3.84/1.69 YES 3.84/1.71 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.84/1.71 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.84/1.71 3.84/1.71 3.84/1.71 Termination w.r.t. Q of the given QTRS could be proven: 3.84/1.71 3.84/1.71 (0) QTRS 3.84/1.71 (1) QTRS Reverse [SOUND, 0 ms] 3.84/1.71 (2) QTRS 3.84/1.71 (3) RFCMatchBoundsTRSProof [EQUIVALENT, 4 ms] 3.84/1.71 (4) YES 3.84/1.71 3.84/1.71 3.84/1.71 ---------------------------------------- 3.84/1.71 3.84/1.71 (0) 3.84/1.71 Obligation: 3.84/1.71 Q restricted rewrite system: 3.84/1.71 The TRS R consists of the following rules: 3.84/1.71 3.84/1.71 active(c) -> mark(f(g(c))) 3.84/1.71 active(f(g(X))) -> mark(g(X)) 3.84/1.71 proper(c) -> ok(c) 3.84/1.71 proper(f(X)) -> f(proper(X)) 3.84/1.71 proper(g(X)) -> g(proper(X)) 3.84/1.71 f(ok(X)) -> ok(f(X)) 3.84/1.71 g(ok(X)) -> ok(g(X)) 3.84/1.71 top(mark(X)) -> top(proper(X)) 3.84/1.71 top(ok(X)) -> top(active(X)) 3.84/1.71 3.84/1.71 The set Q consists of the following terms: 3.84/1.71 3.84/1.71 active(c) 3.84/1.71 active(f(g(x0))) 3.84/1.71 proper(c) 3.84/1.71 proper(f(x0)) 3.84/1.71 proper(g(x0)) 3.84/1.71 f(ok(x0)) 3.84/1.71 g(ok(x0)) 3.84/1.71 top(mark(x0)) 3.84/1.71 top(ok(x0)) 3.84/1.71 3.84/1.71 3.84/1.71 ---------------------------------------- 3.84/1.71 3.84/1.71 (1) QTRS Reverse (SOUND) 3.84/1.71 We applied the QTRS Reverse Processor [REVERSE]. 3.84/1.71 ---------------------------------------- 3.84/1.71 3.84/1.71 (2) 3.84/1.71 Obligation: 3.84/1.71 Q restricted rewrite system: 3.84/1.71 The TRS R consists of the following rules: 3.84/1.71 3.84/1.71 c'(active(x)) -> c'(g(f(mark(x)))) 3.84/1.71 g(f(active(X))) -> g(mark(X)) 3.84/1.71 c'(proper(x)) -> c'(ok(x)) 3.84/1.71 f(proper(X)) -> proper(f(X)) 3.84/1.71 g(proper(X)) -> proper(g(X)) 3.84/1.71 ok(f(X)) -> f(ok(X)) 3.84/1.71 ok(g(X)) -> g(ok(X)) 3.84/1.71 mark(top(X)) -> proper(top(X)) 3.84/1.71 ok(top(X)) -> active(top(X)) 3.84/1.71 3.84/1.71 Q is empty. 3.84/1.71 3.84/1.71 ---------------------------------------- 3.84/1.71 3.84/1.71 (3) RFCMatchBoundsTRSProof (EQUIVALENT) 3.84/1.71 Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 5. This implies Q-termination of R. 3.84/1.71 The following rules were used to construct the certificate: 3.84/1.71 3.84/1.71 c'(active(x)) -> c'(g(f(mark(x)))) 3.84/1.71 g(f(active(X))) -> g(mark(X)) 3.84/1.71 c'(proper(x)) -> c'(ok(x)) 3.84/1.71 f(proper(X)) -> proper(f(X)) 3.84/1.71 g(proper(X)) -> proper(g(X)) 3.84/1.71 ok(f(X)) -> f(ok(X)) 3.84/1.71 ok(g(X)) -> g(ok(X)) 3.84/1.71 mark(top(X)) -> proper(top(X)) 3.84/1.71 ok(top(X)) -> active(top(X)) 3.84/1.71 3.84/1.71 The certificate found is represented by the following graph. 3.84/1.71 The certificate consists of the following enumerated nodes: 3.84/1.71 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86 3.84/1.71 3.84/1.71 Node 43 is start node and node 44 is final node. 3.84/1.71 3.84/1.71 Those nodes are connected through the following edges: 3.84/1.71 3.84/1.71 * 43 to 45 labelled c'_1(0), f_1(0), g_1(0)* 43 to 47 labelled g_1(0)* 43 to 48 labelled proper_1(0), active_1(0)* 43 to 53 labelled c'_1(1)* 43 to 56 labelled proper_1(1)* 43 to 55 labelled g_1(1)* 43 to 64 labelled proper_1(2)* 43 to 69 labelled c'_1(2)* 43 to 83 labelled c'_1(3)* 44 to 44 labelled #_1(0)* 45 to 46 labelled g_1(0)* 45 to 44 labelled ok_1(0)* 45 to 49 labelled f_1(1), g_1(1)* 45 to 50 labelled active_1(1)* 45 to 58 labelled proper_1(1)* 45 to 63 labelled g_1(2)* 45 to 66 labelled proper_1(3)* 45 to 70 labelled proper_1(2)* 46 to 47 labelled f_1(0)* 46 to 57 labelled proper_1(1)* 47 to 44 labelled mark_1(0)* 47 to 50 labelled proper_1(1)* 48 to 44 labelled f_1(0), g_1(0), top_1(0)* 48 to 51 labelled proper_1(1)* 48 to 52 labelled g_1(1)* 48 to 61 labelled proper_1(2)* 49 to 44 labelled ok_1(1)* 49 to 49 labelled f_1(1), g_1(1)* 49 to 50 labelled active_1(1)* 49 to 63 labelled g_1(2)* 49 to 66 labelled proper_1(3)* 49 to 70 labelled proper_1(2)* 50 to 44 labelled top_1(1)* 51 to 44 labelled f_1(1), g_1(1)* 51 to 51 labelled proper_1(1)* 51 to 52 labelled g_1(1)* 51 to 61 labelled proper_1(2)* 52 to 44 labelled mark_1(1)* 52 to 50 labelled proper_1(1)* 53 to 54 labelled g_1(1)* 53 to 58 labelled ok_1(1)* 53 to 67 labelled proper_1(2)* 53 to 68 labelled g_1(2)* 53 to 66 labelled ok_1(1)* 53 to 70 labelled ok_1(1)* 53 to 73 labelled f_1(2), g_1(2)* 53 to 76 labelled g_1(3)* 53 to 79 labelled proper_1(4)* 54 to 55 labelled f_1(1)* 54 to 65 labelled proper_1(2)* 55 to 50 labelled mark_1(1)* 55 to 62 labelled proper_1(2)* 56 to 50 labelled g_1(1)* 56 to 58 labelled f_1(1), g_1(1)* 56 to 66 labelled f_1(1), g_1(1)* 56 to 70 labelled f_1(1), g_1(1)* 57 to 50 labelled f_1(1)* 58 to 57 labelled g_1(1)* 61 to 50 labelled g_1(2)* 62 to 44 labelled top_1(2)* 63 to 50 labelled mark_1(2)* 63 to 62 labelled proper_1(2)* 64 to 62 labelled g_1(2)* 65 to 62 labelled f_1(2)* 66 to 62 labelled g_1(3)* 67 to 65 labelled g_1(2)* 68 to 57 labelled ok_1(2)* 68 to 71 labelled f_1(2)* 68 to 62 labelled ok_1(2)* 68 to 75 labelled active_1(3)* 69 to 67 labelled ok_1(2)* 69 to 72 labelled g_1(3)* 69 to 80 labelled g_1(4)* 69 to 79 labelled ok_1(2)* 69 to 82 labelled proper_1(5)* 70 to 66 labelled f_1(2), g_1(2)* 70 to 70 labelled f_1(2), g_1(2)* 71 to 50 labelled ok_1(2)* 71 to 62 labelled active_1(2)* 72 to 65 labelled ok_1(3)* 72 to 74 labelled f_1(3)* 72 to 75 labelled ok_1(3)* 72 to 81 labelled active_1(4)* 73 to 66 labelled ok_1(2)* 73 to 70 labelled ok_1(2)* 73 to 74 labelled g_1(3)* 73 to 77 labelled f_1(3), g_1(3)* 74 to 62 labelled ok_1(3)* 74 to 75 labelled active_1(3)* 75 to 44 labelled top_1(3)* 76 to 62 labelled mark_1(3)* 76 to 75 labelled proper_1(3)* 77 to 66 labelled ok_1(3)* 77 to 70 labelled ok_1(3)* 77 to 78 labelled g_1(4)* 77 to 77 labelled f_1(3), g_1(3)* 78 to 62 labelled ok_1(4)* 78 to 75 labelled active_1(3)* 79 to 75 labelled g_1(4)* 80 to 75 labelled mark_1(4)* 80 to 81 labelled proper_1(4)* 81 to 44 labelled top_1(4)* 82 to 81 labelled g_1(5)* 83 to 82 labelled ok_1(3)* 83 to 85 labelled g_1(4)* 85 to 81 labelled ok_1(4)* 85 to 86 labelled active_1(5)* 86 to 44 labelled top_1(5) 3.84/1.71 3.84/1.71 3.84/1.71 ---------------------------------------- 3.84/1.71 3.84/1.71 (4) 3.84/1.71 YES 3.96/1.74 EOF