Lambda calculus expression evaluation. I ran across the following lambda calculus example problem
plusTwo = n.successor(successor n)
4 plusTwo 2 = 10

I'm having trouble understanding how the answer is 10 and not 8.
Would it not break down as follows:
4 n.successor(successor n)2 
4 successor(successor 2) 
4 successor(3) 
4 4
8

Is the answer I'm given incorrect? Am I missing something obvious? 
Maybe, the answer comes from calculating with the entire 4 plusTwo 2 expression
4 plusTwo 2 = 8 

n.successor(successor n)8
successor(successor 8)
successor(9)
10

 A: In the $\lambda$-calculus, application is left-associative, which means that a term of the form $MNL$ must be read as $(MN)L$.
Your first reduction sequence (the one ending in $\underline{8}$) is wrong because at the beginning you read the term $\underline{4} \, \mathrm{plusTwo} \, \underline{2}$ as the term $\underline{4} \, (\mathrm{plusTwo} \, \underline{2})$ (instead of $(\underline{4} \, \mathrm{plusTwo}) \, \underline{2}$), breaking the aforementioned convention.
A correct reduction sequence is the following.
\begin{align}
 \underline{4} \, \mathrm{plusTwo} \, \underline{2} &= \big(\lambda f.\lambda x.f(f(f(fx)) \big) \mathrm{plusTwo} \, \underline{2}
\\
& \to_\beta \big(\lambda x. \mathrm{plusTwo} \, (\mathrm{plusTwo} \,(\mathrm{plusTwo} \, (\mathrm{plusTwo} \, x))) \big) \underline{2}
\\
& \to_\beta \mathrm{plusTwo} \, (\mathrm{plusTwo} \, (\mathrm{plusTwo}  \, (\mathrm{plusTwo} \, \underline{2})))
\\
&\to_\beta^* \mathrm{plusTwo} \, (\mathrm{plusTwo} \, \underline{6})
\\
&\to_\beta^* \mathrm{plusTwo} \, \underline{8}
\\
&\to_\beta^* \underline{10}
\end{align}
where I used that fact that for every natural number $m$, 
\begin{align}
\mathrm{plusTwo} \, \underline{m} &= (\lambda n . \mathrm{succ} \, (\mathrm{succ} \, n))  \underline{m}
\\ 
&\to_\beta \mathrm{succ} \, (\mathrm{succ} \, \underline{m})
\\ 
&\to_\beta \mathrm{succ} \, \underline{m+1}
\\ 
&\to_\beta \underline{m+2}.
\end{align}

Your second reduction sequence is meaningless, I don't understand the relation between its first line and its second line.
