2.64/1.28 YES 2.64/1.28 2.64/1.28 Proof: 2.64/1.28 This system is confluent. 2.64/1.28 By \cite{ALS94}, Theorem 4.1. 2.64/1.28 This system is of type 3 or smaller. 2.64/1.28 This system is strongly deterministic. 2.64/1.28 This system is quasi-decreasing. 2.64/1.28 By \cite{O02}, p. 214, Proposition 7.2.50. 2.64/1.28 This system is of type 3 or smaller. 2.64/1.28 This system is deterministic. 2.64/1.28 System R transformed to optimized U(R). 2.64/1.28 This system is terminating. 2.64/1.28 Call external tool: 2.64/1.28 ./ttt2.sh 2.64/1.28 Input: 2.64/1.28 f(x) -> ?1(a, x) 2.64/1.28 ?1(x, x) -> a 2.64/1.28 f(x) -> ?2(b, x) 2.64/1.28 ?2(x, x) -> b 2.64/1.28 2.64/1.29 Polynomial Interpretation Processor: 2.64/1.29 dimension: 1 2.64/1.29 interpretation: 2.64/1.29 [?2](x0, x1) = 2x0 + 2x1 + 7x0x0 + 1, 2.64/1.29 2.64/1.29 [b] = 0, 2.64/1.29 2.64/1.29 [?1](x0, x1) = 2x1 + 6x0x0 + 1, 2.64/1.29 2.64/1.29 [a] = 0, 2.64/1.29 2.64/1.29 [f](x0) = 6x0 + 2x0x0 + 6 2.64/1.29 orientation: 2.64/1.29 f(x) = 6x + 2x*x + 6 >= 2x + 1 = ?1(a(),x) 2.64/1.29 2.64/1.29 ?1(x,x) = 2x + 6x*x + 1 >= 0 = a() 2.64/1.29 2.64/1.29 f(x) = 6x + 2x*x + 6 >= 2x + 1 = ?2(b(),x) 2.64/1.29 2.64/1.29 ?2(x,x) = 4x + 7x*x + 1 >= 0 = b() 2.64/1.30 problem: 2.64/1.30 2.64/1.30 Qed 2.64/1.30 All 2 critical pairs are joinable. 2.64/1.30 Overlap: (rule1: f(y) -> b <= b = y, rule2: f(z) -> a <= a = z, pos: ε, mgu: {(z,y)}) 2.64/1.30 CP: a = b <= b = y, a = y 2.64/1.30 This critical pair is context-joinable. 2.64/1.30 Overlap: (rule1: f(y) -> a <= a = y, rule2: f(z) -> b <= b = z, pos: ε, mgu: {(z,y)}) 2.64/1.30 CP: b = a <= a = y, b = y 2.64/1.30 This critical pair is context-joinable. 2.64/1.30 2.90/1.32 EOF