YES proof of Transformed_CSR_04_Ex23_Luc06_C.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 26 ms] (4) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(f(f(a))) -> mark(c(f(g(f(a))))) active(f(X)) -> f(active(X)) active(g(X)) -> g(active(X)) f(mark(X)) -> mark(f(X)) g(mark(X)) -> mark(g(X)) proper(f(X)) -> f(proper(X)) proper(a) -> ok(a) proper(c(X)) -> c(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) c(ok(X)) -> ok(c(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Q is empty. ---------------------------------------- (1) QTRS Reverse (EQUIVALENT) We applied the QTRS Reverse Processor [REVERSE]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a'(f(f(active(x)))) -> a'(f(g(f(c(mark(x)))))) f(active(X)) -> active(f(X)) g(active(X)) -> active(g(X)) mark(f(X)) -> f(mark(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) c(proper(X)) -> proper(c(X)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(ok(X)) ok(c(X)) -> c(ok(X)) ok(g(X)) -> g(ok(X)) mark(top(X)) -> proper(top(X)) ok(top(X)) -> active(top(X)) Q is empty. ---------------------------------------- (3) RFCMatchBoundsTRSProof (EQUIVALENT) Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 6. This implies Q-termination of R. The following rules were used to construct the certificate: a'(f(f(active(x)))) -> a'(f(g(f(c(mark(x)))))) f(active(X)) -> active(f(X)) g(active(X)) -> active(g(X)) mark(f(X)) -> f(mark(X)) mark(g(X)) -> g(mark(X)) f(proper(X)) -> proper(f(X)) a'(proper(x)) -> a'(ok(x)) c(proper(X)) -> proper(c(X)) g(proper(X)) -> proper(g(X)) ok(f(X)) -> f(ok(X)) ok(c(X)) -> c(ok(X)) ok(g(X)) -> g(ok(X)) mark(top(X)) -> proper(top(X)) ok(top(X)) -> active(top(X)) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 44, 45, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 71, 74, 75, 76, 80, 81, 82, 83, 87, 88, 92, 93, 94, 97, 99, 100 Node 21 is start node and node 22 is final node. Those nodes are connected through the following edges: * 21 to 23 labelled a'_1(0), f_1(0), c_1(0), g_1(0)* 21 to 28 labelled active_1(0), proper_1(0)* 21 to 27 labelled f_1(0), g_1(0)* 21 to 44 labelled active_1(1), proper_1(1)* 21 to 51 labelled a'_1(1)* 21 to 74 labelled a'_1(2)* 22 to 22 labelled #_1(0)* 23 to 24 labelled f_1(0)* 23 to 22 labelled ok_1(0)* 23 to 30 labelled f_1(1), c_1(1), g_1(1)* 23 to 31 labelled active_1(1)* 23 to 49 labelled active_1(2)* 23 to 58 labelled proper_1(1)* 24 to 25 labelled g_1(0)* 24 to 56 labelled proper_1(1)* 25 to 26 labelled f_1(0)* 25 to 50 labelled proper_1(1)* 26 to 27 labelled c_1(0)* 26 to 45 labelled proper_1(1)* 27 to 22 labelled mark_1(0)* 27 to 32 labelled f_1(1), g_1(1)* 27 to 31 labelled proper_1(1)* 27 to 49 labelled proper_1(2)* 28 to 22 labelled f_1(0), g_1(0), c_1(0), top_1(0)* 28 to 33 labelled active_1(1), proper_1(1)* 30 to 22 labelled ok_1(1)* 30 to 30 labelled f_1(1), c_1(1), g_1(1)* 30 to 31 labelled active_1(1)* 30 to 49 labelled active_1(2)* 31 to 22 labelled top_1(1)* 32 to 22 labelled mark_1(1)* 32 to 32 labelled f_1(1), g_1(1)* 32 to 31 labelled proper_1(1)* 32 to 49 labelled proper_1(2)* 33 to 22 labelled f_1(1), g_1(1), c_1(1)* 33 to 33 labelled active_1(1), proper_1(1)* 44 to 31 labelled f_1(1), g_1(1)* 44 to 49 labelled f_1(1), g_1(1)* 44 to 58 labelled f_1(1), c_1(1), g_1(1)* 45 to 31 labelled c_1(1)* 45 to 49 labelled c_1(1)* 49 to 31 labelled f_1(2), g_1(2)* 49 to 49 labelled f_1(2), g_1(2)* 50 to 45 labelled f_1(1)* 51 to 52 labelled f_1(1)* 51 to 58 labelled ok_1(1)* 51 to 63 labelled f_1(2)* 51 to 68 labelled proper_1(2)* 52 to 53 labelled g_1(1)* 52 to 64 labelled proper_1(2)* 53 to 54 labelled f_1(1)* 53 to 61 labelled proper_1(2)* 54 to 55 labelled c_1(1)* 54 to 59 labelled proper_1(2)* 55 to 31 labelled mark_1(1)* 55 to 57 labelled proper_1(2)* 55 to 49 labelled mark_1(1)* 55 to 60 labelled f_1(2), g_1(2)* 55 to 65 labelled proper_1(3)* 56 to 50 labelled g_1(1)* 57 to 22 labelled top_1(2)* 58 to 56 labelled f_1(1)* 59 to 57 labelled c_1(2)* 59 to 65 labelled c_1(2)* 60 to 31 labelled mark_1(2)* 60 to 49 labelled mark_1(2)* 60 to 57 labelled proper_1(2)* 60 to 62 labelled f_1(3), g_1(3)* 60 to 65 labelled proper_1(3)* 60 to 75 labelled proper_1(4)* 61 to 59 labelled f_1(2)* 62 to 31 labelled mark_1(3)* 62 to 49 labelled mark_1(3)* 62 to 57 labelled proper_1(2)* 62 to 62 labelled f_1(3), g_1(3)* 62 to 65 labelled proper_1(3)* 62 to 75 labelled proper_1(4)* 63 to 56 labelled ok_1(2)* 63 to 71 labelled g_1(2)* 64 to 61 labelled g_1(2)* 65 to 57 labelled f_1(3), g_1(3)* 65 to 65 labelled f_1(3), g_1(3)* 65 to 75 labelled f_1(3), g_1(3)* 68 to 64 labelled f_1(2)* 71 to 50 labelled ok_1(2)* 71 to 76 labelled f_1(2)* 74 to 68 labelled ok_1(2)* 74 to 80 labelled f_1(3)* 75 to 65 labelled f_1(4), g_1(4)* 75 to 75 labelled f_1(4), g_1(4)* 76 to 45 labelled ok_1(2)* 76 to 81 labelled c_1(2)* 80 to 64 labelled ok_1(3)* 80 to 83 labelled g_1(3)* 81 to 31 labelled ok_1(2)* 81 to 49 labelled ok_1(2)* 81 to 57 labelled active_1(2)* 81 to 82 labelled f_1(3), g_1(3)* 81 to 65 labelled active_1(3)* 81 to 75 labelled active_1(4)* 82 to 31 labelled ok_1(3)* 82 to 49 labelled ok_1(3)* 82 to 57 labelled active_1(2)* 82 to 82 labelled f_1(3), g_1(3)* 82 to 65 labelled active_1(3)* 82 to 75 labelled active_1(4)* 83 to 61 labelled ok_1(3)* 83 to 87 labelled f_1(3)* 87 to 59 labelled ok_1(3)* 87 to 88 labelled c_1(3)* 88 to 57 labelled ok_1(3)* 88 to 65 labelled ok_1(3)* 88 to 92 labelled active_1(3)* 88 to 93 labelled f_1(4), g_1(4)* 88 to 97 labelled active_1(4)* 88 to 99 labelled active_1(5)* 92 to 22 labelled top_1(3)* 93 to 57 labelled ok_1(4)* 93 to 65 labelled ok_1(4)* 93 to 75 labelled ok_1(4)* 93 to 92 labelled active_1(3)* 93 to 93 labelled f_1(4), g_1(4)* 93 to 94 labelled f_1(5), g_1(5)* 93 to 97 labelled active_1(4)* 93 to 99 labelled active_1(5)* 93 to 100 labelled active_1(6)* 94 to 65 labelled ok_1(5)* 94 to 75 labelled ok_1(5)* 94 to 93 labelled f_1(4), g_1(4)* 94 to 94 labelled f_1(5), g_1(5)* 94 to 97 labelled active_1(4)* 94 to 99 labelled active_1(5)* 94 to 100 labelled active_1(6)* 97 to 92 labelled f_1(4), g_1(4)* 99 to 97 labelled f_1(5), g_1(5)* 99 to 99 labelled f_1(5), g_1(5)* 99 to 100 labelled f_1(5), g_1(5)* 100 to 99 labelled f_1(6), g_1(6)* 100 to 100 labelled f_1(6), g_1(6) ---------------------------------------- (4) YES