-there are no variables in Combiantory Logic, there is no need to worry
-about variable collision.
-
-Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. As we said, the most common system uses S, K, and I as defined here.
-
-###The equivalence of the untyped lambda calculus and combinatory logic###
-
-We've claimed that Combinatory Logic is equivalent to the lambda
-calculus. If that's so, then S, K, and I must be enough to accomplish
-any computational task imaginable. Actually, S and K must suffice,
-since we've just seen that we can simulate I using only S and K. In
-order to get an intuition about what it takes to be Turing complete,
-recall our discussion of the lambda calculus in terms of a text editor.
-A text editor has the power to transform any arbitrary text into any other arbitrary text. The way it does this is by deleting, copying, and reordering characters. We've already seen that K deletes its second argument, so we have deletion covered. S duplicates and reorders, so we have some reason to hope that S and K are enough to define arbitrary functions.
-
-We've already established that the behavior of combinatory terms can
-be perfectly mimicked by lambda terms: just replace each combinator
-with its equivalent lambda term, i.e., replace I with `\x.x`, replace
-K with `\fxy.x`, and replace S with `\fgx.fx(gx)`. So the behavior of
-any combination of combinators in Combinatory Logic can be exactly
-reproduced by a lambda term.
-
-How about the other direction? Here is a method for converting an
-arbitrary lambda term into an equivalent Combinatory Logic term using
-only S, K, and I. Besides the intrinsic beauty of this mapping, and
-the importance of what it says about the nature of binding and
-computation, it is possible to hear an echo of computing with
-continuations in this conversion strategy (though you wouldn't be able
-to hear these echos until we've covered a considerable portion of the
-rest of the course). In addition, there is a direct linguistic
-appliction of this mapping in chapter 17 of Barker and Shan 2014,
-where it is used to establish a correpsondence between two natural
-language grammars, one of which is based on lambda-like abstraction,
-the other of which is based on Combinatory Logic like manipulations.
-
-Assume that for any lambda term T, [T] is the equivalent combinatory logic term. The we can define the [.] mapping as follows:
+there are no variables in Combinatory Logic, there is no need to worry
+about variables colliding when we substitute.
+
+Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. As we said, the most common system uses `S`, `K`, and `I` as defined here.
+
+###The equivalence of the untyped Lambda Calculus and Combinatory Logic###
+
+We've claimed that Combinatory Logic is "equivalent to" the Lambda Calculus. If
+that's so, then `S`, `K`, and `I` must be enough to accomplish any computational task
+imaginable. Actually, `S` and `K` must suffice, since we've just seen that we can
+simulate `I` using only `S` and `K`. In order to get an intuition about what it
+takes to be Turing Complete, <!-- FIXME -->
+recall our discussion of the Lambda Calculus in
+terms of a text editor. A text editor has the power to transform any arbitrary
+text into any other arbitrary text.
+The way it does this is by deleting, copying, and reordering characters. We've
+already seen that `K` deletes its second argument, so we have deletion covered.
+`S` duplicates and reorders, so we have some reason to hope that `S` and `K` are
+enough to define arbitrary functions.
+
+We've already established that the behavior of combinatory terms can be
+perfectly mimicked by lambda terms: just replace each combinator with its
+equivalent lambda term, i.e., replace `I` with `\x. x`, replace `K` with `\x y. x`,
+and replace `S` with `\f g x. f x (g x)`. So the behavior of any combination of
+combinators in Combinatory Logic can be exactly reproduced by a lambda term.
+
+How about the other direction? Here is a method for converting an arbitrary
+lambda term into an equivalent Combinatory Logic term using only `S`, `K`, and `I`.
+Besides the intrinsic beauty of this mapping, and the importance of what it
+says about the nature of binding and computation, it is possible to hear an
+echo of computing with continuations in this conversion strategy (though you
+wouldn't be able to hear these echos until we've covered a considerable portion
+of the rest of the course). In addition, there is a direct linguistic
+application of this mapping in chapter 17 of Barker and Shan 2014, where it is
+used to establish a correspondence between two natural language grammars, one
+of which is based on lambda-like abstraction, the other of which is based on
+Combinatory Logic-like manipulations.
+
+Assume that for any lambda term T, [T] is the equivalent Combinatory Logic term. Then we can define the [.] mapping as follows: