3.11/1.57 YES 3.11/1.58 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 3.11/1.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.11/1.58 3.11/1.58 3.11/1.58 Termination w.r.t. Q of the given QTRS could be proven: 3.11/1.58 3.11/1.58 (0) QTRS 3.11/1.58 (1) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] 3.11/1.58 (2) YES 3.11/1.58 3.11/1.58 3.11/1.58 ---------------------------------------- 3.11/1.58 3.11/1.58 (0) 3.11/1.58 Obligation: 3.11/1.58 Q restricted rewrite system: 3.11/1.58 The TRS R consists of the following rules: 3.11/1.58 3.11/1.58 a__f(X) -> g(h(f(X))) 3.11/1.58 mark(f(X)) -> a__f(mark(X)) 3.11/1.58 mark(g(X)) -> g(X) 3.11/1.58 mark(h(X)) -> h(mark(X)) 3.11/1.58 a__f(X) -> f(X) 3.11/1.58 3.11/1.58 The set Q consists of the following terms: 3.11/1.58 3.11/1.58 a__f(x0) 3.11/1.58 mark(f(x0)) 3.11/1.58 mark(g(x0)) 3.11/1.58 mark(h(x0)) 3.11/1.58 3.11/1.58 3.11/1.58 ---------------------------------------- 3.11/1.58 3.11/1.58 (1) RFCMatchBoundsTRSProof (EQUIVALENT) 3.11/1.58 Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R. 3.11/1.58 The following rules were used to construct the certificate: 3.11/1.58 3.11/1.58 a__f(X) -> g(h(f(X))) 3.11/1.58 mark(f(X)) -> a__f(mark(X)) 3.11/1.58 mark(g(X)) -> g(X) 3.11/1.58 mark(h(X)) -> h(mark(X)) 3.11/1.58 a__f(X) -> f(X) 3.11/1.58 3.11/1.58 The certificate found is represented by the following graph. 3.11/1.58 The certificate consists of the following enumerated nodes: 3.11/1.58 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 3.11/1.58 3.11/1.58 Node 1 is start node and node 2 is final node. 3.11/1.58 3.11/1.58 Those nodes are connected through the following edges: 3.11/1.58 3.11/1.58 * 1 to 3 labelled g_1(0)* 1 to 5 labelled a__f_1(0), h_1(0), f_1(1)* 1 to 2 labelled g_1(0), f_1(0)* 1 to 6 labelled g_1(1)* 2 to 2 labelled #_1(0)* 3 to 4 labelled h_1(0)* 4 to 2 labelled f_1(0)* 5 to 2 labelled mark_1(0), g_1(1)* 5 to 8 labelled a__f_1(1), h_1(1), f_1(2)* 5 to 9 labelled g_1(2)* 6 to 7 labelled h_1(1)* 7 to 5 labelled f_1(1)* 8 to 2 labelled mark_1(1), g_1(1)* 8 to 8 labelled a__f_1(1), h_1(1), f_1(2)* 8 to 9 labelled g_1(2)* 9 to 10 labelled h_1(2)* 10 to 8 labelled f_1(2) 3.11/1.58 3.11/1.58 3.11/1.58 ---------------------------------------- 3.11/1.58 3.11/1.58 (2) 3.11/1.58 YES 3.37/1.64 EOF