Update README

README.md

@@ -30,13 +30,20 @@ See [KarinaTyulebaeva/lambda-n-ways](https://github.com/KarinaTyulebaeva/lambda-
 
 The Haskell code is organized into multiple modules as follows:
 
-- [`Control.Monad.Foil`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html) provides basic definitions for the foil[^1], including safe scopes, sinking, name maps, and scope constraints. In addition to the standard definitions, we contribute [`CoSinkable`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html#t:CoSinkable) class (dual to [`Sinkable`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html#t:Sinkable), for patterns that generalize `NameBinder`) and a general total [`NameMap`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html#g:4) which is useful for some implementations (e.g. conversion functions).
+- [`Control.Monad.Foil`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html) provides basic definitions for the foil[^1], including safe scopes, sinking, name maps, and scope constraints. In addition to the standard definitions, we contribute
+
+  - [`CoSinkable`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html#t:CoSinkable) class (dual to [`Sinkable`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html#t:Sinkable), for patterns that generalize `NameBinder`)
+  - a general total [`NameMap`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil.html#g:4) which is useful for some implementations (e.g. conversion functions)
+  - unification of binders, which is useful for efficient α-equivalence implementations
 
   - [`Control.Monad.Foil.TH`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil-TH.html) provides Template Haskell functions that generate scope-safe representation types together with some conversion functions and helpers for patterns. All of this is generated from a raw recursive representation of syntax, that has to be split into 4 types: type of variable identifiers, type of terms, type of scoped terms (to clearly understand which parts get under binders), and type of patterns.
 
   - [`Control.Monad.Foil.Relative`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil-Relative.html) defines a variation of a relative monad typeclass specifcially indexed by scope type variables of kind `S`. This is merely for the demonstration that term type constructors are monads relative to [`Name`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Foil-Internal.html#t:Name).
 
-- [`Control.Monad.Free.Foil`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Free-Foil.html) defines a variation of free scoped (relative) monads relying on the foil for the scope-safe efficient handling of the binders. This module also defines generic substitution for this generic representation of syntax with binders.
+- [`Control.Monad.Free.Foil`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Free-Foil.html) defines a variation of free scoped (relative) monads relying on the foil for the scope-safe efficient handling of the binders. This module also defines
+
+  - generic substitution for this generic representation of syntax with binders
+  - generic α-normalization and α-equivalence checks
 
 - [`Language.LambdaPi.Impl.Foil`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Language-LambdaPi-Impl-Foil.html) defines a simple interpreter for λΠ-calculus with pairs, using the manually constructed scope-safe representation and conversion to and from the raw representation generated by BNFC.
 
@@ -52,7 +59,7 @@ Scala implementation is not documented yet.
 
 In Haskell:
 
-1. We do not (yet) provide α-equivalence for scopes. That said, we do have a tested (but undocumented) implementation for the foil in the benchmarks (see [`unsafeAeq`](https://github.com/KarinaTyulebaeva/lambda-n-ways/blob/09a995f9644f6bf29db1803eef004ddd382b823d/lib/FreeScoped/Foil.hs#L336-L401)).
+1. We do not (yet) generate α-equivalence via Template Haskell or `GHC.Generics`. We do provide a generic α-equivalence functions for `AST sig n`, and demonstrate how to implement the check manually for λΠ-terms using the foil, but in generating this from Template Haskell or using GHC.Generics is left for future work.
 2. We do not (yet) demonstrate an implementation of the typechecker for λΠ in these representations. While it is largely straightforward the main non-trivial part is the equality for the Π-types, which relies on the α-equivalence from the previous item.
 3. The free foil does not (yet) support patterns, only single-variable binders. While we think it should be sufficient to parametrize [`AST`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Free-Foil.html#t:AST) and [`ScopedAST`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Free-Foil.html#t:ScopedAST) with a `pattern` type constructor (e.g. literally the same [`Pattern`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Language-LambdaPi-Impl-Foil.html#t:Pattern) type used in the foil examples), we did not check if everything generalizes well enough.
 4. Template Haskell generates the foil, but not (yet) free foil. We believe there should be no problems in implementing TH-generation for the free foil, achieving the best of both worlds: generating the signature bifunctor together with conversion functions while getting [`substitute`](https://fizruk.github.io/free-foil/haddock/free-foil-0.0.1/Control-Monad-Free-Foil.html#v:substitute) (and, potentially, other useful algorithms[^3]) for free (except, as our benchmarks show, a moderate slowdown in performance should be expected compared to the foil).