YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 0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: a(c(a(x1))) -> c(a(c(x1))) a(a(b(x1))) -> a(d(b(x1))) a(b(x1)) -> b(a(a(x1))) d(d(x1)) -> a(d(b(x1))) b(b(x1)) -> b(c(x1)) a(d(c(x1))) -> c(a(x1)) b(c(x1)) -> a(a(a(x1))) 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(c(a(x1))) -> c(a(c(x1))) b(a(a(x1))) -> b(d(a(x1))) b(a(x1)) -> a(a(b(x1))) d(d(x1)) -> b(d(a(x1))) b(b(x1)) -> c(b(x1)) c(d(a(x1))) -> a(c(x1)) c(b(x1)) -> a(a(a(x1))) 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: a(c(a(x1))) -> c(a(c(x1))) b(a(a(x1))) -> b(d(a(x1))) b(a(x1)) -> a(a(b(x1))) d(d(x1)) -> b(d(a(x1))) b(b(x1)) -> c(b(x1)) c(d(a(x1))) -> a(c(x1)) c(b(x1)) -> a(a(a(x1))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 124, 125 Node 35 is start node and node 36 is final node. Those nodes are connected through the following edges: * 35 to 37 labelled c_1(0)* 35 to 39 labelled b_1(0)* 35 to 41 labelled a_1(0)* 35 to 42 labelled c_1(0)* 35 to 38 labelled a_1(0), c_1(1)* 35 to 43 labelled a_1(1)* 35 to 52 labelled c_1(1)* 35 to 58 labelled c_1(2)* 36 to 36 labelled #_1(0)* 37 to 38 labelled a_1(0), c_1(1)* 37 to 52 labelled c_1(1)* 38 to 36 labelled c_1(0)* 38 to 45 labelled a_1(1)* 38 to 43 labelled a_1(1)* 38 to 38 labelled c_1(1)* 38 to 54 labelled c_1(2)* 38 to 58 labelled c_1(2)* 38 to 56 labelled c_1(2)* 38 to 60 labelled c_1(2)* 39 to 40 labelled d_1(0)* 40 to 36 labelled a_1(0)* 40 to 38 labelled c_1(1)* 41 to 42 labelled a_1(0)* 41 to 52 labelled c_1(1)* 42 to 36 labelled b_1(0), a_1(0)* 42 to 46 labelled b_1(1)* 42 to 48 labelled a_1(1)* 42 to 49 labelled c_1(1)* 42 to 38 labelled c_1(1)* 42 to 50 labelled a_1(2)* 42 to 60 labelled c_1(2)* 42 to 62 labelled c_1(3)* 43 to 44 labelled a_1(1)* 43 to 54 labelled c_1(2)* 44 to 36 labelled a_1(1)* 44 to 46 labelled a_1(1)* 44 to 38 labelled c_1(1)* 45 to 36 labelled c_1(1)* 45 to 45 labelled a_1(1)* 45 to 43 labelled a_1(1)* 45 to 53 labelled c_1(1)* 45 to 38 labelled c_1(1)* 45 to 54 labelled c_1(2)* 45 to 58 labelled c_1(2)* 45 to 55 labelled c_1(1)* 45 to 59 labelled c_1(1)* 45 to 56 labelled c_1(2)* 45 to 57 labelled c_1(1)* 45 to 60 labelled c_1(2)* 46 to 47 labelled d_1(1)* 47 to 36 labelled a_1(1)* 47 to 38 labelled c_1(1)* 48 to 49 labelled a_1(1)* 48 to 56 labelled c_1(2)* 49 to 36 labelled b_1(1)* 49 to 46 labelled b_1(1)* 49 to 48 labelled a_1(1)* 49 to 49 labelled c_1(1)* 49 to 50 labelled a_1(2)* 49 to 60 labelled c_1(2)* 49 to 62 labelled c_1(3)* 50 to 51 labelled a_1(2)* 50 to 54 labelled c_1(2)* 51 to 36 labelled a_1(2)* 51 to 38 labelled c_1(1)* 51 to 46 labelled a_1(2)* 52 to 53 labelled a_1(1)* 52 to 54 labelled c_1(2)* 53 to 45 labelled c_1(1)* 53 to 43 labelled c_1(1)* 53 to 48 labelled c_1(1)* 53 to 50 labelled c_1(1)* 53 to 61 labelled c_1(1)* 53 to 63 labelled c_1(1)* 54 to 55 labelled a_1(2)* 54 to 54 labelled c_1(2)* 54 to 38 labelled c_1(1)* 54 to 56 labelled c_1(2)* 54 to 64 labelled c_1(3)* 54 to 62 labelled c_1(3)* 55 to 45 labelled c_1(2), a_1(1)* 55 to 43 labelled c_1(2), a_1(1)* 55 to 44 labelled c_1(2)* 55 to 49 labelled c_1(2)* 55 to 36 labelled c_1(2)* 55 to 46 labelled c_1(2)* 55 to 50 labelled a_1(2)* 55 to 59 labelled c_1(2)* 55 to 51 labelled c_1(2)* 55 to 55 labelled a_1(2)* 55 to 38 labelled c_1(1)* 55 to 54 labelled c_1(2)* 55 to 58 labelled c_1(2)* 55 to 62 labelled c_1(3)* 55 to 56 labelled c_1(2)* 55 to 64 labelled c_1(3)* 55 to 60 labelled c_1(2)* 56 to 57 labelled a_1(2)* 56 to 54 labelled c_1(2)* 56 to 66 labelled c_1(3)* 57 to 48 labelled c_1(2)* 57 to 50 labelled c_1(2)* 57 to 61 labelled c_1(2)* 57 to 63 labelled c_1(2)* 58 to 59 labelled a_1(2)* 58 to 54 labelled c_1(2)* 58 to 64 labelled c_1(3)* 58 to 62 labelled c_1(3)* 59 to 55 labelled c_1(2)* 60 to 61 labelled a_1(2)* 61 to 57 labelled c_1(2)* 62 to 63 labelled a_1(3)* 62 to 54 labelled c_1(2)* 62 to 64 labelled c_1(3)* 62 to 62 labelled c_1(3)* 63 to 55 labelled c_1(3)* 64 to 65 labelled a_1(3)* 64 to 66 labelled c_1(3)* 64 to 38 labelled c_1(1)* 64 to 54 labelled c_1(2)* 64 to 62 labelled c_1(3)* 64 to 64 labelled c_1(3)* 64 to 124 labelled c_1(4)* 65 to 50 labelled c_1(3)* 65 to 36 labelled c_1(3)* 65 to 46 labelled c_1(3)* 65 to 45 labelled a_1(1)* 65 to 43 labelled a_1(1)* 65 to 59 labelled c_1(3)* 65 to 57 labelled c_1(3)* 65 to 63 labelled c_1(3)* 65 to 65 labelled c_1(3)* 65 to 55 labelled a_1(2)* 65 to 38 labelled c_1(1)* 65 to 54 labelled c_1(2)* 65 to 58 labelled c_1(2)* 65 to 62 labelled c_1(3)* 65 to 64 labelled c_1(3)* 65 to 60 labelled c_1(2)* 65 to 56 labelled c_1(2)* 65 to 61 labelled c_1(3)* 66 to 67 labelled a_1(3)* 66 to 64 labelled c_1(3)* 67 to 51 labelled c_1(3)* 124 to 125 labelled a_1(4)* 124 to 54 labelled c_1(2)* 124 to 62 labelled c_1(3)* 125 to 63 labelled c_1(4)* 125 to 65 labelled c_1(4) ---------------------------------------- (4) YES