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Composite proofs

Composite proofs are used to link together multiple inference steps in such a way that the conclusions of earlier deductions become available in subsequent deductions. Specifically, to evaluate a composite proof \bgroup\color{red}$D_1;D_2$\egroup in some assumption base \bgroup\color{red}$\beta$\egroup, we first evaluate \bgroup\color{red}$D_1$\egroup in \bgroup\color{red}$\beta$\egroup, obtaining some conclusion \bgroup\color{red}$F_1$\egroup; and we then evaluate \bgroup\color{red}$D_2$\egroup in the augmented assumption base \bgroup\color{red}$\mbox{$\beta$}\cup \{F_1\}$\egroup. This means that the conclusion of \bgroup\color{red}$D_1$\egroup becomes available (serves as a ``lemma'') inside \bgroup\color{red}$D_2$\egroup. The conclusion we obtain by evaluating \bgroup\color{red}$D_2$\egroup in \bgroup\color{red}$\mbox{$\beta$}\cup \{F_1\}$\egroup becomes the conclusion of the entire composition \bgroup\color{red}$D_1;D_2$\egroup. This is formally specified as follows:
\bgroup\color{red}$\;\;$\egroup \bgroup\color{red}$\mbox{$\beta$}\mbox{$\:\vdash\:$}D_1 \mbox{$\:\longrightarrow...
...eta$}\cup \{F_1\} \mbox{$\:\vdash\:$}D_2 \mbox{$\:\longrightarrow\;$}F_2$\egroup \bgroup\color{red}$\;$\egroup
\bgroup\color{red}$\;\:$\egroup
\bgroup\color{red}$\;\;$\egroup \bgroup\color{red}$\mbox{$\beta$}\mbox{$\:\vdash\:$}D_1;D_2 \mbox{$\:\longrightarrow\;$}F_2$\egroup \bgroup\color{red}$\;$\egroup
As an example, suppose we have the premise and we want to derive the conclusion C. This can be done in three steps: first we use right-and to obtain , then we use right-and again to obtain , and finally we use double-negation to get C. These three steps can be put together using the semi-colon proof-composition operator:
right-and A & B & ~~C;  // this puts B & ~~C in the a.b.
right-and B & ~~C;      // this puts ~~C in the a.b.
double-negation ~~C     // and finally we get C
Note that the convention that ; associates to the right is important in parsing the above deduction correctly.
next up previous contents
Next: Conditional proofs Up: Proof semantics Previous: Rules dealing with equality   Contents
2004-08-06