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# Differences

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sav08:introductory_remarks_on_smt_provers [2009/05/06 09:45]
vkuncak
sav08:introductory_remarks_on_smt_provers [2015/04/21 17:30] (current)
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Note: formulas that are valid in the combination of quantifiers,​ linear integers, and uninterpreted functions are not enumerable (and neither are formulas that are not valid). ​ The situation is worse than in pure first-order logic. Note: formulas that are valid in the combination of quantifiers,​ linear integers, and uninterpreted functions are not enumerable (and neither are formulas that are not valid). ​ The situation is worse than in pure first-order logic.
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===== Example ===== ===== Example =====

Consider the following formula, where $f$ denotes functions from integers to integers and $a,b$ are integers. Consider the following formula, where $f$ denotes functions from integers to integers and $a,b$ are integers.
-$\begin{array}{l}+\begin{equation*}\begin{array}{l} ​(\forall x.\forall y.\ x \le y \rightarrow f(x) \le f(y))\ \land \\ ​(\forall x.\forall y.\ x \le y \rightarrow f(x) \le f(y))\ \land \\ ​(\forall x.\forall y. f(x)=f(y) \rightarrow x=y)\ \land\\ ​(\forall x.\forall y. f(x)=f(y) \rightarrow x=y)\ \land\\ Line 35: Line 36: b < a b < a \end{array} \end{array} -$ +\end{equation*}
-Is this formula satisfiable?​ ++|No. Why?+++Is this formula satisfiable?​
+  * over integers  ​++|No. Why?++
+  * over reals  ++|Yes. Why?++