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sav08:interpretation_quotient_under_congruence [2008/04/02 22:20] vkuncak |
sav08:interpretation_quotient_under_congruence [2008/04/02 22:36] vkuncak |
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\] | \] | ||
- | **Lemma 1:** $I_Q = ([D],I_Q)$ is a well-defined interpretation, that is, | + | **Lemma 1:** For each function symbol $f$ with $ar(f)=n$, the relation $\alpha_Q(f)$ is a total function $[D]^n \to [D]$ and for all $x_1,\ldots,x_n \in D$, |
- | * $[D]$ is non-empty; | + | \[ |
- | * for each function symbol $f$ with $ar(f)=n$, the relation $\alpha_Q(f)$ is a total function $[D]^n \to [D]$. | + | \alpha_Q(f)([x_1],\ldots,[x_n]) = [\alpha(f)(x_1,\ldots,x_n)] |
+ | \] | ||
**Lemma 2:** For each term $t$ we have $e_T(t)(I_Q) = [e_T(t)(I)]$. | **Lemma 2:** For each term $t$ we have $e_T(t)(I_Q) = [e_T(t)(I)]$. |