Handout for Lecture 8 Dale Miller , 21 September 2011
نویسنده
چکیده
Here, the selection of the formula D for the focus can be taken as selecting among several macro-rules: this derivation illustrates one such macro-rule: the inference rule with conclusion ` Θ,D ⇑ · and with n ≥ 0 premises ` Θ,D ⇑ N1, . . . , ` Θ,D ⇑ Nn (where N1, . . . ,Nn are negative formulas). We shall say that this macro-rule is positive. Similarly, there is a corresponding negative macro-rule with conclusion, say, ` Θ,D ⇑ Ni, and with m ≥ 0 premises of the form ` Θ,D,C ⇑ ·, where C is a multiset of positive formulas or negative literals. In this way, focused proofs allow us to view the construction of proofs from conclusions of the form ` Θ ⇑ · as first attaching a positive macro rule (by focusing on some formula in Θ) and then attaching negative inference rules to the resulting premises until one is again to sequents of the form ` Θ′ ⇑ ·. Such a combination of a positive macro rule below negative macro rules is often called a bipole [1]. Focusing can be broken at any point via delays. Within LKF, we can define the delaying operators
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