bump lambda calculus examples

003_lambda_calculus.md

@@ -400,6 +400,23 @@ In later variations of the lambda calculus let expressions will have different
 semantics and will differ from applied lambda expressions. More on this will be
 discussed in the section on Hindley-Milner inference.
 
+Everything Can Be a $\lambda$ term
+----------------------------------
+
+* 0
+* 1
+* 2
+
+* succ
+* pred
+
+* not
+* and
+* or
+
+* add
+* mul
+
 Recursion
 ---------
 
@@ -407,23 +424,29 @@ Probably the most famous combinator is Curry's Y combinator. Within an untyped
 lambda calculus, Y can be used to allow an expression to contain a reference to
 itself and reduce on itself permitting recursion and looping logic.
 
-$$\textbf{Y} = \lambda f.(\lambda x.(f (x x)) \lambda x.(f (x x)))$$
+$$
+f = \textbf{YR}
+$$
 
-The Y combinator satisfies:
+The $\textbf{Y}$ combinator is one of many so called fixed point combinators.
 
-$$\textbf{Y} f  = f (\textbf{Y} f) $$
+$$
+\textbf{Y} = \lambda R.(\lambda x.(R (x x)) \lambda x.(R (x x)))
+$$
 
-since
+The combinator Y is called the fixed point combinator. Given R It returns the
+fixed point of R.
 
 $$
 \begin{aligned}
-\textbf{Y} f & = (\lambda f.(\lambda x.(f(x x)) \lambda x.(f(x x)))) f 
-& = (\lambda x.(f(x x))) (\lambda x.(f(x x)))
-& = f((\lambda x.(f (x x))) (\lambda x.(f (x x))))
-& = f(\textbf{Y} f)
+\textbf{YR} &= \lambda f.(\lambda x.(f (x x)) \lambda x.(f (x x))) R \\
+& = (\lambda x.(\textbf{R} (x x)) \lambda x.(\textbf{R} (x x))) \\
+& = \textbf{R} (\lambda x.(\textbf{R} (x x)) \lambda x.(\textbf{R} (x x))) \\
+& = \textbf{R Y R}
 \end{aligned}
 $$
 
+
 For fun one can prove that the Y-combinator can be expressed in terms of the S
 and K combinators.
 

chapter4/untyped/README.md

@@ -9,6 +9,17 @@ To compile and run:
 $ cabal run
 ```
 
+Example
+-------
+
+```bash
+Untyped> (\x. x) 1
+ => \x . x
+ => 1
+ => x
+1
+```
+
 License
 =======
 

chapter4/untyped/stack.yaml

@@ -0,0 +1,37 @@
+# For more information, see: https://github.com/commercialhaskell/stack/blob/release/doc/yaml_configuration.md
+
+# Specifies the GHC version and set of packages available (e.g., lts-3.5, nightly-2015-09-21, ghc-7.10.2)
+resolver: ghc-7.8.4
+
+# Local packages, usually specified by relative directory name
+packages:
+- '.'
+
+# Packages to be pulled from upstream that are not in the resolver (e.g., acme-missiles-0.3)
+extra-deps:
+- mtl-2.1.3.1
+- parsec-3.1.9
+- text-1.2.1.3
+
+# Override default flag values for local packages and extra-deps
+flags:
+  text:
+    integer-simple: false
+
+# Extra package databases containing global packages
+extra-package-dbs: []
+
+# Control whether we use the GHC we find on the path
+# system-ghc: true
+
+# Require a specific version of stack, using version ranges
+# require-stack-version: -any # Default
+# require-stack-version: >= 0.1.4.0
+
+# Override the architecture used by stack, especially useful on Windows
+# arch: i386
+# arch: x86_64
+
+# Extra directories used by stack for building
+# extra-include-dirs: [/path/to/dir]
+# extra-lib-dirs: [/path/to/dir]

chapter4/untyped/untyped.cabal

@@ -10,7 +10,7 @@ cabal-version:       >=1.10
 
 executable untyped
   build-depends:       
-      base       >= 4.6 && <4.7
+      base       >= 4.6 && <4.8
     , pretty     >= 1.1 && <1.2
     , parsec     >= 3.1 && <3.2
     , containers >= 0.5 && <0.6