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sav08:substitutions_for_first-order_logic [2008/03/19 10:31] vkuncak |
sav08:substitutions_for_first-order_logic [2008/03/19 15:59] vkuncak |
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| We define naive substitution recursively, first for terms: | We define naive substitution recursively, first for terms: | ||
| - | \[ | + | $ subst(\sigma)( x ) = \sigma( x ),\ \sigma {\rm defined at } x $ |
| - | \begin{array}{rcl} | + | |
| - | subst(\sigma)(x) &=& \sigma(x), \mbox{ if $\sigma$ defined at $x$} \\ | + | |
| - | subst(\sigma)(x) &=& x, \mbox{ if $\sigma$ not defined at $x$} \\ | + | |
| - | subst(\sigma)(f(t_1,\ldots,t_n)) &=& f(subst(\sigma)(t_1),\ldots,subst(\sigma)(t_n)) | + | |
| - | \end{array} | + | |
| - | \] | + | |
| - | then for formulas: | + | $ subst(\sigma)( x ) = x,\ \sigma {\rm not defined at } x $ |
| + | |||
| + | $subst(\sigma)(f(t_1,\ldots,t_n)) = f(subst(\sigma)(t_1),\ldots,subst(\sigma)(t_n))$ | ||
| + | |||
| + | and then for formulas: | ||
| \[\begin{array}{rcl} | \[\begin{array}{rcl} | ||
| nsubst(\sigma)(R(t_1,\ldots,t_n)) &=& R(nsubst(\sigma)(t_1),\ldots,nsubst(\sigma)(t_n)) \\ | nsubst(\sigma)(R(t_1,\ldots,t_n)) &=& R(nsubst(\sigma)(t_1),\ldots,nsubst(\sigma)(t_n)) \\ | ||