4.17/1.75 YES 4.49/1.78 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 4.49/1.78 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.49/1.78 4.49/1.78 4.49/1.78 Termination w.r.t. Q of the given QTRS could be proven: 4.49/1.78 4.49/1.78 (0) QTRS 4.49/1.78 (1) QTRS Reverse [EQUIVALENT, 0 ms] 4.49/1.78 (2) QTRS 4.49/1.78 (3) RFCMatchBoundsTRSProof [EQUIVALENT, 1 ms] 4.49/1.78 (4) YES 4.49/1.78 4.49/1.78 4.49/1.78 ---------------------------------------- 4.49/1.78 4.49/1.78 (0) 4.49/1.78 Obligation: 4.49/1.78 Q restricted rewrite system: 4.49/1.78 The TRS R consists of the following rules: 4.49/1.78 4.49/1.78 c(c(x1)) -> a(b(x1)) 4.49/1.78 b(x1) -> a(a(x1)) 4.49/1.78 b(b(b(x1))) -> a(c(b(x1))) 4.49/1.78 a(c(a(x1))) -> a(c(c(x1))) 4.49/1.78 4.49/1.78 Q is empty. 4.49/1.78 4.49/1.78 ---------------------------------------- 4.49/1.78 4.49/1.78 (1) QTRS Reverse (EQUIVALENT) 4.49/1.78 We applied the QTRS Reverse Processor [REVERSE]. 4.49/1.78 ---------------------------------------- 4.49/1.78 4.49/1.78 (2) 4.49/1.78 Obligation: 4.49/1.78 Q restricted rewrite system: 4.49/1.78 The TRS R consists of the following rules: 4.49/1.78 4.49/1.78 c(c(x1)) -> b(a(x1)) 4.49/1.78 b(x1) -> a(a(x1)) 4.49/1.78 b(b(b(x1))) -> b(c(a(x1))) 4.49/1.78 a(c(a(x1))) -> c(c(a(x1))) 4.49/1.78 4.49/1.78 Q is empty. 4.49/1.78 4.49/1.78 ---------------------------------------- 4.49/1.78 4.49/1.78 (3) RFCMatchBoundsTRSProof (EQUIVALENT) 4.49/1.78 Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 6. This implies Q-termination of R. 4.49/1.78 The following rules were used to construct the certificate: 4.49/1.78 4.49/1.78 c(c(x1)) -> b(a(x1)) 4.49/1.78 b(x1) -> a(a(x1)) 4.49/1.78 b(b(b(x1))) -> b(c(a(x1))) 4.49/1.78 a(c(a(x1))) -> c(c(a(x1))) 4.49/1.78 4.49/1.78 The certificate found is represented by the following graph. 4.49/1.78 The certificate consists of the following enumerated nodes: 4.49/1.78 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, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91 4.49/1.78 4.49/1.78 Node 42 is start node and node 43 is final node. 4.49/1.78 4.49/1.78 Those nodes are connected through the following edges: 4.49/1.78 4.49/1.78 * 42 to 44 labelled b_1(0), a_1(0)* 42 to 45 labelled b_1(0), c_1(0)* 42 to 47 labelled a_1(1)* 42 to 48 labelled b_1(1)* 42 to 51 labelled a_1(2)* 42 to 62 labelled c_1(1)* 42 to 64 labelled c_1(2)* 42 to 66 labelled b_1(1)* 42 to 68 labelled b_1(2)* 42 to 69 labelled b_1(3)* 42 to 70 labelled a_1(3)* 42 to 71 labelled a_1(4)* 43 to 43 labelled #_1(0)* 44 to 43 labelled a_1(0)* 44 to 49 labelled c_1(1)* 44 to 52 labelled b_1(2)* 44 to 54 labelled a_1(3)* 45 to 46 labelled c_1(0)* 45 to 53 labelled b_1(1)* 45 to 55 labelled a_1(2)* 45 to 72 labelled b_1(2)* 45 to 80 labelled a_1(3)* 46 to 43 labelled a_1(0)* 46 to 49 labelled c_1(1)* 46 to 52 labelled b_1(2)* 46 to 54 labelled a_1(3)* 47 to 44 labelled a_1(1)* 47 to 45 labelled a_1(1)* 47 to 49 labelled c_1(1)* 47 to 52 labelled b_1(2)* 47 to 54 labelled a_1(3)* 47 to 64 labelled c_1(2)* 47 to 69 labelled b_1(3)* 47 to 71 labelled a_1(4)* 48 to 46 labelled a_1(1)* 48 to 64 labelled c_1(2)* 48 to 69 labelled b_1(3)* 48 to 71 labelled a_1(4)* 49 to 50 labelled c_1(1)* 49 to 56 labelled b_1(2)* 49 to 59 labelled a_1(3)* 49 to 74 labelled b_1(3)* 49 to 81 labelled a_1(4)* 50 to 43 labelled a_1(1)* 50 to 49 labelled c_1(1)* 50 to 52 labelled b_1(2)* 50 to 54 labelled a_1(3), a_1(1)* 51 to 48 labelled a_1(2)* 51 to 66 labelled a_1(2)* 51 to 76 labelled c_1(2)* 51 to 82 labelled b_1(3)* 51 to 72 labelled c_1(2)* 51 to 84 labelled a_1(4)* 52 to 50 labelled a_1(2)* 52 to 64 labelled c_1(2)* 52 to 69 labelled b_1(3)* 52 to 71 labelled a_1(4)* 53 to 49 labelled a_1(1)* 53 to 57 labelled c_1(2)* 53 to 60 labelled b_1(3)* 53 to 61 labelled a_1(4)* 54 to 52 labelled a_1(3)* 55 to 53 labelled a_1(2)* 55 to 78 labelled c_1(3)* 55 to 83 labelled b_1(4)* 55 to 85 labelled a_1(5)* 56 to 49 labelled a_1(2)* 56 to 57 labelled c_1(2)* 56 to 60 labelled b_1(3)* 56 to 61 labelled a_1(4)* 57 to 58 labelled c_1(2)* 57 to 56 labelled b_1(2)* 57 to 59 labelled a_1(3)* 57 to 74 labelled b_1(3)* 57 to 81 labelled a_1(4)* 58 to 43 labelled a_1(2)* 58 to 49 labelled c_1(1)* 58 to 52 labelled b_1(2)* 58 to 54 labelled a_1(2), a_1(3)* 59 to 56 labelled a_1(3)* 59 to 78 labelled c_1(3)* 59 to 83 labelled b_1(4)* 59 to 85 labelled a_1(5)* 60 to 58 labelled a_1(3)* 60 to 64 labelled c_1(2)* 60 to 69 labelled b_1(3)* 60 to 71 labelled a_1(4)* 61 to 60 labelled a_1(4)* 62 to 63 labelled c_1(1)* 63 to 59 labelled a_1(1)* 63 to 81 labelled a_1(1)* 64 to 65 labelled c_1(2)* 65 to 59 labelled a_1(2)* 65 to 81 labelled a_1(2)* 66 to 67 labelled c_1(1)* 67 to 60 labelled a_1(1)* 67 to 69 labelled a_1(1)* 68 to 63 labelled a_1(2)* 69 to 65 labelled a_1(3)* 70 to 68 labelled a_1(3)* 71 to 69 labelled a_1(4)* 72 to 73 labelled c_1(2)* 73 to 69 labelled a_1(2)* 74 to 75 labelled c_1(3)* 75 to 69 labelled a_1(3)* 76 to 77 labelled c_1(2)* 77 to 60 labelled a_1(2)* 78 to 79 labelled c_1(3)* 79 to 59 labelled a_1(3)* 79 to 81 labelled a_1(3)* 80 to 72 labelled a_1(3)* 80 to 74 labelled c_1(3)* 80 to 88 labelled b_1(4)* 80 to 90 labelled a_1(5)* 81 to 74 labelled a_1(4)* 81 to 86 labelled c_1(4)* 81 to 89 labelled b_1(5)* 81 to 91 labelled a_1(6)* 82 to 77 labelled a_1(3)* 82 to 73 labelled a_1(3)* 83 to 79 labelled a_1(4)* 84 to 82 labelled a_1(4)* 85 to 83 labelled a_1(5)* 86 to 87 labelled c_1(4)* 87 to 69 labelled a_1(4)* 88 to 75 labelled a_1(4)* 89 to 87 labelled a_1(5)* 90 to 88 labelled a_1(5)* 91 to 89 labelled a_1(6) 4.49/1.78 4.49/1.78 4.49/1.78 ---------------------------------------- 4.49/1.78 4.49/1.78 (4) 4.49/1.78 YES 4.54/1.82 EOF