Termination Proof Script

• Consider the SCC {11}. There are no usable rules. By taking the AF π with π(PLUS) = 1 together with the lexicographic path order with empty precedence, rule 11 is strictly decreasing.


• Consider the SCC {8,12}. The usable rules are {5,6}. By taking the AF π with π(MINUS) = π(s) = 1, π(plus) = 2 and π(minus) = [1] together with the lexicographic path order with empty precedence, the rules in {5,6,8} are weakly decreasing and rule 12 is strictly decreasing. There is one new SCC.
  • Consider the SCC {8}. There are no usable rules. By taking the AF π with π(MINUS) = 1 together with the lexicographic path order with empty precedence, rule 8 is strictly decreasing.


• Consider the SCC {9}. The usable rules are {1,2,5-7}. By taking the AF π with π(minus) = π(QUOT) = 1 together with the lexicographic path order with precedence plus ≻ s, the rules in {1,7} are weakly decreasing and the rules in {2,5,6,9} are strictly decreasing.