Prefix of WFF of PropLog is not WFF

Theorem
Let $\mathbf A$ be a WFF of propositional logic.

Let $\mathbf S$ be an initial part of $\mathbf A$.

Then $\mathbf S$ is not a WFF of propositional logic.

Proof
The proof proceeds by strong induction on the length of a WFF of propositional logic.

Let $\map l {\mathbf Q}$ denote the length of a string $\mathbf Q$.

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * An initial part of $\mathbf A$ such that $\map l {\mathbf A} = n$ is not a WFF of propositional logic.

By definition, $\mathbf S$ is an initial part of $\mathbf A$ $\mathbf A = \mathbf {S T}$ for some non-null string $\mathbf T$.

Thus we note that $\map l {\mathbf S} < \map l {\mathbf A}$.

Basis for the Induction
Let $\mathbf A$ be a WFF such that $\map l {\mathbf A} = 1$.

Then for an initial part $\mathbf S$:
 * $\map l {\mathbf S} < 1 = 0$

That is, $\mathbf S$ must be the null string, which is not a WFF.

So the result holds for WFFs of length $1$.

That is, $\map P 1$ is true.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that for $k \ge 1$, if $\map P j$ is true for all $j \le k$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * An initial part of $\mathbf A$ such that $\map l {\mathbf A} = k$ is not a WFF of propositional logic.

Then we need to show:
 * An initial part of $\mathbf A$ such that $\map l {\mathbf A} = k + 1$ is not a WFF of propositional logic.

Induction Step
This is our induction step:

Let $\mathbf A$ be a WFF such that $\map l {\mathbf A} = k + 1$.

Suppose $\mathbf D$ is an initial part of $\mathbf A$ which happens to be a WFF.

That is, $\mathbf A = \mathbf {D T}$ where:
 * $\mathbf D$ is a WFF
 * $\mathbf T$ is non-null.

There are two cases:

$(1): \quad \mathbf A = \neg \mathbf B$

where $\mathbf B$ is a WFF of length $k$.

Thus $\mathbf D$ is a WFF starting with $\neg$.

So:
 * $\mathbf D = \neg \mathbf E$

where $\mathbf E$ is also a WFF.

We remove the initial $\neg$ from $\mathbf A = \mathbf {D T}$ to get:
 * $\mathbf B = \mathbf {E T}$

But then $\mathbf B$ is a WFF of length $k$ which has $\mathbf E$ as an initial part which is itself a WFF.

This contradicts the induction hypothesis.

Therefore no initial part of $\mathbf A = \neg \mathbf B$ can be a WFF.

$(2): \quad \mathbf A = \paren {\mathbf B \circ \mathbf C}$

where $\circ$ is one of the binary connectives.

In this case, $\mathbf D$ is a WFF starting with $($, so:
 * $\mathbf D = \paren {\mathbf E * \mathbf F}$

for some binary connectives $*$ and some WFFs $\mathbf E$ and $\mathbf F$.

Thus:
 * $\mathbf B \circ \mathbf C) = \mathbf E * \mathbf F) \mathbf T$.

Both $\mathbf B$ and $\mathbf E$ are WFFs of length less than $k + 1$.

Therefore, by the induction hypothesis, neither $\mathbf B$ nor $\mathbf E$ can be an initial part of the other.

But since both $\mathbf B$ and $\mathbf E$ start at the same place in $\mathbf A$, they must be the same:
 * $\mathbf B = \mathbf E$

Therefore:
 * $\mathbf B \circ \mathbf C) = \mathbf B * \mathbf F) \mathbf T$

So $\circ = *$ and:
 * $\mathbf C) = \mathbf F) \mathbf T$

But then the WFF $\mathbf F$ is an initial part of the WFF $\mathbf C$ of length less than $k + 1$.

This contradicts our induction hypothesis.

Therefore no initial part of $\mathbf A = \paren {\mathbf B \circ \mathbf C}$ can be a WFF.

As $\mathbf A$ is arbitrary, it follows that no initial part of any WFF of length $k + 1$ can be a WFF.

So $\map P k \implies \map P {k + 1}$ and the result follows by strong induction.

Therefore, for all $n \in \N_{> 0}$, the initial part of $\mathbf A$ such that $\map l {\mathbf A} = n$ is not a WFF of propositional logic.

Hence the result.