3.55/1.80 YES 3.92/2.53 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.92/2.53 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.92/2.53 3.92/2.53 3.92/2.53 Termination w.r.t. Q of the given QTRS could be proven: 3.92/2.53 3.92/2.53 (0) QTRS 3.92/2.53 (1) RFCMatchBoundsTRSProof [EQUIVALENT, 25 ms] 3.92/2.53 (2) YES 3.92/2.53 3.92/2.53 3.92/2.53 ---------------------------------------- 3.92/2.53 3.92/2.53 (0) 3.92/2.53 Obligation: 3.92/2.53 Q restricted rewrite system: 3.92/2.53 The TRS R consists of the following rules: 3.92/2.53 3.92/2.53 active(f(X)) -> mark(g(h(f(X)))) 3.92/2.53 mark(f(X)) -> active(f(mark(X))) 3.92/2.53 mark(g(X)) -> active(g(X)) 3.92/2.53 mark(h(X)) -> active(h(mark(X))) 3.92/2.53 f(mark(X)) -> f(X) 3.92/2.53 f(active(X)) -> f(X) 3.92/2.53 g(mark(X)) -> g(X) 3.92/2.53 g(active(X)) -> g(X) 3.92/2.53 h(mark(X)) -> h(X) 3.92/2.53 h(active(X)) -> h(X) 3.92/2.53 3.92/2.53 The set Q consists of the following terms: 3.92/2.53 3.92/2.53 active(f(x0)) 3.92/2.53 mark(f(x0)) 3.92/2.53 mark(g(x0)) 3.92/2.53 mark(h(x0)) 3.92/2.53 f(mark(x0)) 3.92/2.53 f(active(x0)) 3.92/2.53 g(mark(x0)) 3.92/2.53 g(active(x0)) 3.92/2.53 h(mark(x0)) 3.92/2.53 h(active(x0)) 3.92/2.53 3.92/2.53 3.92/2.53 ---------------------------------------- 3.92/2.53 3.92/2.53 (1) RFCMatchBoundsTRSProof (EQUIVALENT) 3.92/2.53 Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. This implies Q-termination of R. 3.92/2.53 The following rules were used to construct the certificate: 3.92/2.53 3.92/2.53 active(f(X)) -> mark(g(h(f(X)))) 3.92/2.53 mark(f(X)) -> active(f(mark(X))) 3.92/2.53 mark(g(X)) -> active(g(X)) 3.92/2.53 mark(h(X)) -> active(h(mark(X))) 3.92/2.53 f(mark(X)) -> f(X) 3.92/2.53 f(active(X)) -> f(X) 3.92/2.53 g(mark(X)) -> g(X) 3.92/2.53 g(active(X)) -> g(X) 3.92/2.53 h(mark(X)) -> h(X) 3.92/2.53 h(active(X)) -> h(X) 3.92/2.53 3.92/2.53 The certificate found is represented by the following graph. 3.92/2.53 The certificate consists of the following enumerated nodes: 3.92/2.53 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 3.92/2.53 3.92/2.53 Node 1 is start node and node 2 is final node. 3.92/2.53 3.92/2.53 Those nodes are connected through the following edges: 3.92/2.53 3.92/2.53 * 1 to 3 labelled mark_1(0)* 1 to 6 labelled active_1(0)* 1 to 8 labelled active_1(0)* 1 to 2 labelled f_1(0), g_1(0), h_1(0), f_1(1), g_1(1), h_1(1)* 1 to 19 labelled active_1(1)* 1 to 20 labelled mark_1(1)* 1 to 30 labelled active_1(2)* 2 to 2 labelled #_1(0)* 3 to 4 labelled g_1(0)* 4 to 5 labelled h_1(0)* 5 to 2 labelled f_1(0), f_1(1)* 6 to 7 labelled f_1(0)* 6 to 2 labelled g_1(0), g_1(1), f_1(1)* 6 to 23 labelled f_1(1)* 6 to 6 labelled f_1(1)* 6 to 25 labelled f_1(1)* 6 to 20 labelled f_1(1)* 6 to 27 labelled f_1(1)* 6 to 30 labelled f_1(1)* 6 to 31 labelled f_1(1)* 7 to 2 labelled mark_1(0)* 7 to 23 labelled active_1(1)* 7 to 6 labelled active_1(1)* 7 to 25 labelled active_1(1)* 7 to 20 labelled mark_1(1)* 7 to 27 labelled mark_1(2)* 7 to 30 labelled active_1(2)* 7 to 31 labelled active_1(3)* 8 to 9 labelled h_1(0)* 8 to 2 labelled h_1(1)* 8 to 23 labelled h_1(1)* 8 to 6 labelled h_1(1)* 8 to 25 labelled h_1(1)* 8 to 20 labelled h_1(1)* 8 to 27 labelled h_1(1)* 8 to 30 labelled h_1(1)* 8 to 31 labelled h_1(1)* 9 to 2 labelled mark_1(0)* 9 to 23 labelled active_1(1)* 9 to 6 labelled active_1(1)* 9 to 25 labelled active_1(1)* 9 to 20 labelled mark_1(1)* 9 to 27 labelled mark_1(2)* 9 to 30 labelled active_1(2)* 9 to 31 labelled active_1(3)* 19 to 4 labelled g_1(1)* 20 to 21 labelled g_1(1)* 21 to 22 labelled h_1(1)* 22 to 7 labelled f_1(1)* 22 to 2 labelled f_1(1)* 22 to 23 labelled f_1(2), f_1(1)* 22 to 6 labelled f_1(2), f_1(1)* 22 to 25 labelled f_1(2), f_1(1)* 22 to 20 labelled f_1(2), f_1(1)* 22 to 27 labelled f_1(2), f_1(1)* 22 to 30 labelled f_1(2), f_1(1)* 22 to 31 labelled f_1(2), f_1(1)* 23 to 24 labelled f_1(1)* 23 to 2 labelled f_1(2), f_1(1)* 23 to 23 labelled f_1(2)* 23 to 6 labelled f_1(2)* 23 to 25 labelled f_1(2)* 23 to 20 labelled f_1(2)* 23 to 27 labelled f_1(2)* 23 to 30 labelled f_1(2)* 23 to 31 labelled f_1(2)* 24 to 2 labelled mark_1(1)* 24 to 23 labelled active_1(1)* 24 to 6 labelled active_1(1)* 24 to 25 labelled active_1(1)* 24 to 20 labelled mark_1(1)* 24 to 27 labelled mark_1(2)* 24 to 30 labelled active_1(2)* 24 to 31 labelled active_1(3)* 25 to 26 labelled h_1(1)* 25 to 2 labelled h_1(2), h_1(1)* 25 to 23 labelled h_1(2)* 25 to 6 labelled h_1(2)* 25 to 25 labelled h_1(2)* 25 to 20 labelled h_1(2)* 25 to 27 labelled h_1(2)* 25 to 30 labelled h_1(2)* 25 to 31 labelled h_1(2)* 26 to 2 labelled mark_1(1)* 26 to 23 labelled active_1(1)* 26 to 6 labelled active_1(1)* 26 to 25 labelled active_1(1)* 26 to 20 labelled mark_1(1)* 26 to 27 labelled mark_1(2)* 26 to 30 labelled active_1(2)* 26 to 31 labelled active_1(3)* 27 to 28 labelled g_1(2)* 28 to 29 labelled h_1(2)* 29 to 2 labelled f_1(2), f_1(1)* 29 to 24 labelled f_1(2)* 29 to 23 labelled f_1(2)* 29 to 6 labelled f_1(2)* 29 to 25 labelled f_1(2)* 29 to 20 labelled f_1(2)* 29 to 27 labelled f_1(2), f_1(3)* 29 to 30 labelled f_1(2), f_1(3)* 29 to 31 labelled f_1(2), f_1(3)* 30 to 21 labelled g_1(2)* 31 to 28 labelled g_1(3) 3.92/2.53 3.92/2.53 3.92/2.53 ---------------------------------------- 3.92/2.53 3.92/2.53 (2) 3.92/2.53 YES 3.95/2.58 EOF