By Jean Trignan

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In fact, a canonical calculus is a finite supply of rules (permitting some rules without any input). But the more detailed explanations will follow in the next section where the connections with the inductive definitions will be cleared up exactly. 1. DEFINITION. Let C be an alphabet and I ~' a character not occurring in C. We define inductively the notion ofa c-rule by the following two stipulations: (i) Iff eC~ f is a C-rule. (ii) Ifr is a c-rule, and f eC O then (Remark. Note that by (i), (0 uf~ r'' is a c-rule.

Finally, if / does not occur in 8, go further. Step 3, and, in general, Step 2n+l (n ~ 1). Apply the second (the n-th) substitution of variables. Extend 8 by adding the arrow-free words resulting from the rules of K by this substitution, and P by the other words. Step 4, and, in general, Step 2n. The s~me as Step 2. The sequence marked by '8' will be extended in each step of uneven number, and, sometimes, in steps of even number. The other sequence marked by 'P' contains 51. u {~} -words, it increases in steps of uneven number and may change or even de- crease in steps of even number.

Mula), and the letters 'u' and 'v' as calculus variables. Now our calculus called KpL consists of the followingrules: K pL : 1. 2. 3. 4. 5. 5*. 10 Iu ~ Iut Iu ~ F1tu Fu ~ F-u Fu ~ Fv ~ F(u::) v) Fu~ u (The numbering of rules does not belong to the calculus. ) Comments. Rule 1 tells that the empty word is an index. , if '0' is a rule. Thus, rule 1 can be written simply as 'I'. - Rules 1 and 2 together define the class of indices; one sees that indices are just the {t}-words. - Rule 3 defines the atomic formulas, rules 4 and 5 define compound formulas.

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