YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms] (2) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) Q is empty. ---------------------------------------- (1) 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(a(x1)) -> a(b(b(b(x1)))) b(a(x1)) -> b(b(c(x1))) a(b(b(c(x1)))) -> a(a(a(b(x1)))) The certificate found is represented by the following graph. The certificate consists of the following enumerated nodes: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48 Node 1 is start node and node 2 is final node. Those nodes are connected through the following edges: * 1 to 3 labelled a_1(0)* 1 to 6 labelled b_1(0)* 1 to 8 labelled a_1(0)* 1 to 11 labelled a_1(1)* 2 to 2 labelled #_1(0)* 3 to 4 labelled b_1(0)* 4 to 5 labelled b_1(0)* 5 to 2 labelled b_1(0)* 5 to 14 labelled b_1(1)* 6 to 7 labelled b_1(0)* 7 to 2 labelled c_1(0)* 8 to 9 labelled a_1(0)* 8 to 16 labelled a_1(1)* 9 to 10 labelled a_1(0)* 9 to 19 labelled a_1(1)* 9 to 24 labelled a_1(2)* 10 to 2 labelled b_1(0)* 10 to 14 labelled b_1(1)* 11 to 12 labelled b_1(1)* 12 to 13 labelled b_1(1)* 13 to 9 labelled b_1(1)* 13 to 16 labelled b_1(1)* 13 to 22 labelled b_1(2)* 14 to 15 labelled b_1(1)* 15 to 2 labelled c_1(1)* 16 to 17 labelled b_1(1)* 17 to 18 labelled b_1(1)* 17 to 10 labelled c_1(1)* 18 to 10 labelled b_1(1)* 18 to 19 labelled b_1(1)* 18 to 24 labelled b_1(1)* 18 to 30 labelled b_1(2)* 19 to 20 labelled a_1(1)* 19 to 27 labelled a_1(2)* 20 to 21 labelled a_1(1)* 20 to 19 labelled a_1(1)* 20 to 24 labelled a_1(2)* 20 to 32 labelled a_1(2)* 20 to 39 labelled a_1(3)* 21 to 2 labelled b_1(1)* 21 to 14 labelled b_1(1)* 22 to 23 labelled b_1(2)* 23 to 19 labelled c_1(2)* 23 to 24 labelled c_1(2)* 24 to 25 labelled b_1(2)* 25 to 26 labelled b_1(2)* 26 to 20 labelled b_1(2)* 26 to 27 labelled b_1(2)* 26 to 22 labelled b_1(2)* 26 to 37 labelled b_1(3)* 27 to 28 labelled b_1(2)* 28 to 29 labelled b_1(2)* 28 to 21 labelled c_1(2)* 29 to 21 labelled b_1(2)* 29 to 19 labelled b_1(2)* 29 to 30 labelled b_1(2)* 29 to 35 labelled b_1(3)* 29 to 24 labelled b_1(2)* 29 to 32 labelled b_1(2)* 29 to 39 labelled b_1(2)* 30 to 31 labelled b_1(2)* 31 to 20 labelled c_1(2)* 31 to 27 labelled c_1(2)* 32 to 33 labelled a_1(2)* 32 to 42 labelled a_1(3)* 32 to 27 labelled a_1(2)* 33 to 34 labelled a_1(2)* 33 to 19 labelled a_1(1)* 33 to 24 labelled a_1(2)* 33 to 32 labelled a_1(2)* 33 to 39 labelled a_1(3)* 34 to 2 labelled b_1(2)* 34 to 14 labelled b_1(1)* 35 to 36 labelled b_1(3)* 36 to 27 labelled c_1(3)* 36 to 33 labelled c_1(3)* 36 to 42 labelled c_1(3)* 37 to 38 labelled b_1(3)* 38 to 24 labelled c_1(3)* 38 to 32 labelled c_1(3)* 38 to 39 labelled c_1(3)* 39 to 40 labelled b_1(3)* 40 to 41 labelled b_1(3)* 41 to 33 labelled b_1(3)* 41 to 42 labelled b_1(3)* 41 to 22 labelled b_1(2)* 41 to 27 labelled b_1(3)* 41 to 37 labelled b_1(3)* 41 to 47 labelled b_1(4)* 42 to 43 labelled b_1(3)* 43 to 44 labelled b_1(3)* 43 to 34 labelled c_1(3)* 44 to 34 labelled b_1(3)* 44 to 24 labelled b_1(3)* 44 to 32 labelled b_1(3)* 44 to 35 labelled b_1(3)* 44 to 45 labelled b_1(4)* 44 to 39 labelled b_1(3)* 45 to 46 labelled b_1(4)* 46 to 42 labelled c_1(4)* 47 to 48 labelled b_1(4)* 48 to 39 labelled c_1(4) ---------------------------------------- (2) YES