116.14/30.48 MAYBE 116.14/30.48 116.14/30.48 Proof: 116.14/30.48 ConCon could not decide confluence of the system. 116.14/30.48 \cite{ALS94}, Theorem 4.1 does not apply. 116.14/30.48 This system is of type 3 or smaller. 116.14/30.48 This system is strongly deterministic. 116.14/30.48 This system is of type 3 or smaller. 116.14/30.48 This system is deterministic. 116.14/30.48 ConCon could not decide if this system is quasi-decreasing. 116.14/30.48 \cite{O02}, p. 214, Proposition 7.2.50 does not apply. 116.14/30.48 This system is of type 3 or smaller. 116.14/30.48 This system is deterministic. 116.14/30.48 System R transformed to optimized U(R). 116.14/30.48 The external tool could not decide termination of the system. 116.14/30.48 Call external tool: 116.14/30.48 ./ttt2.sh 116.14/30.48 Input: 116.14/30.48 mult(s(x), y) -> add(mult(x, y), y) 116.14/30.48 add(s(x), y) -> s(add(x, y)) 116.14/30.48 lte(s(x), 0) -> false 116.14/30.48 minus(0, s(y)) -> 0 116.14/30.48 div(s(x), s(y)) -> ?2(lte(s(x), y), x, y) 116.14/30.48 ?2(false, x, y) -> ?3(div(minus(x, y), s(y)), x, y) 116.14/30.48 ?3(q, x, y) -> s(q) 116.14/30.48 div(0, s(x)) -> 0 116.14/30.48 mod(x, s(y)) -> ?1(lte(s(y), x), x, y) 116.14/30.48 ?1(false, x, y) -> x 116.14/30.48 mod(x, s(y)) -> ?1(lte(s(y), x), x, y) 116.14/30.48 ?1(true, x, y) -> mod(minus(x, s(y)), s(y)) 116.14/30.48 minus(s(x), s(y)) -> minus(x, y) 116.14/30.48 div(s(x), s(y)) -> ?2(lte(s(x), y), x, y) 116.14/30.48 ?2(true, x, y) -> 0 116.14/30.48 power(x, n) -> ?4(mod(n, s(s(0))), x, n) 116.14/30.48 ?4(s(z), x, n) -> ?6(power(x, div(n, s(s(0)))), x, n, z) 116.14/30.48 ?6(y, x, n, z) -> mult(mult(y, y), x) 116.14/30.48 mult(0, y) -> 0 116.14/30.48 power(x, 0) -> s(0) 116.14/30.48 mod(x, 0) -> x 116.14/30.48 lte(s(x), s(y)) -> lte(x, y) 116.14/30.48 add(0, x) -> x 116.14/30.48 mod(0, y) -> 0 116.14/30.48 minus(x, 0) -> x 116.14/30.48 lte(0, y) -> true 116.14/30.48 power(x, n) -> ?4(mod(n, s(s(0))), x, n) 116.14/30.48 ?4(0, x, n) -> ?5(power(x, div(n, s(s(0)))), x, n) 116.14/30.48 ?5(y, x, n) -> mult(mult(y, y), s(0)) 116.14/30.48 116.14/30.48 116.76/30.51 EOF