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sav08:substitutions_for_first-order_logic [2008/03/19 08:39] vkuncak |
sav08:substitutions_for_first-order_logic [2008/03/19 10:30] vkuncak |
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| \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 31: | Line 31: | ||
| \end{array} | \end{array} | ||
| \] | \] | ||
| + | |||
| then for formulas: | then for formulas: | ||
| \[\begin{array}{rcl} | \[\begin{array}{rcl} | ||
| Line 39: | Line 40: | ||
| nsubst(\sigma)(\forall x.F) &=& \\ | nsubst(\sigma)(\forall x.F) &=& \\ | ||
| nsubst(\sigma)(\exists x.F) &=& | nsubst(\sigma)(\exists x.F) &=& | ||
| + | \end{array} | ||
| \] | \] | ||
| Line 64: | Line 66: | ||
| **Lemma:** $(\forall x.F) \models sfsubst(\{x \mapsto t\}(F)$. | **Lemma:** $(\forall x.F) \models sfsubst(\{x \mapsto t\}(F)$. | ||
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