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sav08:axioms_for_equality [2009/05/05 23:21] vkuncak |
sav08:axioms_for_equality [2015/04/21 17:30] (current) |
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| ====== Axioms for Equality ====== | ====== Axioms for Equality ====== | ||
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| + | //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*} |
| ++ | ++ | ||