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sav08:axioms_for_equality [2009/05/06 09:52]
vkuncak
sav08:axioms_for_equality [2015/04/21 17:30] (current)
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====== Axioms for Equality ====== ====== Axioms for Equality ======

-The following definitions are useful when axiomatizing equality in a logic that does not have equality built in. It is also useful when discussing algorithms that automate reasoning about equality.+//The following definitions are useful when axiomatizing equality in a logic that does not have equality built in. It is also useful when discussing algorithms that automate reasoning about equality.//

For language ${\cal L}$ and a relation symbol $eq \notin {\cal L}$, the theory of equality, denoted AxEq, is the following set of formulas: For language ${\cal L}$ and a relation symbol $eq \notin {\cal L}$, the theory of equality, denoted AxEq, is the following set of formulas:
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* Transitivity:​ ++| $\forall x. \forall y. \forall z.\ eq(x,y) \land eq(y,z) \rightarrow eq(x,z)$ ++   * Transitivity:​ ++| $\forall x. \forall y. \forall z.\ eq(x,y) \land eq(y,z) \rightarrow eq(x,z)$ ++
* Congruence for function symbols: for $f \in {\cal L}$ function symbol with $ar(f)=n$, ++ |   * Congruence for function symbols: for $f \in {\cal L}$ function symbol with $ar(f)=n$, ++ |
-$+\begin{equation*} ​\forall x_1,​\ldots,​x_n,​ y_1,​\ldots,​y_n.\ (\bigwedge_{i=1}^n eq(x_i,​y_i)) \rightarrow eq(f(x_1,​\ldots,​x_n),​f(y_1,​\ldots,​y_n)) ​\forall x_1,​\ldots,​x_n,​ y_1,​\ldots,​y_n.\ (\bigwedge_{i=1}^n eq(x_i,​y_i)) \rightarrow eq(f(x_1,​\ldots,​x_n),​f(y_1,​\ldots,​y_n)) -$+\end{equation*}
++ ++
* Congruence for relation symbols: for $R \in {\cal L}$ relation symbol with $ar(R)=n$, ++ |   * Congruence for relation symbols: for $R \in {\cal L}$ relation symbol with $ar(R)=n$, ++ |
-$+\begin{equation*} ​\forall x_1,​\ldots,​x_n,​ y_1,​\ldots,​y_n.\ (\bigwedge_{i=1}^n eq(x_i,​y_i)) \rightarrow (R(x_1,​\ldots,​x_n) \leftrightarrow R(y_1,​\ldots,​y_n)) ​\forall x_1,​\ldots,​x_n,​ y_1,​\ldots,​y_n.\ (\bigwedge_{i=1}^n eq(x_i,​y_i)) \rightarrow (R(x_1,​\ldots,​x_n) \leftrightarrow R(y_1,​\ldots,​y_n)) -$+\end{equation*}
++ ++