Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
sav08:substitutions_for_first-order_logic [2008/03/19 00:12] vkuncak |
sav08:substitutions_for_first-order_logic [2008/03/19 10:27] vkuncak |
||
---|---|---|---|
Line 17: | Line 17: | ||
\forall x_1. \exists y_1. x_1 < y_1 | \forall x_1. \exists y_1. x_1 < y_1 | ||
\] | \] | ||
- | and then after substitution we obtain $\exists y_1. y + 1 < y_1$, which is a correct consequence. | + | and then after substitution $\{x_1 \mapsto y+1\}$ we obtain $\exists y_1. y + 1 < y_1$, which is a correct consequence of $\forall x. \exists y. x < y$. |
===== Naive and Safe Substitutions ===== | ===== Naive and Safe Substitutions ===== | ||
Line 38: | Line 38: | ||
nsubst(\sigma)(F_1 \land F_2) &=& \\ | nsubst(\sigma)(F_1 \land F_2) &=& \\ | ||
nsubst(\sigma)(\forall x.F) &=& \\ | nsubst(\sigma)(\forall x.F) &=& \\ | ||
- | nsubst(\sigma)(\forall x.F) &=& | + | nsubst(\sigma)(\exists x.F) &=& |
\] | \] | ||
Line 53: | Line 53: | ||
sfsubst(\sigma)(\lnot F) &=& \\ | sfsubst(\sigma)(\lnot F) &=& \\ | ||
sfsubst(\sigma)(F_1 \land F_2) &=& \\ | sfsubst(\sigma)(F_1 \land F_2) &=& \\ | ||
- | sfsubst(\sigma)(\forall x.F) &=& | + | sfsubst(\sigma)(\forall x.F) &=& \\ |
- | sfsubst(\sigma)(\forall x.F) &=& | + | sfsubst(\sigma)(\exists x.F) &=& |
\end{array} | \end{array} | ||
\] | \] | ||
Line 64: | Line 64: | ||
**Lemma:** $(\forall x.F) \models sfsubst(\{x \mapsto t\}(F)$. | **Lemma:** $(\forall x.F) \models sfsubst(\{x \mapsto t\}(F)$. | ||
- |