LK推論規則での包含関係

LK推論規則での包含関係
\(P,Q,R,S\)は命題変数とする。

(1)WL,WR

\[ P\rightarrow Q\Rightarrow\left(P\land R\right)\rightarrow\left(Q\lor S\right) \]

(2)\(\lor L\)

\[ \left(P\rightarrow Q\right)\land\left(R\rightarrow S\right)\Rightarrow\left(P\lor R\right)\rightarrow\left(Q\land S\right) \]

(3)\(\land R\)

\[ \left(P\rightarrow Q\right)\land\left(R\rightarrow S\right)\Rightarrow\left(P\land R\right)\rightarrow\left(Q\land S\right) \]

(4)\(\rightarrow L\)

\[ \left(P\rightarrow Q\right)\land\left(R\rightarrow S\right)\Rightarrow\left\{ P\land\left(Q\rightarrow R\right)\right\} \rightarrow S \]

(5)\(\rightarrow R\)

\[ P\land R\rightarrow S\lor Q\Leftrightarrow P\rightarrow\left\{ S\lor\left(R\rightarrow Q\right)\right\} \]

(1)

\begin{align*} P\rightarrow Q & \Leftrightarrow\lnot P\lor Q\\ & \Rightarrow\lnot P\lor Q\lor\lnot R\lor S\\ & \Leftrightarrow\lnot\left(P\land R\right)\lor\left(Q\lor S\right)\\ & \Leftrightarrow\left(P\land R\right)\rightarrow\left(Q\lor S\right) \end{align*}

(2)

\begin{align*} \left(P\rightarrow Q\right)\land\left(R\rightarrow S\right) & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left(\lnot R\lor S\right)\\ & \Leftrightarrow\left\{ \left(\lnot P\lor Q\right)\land\lnot R\right\} \lor\left\{ \left(\lnot P\lor Q\right)\land S\right\} \\ & \Leftrightarrow\left\{ \left(Q\lor\lnot P\right)\land\lnot R\right\} \lor\left\{ \left(\lnot P\lor Q\right)\land S\right\} \\ & \Rightarrow\left\{ Q\lor\left(\lnot P\land\lnot R\right)\right\} \lor\left\{ \left(\lnot P\lor Q\right)\land S\right\} \\ & \Rightarrow\left\{ Q\lor\left(\lnot P\land\lnot R\right)\right\} \lor S\\ & \Leftrightarrow\left(\lnot P\land\lnot R\right)\lor Q\lor S\\ & \Leftrightarrow\lnot\left(P\lor R\right)\lor Q\lor S\\ & \Leftrightarrow\left(P\lor R\right)\rightarrow\left(Q\lor S\right) \end{align*}

(3)

\begin{align*} \left(P\rightarrow Q\right)\land\left(R\rightarrow S\right) & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left(\lnot R\lor S\right)\\ & \Rightarrow\lnot P\lor\left\{ Q\land\left(\lnot R\lor S\right)\right\} \\ & \Rightarrow\lnot P\lor\lnot R\lor\left(Q\land S\right)\\ & \Leftrightarrow\lnot\left(P\land R\right)\lor\left(Q\land S\right)\\ & \Leftrightarrow\left(P\land R\right)\rightarrow\left(Q\land S\right) \end{align*}

(4)

\begin{align*} \left(P\rightarrow Q\right)\land\left(R\rightarrow S\right) & \Leftrightarrow\left(\lnot P\lor Q\right)\land\left(\lnot R\lor S\right)\\ & \Rightarrow\left\{ \left(\lnot P\lor Q\right)\land\lnot R\right\} \lor S\\ & \Rightarrow\left\{ \lnot P\lor\left(Q\land\lnot R\right)\right\} \lor S\\ & \Leftrightarrow\lnot\left\{ P\land\left(\lnot Q\lor R\right)\right\} \lor S\\ & \Leftrightarrow\left\{ P\land\left(Q\rightarrow R\right)\right\} \rightarrow S \end{align*}

(5)

\begin{align*} \left(P\land Q\right)\rightarrow\left(R\lor S\right) & \Leftrightarrow P\rightarrow\left(R\lor S\lor\lnot Q\right)\\ & \Leftrightarrow P\rightarrow\left\{ R\lor\left(Q\rightarrow S\right)\right\} \end{align*}

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LK推論規則での包含関係

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