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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 10:31] 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 ===== | ||
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We define naive substitution recursively, first for terms: | We define naive substitution recursively, first for terms: | ||
+ | |||
\[ | \[ | ||
\begin{array}{rcl} | \begin{array}{rcl} | ||
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\end{array} | \end{array} | ||
\] | \] | ||
+ | |||
then for formulas: | then for formulas: | ||
\[\begin{array}{rcl} | \[\begin{array}{rcl} | ||
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nsubst(\sigma)(\forall x.F) &=& \\ | nsubst(\sigma)(\forall x.F) &=& \\ | ||
nsubst(\sigma)(\exists x.F) &=& | nsubst(\sigma)(\exists x.F) &=& | ||
+ | \end{array} | ||
\] | \] | ||
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**Lemma:** $(\forall x.F) \models sfsubst(\{x \mapsto t\}(F)$. | **Lemma:** $(\forall x.F) \models sfsubst(\{x \mapsto t\}(F)$. | ||
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