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sav08:axioms_of_hol [2008/05/27 17:25] vkuncak created |
sav08:axioms_of_hol [2015/04/21 17:30] (current) |
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| - | ====== Axioms of Classical HOL ====== | + | ====== Axioms and Inference Rules of Classical HOL ====== |
| - | Sound, incomplete. | + | Axiom system that is sound (but necessarily incomplete with respect to standard semantics) is the following: |
| + | - $(g\ true\ \land\ g\ false) = (\forall x. g x)$ (domain 'o' has only true,false as elements) | ||
| + | - $(x=y) \rightarrow (fx = fy)$ (congruence) | ||
| + | - $(\lambda x.F_1)F_2 = F_1[x:=F_2]$ (beta reduction) | ||
| + | - $\iota(\lambda y.(x=y)) = x$ (selecting element from $\{x\}$ gives $x$) | ||
| + | |||
| + | Inference rule: | ||
| + | \begin{equation*} | ||
| + | \frac{C\ \ ; \ \ A=B} | ||
| + | {C[A:=B]} | ||
| + | \end{equation*} | ||
| + | which replaces one occurrence of $A$ in $C$ with $B$ (we cannot replace variables under $\lambda$) | ||