Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas

Theorem
Let $\mathscr G$ be instance 1 of a Gentzen proof system. Let $U$ be a set of WFFs of propositional logic.

Then $U$ is a $\mathscr G$-theorem iff:


 * $\bar U$ has a closed semantic tableau

where $\bar U$ is the set comprising the logical complements of all WFFs in $U$.

Proof
Denote with $\bar{\mathbf A}$ and $\bar U$ the logical complement of a WFF $\mathbf A$ and set $U$, respectively.

Necessary Condition
We aim to prove this result using the Second Principle of Mathematical Induction, applied to the minimal length of a formal proof of $U$.

Suppose first that $U$ is an axiom of $\mathscr G$.

Then $U$ contains a complementary pair of literals.

Hence, so does $\bar U$, by definition of logical complement.

Therefore, the semantic tableau comprising only a root node labeled $\bar U$ is closed.

Next, suppose that the last step in proving $U$ was an instance of the $\alpha$-rule.

That is, for some $\alpha$-formula $\mathbf A$ and corresponding $\mathbf A_1, \mathbf A_2$:


 * $U = U_1 \cup U_2 \cup \left\{{\mathbf A}\right\}$

where $U_1 \cup \left\{{\mathbf A_1}\right\}$ and $U_2 \cup \left\{{\mathbf A_2}\right\}$ are $\mathscr G$-theorems.

By induction hypothesis, the logical complements:


 * $\bar U_1 \cup \left\{{\bar{\mathbf A_1}}\right\}$
 * $\bar U_2 \cup \left\{{\bar{\mathbf A_2}}\right\}$

of these theorems have closed tableaus.

From the Soundness Theorem for Semantic Tableaus, it follows that these sets are unsatisfiable.

Now by Superset of Unsatisfiable Set is Unsatisfiable, so are:


 * $\bar U' := \bar U_1 \cup \bar U_2 \cup \left\{{\bar{\mathbf A_1}}\right\}$
 * $\bar U'' := \bar U_1 \cup \bar U_2 \cup \left\{{\bar{\mathbf A_2}}\right\}$

By the Completeness Theorem for Semantic Tableaus, these have closed tableaus.

Since $\mathbf A$ is an $\alpha$-formula, it is equivalent to $\mathbf A_1 \land \mathbf A_2$.

By De Morgan's Laws, it follows that:


 * $\bar{\mathbf A}$ is equivalent to $\bar{\mathbf A_1} \lor \bar{\mathbf A_2}$.

Therefore, $\bar{\mathbf A}$ is a $\beta$-formula, and $\bar{\mathbf A_1}, \bar{\mathbf A_2}$ correspond to it as in the table of $\beta$-formulas.

This allows the Semantic Tableau Algorithm to expand a leaf labeled $\bar U$ into two leaves labeled $\bar U'$ and $\bar U''$.

Because $\bar U'$ and $\bar U''$ have closed tableaus, so does $\bar U$.

Finally, suppose that the last step in proving $U$ was an instance of the $\beta$-rule.

That is, for some $\beta$-formula $\mathbf B$ and corresponding $\mathbf B_1, \mathbf B_2$:


 * $U = U_1 \cup \left\{{\mathbf B}\right\}$

where $U' := U_1 \cup \left\{{\mathbf B_1, \mathbf B_2}\right\}$ is a $\mathscr G$-theorem.

By induction hypothesis, the logical complement:


 * $\bar U' = \bar U_1 \cup \left\{{\bar{\mathbf B_1}, \bar{\mathbf B_2}}\right\}$

of this $\mathscr G$-theorem has a closed tableau.

Since $\mathbf B$ is a $\beta$-formula, it is equivalent to $\mathbf B_1 \lor \mathbf B_2$.

By De Morgan's Laws, it follows that:


 * $\bar{\mathbf B}$ is equivalent to $\bar{\mathbf B_1} \land \bar{\mathbf B_2}$.

Therefore, $\bar{\mathbf B}$ is an $\alpha$-formula, and $\bar{\mathbf B_1}, \bar{\mathbf B_2}$ correspond to it as in the table of $\alpha$-formulas.

This allows the Semantic Tableau Algorithm to expand a leaf labeled $\bar U$ into a leaf labeled $\bar U'$.

Because $\bar U'$ has a closed tableau, so does $\bar U$.

The result follows by the Second Principle of Mathematical Induction.