Definition:Constructed Semantics/Instance 4/Factor Principle

Theorem
The Factor Principle:


 * $\paren {p \implies q} \implies \paren {\paren {r \lor p} \implies \paren {r \lor q} }$

is a tautology in Instance 4 of constructed semantics.

Proof
By the definitional abbreviation for the conditional:


 * $\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$

the Factor Principle can be written as:


 * $\neg \paren {\neg p \lor q} \lor \paren {\neg \paren {r \lor p} \lor \paren {r \lor q} }$

This evaluates as follows:


 * $\begin{array}{|ccccc|c|cccccccc|} \hline

\neg & (\neg & p & \lor & q) & \lor & (\neg & (r & \lor & p) & \lor & (r & \lor & q)) \\ \hline 1 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 2 & 0 & 0 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 0 \\ 1 & 2 & 3 & 0 & 0 & 0 & 1 & 0 & 0 & 3 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 2 & 0 & 0 & 0 & 0 & 1 & 2 & 2 & 0 & 1 & 0 & 0 \\ 1 & 2 & 3 & 0 & 0 & 0 & 2 & 1 & 3 & 3 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 2 & 0 & 0 & 0 & 2 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 0 & 0 \\ 1 & 0 & 2 & 0 & 0 & 0 & 0 & 2 & 2 & 2 & 0 & 2 & 0 & 0 \\ 1 & 2 & 3 & 0 & 0 & 0 & 1 & 2 & 0 & 3 & 0 & 2 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 & 0 & 1 & 3 & 0 & 0 & 0 & 3 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 2 & 3 & 3 & 1 & 0 & 3 & 0 & 0 \\ 1 & 0 & 2 & 0 & 0 & 0 & 2 & 3 & 3 & 2 & 0 & 3 & 0 & 0 \\ 1 & 2 & 3 & 0 & 0 & 0 & 2 & 3 & 3 & 3 & 0 & 3 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 2 & 0 & 1 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 1 \\ 0 & 2 & 3 & 2 & 1 & 0 & 1 & 0 & 0 & 3 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 2 & 0 & 1 & 0 & 0 & 1 & 2 & 2 & 0 & 1 & 1 & 1 \\ 0 & 2 & 3 & 2 & 1 & 0 & 2 & 1 & 3 & 3 & 2 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 2 & 0 & 0 & 1 & 2 & 2 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 2 & 1 \\ 1 & 0 & 2 & 0 & 1 & 0 & 0 & 2 & 2 & 2 & 0 & 2 & 2 & 1 \\ 0 & 2 & 3 & 2 & 1 & 0 & 1 & 2 & 0 & 3 & 1 & 2 & 2 & 1 \\ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 3 & 0 & 0 & 3 & 3 & 3 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 0 & 3 & 1 & 1 & 0 & 3 & 3 & 1 \\ 1 & 0 & 2 & 0 & 1 & 0 & 2 & 3 & 3 & 2 & 0 & 3 & 3 & 1 \\ 0 & 2 & 3 & 2 & 1 & 0 & 2 & 3 & 3 & 3 & 0 & 3 & 3 & 1 \\ 0 & 1 & 0 & 2 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \\ 1 & 0 & 1 & 0 & 2 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 2 \\ 1 & 0 & 2 & 0 & 2 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 2 \\ 0 & 2 & 3 & 2 & 2 & 0 & 1 & 0 & 0 & 3 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 & 2 & 0 & 1 & 1 & 0 & 0 & 2 & 1 & 2 & 2 \\ 1 & 0 & 1 & 0 & 2 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 2 & 2 \\ 1 & 0 & 2 & 0 & 2 & 0 & 0 & 1 & 2 & 2 & 0 & 1 & 2 & 2 \\ 0 & 2 & 3 & 2 & 2 & 0 & 2 & 1 & 3 & 3 & 2 & 1 & 2 & 2 \\ 0 & 1 & 0 & 2 & 2 & 0 & 1 & 2 & 0 & 0 & 2 & 2 & 2 & 2 \\ 1 & 0 & 1 & 0 & 2 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 2 & 2 \\ 1 & 0 & 2 & 0 & 2 & 0 & 0 & 2 & 2 & 2 & 0 & 2 & 2 & 2 \\ 0 & 2 & 3 & 2 & 2 & 0 & 1 & 2 & 0 & 3 & 2 & 2 & 2 & 2 \\ 0 & 1 & 0 & 2 & 2 & 0 & 1 & 3 & 0 & 0 & 3 & 3 & 3 & 2 \\ 1 & 0 & 1 & 0 & 2 & 0 & 2 & 3 & 3 & 1 & 0 & 3 & 3 & 2 \\ 1 & 0 & 2 & 0 & 2 & 0 & 2 & 3 & 3 & 2 & 0 & 3 & 3 & 2 \\ 0 & 2 & 3 & 2 & 2 & 0 & 2 & 3 & 3 & 3 & 0 & 3 & 3 & 2 \\ 2 & 1 & 0 & 3 & 3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\ 1 & 0 & 1 & 0 & 3 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 3 \\ 1 & 0 & 2 & 0 & 3 & 0 & 1 & 0 & 0 & 2 & 0 & 0 & 0 & 3 \\ 1 & 2 & 3 & 0 & 3 & 0 & 1 & 0 & 0 & 3 & 0 & 0 & 0 & 3 \\ 2 & 1 & 0 & 3 & 3 & 0 & 1 & 1 & 0 & 0 & 3 & 1 & 3 & 3 \\ 1 & 0 & 1 & 0 & 3 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 3 & 3 \\ 1 & 0 & 2 & 0 & 3 & 0 & 0 & 1 & 2 & 2 & 0 & 1 & 3 & 3 \\ 1 & 2 & 3 & 0 & 3 & 0 & 2 & 1 & 3 & 3 & 0 & 1 & 3 & 3 \\ 2 & 1 & 0 & 3 & 3 & 0 & 1 & 2 & 0 & 0 & 0 & 2 & 0 & 3 \\ 1 & 0 & 1 & 0 & 3 & 0 & 0 & 2 & 2 & 1 & 0 & 2 & 0 & 3 \\ 1 & 0 & 2 & 0 & 3 & 0 & 0 & 2 & 2 & 2 & 0 & 2 & 0 & 3 \\ 1 & 2 & 3 & 0 & 3 & 0 & 1 & 2 & 0 & 3 & 0 & 2 & 0 & 3 \\ 2 & 1 & 0 & 3 & 3 & 0 & 1 & 3 & 0 & 0 & 3 & 3 & 3 & 3 \\ 1 & 0 & 1 & 0 & 3 & 0 & 2 & 3 & 3 & 1 & 0 & 3 & 3 & 3 \\ 1 & 0 & 2 & 0 & 3 & 0 & 2 & 3 & 3 & 2 & 0 & 3 & 3 & 3 \\ 1 & 2 & 3 & 0 & 3 & 0 & 2 & 3 & 3 & 3 & 0 & 3 & 3 & 3 \\ \hline \end{array}$