3.45/1.72 YES 3.68/1.74 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 3.68/1.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.68/1.74 3.68/1.74 3.68/1.74 Termination w.r.t. Q of the given QTRS could be proven: 3.68/1.74 3.68/1.74 (0) QTRS 3.68/1.74 (1) RFCMatchBoundsTRSProof [EQUIVALENT, 35 ms] 3.68/1.74 (2) YES 3.68/1.74 3.68/1.74 3.68/1.74 ---------------------------------------- 3.68/1.74 3.68/1.74 (0) 3.68/1.74 Obligation: 3.68/1.74 Q restricted rewrite system: 3.68/1.74 The TRS R consists of the following rules: 3.68/1.74 3.68/1.74 active(f(f(a))) -> mark(c(f(g(f(a))))) 3.68/1.74 mark(f(X)) -> active(f(mark(X))) 3.68/1.74 mark(a) -> active(a) 3.68/1.74 mark(c(X)) -> active(c(X)) 3.68/1.74 mark(g(X)) -> active(g(mark(X))) 3.68/1.74 f(mark(X)) -> f(X) 3.68/1.74 f(active(X)) -> f(X) 3.68/1.74 c(mark(X)) -> c(X) 3.68/1.74 c(active(X)) -> c(X) 3.68/1.74 g(mark(X)) -> g(X) 3.68/1.74 g(active(X)) -> g(X) 3.68/1.74 3.68/1.74 The set Q consists of the following terms: 3.68/1.74 3.68/1.74 active(f(f(a))) 3.68/1.74 mark(f(x0)) 3.68/1.74 mark(a) 3.68/1.74 mark(c(x0)) 3.68/1.74 mark(g(x0)) 3.68/1.74 f(mark(x0)) 3.68/1.74 f(active(x0)) 3.68/1.74 c(mark(x0)) 3.68/1.74 c(active(x0)) 3.68/1.74 g(mark(x0)) 3.68/1.74 g(active(x0)) 3.68/1.74 3.68/1.74 3.68/1.74 ---------------------------------------- 3.68/1.74 3.68/1.74 (1) RFCMatchBoundsTRSProof (EQUIVALENT) 3.68/1.74 Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. This implies Q-termination of R. 3.68/1.74 The following rules were used to construct the certificate: 3.68/1.74 3.68/1.74 active(f(f(a))) -> mark(c(f(g(f(a))))) 3.68/1.74 mark(f(X)) -> active(f(mark(X))) 3.68/1.74 mark(a) -> active(a) 3.68/1.74 mark(c(X)) -> active(c(X)) 3.68/1.74 mark(g(X)) -> active(g(mark(X))) 3.68/1.74 f(mark(X)) -> f(X) 3.68/1.74 f(active(X)) -> f(X) 3.68/1.74 c(mark(X)) -> c(X) 3.68/1.74 c(active(X)) -> c(X) 3.68/1.74 g(mark(X)) -> g(X) 3.68/1.74 g(active(X)) -> g(X) 3.68/1.74 3.68/1.74 The certificate found is represented by the following graph. 3.68/1.74 The certificate consists of the following enumerated nodes: 3.68/1.74 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 38, 39, 40, 41, 42, 57 3.68/1.74 3.68/1.74 Node 1 is start node and node 2 is final node. 3.68/1.74 3.68/1.74 Those nodes are connected through the following edges: 3.68/1.74 3.68/1.74 * 1 to 5 labelled mark_1(0)* 1 to 10 labelled active_1(0)* 1 to 9 labelled active_1(0)* 1 to 12 labelled active_1(0)* 1 to 2 labelled f_1(0), c_1(0), g_1(0), f_1(1), c_1(1), g_1(1)* 1 to 23 labelled active_1(1)* 1 to 28 labelled mark_1(1)* 1 to 33 labelled active_1(2)* 2 to 2 labelled #_1(0)* 5 to 6 labelled c_1(0)* 6 to 7 labelled f_1(0)* 7 to 8 labelled g_1(0)* 8 to 9 labelled f_1(0)* 9 to 2 labelled a(0)* 10 to 11 labelled f_1(0)* 10 to 2 labelled c_1(0), c_1(1), f_1(1)* 10 to 24 labelled f_1(1)* 10 to 10 labelled f_1(1)* 10 to 26 labelled f_1(1)* 10 to 28 labelled f_1(1)* 10 to 33 labelled f_1(1)* 10 to 38 labelled f_1(1)* 10 to 57 labelled f_1(1)* 11 to 2 labelled mark_1(0)* 11 to 24 labelled active_1(1)* 11 to 10 labelled active_1(1)* 11 to 26 labelled active_1(1)* 11 to 28 labelled mark_1(1)* 11 to 33 labelled active_1(2)* 11 to 38 labelled mark_1(2)* 11 to 57 labelled active_1(3)* 12 to 13 labelled g_1(0)* 12 to 2 labelled g_1(1)* 12 to 24 labelled g_1(1)* 12 to 10 labelled g_1(1)* 12 to 26 labelled g_1(1)* 12 to 28 labelled g_1(1)* 12 to 33 labelled g_1(1)* 12 to 38 labelled g_1(1)* 12 to 57 labelled g_1(1)* 13 to 2 labelled mark_1(0)* 13 to 24 labelled active_1(1)* 13 to 10 labelled active_1(1)* 13 to 26 labelled active_1(1)* 13 to 28 labelled mark_1(1)* 13 to 33 labelled active_1(2)* 13 to 38 labelled mark_1(2)* 13 to 57 labelled active_1(3)* 23 to 6 labelled c_1(1)* 24 to 25 labelled f_1(1)* 24 to 2 labelled a(1), f_1(2), f_1(1)* 24 to 24 labelled f_1(2)* 24 to 10 labelled f_1(2)* 24 to 26 labelled f_1(2)* 24 to 28 labelled f_1(2)* 24 to 38 labelled f_1(2)* 24 to 33 labelled f_1(2)* 24 to 57 labelled f_1(2)* 25 to 2 labelled mark_1(1)* 25 to 24 labelled active_1(1)* 25 to 10 labelled active_1(1)* 25 to 26 labelled active_1(1)* 25 to 28 labelled mark_1(1)* 25 to 38 labelled mark_1(2)* 25 to 33 labelled active_1(2)* 25 to 57 labelled active_1(3)* 26 to 27 labelled g_1(1)* 26 to 2 labelled g_1(2), g_1(1)* 26 to 24 labelled g_1(2)* 26 to 10 labelled g_1(2)* 26 to 26 labelled g_1(2)* 26 to 28 labelled g_1(2)* 26 to 38 labelled g_1(2)* 26 to 33 labelled g_1(2)* 26 to 57 labelled g_1(2)* 27 to 2 labelled mark_1(1)* 27 to 24 labelled active_1(1)* 27 to 10 labelled active_1(1)* 27 to 26 labelled active_1(1)* 27 to 28 labelled mark_1(1)* 27 to 38 labelled mark_1(2)* 27 to 33 labelled active_1(2)* 27 to 57 labelled active_1(3)* 28 to 29 labelled c_1(1)* 29 to 30 labelled f_1(1)* 30 to 31 labelled g_1(1)* 31 to 32 labelled f_1(1)* 32 to 2 labelled a(1)* 33 to 29 labelled c_1(2)* 38 to 39 labelled c_1(2)* 39 to 40 labelled f_1(2)* 40 to 41 labelled g_1(2)* 41 to 42 labelled f_1(2)* 42 to 2 labelled a(2)* 57 to 39 labelled c_1(3) 3.68/1.74 3.68/1.74 3.68/1.74 ---------------------------------------- 3.68/1.74 3.68/1.74 (2) 3.68/1.74 YES 3.74/2.94 EOF