3.15/1.57 YES 3.15/1.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.15/1.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.15/1.58 3.15/1.58 3.15/1.58 Termination w.r.t. Q of the given QTRS could be proven: 3.15/1.58 3.15/1.58 (0) QTRS 3.15/1.58 (1) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] 3.15/1.58 (2) YES 3.15/1.58 3.15/1.58 3.15/1.58 ---------------------------------------- 3.15/1.58 3.15/1.58 (0) 3.15/1.58 Obligation: 3.15/1.58 Q restricted rewrite system: 3.15/1.58 The TRS R consists of the following rules: 3.15/1.58 3.15/1.58 a__f(f(a)) -> c(f(g(f(a)))) 3.15/1.58 mark(f(X)) -> a__f(mark(X)) 3.15/1.58 mark(a) -> a 3.15/1.58 mark(c(X)) -> c(X) 3.15/1.58 mark(g(X)) -> g(mark(X)) 3.15/1.58 a__f(X) -> f(X) 3.15/1.58 3.15/1.58 The set Q consists of the following terms: 3.15/1.58 3.15/1.58 mark(f(x0)) 3.15/1.58 mark(a) 3.15/1.58 mark(c(x0)) 3.15/1.58 mark(g(x0)) 3.15/1.58 a__f(x0) 3.15/1.58 3.15/1.58 3.15/1.58 ---------------------------------------- 3.15/1.58 3.15/1.58 (1) RFCMatchBoundsTRSProof (EQUIVALENT) 3.15/1.58 Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R. 3.15/1.58 The following rules were used to construct the certificate: 3.15/1.58 3.15/1.58 a__f(f(a)) -> c(f(g(f(a)))) 3.15/1.58 mark(f(X)) -> a__f(mark(X)) 3.15/1.58 mark(a) -> a 3.15/1.58 mark(c(X)) -> c(X) 3.15/1.58 mark(g(X)) -> g(mark(X)) 3.15/1.58 a__f(X) -> f(X) 3.15/1.58 3.15/1.58 The certificate found is represented by the following graph. 3.15/1.58 The certificate consists of the following enumerated nodes: 3.15/1.58 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 3.15/1.58 3.15/1.58 Node 1 is start node and node 2 is final node. 3.15/1.58 3.15/1.58 Those nodes are connected through the following edges: 3.15/1.58 3.15/1.58 * 1 to 5 labelled c_1(0)* 1 to 9 labelled a__f_1(0), g_1(0), f_1(1)* 1 to 2 labelled a(0), c_1(0), f_1(0)* 1 to 11 labelled c_1(1)* 2 to 2 labelled #_1(0)* 5 to 6 labelled f_1(0)* 6 to 7 labelled g_1(0)* 7 to 8 labelled f_1(0)* 8 to 2 labelled a(0)* 9 to 2 labelled mark_1(0), a(1), c_1(1)* 9 to 10 labelled a__f_1(1), g_1(1), f_1(2)* 9 to 15 labelled c_1(2)* 10 to 2 labelled mark_1(1), a(1), c_1(1)* 10 to 10 labelled a__f_1(1), g_1(1), f_1(2)* 10 to 15 labelled c_1(2)* 11 to 12 labelled f_1(1)* 12 to 13 labelled g_1(1)* 13 to 14 labelled f_1(1)* 14 to 2 labelled a(1)* 15 to 16 labelled f_1(2)* 16 to 17 labelled g_1(2)* 17 to 18 labelled f_1(2)* 18 to 2 labelled a(2) 3.15/1.58 3.15/1.58 3.15/1.58 ---------------------------------------- 3.15/1.58 3.15/1.58 (2) 3.15/1.58 YES 3.15/1.60 EOF