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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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* 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*} |
++ | ++ | ||