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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRS Reverse [EQUIVALENT, 0 ms] (2) QTRS (3) RFCMatchBoundsTRSProof [EQUIVALENT, 28 ms] (4) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: c(c(x1)) -> a(b(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> a(c(b(x1))) a(c(a(x1))) -> a(c(c(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: c(c(x1)) -> b(a(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> b(c(a(x1))) a(c(a(x1))) -> c(c(a(x1))) 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: c(c(x1)) -> b(a(x1)) b(x1) -> a(a(x1)) b(b(b(x1))) -> b(c(a(x1))) a(c(a(x1))) -> c(c(a(x1))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 9, 10, 11, 12, 13, 14, 15, 16, 17, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 71, 72, 75, 76, 77, 78 Node 9 is start node and node 10 is final node. Those nodes are connected through the following edges: * 9 to 11 labelled b_1(0), a_1(0)* 9 to 12 labelled b_1(0), c_1(0)* 9 to 14 labelled a_1(1)* 9 to 15 labelled b_1(1)* 9 to 29 labelled a_1(2)* 9 to 44 labelled c_1(1)* 9 to 46 labelled c_1(2)* 9 to 48 labelled b_1(1)* 9 to 50 labelled b_1(2)* 9 to 51 labelled b_1(3)* 9 to 52 labelled a_1(3)* 9 to 53 labelled a_1(4)* 10 to 10 labelled #_1(0)* 11 to 10 labelled a_1(0)* 11 to 16 labelled c_1(1)* 11 to 30 labelled b_1(2)* 11 to 32 labelled a_1(3)* 12 to 13 labelled c_1(0)* 12 to 31 labelled b_1(1)* 12 to 33 labelled a_1(2)* 12 to 54 labelled b_1(2)* 12 to 62 labelled a_1(3)* 13 to 10 labelled a_1(0)* 13 to 16 labelled c_1(1)* 13 to 30 labelled b_1(2)* 13 to 32 labelled a_1(3)* 14 to 11 labelled a_1(1)* 14 to 12 labelled a_1(1)* 14 to 16 labelled c_1(1)* 14 to 30 labelled b_1(2)* 14 to 32 labelled a_1(3)* 14 to 46 labelled c_1(2)* 14 to 51 labelled b_1(3)* 14 to 53 labelled a_1(4)* 15 to 13 labelled a_1(1)* 15 to 46 labelled c_1(2)* 15 to 51 labelled b_1(3)* 15 to 53 labelled a_1(4)* 16 to 17 labelled c_1(1)* 16 to 34 labelled b_1(2)* 16 to 37 labelled a_1(3)* 16 to 56 labelled b_1(3)* 16 to 63 labelled a_1(4)* 17 to 10 labelled a_1(1)* 17 to 16 labelled c_1(1)* 17 to 30 labelled b_1(2)* 17 to 32 labelled a_1(3), a_1(1)* 29 to 15 labelled a_1(2)* 29 to 48 labelled a_1(2)* 29 to 58 labelled c_1(2)* 29 to 64 labelled b_1(3)* 29 to 54 labelled c_1(2)* 29 to 68 labelled a_1(4)* 30 to 17 labelled a_1(2)* 30 to 46 labelled c_1(2)* 30 to 51 labelled b_1(3)* 30 to 53 labelled a_1(4)* 31 to 16 labelled a_1(1)* 31 to 35 labelled c_1(2)* 31 to 38 labelled b_1(3)* 31 to 43 labelled a_1(4)* 32 to 30 labelled a_1(3)* 33 to 31 labelled a_1(2)* 33 to 60 labelled c_1(3)* 33 to 65 labelled b_1(4)* 33 to 69 labelled a_1(5)* 34 to 16 labelled a_1(2)* 34 to 35 labelled c_1(2)* 34 to 38 labelled b_1(3)* 34 to 43 labelled a_1(4)* 35 to 36 labelled c_1(2)* 35 to 34 labelled b_1(2)* 35 to 37 labelled a_1(3)* 35 to 56 labelled b_1(3)* 35 to 63 labelled a_1(4)* 36 to 10 labelled a_1(2)* 36 to 16 labelled c_1(1)* 36 to 30 labelled b_1(2)* 36 to 32 labelled a_1(2), a_1(3)* 37 to 34 labelled a_1(3)* 37 to 60 labelled c_1(3)* 37 to 65 labelled b_1(4)* 37 to 69 labelled a_1(5)* 38 to 36 labelled a_1(3)* 38 to 46 labelled c_1(2)* 38 to 51 labelled b_1(3)* 38 to 53 labelled a_1(4)* 43 to 38 labelled a_1(4)* 44 to 45 labelled c_1(1)* 45 to 37 labelled a_1(1)* 45 to 63 labelled a_1(1)* 46 to 47 labelled c_1(2)* 47 to 37 labelled a_1(2)* 47 to 63 labelled a_1(2)* 48 to 49 labelled c_1(1)* 49 to 38 labelled a_1(1)* 49 to 51 labelled a_1(1)* 50 to 45 labelled a_1(2)* 51 to 47 labelled a_1(3)* 52 to 50 labelled a_1(3)* 53 to 51 labelled a_1(4)* 54 to 55 labelled c_1(2)* 55 to 51 labelled a_1(2)* 56 to 57 labelled c_1(3)* 57 to 51 labelled a_1(3)* 58 to 59 labelled c_1(2)* 59 to 38 labelled a_1(2)* 60 to 61 labelled c_1(3)* 61 to 37 labelled a_1(3)* 61 to 63 labelled a_1(3)* 62 to 54 labelled a_1(3)* 62 to 56 labelled c_1(3)* 62 to 75 labelled b_1(4)* 62 to 77 labelled a_1(5)* 63 to 56 labelled a_1(4)* 63 to 71 labelled c_1(4)* 63 to 76 labelled b_1(5)* 63 to 78 labelled a_1(6)* 64 to 59 labelled a_1(3)* 64 to 55 labelled a_1(3)* 65 to 61 labelled a_1(4)* 68 to 64 labelled a_1(4)* 69 to 65 labelled a_1(5)* 71 to 72 labelled c_1(4)* 72 to 51 labelled a_1(4)* 75 to 57 labelled a_1(4)* 76 to 72 labelled a_1(5)* 77 to 75 labelled a_1(5)* 78 to 76 labelled a_1(6) ---------------------------------------- (4) YES