\Sigma$ dem endlichen Eingabe-Alphabet, $\Delta$ dem endlichen Ausgabe-Alphabet, $Q$ einer endlichen Menge an Zuständen, $I \subset Q$ der Menge von initialen Zuständen, $F \subset Q$ der Menge von akzeptierenden Endzuständen, und $E \subset Q \times (\Sigma \cup \{ \epsilon \}) \times (\Delta \cup \{ \epsilon \}) \times Q$ der Transitionsrelation.
+
+Semantisch ist ein finite state transducer ein endlicher Automat erweitert um die Fähigkeit bei Zustandsübergängen ein Symbol aus seinem Ausgabe-Alphabet an ein Ausgabe-Wort anzuhängen.
+
+In Haskell lockern wir die Anforderung, dass die Ein- und Ausgabe-Alphabete endlich sein müssen und annotieren sie nur im Typsystem.
+Zudem speichern wir die Transitionsrelation als multimap um effiziente lookups von Zustand-Eingabe-Paaren zu ermöglichen.
+
+\begin{code}
+dFSeata FST state input output = FST
   { stInitial :: Set state
   , stTransition :: Map (state, Maybe input) (Set (state, Maybe output))
   , stAccept :: Set state
   } deriving (Show, Read)
+\end{code}
+\end{defn}
 
+\begin{comment}
+\begin{code}
 instance (Show state, Show input, Show output) => Pretty (FST state input output) where
   pretty FST{..} = PP.vsep
     [ PP.text "Initial states:" PP.</> PP.hang 2 (list . map (PP.text . show) $ Set.toAscList stInitial)
@@ -55,62 +74,164 @@ instance (Show state, Show input, Show output) => Pretty (FST state input output
     ]
     where
       label :: Show a => Maybe a -> PP.Doc
-      label = maybe (PP.text "ɛ") (PP.text . show)
+      label = PP.text . maybe "ɛ" show
       list :: [PP.Doc] -> PP.Doc
       list = PP.encloseSep (PP.lbracket PP.<> PP.space) (PP.space PP.<> PP.rbracket) (PP.comma PP.<> PP.space)
+\end{code}
+\end{comment}
 
-runFST :: forall input output state. (Ord input, Ord output, Ord state) => FST state input output -> Seq input -> [Seq output]
-runFST = fmap (map $ catMaybes . fmap (view _2) . view _2) . runFST'
-  where
-    catMaybes = fmap fromJust . Seq.filter isJust
+Wir definieren die Auswertung von finite state transducers induktiv indem wir zunächst angeben wie ein einzelner Auswertungs-Schritt erfolgt.
+
+Hierzu kommentieren wir die Haskell-Implementierung eines Auswertungs-Schritts.
+Notwendigerweise ist die Auswertung eines FSTs nicht deterministisch, wir produzieren daher eine Liste von möglichen Resultaten in keiner besonderen Reihenfolge.
+
+\begin{code}
+step :: forall input output state. (Ord input, Ord output, Ord state)
+     => FST state input output
+     -> Maybe state -- ^ Current state
+     -> Maybe input -- ^ Head of remaining input
+     -> [(Maybe input, state, Maybe output)] -- ^ Tuples of unconsumed input, next state, and produced output
+step FST{..} Nothing inS = (\s -> (inS, s, Nothing)) <$> Set.toList stInitial
+\end{code}
+Ist kein vorheriger Schritt erfolgt so wählen wir einen initialen Zustand, konsumieren keine Eingabe, und produzieren keine Ausgabe.
+
+\begin{code}
+step FST{..} (Just c) inS = let
+  consuming = fromMaybe Set.empty $ Map.lookup (c, inS) stTransition
+  unconsuming = fromMaybe Set.empty $ Map.lookup (c, Nothing) stTransition
+  in Set.toList $ Set.map (\(n, mOut) -> (Nothing, n, mOut)) consuming `Set.union` Set.map (\(n, mOut) -> (inS, n, mOut)) unconsuming
+\end{code}
+Ansonsten wählen wir einen Eintrag aus der Transitionstabelle für den aktuellen Zustand, der entweder keine oder die gegebene Eingabe konsumiert.
+Im Ergebnis geben wir den nächsten Zustand, die Ausgabe aus der Transitionstabelle, und ob die Eingabe konsumiert wurde an.
 
+\begin{code}
 runFST' :: forall input output state. (Ord input, Ord output, Ord state)
         => FST state input output
         -> Seq input
         -> [(state, Seq (state, Maybe output))] -- ^ Tuples of initial state and chosen transitions; not neccessarily finite
 -- ^ Compute all possible runs on the given input
-runFST' fst Seq.Empty = guardAccept $ (\(_, st, _) -> (st, Seq.Empty)) <$> step fst Nothing Nothing
-runFST' fst cs = guardAccept $ do
-  initial <- view _2 <$> step fst Nothing Nothing
-  go (initial, Seq.Empty) cs
+runFST' fst@FST{..} cs = do
+  initial <- view _2 <$> step fst Nothing Nothing -- Nondeterministically choose an initial state
+  go (initial, Seq.Empty) cs -- Recursively extend the run consisting of only the initial state
   where
-    guardAccept res = do
-      (initial, path) <- res
-      let
-        finalState
-          | (_ :> (st, _)) <- path = st
-          | otherwise              = initial
-      guard $ finalState `Set.member` stAccept
-      return res
-
     go :: (state, Seq (state, Maybe output)) -> Seq input-> [(state, Seq (state, Maybe output))]
+    -- ^ Uses `step` on last state of run and nondeterministically chooses between alternatives given
+\end{code}
+
+Um alle möglichen Läufe auf einer gegebenen Eingabe zu berechnen wenden wir
+rekursiv \texttt{step} auf den letzten Zustand des Laufs (und der verbleibenden
+Eingabe) an bis keine Eingabe verbleibt und der letzte Zustand in der Menge der
+akzeptierenden Endzustände liegt.
+
+\begin{comment}
+\begin{code}
     go (initial, path) cs = do
       let
+        -- | Determine last state of the run
         current
           | (_ :> (st, _)) <- path = st
           | otherwise              = initial
-      (head, next, out) <- step fst (Just current) (Seq.lookup 0 cs)
-      let
-        nPath = path :> (next, out)
-        ncs = maybe id (:<) head cs
-      go (initial, nPath) ncs
-      
+      case step fst (Just current) (Seq.lookup 0 cs) of
+        [] -> do
+          guard $ current `Set.member` stAccept && Seq.null cs
+          return (initial, path)
+        xs -> do
+          (head, next, out) <- xs
+          let
+            nPath = path :> (next, out)
+            ncs
+              | (_ :< cs') <- cs = maybe id (:<) head cs'
+              | otherwise = Seq.Empty
+          go (initial, nPath) ncs
+\end{code}
+\end{comment}
+
+Es ist gelegentlich nützlich nur die möglichen Ausgaben eines FST auf gegebener
+Eingabe zu bestimmen, wir führen eine Hilfsfunktion auf Basis von
+{\ttfamily runFST'} ein:
+
+\begin{code}
+runFST :: forall input output state. (Ord input, Ord output, Ord state) => FST state input output -> Seq input -> [Seq output]
+-- ^ Compute all possible runs on the given input and return only their output
+\end{code}
+\begin{comment}
+\begin{code}
+runFST = fmap (map $ catMaybes . fmap (view _2) . view _2) . runFST'
+  where
+    catMaybes = fmap fromJust . Seq.filter isJust
+\end{code}
+\end{comment}
+
+Wir können das Produkt zweier FSTs definieren.
+Intuitiv wollen wir beide FSTs gleichzeitig ausführen und dabei sicherstellen, dass Ein- und Ausgabe der FSTs übereinstimmen\footnote{Da wir $\epsilon$-Transitionen in FSTs erlauben müssen wir uns festlegen wann eine $\epsilon$-Transition übereinstimmt mit einer anderen Transition. Wir definieren, dass $\epsilon$ als Eingabe mit jeder anderen Eingabe (inkl. einem weiteren $\epsilon$) übereinstimmt.}.
+
+Hierfür berechnen wir das Graphen-Produkt der FSTs:
+
+\begin{defn}[FST-Produkt]
+  Gegeben zwei finite state transducer $T = (\Sigma, \Delta, Q, I, F, E)$ und $T^\prime = (\Sigma^\prime, \Delta^\prime, Q^\prime, I^\prime, F^\prime, E^\prime)$ nennen wir $T^\times = (\Sigma^\times, \Delta^\times, Q^\times, I^\times, F^\times, E^\times)$ das Produkt $T^\times = T \times T^\prime$ von $T$ und $T^\prime$.
+
+  $T^\times$ bestimmt sich als das Graphenprodukt der beiden, die FSTs unterliegenden Graphen, wobei wir die Zustandsübergänge als Kanten mit Gewichten aus dem Boolschen Semiring auffassen:
+
+  \begin{align*}
+    \Sigma^\times & = \Sigma \cap \Sigma^\prime \\
+    \Delta^\times & = \Delta \cap \Delta^\prime \\
+    Q^\times & = Q \times Q^\prime \\
+    I^\times & = I \times I^\prime \\
+    F^\times & = F \times F^\prime \\
+    E^\times & \subset Q^\times \times (\Sigma^\times \cup \{ \epsilon \}) \times (\Delta^\times \cup \{ \epsilon \}) \times Q^\times \\
+             & = \left \{ ((q, q^\prime), \sigma, \delta, (\bar{q}, \bar{q^\prime})) \colon (q, \sigma, \delta, \bar{q}) \in E, (q^\prime, \sigma^\prime, \delta^\prime, \bar{q^\prime}) \in E^\prime, \sigma = \sigma^\prime, \delta = \delta^\prime \right \}
+  \end{align*}
+\end{defn}
 
-step :: forall input output state. (Ord input, Ord output, Ord state)
-     => FST state input output
-     -> Maybe state -- ^ Current state
-     -> Maybe input -- ^ Head of remaining input
-     -> [(Maybe input, state, Maybe output)] -- ^ Tuples of unconsumed input, next state, and produced output
-step FST{..} Nothing inS = (\s -> (inS, s, Nothing)) <$> Set.toList stInitial
-step FST{..} (Just c) inS = let
-  consuming = fromMaybe Set.empty $ Map.lookup (c, inS) stTransition
-  unconsuming = fromMaybe Set.empty $ Map.lookup (c, Nothing) stTransition
-  in Set.toList $ Set.map (\(n, mOut) -> (Nothing, n, mOut)) consuming `Set.union` Set.map (\(n, mOut) -> (inS, n, mOut)) unconsuming
   
+\begin{code}
+productFST :: forall state1 state2 input output. (Ord state1, Ord state2, Ord input, Ord output) => FST state1 input output -> FST state2 input output -> FST (state1, state2) input output
+-- ^ Cartesian product on states, logical conjunction on transitions and state-properties (initial and accept)
+--
+-- This is the "natural" (that is component-wise) product when considering FSTs to be weighted in the boolean semiring.
+--
+-- Intuitively this corresponds to running both FSTs at the same time requiring them to produce the same output and agree on their input.
+\end{code}
 
-wordFST :: forall input output. Seq output -> FST Natural input output
--- ^ @wordFST str@ is the linear FST generating @str@ as output when given no input
-wordFST outs = FST
+\begin{comment}
+\begin{code}
+productFST fst1 fst2 = FST
+  { stInitial    = Set.fromDistinctAscList $ stInitial fst1 `setProductList` stInitial fst2
+  , stAccept     = Set.fromDistinctAscList $ stAccept fst1 `setProductList` stAccept fst2
+  , stTransition = Map.fromSet transitions . Set.fromDistinctAscList . mapMaybe filterTransition $ Map.keysSet (stTransition fst1) `setProductList` Map.keysSet (stTransition fst2)
+  }
+  where
+    setProductList :: forall a b. Set a -> Set b -> [(a, b)]
+    setProductList as bs = (,) <$> Set.toAscList as <*> Set.toAscList bs
+    filterTransition :: forall label. Eq label => ((state1, Maybe label), (state2, Maybe label)) -> Maybe ((state1, state2), Maybe label)
+    filterTransition ((st1, l1), (st2, l2))
+      | l1 == l2  = Just ((st1, st2), l1)
+      | otherwise = Nothing
+    transitions :: ((state1, state2), Maybe input) -> Set ((state1, state2), Maybe output)
+    transitions ((st1, st2), inS) = Set.fromDistinctAscList . mapMaybe filterTransition $ out1 `setProductList` out2
+      where
+        out1 = fromMaybe Set.empty (stTransition fst1 !? (st1, inS)) `Set.union` fromMaybe Set.empty (stTransition fst1 !? (st1, Nothing))
+        out2 = fromMaybe Set.empty (stTransition fst2 !? (st2, inS)) `Set.union` fromMaybe Set.empty (stTransition fst2 !? (st2, Nothing))
+\end{code}
+\end{comment}
+
+Es ist später erforderlich einen FST derart einzuschränken, dass er eine gegebene Ausgabe produziert.
+
+Hierzu nehmen wir das FST-Produkt mit einem FST, der, ungeachtet der Eingabe, immer die gegebene Ausgabe produziert.
+Da die Ausgaben der beiden FSTs übereinstimmen müssen produziert das Produkt mit einem derartigen FST (solange dessen Ausgabe in keinem Sinne von der Eingabe abhängt) die gewünschte Ausgabe.
+
+Zur Konstruktion eines derartigen \emph{Wort-FST}s nehmen wir Indizes im Ausgabe-Wort (natürliche Zahlen) als Zustände.
+Übergänge sind immer entweder der Form $n \rightarrow \text{succ}(n)$, konsumieren keine Eingabe ($\epsilon$) und produzieren als Ausgabe das Zeichen am Index $n$ im Ausgabe-Wort, oder der Form $n \overset{(i, \epsilon)}{\rightarrow} n$, für jedes Eingabesymbol $i$ (um die Unabhängigkeit von der Eingabe sicherzustellen).
+Weiter ist $0$ initial und $\text{length}(\text{Ausgabe})$ der einzige akzeptierende Endzustand.
+
+\begin{code}
+wordFST :: forall input output. (Ord input, Ord output) => Set input -> Seq output -> FST Natural input output
+-- ^ @wordFST inps str@ is the linear FST generating @str@ as output when given any input with symbols in @inps@
+\end{code}
+  
+\begin{comment}
+\begin{code}
+wordFST inps outs = FST
   { stInitial    = Set.singleton 0
   , stAccept     = Set.singleton l
   , stTransition = Map.fromSet next states
@@ -119,36 +240,50 @@ wordFST outs = FST
     l :: Natural
     l = fromIntegral $ Seq.length outs
     states :: Set (Natural, Maybe input)
-    states = Set.fromDistinctAscList [ (n, Nothing) | n <- [0..pred l] ]
+    states
+      | Seq.null outs = Set.empty
+      | otherwise = Set.fromDistinctAscList [ (n, inp) | n <- [0..pred l], inp <- Nothing : map Just (Set.toList inps) ]
     next :: (Natural, Maybe input) -> Set (Natural, Maybe output)
-    next (i, _) = Set.singleton (succ i, Just . Seq.index outs $ fromIntegral i)
+    next (i, _) = Set.fromList
+      [ (succ i, Just . Seq.index outs $ fromIntegral i)
+      , (i, Nothing)
+      ]
+\end{code}
+\end{comment}
 
-productFST :: forall state1 state2 input output. (Ord state1, Ord state2, Ord input, Ord output) => FST state1 input output -> FST state2 input output -> FST (state1, state2) input output
--- ^ Cartesian product on states, logical conjunction on transitions and state-properties (initial and accept)
---
--- This is the "natural" (that is component-wise) product when considering FSTs to be weighted in the boolean semiring.
---
--- Intuitively this corresponds to running both FSTs at the same time requiring them to produce the same output and "agree" (epsilon agreeing with every character) on their input.
-productFST fst1 fst2 = FST
-  { stInitial    = stInitial fst1 `setProduct` stInitial fst2
-  , stAccept     = stAccept fst1 `setProduct` stAccept fst2
-  , stTransition = Map.fromSet transitions . Set.fromList . mapMaybe filterTransition . Set.toAscList $ Map.keysSet (stTransition fst1) `setProduct` Map.keysSet (stTransition fst2)
+Da \texttt{wordFST} zur Konstruktion eine komprehensive Menge aller Eingabesymbole benötigt verwenden wir im folgenden eine optimierte Variante des Produkts mit einem Wort-FST.
+
+\begin{code}
+restrictOutput :: forall state input output. (Ord state, Ord input, Ord output) => Seq output -> FST state input output -> FST (Natural, state) input output
+-- ^ @restrictOutput out@ is equivalent to @productFST (wordFST inps out)@ where @inps@ is a comprehensive set of all input symbols @inp :: input@
+\end{code}
+
+\begin{comment}
+\begin{code}
+restrictOutput out FST{..} = FST
+  { stInitial    = Set.mapMonotonic (0,) stInitial
+  , stAccept     = Set.mapMonotonic (l,) stAccept
+  , stTransition = Map.filter (not . Set.null) $ Map.fromList (concatMap noProgress $ Map.toList stTransition) `Map.union` Map.fromSet transitions (Set.fromDistinctAscList [((wSt, inSt), inSym) | wSt <- Set.toAscList wordStates, (inSt, inSym) <- Set.toAscList $ Map.keysSet stTransition])
   }
   where
-    setProduct :: forall a b. Set a -> Set b -> Set (a, b)
-    setProduct as bs = Set.fromDistinctAscList $ (,) <$> Set.toAscList as <*> Set.toAscList bs
-    filterTransition :: forall label. Eq label => ((state1, Maybe label), (state2, Maybe label)) -> Maybe ((state1, state2), Maybe label)
-    filterTransition ((st1, Nothing ), (st2, in2     )) = Just ((st1, st2), in2)
-    filterTransition ((st1, in1     ), (st2, Nothing )) = Just ((st1, st2), in1)
-    filterTransition ((st1, Just in1), (st2, Just in2))
-      | in1 == in2 = Just ((st1, st2), Just in1)
-      | otherwise  = Nothing
-    transitions :: ((state1, state2), Maybe input) -> Set ((state1, state2), Maybe output)
-    transitions ((st1, st2), inS) = Set.fromList . mapMaybe filterTransition . Set.toAscList $ out1 `setProduct` out2
+    l :: Natural
+    l = fromIntegral $ Seq.length out
+    wordStates :: Set Natural
+    wordStates
+      | Seq.null out = Set.empty
+      | otherwise = Set.fromDistinctAscList [0..pred l]
+    noProgress :: ((state, Maybe input), Set (state, Maybe output)) -> [(((Natural, state), Maybe input), Set ((Natural, state), Maybe output))]
+    noProgress ((inSt, inSym), outs)
+      = [ (((wState, inSt), inSym), Set.mapMonotonic (\(outSt, Nothing) -> ((wState, outSt), Nothing)) noOutput) | wState <- Set.toList wordStates, not $ Set.null noOutput ]
       where
-        out1 = (fromMaybe Set.empty $ stTransition fst1 !? (st1, inS)) `Set.union` (fromMaybe Set.empty $ stTransition fst1 !? (st1, Nothing))
-        out2 = (fromMaybe Set.empty $ stTransition fst2 !? (st2, inS)) `Set.union` (fromMaybe Set.empty $ stTransition fst2 !? (st2, Nothing))
+        noOutput = Set.filter (\(_, outSym) -> isNothing outSym) outs
+    transitions :: ((Natural, state), Maybe input) -> Set ((Natural, state), Maybe output)
+    transitions ((l, inSt), inSym) = Set.fromDistinctAscList [ ((succ l, outSt), outSym) | (outSt, outSym@(Just _)) <- Set.toAscList $ stTransition ! (inSt, inSym), outSym == Seq.lookup (fromIntegral l) out ]
+\end{code}
+\end{comment}
 
+\begin{comment}
+\begin{code}
 restrictFST :: forall state input output. (Ord state, Ord input, Ord output) => Set state -> FST state input output -> FST state input output
 -- ^ @restrictFST states fst@ removes from @fst@ all states not in @states@ including transitions leading to or originating from them
 restrictFST sts FST{..} = FST
@@ -170,7 +305,7 @@ liveFST :: forall state input output. (Ord state, Ord input, Ord output, Show st
 liveFST fst@FST{..} = flip execState Set.empty $ mapM_ (depthSearch Set.empty) stInitial
   where
     stTransition' :: Map state (Set state)
-    stTransition' = Map.map (Set.map $ \(st, _) -> st) $ Map.mapKeysWith Set.union (\(st, _) -> st) stTransition
+    stTransition' = Map.map (Set.map (\(st, _) -> st)) $ Map.mapKeysWith Set.union (\(st, _) -> st) stTransition
     depthSearch :: Set state -> state -> State (Set state) ()
     depthSearch acc curr = do
       let acc' = Set.insert curr acc
@@ -181,3 +316,4 @@ liveFST fst@FST{..} = flip execState Set.empty $ mapM_ (depthSearch Set.empty) s
       alreadyLive <- get
       mapM_ (depthSearch acc') $ next `Set.difference` alreadyLive
 \end{code}
+\end{comment}
diff --git a/edit-lens/src/Control/Lens/Edit.lhs b/edit-lens/src/Control/Lens/Edit.lhs
index 5a60536..5a106c8 100644
--- a/edit-lens/src/Control/Lens/Edit.lhs
+++ b/edit-lens/src/Control/Lens/Edit.lhs
@@ -1,3 +1,4 @@
+\begin{comment}
 \begin{code}
 module Control.Lens.Edit
   ( Module(..)
@@ -8,9 +9,10 @@ module Control.Lens.Edit
 
 import Control.Edit
 \end{code}
+\end{comment}
 
 \begin{defn}[Zustandsbehaftete Monoidhomomorphismen]
-Mit einer Menge von Komplementen $C$ und Monoiden $M$ und $N$ nennen wir eine partielle Funktion $\psi \colon C \times M \to C \times N$ einen zustandsbehafteten Monoidhomomorphismus wenn sie den folgenden Ansprüchen genügt:
+Gegeben eine Menge von Komplementen $C$ und Monoiden $M$ und $N$ nennen wir eine partielle Funktion $\psi \colon C \times M \to C \times N$ einen zustandsbehafteten Monoidhomomorphismus wenn sie den folgenden Ansprüchen genügt:
 
 \begin{itemize}
    \item $\forall c \in C \colon \psi(1_M, c) = (1_N, c)$
@@ -26,7 +28,7 @@ type StateMonoidHom s m n = (s, m) -> (s, n)
 \end{defn}
 
 \begin{defn}[edit-lenses]
-Für Moduln $M$ und $N$ besteht eine symmetrische edit-lens zwischen $M$ und $N$ aus zwei zustandsbehafteten Monoidhomomorphismen $\Rrightarrow \colon C \times \partial M \to C \times \partial N$ und $\Lleftarrow \colon C \time \partial N \to C \times \partial M$, mit kompatiblem Komplement $C$, einem ausgezeichneten Element $\ground_C$ und einer \emph{Konsistenzrelation} $K \subset \Dom M \times C \times \Dom N$ sodass gilt:
+Für Moduln $M$ und $N$ besteht eine symmetrische edit-lens zwischen $M$ und $N$ aus zwei zustandsbehafteten Monoidhomomorphismen $\Rrightarrow \colon C \times \partial M \to C \times \partial N$ und $\Lleftarrow \colon C \times \partial N \to C \times \partial M$, mit kompatiblem Komplement $C$, einem ausgezeichneten Element $\ground_C$ und einer \emph{Konsistenzrelation} $K \subset \Dom M \times C \times \Dom N$ sodass gilt:
 
 \begin{itemize}
   \item $(\init_M, \ground_C, \init_N) \in K$
@@ -41,7 +43,7 @@ Für Moduln $M$ und $N$ besteht eine symmetrische edit-lens zwischen $M$ und $N$
 
 Wir schreiben auch nur \emph{edit-lens} für den symmetrischen Fall\footnote{Für den asymmetrischen Fall siehe \cite{johnson2016unifying}}.
 
-In Haskell erwähnen wir die Konsistenzrelation nicht in der Erwartung, dass $\Rrightarrow$ und $\Lleftarrow$ nur auf konsistente Zustände angewandt werden (und somit auch entweder einen konsistenten Zustand erzeugen oder nichtt definiert sind):
+In Haskell erwähnen wir die Konsistenzrelation nicht in der Erwartung, dass $\Rrightarrow$ und $\Lleftarrow$ nur auf konsistente Zustände angewandt werden (und somit auch entweder einen konsistenten Zustand erzeugen oder nicht definiert sind):
 
 \begin{code}
 data EditLens c m n where
@@ -74,7 +76,7 @@ Für Typen $n$ und $m$ ist eine \emph{lens} $\ell$ von $n$ in $m$ eine Abbildung
 
 $$ \forall f \, \text{Funktor} \colon \left ( \ell \colon \left ( m \to f(m) \right ) \to \left ( n \to f(n) \right ) \right )$$
 
-Durch geschickte Wahl des Funktors\footnote{\texttt{Const} bzw. \texttt{Identity}} $f$ können dann $\searrow \colon m \to n$ und $\nearrow \colon (m \to m) \to (n \to n)$ rekonstruiert werden oder verwandte Strukturen (folds, traversals, …) konstruiert werden.
+Durch geschickte Wahl des Funktors\footnote{\texttt{Const} bzw. \texttt{Identity}} $f$ können dann $\searrow \colon m \to n$ und $\nearrow \colon (m \to m) \to (n \to n)$ oder verwandte Strukturen (folds, traversals, …) konstruiert werden.
 \end{defn}
 
 Es liegt nun nahe $\nearrow \colon (m \to m) \to (n \to n)$ mit $\Rrightarrow \colon \partial m \to \partial n$ zu identifizieren.
diff --git a/edit-lens/test/Control/DFST/LensTest.hs b/edit-lens/test/Control/DFST/LensTest.hs
new file mode 100644
index 0000000..46a1896
--- /dev/null
+++ b/edit-lens/test/Control/DFST/LensTest.hs
@@ -0,0 +1,35 @@
+module Control.DFST.LensTest where
+
+import Prelude hiding (init)
+
+import Control.DFST
+import Control.DFST.Lens
+import Control.FST hiding (stInitial, stTransition, stAccept)
+
+import Data.Set (Set)
+import qualified Data.Set as Set
+
+import Data.Map.Strict (Map)
+import qualified Data.Map.Strict as Map
+
+import Data.Sequence (Seq)
+import qualified Data.Sequence as Seq
+
+import Data.Maybe (maybeToList)
+
+import Test.Tasty
+import Test.Tasty.Hedgehog
+import Test.Tasty.HUnit hiding (assert)
+
+import Hedgehog
+import qualified Hedgehog.Gen as G
+import qualified Hedgehog.Range as R
+
+import Numeric.Natural
+
+import Control.DFSTTest
+
+hprop_applyDivInit :: Property
+hprop_applyDivInit = property $ do
+  word <- Seq.fromList <$> forAll genWord
+  init @(StringEdits Natural) `apply` (divInit word :: StringEdits Natural) === Just word
diff --git a/edit-lens/test/Control/DFSTTest.hs b/edit-lens/test/Control/DFSTTest.hs
new file mode 100644
index 0000000..4d91a03
--- /dev/null
+++ b/edit-lens/test/Control/DFSTTest.hs
@@ -0,0 +1,101 @@
+module Control.DFSTTest where
+
+import Control.DFST
+import Control.FST hiding (stInitial, stTransition, stAccept)
+
+import Data.Set (Set)
+import qualified Data.Set as Set
+
+import Data.Map.Strict (Map)
+import qualified Data.Map.Strict as Map
+
+import Data.Sequence (Seq)
+import qualified Data.Sequence as Seq
+
+import Data.Maybe (maybeToList)
+
+import Test.Tasty
+import Test.Tasty.Hedgehog
+import Test.Tasty.HUnit hiding (assert)
+
+import Hedgehog
+import qualified Hedgehog.Gen as G
+import qualified Hedgehog.Range as R
+
+import Numeric.Natural
+
+import Text.PrettyPrint.Leijen (Pretty(..))
+
+
+dfstId :: Ord a => Set a -> DFST () a a
+dfstId syms = DFST
+  { stInitial = ()
+  , stTransition = Map.fromList
+    [(((), sym), ((), Seq.singleton sym)) | sym <- Set.toList syms]
+  , stAccept = Set.singleton ()
+  }
+
+dfstDouble :: Ord a => Set a -> DFST () a a
+dfstDouble syms = DFST
+  { stInitial = ()
+  , stTransition = Map.fromList 
+    [(((), sym), ((), Seq.fromList [sym, sym])) | sym <- Set.toList syms]
+  , stAccept = Set.singleton ()
+  }
+
+dfstRunLengthDecode :: Ord a
+                    => Set a
+                    -> Natural
+                    -> DFST (Maybe Natural) (Either Natural a) a
+dfstRunLengthDecode syms lim = DFST
+  { stInitial = Nothing
+  , stTransition = Map.fromList . concat $
+    [ [((Nothing, Left n), (Just n, Seq.empty)) | n <- [0..lim]]
+    , [((Just n, Right sym), (Nothing, Seq.replicate (fromIntegral n) sym)) | n <- [0..lim], sym <- Set.toList syms]
+    ]
+  , stAccept = Set.singleton Nothing
+  }
+
+genWord :: Gen [Natural]
+genWord = G.list (R.linear 0 1000) . G.integral $ R.linear 0 100
+
+genDFST :: (Ord input, Ord output) => Set input -> Set output -> Gen (DFST Natural input output)
+genDFST inA outA = do
+  states <- G.set (R.linear 1 1000) . G.integral $ R.linear 0 100
+  stInitial <- G.element $ Set.toList states
+  stAccept <- Set.fromList <$> G.subsequence (Set.toList states)
+  stTransition <- fmap Map.fromList . G.list (R.linear 0 1000) . G.small $ do
+    st <- G.element $ Set.toList states
+    input <- G.element $ Set.toList inA
+    st' <- G.element $ Set.toList states
+    output <- fmap Seq.fromList . G.list (R.linear 0 20) . G.element $ Set.toList outA
+    return ((st, input), (st', output))
+  return DFST{..}
+    
+
+testDFST :: (Show output, Ord output, Show state, Ord state) => (Set Natural -> DFST state Natural output) -> (Seq Natural -> Seq output) -> Property
+testDFST mkDfst f = property $ do
+  input <- forAll genWord
+  let fst = mkDfst $ Set.fromList input
+  Just (f $ Seq.fromList input) === runDFST fst (Seq.fromList input)
+
+hprop_runDFSTId, hprop_runDFSTDouble :: Property
+hprop_runDFSTId = testDFST dfstId id
+hprop_runDFSTDouble = testDFST dfstDouble double
+  where
+    double :: Seq a -> Seq a
+    double Seq.Empty = Seq.Empty
+    double (a Seq.:<| as) = a Seq.:<| a Seq.:<| double as
+
+unit_runLengthDecode :: Assertion
+unit_runLengthDecode = runDFST dfst input @?= Just (Seq.fromList "aaacc")
+  where
+    input = Seq.fromList [Left 3, Right 'a', Left 0, Right 'b', Left 2, Right 'c']
+    dfst = dfstRunLengthDecode (Set.fromList "abc") 3
+
+hprop_toFST :: Property
+hprop_toFST = property $ do
+  input <- forAll genWord
+  dfst <- forAllWith (show . pretty . toFST) $ genDFST (Set.fromList $ input ++ [0..20]) (Set.fromList [0..20] :: Set Natural)
+  
+  runFST (toFST dfst) (Seq.fromList input) === maybeToList (runDFST dfst $ Seq.fromList input)
diff --git a/edit-lens/test/Control/FSTTest.hs b/edit-lens/test/Control/FSTTest.hs
new file mode 100644
index 0000000..f5e02c2
--- /dev/null
+++ b/edit-lens/test/Control/FSTTest.hs
@@ -0,0 +1,187 @@
+module Control.FSTTest where
+
+import Control.FST
+
+import Data.Set (Set)
+import qualified Data.Set as Set
+
+import Data.Map.Strict (Map)
+import qualified Data.Map.Strict as Map
+
+import Data.Sequence (Seq)
+import qualified Data.Sequence as Seq
+
+import Data.Maybe (fromMaybe)
+import Data.Foldable (Foldable(..))
+
+import Control.Monad (when)
+
+import Data.Void
+
+import Test.Tasty
+import Test.Tasty.Hedgehog
+import Test.Tasty.HUnit hiding (assert)
+
+import Hedgehog
+import qualified Hedgehog.Gen as G
+import qualified Hedgehog.Range as R
+
+import Numeric.Natural
+
+import Text.PrettyPrint.Leijen (Pretty(..))
+
+import Control.DeepSeq (force)
+
+
+fstId :: Ord a => Set a -> FST () a a
+fstId syms = FST
+  { stInitial = Set.singleton ()
+  , stTransition = Map.fromList [(((), Just sym), Set.singleton ((), Just sym)) | sym <- Set.toList syms]
+  , stAccept = Set.singleton ()
+  }
+
+fstDouble :: Ord a => Set a -> FST (Maybe a) a a
+fstDouble syms = FST
+  { stInitial = Set.singleton Nothing
+  , stTransition = Map.fromListWith Set.union . concat $
+    [ [((Nothing, Just sym), Set.singleton (Just sym, Just sym)) | sym <- Set.toList syms]
+    , [((Just sym, Nothing), Set.singleton (Nothing, Just sym)) | sym <- Set.toList syms]
+    ]
+  , stAccept = Set.singleton Nothing
+  }
+
+fstRunLengthDecode :: Ord a
+                   => Set a -- ^ Alphabet
+                   -> Natural -- ^ Upper limit to run length
+                   -> FST (Maybe (Natural, Maybe a)) (Either Natural a) a
+fstRunLengthDecode syms lim = FST
+  { stInitial = Set.singleton Nothing
+  , stTransition = Map.fromListWith Set.union . concat $
+    [ [((Nothing, Just (Left n)), Set.singleton (Just (n, Nothing), Nothing)) | n <- [0..lim]]
+    , [((Just (n, Nothing), Just (Right sym)), Set.singleton (Just (n, Just sym), Nothing)) | n <- [0..lim], sym <- Set.toList syms]
+    , [((Just (n, Just sym), Nothing), Set.singleton (Just (pred n, Just sym), Just sym)) | n <- [1..lim], sym <- Set.toList syms]
+    , [((Just (0, Just sym), Nothing), Set.singleton (Nothing, Nothing)) | sym <- Set.toList syms]
+    ]
+  , stAccept = Set.singleton Nothing
+  }
+
+data StRunLengthEncode a = STREInitial
+                         | STRECountUp a Natural
+                         | STRESwitch (Maybe a) a
+                         | STREFinish
+  deriving (Show, Eq, Ord)
+
+fstRunLengthEncode :: Ord a
+                   => Bool -- ^ Generate /all/ run length encodings instead of the best
+                   -> Set a -- ^ Alphabet
+                   -> Natural -- ^ Upper limit to run length
+                   -> FST (StRunLengthEncode a) a (Either Natural a)
+fstRunLengthEncode genAll syms lim = FST
+  { stInitial = Set.singleton STREInitial
+  , stTransition = Map.fromListWith Set.union . concat $
+    [ [((STREInitial, Just sym), Set.singleton (STRECountUp sym 1, Nothing)) | sym <- Set.toList syms]
+    , [((STRECountUp sym n, Just sym), Set.singleton (STRECountUp sym (succ n), Nothing)) | sym <- Set.toList syms, n <- [1..pred lim]]
+    , [((STRECountUp sym n, Just sym'), Set.singleton (STRESwitch (Just sym') sym, Just $ Left n)) | n <- [1..lim], sym <- Set.toList syms, sym' <- Set.toList syms, sym /= sym']
+    , [((STRECountUp sym lim, Just sym), Set.singleton (STRESwitch (Just sym) sym, Just $ Left lim)) | sym <- Set.toList syms]
+    , [((STRECountUp sym n, Nothing), Set.singleton (STRESwitch Nothing sym, Just $ Left n)) | sym <- Set.toList syms, n <- [1..lim]]
+    , [((STRESwitch (Just sym') sym, Nothing), Set.singleton (STRECountUp sym' 1, Just $ Right sym)) | sym <- Set.toList syms, sym' <- Set.toList syms]
+    , [((STRESwitch Nothing sym, Nothing), Set.singleton (STREFinish, Just $ Right sym)) | sym <- Set.toList syms]
+    , [((STRECountUp sym n, Just sym), Set.singleton (STRESwitch (Just sym) sym, Just $ Left n)) | n <- [1..lim], sym <- Set.toList syms, genAll]
+    ]
+  , stAccept = Set.fromList [STREInitial, STREFinish]
+  }
+
+
+genWord :: Gen [Natural]
+genWord = G.list (R.linear 0 1000) . G.integral $ R.linear 0 100
+
+  
+runFSTDet :: (Show output, Ord output, Show state, Ord state) => (Set Natural -> FST state Natural output) -> (Seq Natural -> Seq output) -> Property
+runFSTDet mkFst f = property $ do
+  input <- forAll genWord
+  let fst = mkFst $ Set.fromList input
+  annotateShow $ pretty fst
+  [f $ Seq.fromList input] === runFST fst (Seq.fromList input)
+
+
+hprop_runFSTId, hprop_runFSTDouble :: Property
+hprop_runFSTId = runFSTDet fstId id
+hprop_runFSTDouble = runFSTDet fstDouble double
+  where
+    double :: Seq a -> Seq a
+    double Seq.Empty = Seq.Empty
+    double (a Seq.:<| as) = a Seq.:<| a Seq.:<| double as
+
+hprop_runWordFST :: Property
+hprop_runWordFST = property $ do
+  input <- forAll genWord
+  let fst = wordFST $ Seq.fromList input
+  annotateShow $ pretty fst
+  [Seq.fromList input] === runFST fst (Seq.empty :: Seq Void)
+
+unit_runLengthDecode, unit_runLengthEncode :: Assertion
+unit_runLengthDecode = runFST fst input @?= [Seq.fromList "aaacc"]
+  where
+    input = Seq.fromList [Left 3, Right 'a', Left 0, Right 'b', Left 2, Right 'c']
+    fst = fstRunLengthDecode (Set.fromList "abc") 3
+unit_runLengthEncode = runFST fst input @?= [Seq.fromList [Left 3, Right 'a', Left 2, Right 'c']]
+  where
+    input = Seq.fromList "aaacc"
+    fst = fstRunLengthEncode False (Set.fromList "abc") 3
+
+hprop_runLength :: Property
+hprop_runLength = property $ do
+  input <- forAll genWord
+  let maxRun = fromMaybe 1 $ countMaxRun input
+      alphabet = Set.fromList input
+      encode = fstRunLengthEncode False alphabet maxRun
+      decode = fstRunLengthDecode alphabet maxRun
+  annotateShow $ countMaxRun input
+  annotateShow $ pretty encode
+  annotateShow $ pretty decode
+  [encoded] <- return . runFST encode $ Seq.fromList input
+  annotateShow $ toList encoded
+  [Seq.fromList input] === runFST decode encoded
+
+hprop_runLength' :: Property
+hprop_runLength' = property $ do
+  input <- forAll genWord
+
+  let maxRun = fromMaybe 1 $ countMaxRun input
+      alphabet = Set.fromList input
+      encode = fstRunLengthEncode True alphabet maxRun
+      decode = fstRunLengthDecode alphabet maxRun
+  annotateShow $ countMaxRun input
+  annotateShow $ pretty encode
+  annotateShow $ pretty decode
+
+  let encoded = runFST encode $ Seq.fromList input
+  annotateShow encoded
+  when (maxRun > 1) $
+    assert $ encoded `longerThan` 1
+
+  encoded' <- forAll $ G.element encoded
+  [Seq.fromList input] === runFST decode encoded'
+  where
+    longerThan :: [a] -> Natural -> Bool
+    longerThan []     _ = False
+    longerThan  _     0 = True
+    longerThan (_:xs) n = longerThan xs $ pred n
+
+countMaxRun :: (Eq a, Foldable f) => f a -> Maybe Natural
+countMaxRun = goMaxRun Nothing Nothing . toList
+  where
+    goMaxRun :: Eq a
+             => Maybe (a, Natural) -- ^ Maximum
+             -> Maybe (a, Natural) -- ^ Current
+             -> [a]
+             -> Maybe Natural
+    goMaxRun Nothing Nothing [] = Nothing
+    goMaxRun (Just (_, n)) (Just (_, n')) []
+      | n' > n    = Just n'
+      | otherwise = Just n
+    goMaxRun Nothing Nothing (a:as) = goMaxRun (Just (a, 1)) (Just (a, 1)) as
+    goMaxRun (Just (a, n)) (Just (a', n')) (x:xs)
+      | x == a'   = goMaxRun (Just (a, n)) (Just (a', succ n')) xs
+      | n' > n    = goMaxRun (Just (a', n')) (Just (x, 1)) xs
+      | otherwise = goMaxRun (Just (a, n)) (Just (x, 1)) xs
diff --git a/edit-lens/test/Driver.hs b/edit-lens/test/Driver.hs
new file mode 100644
index 0000000..327adf4
--- /dev/null
+++ b/edit-lens/test/Driver.hs
@@ -0,0 +1 @@
+{-# OPTIONS_GHC -F -pgmF tasty-discover -optF --tree-display #-}
diff --git a/interactive-edit-lens/ChangeLog.md b/interactive-edit-lens/ChangeLog.md
new file mode 100644
index 0000000..8bae309
--- /dev/null
+++ b/interactive-edit-lens/ChangeLog.md
@@ -0,0 +1,5 @@
+# Revision history for edit-lens
+
+## 0.0.0.0
+
+* First version.
diff --git a/interactive-edit-lens/LICENSE b/interactive-edit-lens/LICENSE
new file mode 100644
index 0000000..4522849
--- /dev/null
+++ b/interactive-edit-lens/LICENSE
@@ -0,0 +1,30 @@
+Copyright (c) 2017, Gregor Kleen
+
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions are met:
+
+    * Redistributions of source code must retain the above copyright
+      notice, this list of conditions and the following disclaimer.
+
+    * Redistributions in binary form must reproduce the above
+      copyright notice, this list of conditions and the following
+      disclaimer in the documentation and/or other materials provided
+      with the distribution.
+
+    * Neither the name of Gregor Kleen nor the names of other
+      contributors may be used to endorse or promote products derived
+      from this software without specific prior written permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
diff --git a/interactive-edit-lens/Setup.hs b/interactive-edit-lens/Setup.hs
new file mode 100644
index 0000000..9a994af
--- /dev/null
+++ b/interactive-edit-lens/Setup.hs
@@ -0,0 +1,2 @@
+import Distribution.Simple
+main = defaultMain
diff --git a/interactive-edit-lens/package.yaml b/interactive-edit-lens/package.yaml
new file mode 100644
index 0000000..9bc3ead
--- /dev/null
+++ b/interactive-edit-lens/package.yaml
@@ -0,0 +1,53 @@
+name: interactive-edit-lens
+version: 0.0.0.0
+license: BSD3
+license-file: LICENSE
+author: Gregor Kleen <aethoago@141.li>
+build-type: Simple
+extra-source-files:
+  - ChangeLog.md
+git: https://git.yggdrasil.li/gkleen/pub/bachelor-thesis
+
+default-extensions:
+  - RecordWildCards
+  - MultiParamTypeClasses
+  - FlexibleInstances
+  - FlexibleContexts
+  - FunctionalDependencies
+  - TupleSections
+  - TypeApplications
+  - ViewPatterns
+  - PatternSynonyms
+  - TypeFamilies
+  - TypeOperators
+  - MultiWayIf
+
+other-extensions:
+  - TemplateHaskell
+
+dependencies:
+  - base
+  - lens
+  - containers
+  - edit-lens
+  - brick
+  - vty
+  - text
+  - text-zipper
+  - dyre
+  - mtl
+  - transformers
+
+# ghc-options: [ -O2 ]
+
+library:
+  source-dirs: src
+  exposed-modules:
+    - Interact
+    - Interact.Types
+
+executables:
+  interact:
+    ghc-options: [ -threaded ]
+    source-dirs: src
+    main: Main.hs
diff --git a/interactive-edit-lens/src/Interact.hs b/interactive-edit-lens/src/Interact.hs
new file mode 100644
index 0000000..3aab5c2
--- /dev/null
+++ b/interactive-edit-lens/src/Interact.hs
@@ -0,0 +1,271 @@
+{-# LANGUAGE ScopedTypeVariables
+           , OverloadedStrings
+  #-}
+
+module Interact
+  ( interactiveEditLens
+  , module Interact.Types
+  , module Config.Dyre
+  ) where
+
+import Prelude hiding (init)
+
+import Interact.Types
+
+import Data.Text (Text)
+import qualified Data.Text as Text
+
+import Data.Text.Zipper
+
+import Data.Sequence (Seq)
+import qualified Data.Sequence as Seq
+
+import Control.Lens
+import Numeric.Lens
+import System.IO
+import Control.Monad
+import Control.Monad.RWS hiding (Last(..), (<>))
+import Control.Monad.Trans.Maybe
+import Control.Monad.Trans.Reader (runReaderT)
+
+import Data.Bool (bool)
+import Data.Tuple (swap)
+import Data.Maybe (fromMaybe)
+import Data.List (groupBy)
+import Data.Function (on)
+import Data.Char (isSpace)
+
+import Data.Foldable (Foldable(toList))
+
+import Brick hiding (on)
+import Brick.Focus
+import Brick.Widgets.Center
+import Brick.Widgets.Border
+import Graphics.Vty hiding (showCursor)
+
+import Config.Dyre
+
+import System.IO.Unsafe
+import Debug.Trace
+
+interactiveEditLens :: (Params (InteractConfig c) -> Params (InteractConfig c)) -> InteractConfig c -> IO ()
+interactiveEditLens f = wrapMain . f $ defaultParams
+  { projectName = "interact-edit-lens"
+  , showError   = \s err -> s & compileError .~ Just err
+  , realMain    = interactiveEditLens'
+  }
+
+interactiveEditLens' :: forall c. InteractConfig c -> IO ()
+interactiveEditLens' cfg@InteractConfig{..}
+  | Just err <- icfgCompileError
+  = hPutStrLn stderr err
+  | otherwise
+  = void . defaultMain app $! initialState &?~ do
+      let
+        a :: Lens' (InteractState c) (Last Validity, Last (Seq Char, Int), StringEdits Natural Char)
+        a = case icfgInitial of
+          InitialRight _ -> right
+          _other -> left
+        b :: Lens' (InteractState c) (Last Validity, Last (Seq Char, Int), StringEdits Natural Char)
+        b = case icfgInitial of
+          InitialRight _ -> left
+          _other -> right
+        dir :: InteractDirection
+        dir = case icfgInitial of
+          InitialRight _ -> PropagateLeft
+          _other -> PropagateRight
+        aDom :: Seq Char
+        (view charseq -> aDom) = case icfgInitial of
+          InitialRight t -> t
+          InitialLeft t -> t
+          InitialEmpty -> ""
+      doEdit $ divInit aDom & stringEdits . sePos %~ (fromIntegral :: Natural -> Integer)
+      -- a .= (Last Valid, Last (aDom, 0))
+      -- bEdit <- prop dir $ divInit aDom
+      -- (b %=) . maybe id (<>) <=< runMaybeT $ do
+      --   bDom <- use $ b . _2 . _Wrapped . _1
+      --   bDom' <- MaybeT . return $ bDom `apply` bEdit
+      --   return $ (Last Valid, Last (bDom', 0))
+  where
+    infix 1 &?~
+    
+    (&?~) :: a -> RWS (InteractConfig c) () a b -> a
+    st &?~ act = (\(s, ()) -> s) $ execRWS act cfg st
+
+    actOn = (&?~)
+
+    initialState :: InteractState c
+    initialState = InteractState
+      { istComplement = ground icfgLens
+      , istLeft  = (Last Valid, Last (init @(StringEdits Natural Char), 0), mempty)
+      , istRight = (Last Valid, Last (init @(StringEdits Natural Char), 0), mempty)
+      , istFocus = focusRing [LeftEditor, RightEditor] &
+                     focusSetCurrent (case icfgInitial of InitialRight _ -> RightEditor; _other -> LeftEditor)
+      }
+
+    app :: InteractApp c
+    app = App{..}
+
+    appDraw :: InteractState c -> [Widget InteractName]
+    appDraw InteractState{..} = [ editors ]
+      where
+        editors = hBox
+          [ mbInvalid (withFocusRing istFocus renderEditor') (istLeft `WithName` LeftEditor)
+          , vBorder
+          , mbInvalid (withFocusRing istFocus renderEditor') (istRight `WithName` RightEditor)
+          ]
+        renderEditor' :: Bool -> (Seq Char, Int) `WithName` InteractName -> Widget InteractName
+        renderEditor' foc ((content, cPos) `WithName` n)
+          = txt (review charseq content)
+          & bool id (showCursor n cPos') foc
+          & visibleRegion cPos' (1, 1) 
+          & viewport n Both
+          where
+            (cPrefix, _) = Seq.splitAt cPos content
+            newls = Seq.findIndicesR (== '\n') cPrefix
+            cPos' = case newls of
+              (p:_)    -> Location (pred $ cPos - p, length newls)
+              []       -> Location (cPos, 0)
+        mbInvalid _ ((Last Invalid, _     , _) `WithName` _)
+          = txt "Invalid"
+          & border
+          & center
+        mbInvalid f ((Last Valid  , Last x, _) `WithName` n) = f $ x `WithName` n
+
+    appHandleEvent :: InteractState c -> BrickEvent InteractName InteractEvent -> EventM InteractName (Next (InteractState c))
+    appHandleEvent st@InteractState{..} (VtyEvent ev) = case ev of
+      EvKey  KEsc        []      -> halt st
+      EvKey (KChar 'c')  [MCtrl] -> halt st
+      EvKey (KChar '\t') []      -> continue $ st & focus %~ focusNext
+      EvKey  KBackTab    []      -> continue $ st & focus %~ focusPrev
+      EvKey (KChar 'a')  [MCtrl] -> continue $ st &?~ doMove
+        (moveSplit (== '\n') $ \(c, (l, p)) -> if any (== '\n') (c l) || Seq.null (c l) then (pred l, 0) else (l, 0))
+      EvKey (KChar 'e')  [MCtrl] -> continue $ st &?~ doMove
+        (moveSplit (== '\n') $ \(c, (l, p)) -> if any (== '\n') $ c l then (pred l, Seq.length . c $ pred l) else (l, Seq.length $ c l))
+      EvKey  KLeft       [MCtrl] -> continue $ st &?~ doMove
+        (moveSplit isSpace   $ \(c, (l, _)) -> if any isSpace (c l) || Seq.null (c l) then (pred l, 0) else (l - 2, 0))
+      EvKey  KRight      [MCtrl] -> continue $ st &?~ doMove
+        (moveSplit isSpace   $ \(c, (l, _)) -> if any isSpace $ c l then (succ l, 0) else (l + 2, 0))
+      EvKey  KUp         [MCtrl] -> continue $ st &?~ doMove
+        (moveSplit (== '\n') $ \(c, (l, p)) -> if any (== '\n') (c l) || Seq.null (c l) then (pred l, p) else (l - 2, p))
+      EvKey  KDown       [MCtrl] -> continue $ st &?~ doMove
+        (moveSplit (== '\n') $ \(c, (l, p)) -> if any (== '\n') (c l) then (succ l, p) else (l + 2, p))
+      EvKey  KLeft       []      -> continue $ st &?~ doMove moveLeft
+      EvKey  KRight      []      -> continue $ st &?~ doMove moveRight
+      EvKey  KDel        []      -> continue $ st &?~ doEdit (delete 0)
+      EvKey  KBS         []      -> continue . actOn st $ do
+        focused' <- preuse $ focused . _2 . _Wrapped
+        doEdit . delete $ -1
+        unless (maybe False ((==) <$> view _2 <*> view (_1 . to Seq.length)) focused') $
+          doMove moveLeft
+      EvKey (KChar c)    []      -> continue . actOn st $ do
+        doEdit $ insert 0 c
+        doMove moveRight
+      EvKey KEnter       []      -> continue . actOn st $ do
+        doEdit $ insert 0 '\n'
+        doMove moveRight
+      other                      -> suspendAndResume $ do
+        traceIO $ "Unhandled event:\n\t" ++ show other
+        return st
+      -- where
+      --   editorMovement f = continue $ st & focused . _Just . editContentsL %~ f
+    appHandleEvent st _ = continue st
+
+    doMove = zoom $ focused . _2 . _Wrapped
+
+    moveLeft, moveRight :: MonadState (Seq Char, Int) m => m ()
+    moveLeft = modify $ \now@(_, nowP) -> if
+      | nowP > 0  -> now & _2 %~ pred
+      | otherwise -> now
+    moveRight = modify $ \now@(contents, nowP) -> if
+      | nowP < length contents -> now & _2 %~ succ
+      | otherwise              -> now
+
+    moveSplit :: MonadState (Seq Char, Int) m
+              => (Char -> Bool) -- ^ Separator predicate
+              -> (((Int -> Seq Char), (Int, Int)) -> (Int, Int)) -- ^ Move in split coordinates (e.g. @(line, charInLine)@) with access to the focused fragment
+              -> m ()
+    moveSplit splitPred relMove = modify $ \now@(toList -> contentsStr, nowP)
+        -> let splitContents = groupBy ((==) `on` splitPred) contentsStr
+               traceShow x y = flip seq y . unsafePerformIO . appendFile "interact.log" . (<> "\n\n") $ show x
+               (before, mCurrent, after) = snd . (\x -> traceShow (nowP, x) x) $ foldl go (0, ([], Nothing, [])) splitContents
+               go acc@(i, st) cGroup
+                 | i <= nowP, nowP < i + length cGroup = (i + length cGroup, st & _2 .~ Just cGroup)
+                 | i + length cGroup <= nowP = (i + length cGroup, st & _1 %~ (flip snoc cGroup))
+                 | otherwise = (i + length cGroup, st & _3 %~ (flip snoc cGroup))
+               relPos = (length before, nowP - sum (map length before))
+               (newL, newS) = relMove (\i -> if 0 <= i && i < length splitContents then Seq.fromList $ splitContents !! i else Seq.empty, relPos)
+               newPos
+                 | null splitContents
+                 , newL /= 0 || newS /= 0 = (0, 0)
+                 | newL >= length splitContents = (pred $ length splitContents, length $ last splitContents)
+                 | newL < 0 = (0, 0)
+                 | newS < 0 = (newL, 0)
+                 | newS > length (splitContents !! newL) = (newL, length $ splitContents !! newL)
+                 | otherwise = (newL, newS)
+            in now & _2 .~ sum (map length $ take (fst newPos) splitContents) + snd newPos
+
+    appStartEvent :: InteractState c -> EventM InteractName (InteractState c)
+    appStartEvent = return
+
+    appAttrMap :: InteractState c -> AttrMap
+    appAttrMap = const $ attrMap defAttr []
+
+    appChooseCursor :: InteractState c -> [CursorLocation InteractName] -> Maybe (CursorLocation InteractName)
+    appChooseCursor = focusRingCursor istFocus
+
+prop :: forall st cfg m.
+        ( MonadState st m
+        , MonadReader cfg m
+        , HasComplement st (Complement cfg)
+        , HasEditLens cfg (StringEdits Natural Char) (StringEdits Natural Char)
+        )
+     => InteractDirection -> StringEdits Natural Char -> m (StringEdits Natural Char)
+prop dir edits = do
+  propD <- case dir of
+    PropagateRight -> asks propR
+    PropagateLeft  -> asks propL
+  (c, res) <- propD . (, edits) <$> use complement
+  unsafePerformIO . fmap return . appendFile "interact.log" . (<> "\n\n") $ show (edits, dir, res)
+  res <$ assign complement c
+
+doEdit :: forall m c.
+          ( MonadState (InteractState c) m
+          , MonadReader (InteractConfig c) m
+          )
+       => StringEdits Integer Char -> m ()
+doEdit relativeEdit = void . runMaybeT $ do
+  currentFocus <- MaybeT $ uses focus focusGetCurrent
+  let direction
+        | RightEditor <- currentFocus = PropagateLeft
+        | otherwise                   = PropagateRight
+      aL :: Lens' (InteractState c) (Last Validity, Last (Seq Char, Int), StringEdits Natural Char)
+      aL | PropagateRight <- direction = left
+         | PropagateLeft  <- direction = right
+      bL :: Lens' (InteractState c) (Last Validity, Last (Seq Char, Int), StringEdits Natural Char)
+      bL | PropagateRight <- direction = right
+         | PropagateLeft  <- direction = left
+      (aN, bN) = bool swap id (direction == PropagateRight) (LeftEditor, RightEditor)
+  currentZipper <- use $ aL . _2 . _Wrapped
+  let currentPos = currentZipper ^. _2
+  absoluteEdit <- MaybeT . return $ do
+    let minOffset = minimumOf (stringEdits . sePos) relativeEdit
+    guard $ maybe True (\o -> 0 <= currentPos + fromIntegral o) minOffset
+    return $ relativeEdit & stringEdits . sePos %~ (\n -> fromIntegral $ currentPos + fromIntegral n)
+  newContent <- MaybeT . return $ view _1 currentZipper `apply` absoluteEdit
+  let currentPos'
+        | currentPos < 0 = 0
+        | currentPos > length newContent = length newContent
+        | otherwise = currentPos
+  aL . _2 %= (<> Last (newContent, currentPos'))
+  absoluteEdit' <- uses (aL . _3) (absoluteEdit `mappend`)
+  bEdits <- prop direction absoluteEdit'
+  bDom <- use $ bL . _2 . _Wrapped . _1
+  case bDom `apply` bEdits of
+    Nothing -> do
+      bL . _1 %= (<> Last Invalid)
+      aL . _3 .= absoluteEdit'
+    Just bDom' -> do
+      bL . _1 %= (<> Last Valid)
+      bL . _2 . _Wrapped . _1 .= bDom'
+      aL . _3 .= mempty
diff --git a/interactive-edit-lens/src/Interact/Types.hs b/interactive-edit-lens/src/Interact/Types.hs
new file mode 100644
index 0000000..a4d08ac
--- /dev/null
+++ b/interactive-edit-lens/src/Interact/Types.hs
@@ -0,0 +1,120 @@
+{-# LANGUAGE TemplateHaskell #-}
+{-# OPTIONS_GHC -fno-warn-orphans #-}
+
+module Interact.Types
+  ( InteractName(..)
+  , _LeftEditor, _RightEditor, _PrimitiveName
+  , Validity, pattern Valid, pattern Invalid
+  , InteractState(..)
+  , HasLeft(..), HasRight(..), HasComplement(..), HasFocus(..), HasFocused(..)
+  , InteractInitial(..)
+  , _InitialLeft, _InitialRight, _InitialEmpty
+  , InteractConfig(..)
+  , HasInitial(..), HasLens(..), HasCompileError(..)
+  , InteractEvent
+  , InteractApp
+  , InteractDirection(..)
+  , charseq
+  , WithName(..)
+  , module Control.Edit
+  , module Control.Lens.Edit
+  , module Control.DFST.Lens
+  , module Data.Semigroup
+  , module Numeric.Natural
+  ) where
+
+import Data.Text (Text)
+import qualified Data.Text as Text
+import Data.Text.Lens
+
+import Data.Sequence (Seq(..))
+import qualified Data.Sequence as Seq
+
+import qualified Data.Foldable as Foldable
+
+import Data.Semigroup (Semigroup(..), Last(..))
+import Numeric.Natural
+
+import Brick
+import Brick.Focus
+import Brick.Widgets.Edit
+
+import Control.Lens
+import Control.Lens.TH
+import Control.Edit
+import Control.Lens.Edit
+import Control.DFST.Lens
+
+import Data.Text.Zipper.Generic
+
+
+data InteractName
+  = LeftEditor
+  | RightEditor
+  | PrimitiveName !Text
+  deriving (Eq, Ord, Show, Read)
+
+makePrisms ''InteractName
+
+type Validity = Bool
+pattern Valid = True
+pattern Invalid = False
+
+data InteractState c = InteractState
+  { istLeft, istRight :: (Last Validity, Last (Seq Char, Int), StringEdits Natural Char)
+  , istComplement :: c
+  , istFocus :: FocusRing InteractName
+  }
+
+makeLensesWith abbreviatedFields ''InteractState
+
+class HasFocused s a | s -> a where
+  focused :: Traversal' s a
+
+instance HasFocused (InteractState c) (Last Validity, Last (Seq Char, Int), StringEdits Natural Char) where
+  focused f st@InteractState{..} = case focusGetCurrent istFocus of
+    Just LeftEditor  -> left f st
+    Just RightEditor -> right f st
+    _other           -> pure st
+
+data InteractInitial
+  = InitialLeft Text
+  | InitialRight Text
+  | InitialEmpty
+  deriving (Eq, Ord, Show, Read)
+
+makePrisms ''InteractInitial
+
+data InteractConfig c = InteractConfig
+  { icfgInitial :: InteractInitial
+  , icfgLens :: EditLens c (StringEdits Natural Char) (StringEdits Natural Char)
+  , icfgCompileError :: Maybe String
+  }
+
+instance HasEditLens (InteractConfig c) (StringEdits Natural Char) (StringEdits Natural Char) where
+  type Complement (InteractConfig c) = c
+  ground = ground . icfgLens
+  propR = propR . icfgLens
+  propL = propL . icfgLens
+
+makeLensesWith abbreviatedFields ''InteractConfig
+
+charseq :: Iso' Text (Seq Char)
+charseq = iso Text.unpack Text.pack . iso Seq.fromList Foldable.toList
+
+type InteractEvent = ()
+
+type InteractApp c = App (InteractState c) InteractEvent InteractName
+
+data InteractDirection = PropagateLeft | PropagateRight
+  deriving (Eq, Ord, Enum, Bounded, Show, Read)
+
+makePrisms ''InteractDirection
+
+
+infixr 1 `WithName`
+data WithName x n = WithName x n
+  deriving (Eq, Ord, Show, Read)
+
+instance Named (x `WithName` n) n where
+  getName (_ `WithName` n) = n
diff --git a/interactive-edit-lens/src/Main.hs b/interactive-edit-lens/src/Main.hs
new file mode 100644
index 0000000..f7bc806
--- /dev/null
+++ b/interactive-edit-lens/src/Main.hs
@@ -0,0 +1,94 @@
+{-# LANGUAGE OverloadedStrings
+           , ExistentialQuantification
+  #-}
+
+module Main where
+
+import Interact
+import Control.DFST.Lens
+import Control.DFST
+
+import Data.Map (Map)
+import qualified Data.Map as Map
+
+import Data.Set (Set)
+import qualified Data.Set as Set
+
+import Data.Sequence (Seq)
+import qualified Data.Sequence as Seq
+
+import Data.Char
+
+import System.Environment
+import System.Exit
+
+import Debug.Trace
+
+data SomeDFST = forall state. (Ord state, Show state) => SomeDFST { someDFST :: DFST state Char Char }
+
+data JsonNewlState = JNUndeterminedStructure | JNOutsideStructure | JNInsideStructure | JNInsideString | JNEscape
+  deriving (Eq, Ord, Show, Read)
+
+dfstMap :: String -> Maybe SomeDFST
+dfstMap "double" = Just . SomeDFST $ DFST
+  { stInitial = ()
+  , stTransition = mconcat
+    [ Map.fromList 
+        [(((), sym), ((), Seq.fromList [sym, sym])) | sym <- ['a'..'z'] ++ ['A'..'Z'] ++ ['!', ' ']] -- sym <- [minBound..maxBound], isPrint sym]
+    , Map.singleton ((), '\n') ((), Seq.singleton '\n')
+    ]
+  , stAccept = Set.singleton ()
+  }
+dfstMap "id" = Just . SomeDFST $ DFST
+  { stInitial = ()
+  , stTransition = Map.fromList 
+      [(((), sym), ((), Seq.singleton sym)) | sym <- ['a'..'z'] ++ ['A'..'Z'] ++ ['!', ' ']] -- sym <- [minBound..maxBound], isPrint sym]
+  , stAccept = Set.singleton ()
+  }
+dfstMap "alternate" = Just . SomeDFST $ DFST
+  { stInitial = 0 :: Int
+  , stTransition = mconcat
+    [ Map.fromList [((0, sym), (1, Seq.singleton sym)) | sym <- ['a'..'z'] ++ ['A'..'Z'] ++ ['!', ' ']] -- sym <- [minBound..maxBound], isPrint sym]
+    , Map.fromList [((1, sym), (0, Seq.fromList [toUpper sym, toUpper sym])) | sym <- ['a'..'z'] ++ ['A'..'Z'] ++ ['!', ' ']] -- sym <- [minBound..maxBound], isPrint sym]
+    ]
+  , stAccept = Set.fromList [0]
+  }
+dfstMap "json-newl" = Just . SomeDFST $ DFST
+  { stInitial = JNOutsideStructure
+  , stTransition = mconcat
+    [ Map.fromList [((jnOutside, sym), (jnOutside, Seq.empty)) | sym <- whitespace, jnOutside <- [JNOutsideStructure, JNUndeterminedStructure]]
+    , Map.fromList [((jnOutside, sym), (JNInsideStructure, Seq.fromList [sym, ' '])) | sym <- "[{", jnOutside <- [JNOutsideStructure, JNInsideStructure, JNUndeterminedStructure]]
+    , Map.fromList [((JNInsideStructure, sym), (JNInsideStructure, Seq.empty)) | sym <- whitespace]
+    , Map.fromList [((jnInside, sym), (JNUndeterminedStructure, Seq.fromList ['\n', sym])) | sym <- "]}", jnInside <- [JNInsideStructure, JNUndeterminedStructure]]
+    , Map.fromList [((jnInside, ','), (JNInsideStructure, Seq.fromList "\n, ")) | jnInside <- [JNInsideStructure, JNUndeterminedStructure] ]
+    , Map.fromList [((jnInside, ':'), (JNInsideStructure, Seq.fromList " : ")) | jnInside <- [JNInsideStructure, JNUndeterminedStructure] ]
+    , Map.fromList [((jn, '"'), (JNInsideString, Seq.singleton '"')) | jn <- [JNUndeterminedStructure, JNInsideStructure]]
+    , Map.fromList [((JNInsideString, sym), (JNInsideString, Seq.singleton sym)) | sym <- ['a'..'z'] ++ ['A'..'Z'] ++ ",.!?: "]
+    , Map.singleton (JNInsideString, '"') (JNUndeterminedStructure, Seq.singleton '"')
+    , Map.singleton (JNInsideString, '\\') (JNEscape, Seq.singleton '\\')
+    , Map.singleton (JNEscape, '"') (JNInsideString, Seq.singleton '"')
+    ]
+  , stAccept = Set.fromList [JNOutsideStructure, JNUndeterminedStructure]
+  }
+  where
+    whitespace = toEnum <$> [0x0020, 0x0009, 0x000a, 0x000d]
+dfstMap _ = Nothing
+
+main :: IO ()
+main = do
+  args <- getArgs
+
+  dfst <- case args of
+    [name] | Just dfst <- dfstMap name
+      -> return dfst
+    _ -> exitWith $ ExitFailure 2
+
+  interactiveEditLens' dfst
+
+interactiveEditLens' :: SomeDFST -> IO ()
+interactiveEditLens' (SomeDFST dfst)
+  = interactiveEditLens id $ InteractConfig
+      { icfgInitial = InitialEmpty
+      , icfgLens = dfstLens dfst
+      , icfgCompileError = Nothing
+      }
diff --git a/literature.meta.yml b/literature.meta.yml
new file mode 100644
index 0000000..9eb6cc7
--- /dev/null
+++ b/literature.meta.yml
@@ -0,0 +1,5 @@
+---
+lang: de-de
+link-citations: true
+bibliography: literature.bibtex
+...
diff --git a/stack.nix b/stack.nix
index eec431d..dba6b1d 100644
--- a/stack.nix
+++ b/stack.nix
@@ -1,4 +1,4 @@
-{ ghc, nixpkgs ? (import <nixpkgs> {}) }:
+{ ghc, nixpkgs ? (import <nixos> {}) }:
 
 let
   inherit (nixpkgs) haskell pkgs;
diff --git a/stack.yaml b/stack.yaml
index f36abc3..92dd54f 100644
--- a/stack.yaml
+++ b/stack.yaml
@@ -5,7 +5,11 @@ packages:
       git: https://github.com/pngwjpgh/composition-tree
       commit: c9c1c11f6820bbbe1ac96513a66609599483bdb6
     extra-dep: true
+  - interactive-edit-lens
 nix:
   packages: []
   pure: false
   shell-file: ./stack.nix
+build:
+  library-profiling: true
+  executable-profiling: true
diff --git a/thesis.pdf.gup b/thesis.pdf.gup
index 2167f7b..6e2fa65 100755
--- a/thesis.pdf.gup
+++ b/thesis.pdf.gup
@@ -1,10 +1,13 @@
 #!/usr/bin/env zsh
 
+gup -u edit-lens/src/Control/Edit.lhs
 gup -u edit-lens/src/Control/Lens/Edit.lhs
-gup -u edit-lens/src/Control/Lens/Edit/Compose.lhs
+gup -u edit-lens/src/Control/FST.lhs
+gup -u edit-lens/src/Control/DFST.lhs
+gup -u edit-lens/src/Control/DFST/Lens.lhs
 
 bDir=$(pwd)
 
 cd ..
 
-exec gup/pdf.gup $1 ${bDir}/$2
\ No newline at end of file
+exec gup/pdf.gup $1 ${bDir}/$2
diff --git a/thesis.tex b/thesis.tex
index 6b94318..cf5c489 100644
--- a/thesis.tex
+++ b/thesis.tex
@@ -1,5 +1,19 @@
+\section{Edit-lenses}
+
 Ziel ist es zunächst edit-lenses alá \cite{hofmann2012edit} in Haskell zur Verfügung zu stellen.
 Dabei werden wir die Definitionen aus \cite{hofmann2012edit} sowohl in natürlicher Sprache als auch in lauffähigem Haskell vorstellen:
 
 \input{./edit-lens/src/Control/Edit.lhs}
 \input{./edit-lens/src/Control/Lens/Edit.lhs}
+
+\section{Finite state transducers}
+
+\input{./edit-lens/src/Control/FST.lhs}
+\input{./edit-lens/src/Control/DFST.lhs}
+
+\subsection{Edit-lenses für deterministic finite state transducers}
+\input{./edit-lens/src/Control/DFST/Lens.lhs}
+
+% \section{Container}
+
+% TODO: recover from git