YES proof of Transformed_CSR_04_ExConc_Zan97_C.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] (4) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(f(X)) -> mark(g(h(f(X)))) active(f(X)) -> f(active(X)) active(h(X)) -> h(active(X)) f(mark(X)) -> mark(f(X)) h(mark(X)) -> mark(h(X)) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) proper(h(X)) -> h(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) h(ok(X)) -> ok(h(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: f(active(X)) -> f(h(g(mark(X)))) f(active(X)) -> active(f(X)) h(active(X)) -> active(h(X)) mark(f(X)) -> f(mark(X)) mark(h(X)) -> h(mark(X)) f(proper(X)) -> proper(f(X)) g(proper(X)) -> proper(g(X)) h(proper(X)) -> proper(h(X)) ok(f(X)) -> f(ok(X)) ok(g(X)) -> g(ok(X)) ok(h(X)) -> h(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 4. This implies Q-termination of R. The following rules were used to construct the certificate: f(active(X)) -> f(h(g(mark(X)))) f(active(X)) -> active(f(X)) h(active(X)) -> active(h(X)) mark(f(X)) -> f(mark(X)) mark(h(X)) -> h(mark(X)) f(proper(X)) -> proper(f(X)) g(proper(X)) -> proper(g(X)) h(proper(X)) -> proper(h(X)) ok(f(X)) -> f(ok(X)) ok(g(X)) -> g(ok(X)) ok(h(X)) -> h(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: 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75 Node 36 is start node and node 37 is final node. Those nodes are connected through the following edges: * 36 to 38 labelled f_1(0), g_1(0), h_1(0)* 36 to 41 labelled active_1(0), proper_1(0)* 36 to 40 labelled f_1(0), h_1(0)* 36 to 49 labelled f_1(1)* 36 to 52 labelled active_1(1), proper_1(1)* 36 to 69 labelled proper_1(2)* 37 to 37 labelled #_1(0)* 38 to 39 labelled h_1(0)* 38 to 37 labelled ok_1(0)* 38 to 42 labelled f_1(1), g_1(1), h_1(1)* 38 to 43 labelled active_1(1)* 38 to 55 labelled proper_1(1)* 38 to 56 labelled f_1(2)* 38 to 59 labelled active_1(2)* 38 to 73 labelled proper_1(3)* 38 to 74 labelled proper_1(2)* 39 to 40 labelled g_1(0)* 39 to 53 labelled proper_1(1)* 40 to 37 labelled mark_1(0)* 40 to 44 labelled f_1(1), h_1(1)* 40 to 43 labelled proper_1(1)* 40 to 59 labelled proper_1(2)* 41 to 37 labelled f_1(0), h_1(0), g_1(0), top_1(0)* 41 to 45 labelled f_1(1)* 41 to 48 labelled active_1(1), proper_1(1)* 41 to 65 labelled proper_1(2)* 42 to 37 labelled ok_1(1)* 42 to 42 labelled f_1(1), g_1(1), h_1(1)* 42 to 43 labelled active_1(1)* 42 to 56 labelled f_1(2)* 42 to 59 labelled active_1(2)* 42 to 73 labelled proper_1(3)* 42 to 74 labelled proper_1(2)* 43 to 37 labelled top_1(1)* 44 to 37 labelled mark_1(1)* 44 to 44 labelled f_1(1), h_1(1)* 44 to 43 labelled proper_1(1)* 44 to 59 labelled proper_1(2)* 45 to 46 labelled h_1(1)* 45 to 62 labelled proper_1(2)* 46 to 47 labelled g_1(1)* 46 to 60 labelled proper_1(2)* 47 to 37 labelled mark_1(1)* 47 to 44 labelled f_1(1), h_1(1)* 47 to 43 labelled proper_1(1)* 47 to 59 labelled proper_1(2)* 48 to 37 labelled f_1(1), h_1(1), g_1(1)* 48 to 45 labelled f_1(1)* 48 to 48 labelled active_1(1), proper_1(1)* 48 to 65 labelled proper_1(2)* 49 to 50 labelled h_1(1)* 49 to 66 labelled proper_1(2)* 50 to 51 labelled g_1(1)* 50 to 63 labelled proper_1(2)* 51 to 43 labelled mark_1(1)* 51 to 61 labelled proper_1(2)* 51 to 59 labelled mark_1(1)* 51 to 58 labelled f_1(2), h_1(2)* 51 to 70 labelled proper_1(3)* 52 to 43 labelled f_1(1), h_1(1)* 52 to 55 labelled f_1(1), g_1(1), h_1(1)* 52 to 59 labelled f_1(1), h_1(1)* 52 to 73 labelled f_1(1), g_1(1), h_1(1)* 52 to 74 labelled f_1(1), g_1(1), h_1(1)* 53 to 43 labelled g_1(1)* 53 to 59 labelled g_1(1)* 55 to 53 labelled h_1(1)* 56 to 57 labelled h_1(2)* 56 to 71 labelled proper_1(3)* 57 to 58 labelled g_1(2)* 57 to 68 labelled proper_1(3)* 58 to 43 labelled mark_1(2)* 58 to 59 labelled mark_1(2)* 58 to 61 labelled proper_1(2)* 58 to 67 labelled f_1(3), h_1(3)* 58 to 70 labelled proper_1(3)* 58 to 75 labelled proper_1(4)* 59 to 43 labelled f_1(2), h_1(2)* 59 to 59 labelled f_1(2), h_1(2)* 60 to 43 labelled g_1(2)* 60 to 59 labelled g_1(2)* 61 to 37 labelled top_1(2)* 62 to 60 labelled h_1(2)* 63 to 61 labelled g_1(2)* 63 to 70 labelled g_1(2)* 65 to 62 labelled f_1(2)* 66 to 63 labelled h_1(2)* 67 to 43 labelled mark_1(3)* 67 to 59 labelled mark_1(3)* 67 to 61 labelled proper_1(2)* 67 to 67 labelled f_1(3), h_1(3)* 67 to 70 labelled proper_1(3)* 67 to 75 labelled proper_1(4)* 68 to 61 labelled g_1(3)* 68 to 70 labelled g_1(3)* 68 to 75 labelled g_1(3)* 69 to 66 labelled f_1(2)* 70 to 61 labelled f_1(3), h_1(3)* 70 to 70 labelled f_1(3), h_1(3)* 70 to 75 labelled f_1(3), h_1(3)* 71 to 68 labelled h_1(3)* 73 to 71 labelled f_1(3)* 74 to 73 labelled f_1(2), g_1(2), h_1(2)* 74 to 74 labelled f_1(2), g_1(2), h_1(2)* 75 to 70 labelled f_1(4), h_1(4)* 75 to 75 labelled f_1(4), h_1(4) ---------------------------------------- (4) YES