This is something all introductory texts seem to avoid proving, and many even avoid stating.
We consider untyped $\lambda$-terms on some countably infinite alphabet. If $x$ is a variable and $p$ is an $\alpha$-equivalence class of a term, then we define an $\alpha$-equivalence class $p\left[s/x\right]$ as follows:
begin definition of $\left[s/x\right]$
Let $t$ be a term such that $p=\overline{t}$ (where $\overline{t}$ means the $\alpha$-equivalence class of $t$).
If $t=x$, set $p\left[s/x\right] = \overline{s}$.
If $t=y$ for a variable $y\neq x$, set $p\left[s/x\right] = \overline{y}$.
If $t=t_1t_2$ for two terms $t_1$ and $t_2$, set $p\left[s/x\right] = \overline{\rho\left(t_1\left[s/x\right]\right) \rho\left(t_2\left[s/x\right]\right)}$. Here, $\rho\left(u\right)$ means any representative of the $\alpha$-equivalence class $u$.
If $t=\lambda yr$ for some variable $y\neq x$ and term $r$, and if $y$ appears as a free variable in $s$, set $p\left[s/x\right] = \overline{\lambda y^{\prime}.\rho\left(r\left[y^{\prime}/y\right]\left[s/x\right]\right)}$, where $y^{\prime}$ is some fresh (unused) variable.
If $t=\lambda yr$ for some variable $y\neq x$ and term $r$, and if $y$ does not appear as a free variable in $s$, set $p\left[s/x\right] = \overline{\lambda y .\rho\left(r\left[s/x\right]\right)}$.
If $t=\lambda xr$ for some term $r$, set $p\left[s/x\right] = \overline{\lambda x.\rho\left(r\right)}$.
end definition of $\left[s/x\right]$
How can I prove that this definition makes sense? I. e., that the result never depends on the choice of representatives and on the choice of the free variable $\rho^{\prime}$ ? I know that people like to call things like these intuitively obvious, but to me the definition looks much too complex to speak of triviality. It is like claiming that a code does what one expects it to – it is trivial until one finds the first bug. I have tried the usual structural induction, but I got lost in the casebash (there are much more cases than usually, and one has to prove lemmata about substitution). Is there a readable proof anywhere?
Best Answer
I think this is proved in H. B. Curry and R. Feys. Combinatory Logic, Volume I. North-Holland Co., Amsterdam, Theorem 2a on page 95. The proof, with all the auxiliary results and some consequences, occupies pages 96-103.