30.57/8.57 YES 30.62/8.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 30.62/8.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.62/8.58 30.62/8.58 30.62/8.58 Termination w.r.t. Q of the given QTRS could be proven: 30.62/8.58 30.62/8.58 (0) QTRS 30.62/8.58 (1) QTRSRRRProof [EQUIVALENT, 75 ms] 30.62/8.58 (2) QTRS 30.62/8.58 (3) DependencyPairsProof [EQUIVALENT, 25 ms] 30.62/8.58 (4) QDP 30.62/8.58 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 30.62/8.58 (6) AND 30.62/8.58 (7) QDP 30.62/8.58 (8) MNOCProof [EQUIVALENT, 0 ms] 30.62/8.58 (9) QDP 30.62/8.58 (10) UsableRulesProof [EQUIVALENT, 0 ms] 30.62/8.58 (11) QDP 30.62/8.58 (12) QReductionProof [EQUIVALENT, 0 ms] 30.62/8.58 (13) QDP 30.62/8.58 (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.62/8.58 (15) YES 30.62/8.58 (16) QDP 30.62/8.58 (17) MNOCProof [EQUIVALENT, 0 ms] 30.62/8.58 (18) QDP 30.62/8.58 (19) UsableRulesProof [EQUIVALENT, 0 ms] 30.62/8.58 (20) QDP 30.62/8.58 (21) QReductionProof [EQUIVALENT, 0 ms] 30.62/8.58 (22) QDP 30.62/8.58 (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.62/8.58 (24) YES 30.62/8.58 (25) QDP 30.62/8.58 (26) UsableRulesProof [EQUIVALENT, 0 ms] 30.62/8.58 (27) QDP 30.62/8.58 (28) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.62/8.58 (29) YES 30.62/8.58 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (0) 30.62/8.58 Obligation: 30.62/8.58 Q restricted rewrite system: 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a12(x1)) -> x1 30.62/8.58 a13(a13(x1)) -> x1 30.62/8.58 a14(a14(x1)) -> x1 30.62/8.58 a15(a15(x1)) -> x1 30.62/8.58 a16(a16(x1)) -> x1 30.62/8.58 a23(a23(x1)) -> x1 30.62/8.58 a24(a24(x1)) -> x1 30.62/8.58 a25(a25(x1)) -> x1 30.62/8.58 a26(a26(x1)) -> x1 30.62/8.58 a34(a34(x1)) -> x1 30.62/8.58 a35(a35(x1)) -> x1 30.62/8.58 a36(a36(x1)) -> x1 30.62/8.58 a45(a45(x1)) -> x1 30.62/8.58 a46(a46(x1)) -> x1 30.62/8.58 a56(a56(x1)) -> x1 30.62/8.58 a13(x1) -> a12(a23(a12(x1))) 30.62/8.58 a14(x1) -> a12(a23(a34(a23(a12(x1))))) 30.62/8.58 a15(x1) -> a12(a23(a34(a45(a34(a23(a12(x1))))))) 30.62/8.58 a16(x1) -> a12(a23(a34(a45(a56(a45(a34(a23(a12(x1))))))))) 30.62/8.58 a24(x1) -> a23(a34(a23(x1))) 30.62/8.58 a25(x1) -> a23(a34(a45(a34(a23(x1))))) 30.62/8.58 a26(x1) -> a23(a34(a45(a56(a45(a34(a23(x1))))))) 30.62/8.58 a35(x1) -> a34(a45(a34(x1))) 30.62/8.58 a36(x1) -> a34(a45(a56(a45(a34(x1))))) 30.62/8.58 a46(x1) -> a45(a56(a45(x1))) 30.62/8.58 a12(a23(a12(a23(a12(a23(x1)))))) -> x1 30.62/8.58 a23(a34(a23(a34(a23(a34(x1)))))) -> x1 30.62/8.58 a34(a45(a34(a45(a34(a45(x1)))))) -> x1 30.62/8.58 a45(a56(a45(a56(a45(a56(x1)))))) -> x1 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 Q is empty. 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (1) QTRSRRRProof (EQUIVALENT) 30.62/8.58 Used ordering: 30.62/8.58 Polynomial interpretation [POLO]: 30.62/8.58 30.62/8.58 POL(a12(x_1)) = 1 + x_1 30.62/8.58 POL(a13(x_1)) = 4 + x_1 30.62/8.58 POL(a14(x_1)) = 6 + x_1 30.62/8.58 POL(a15(x_1)) = 8 + x_1 30.62/8.58 POL(a16(x_1)) = 10 + x_1 30.62/8.58 POL(a23(x_1)) = 1 + x_1 30.62/8.58 POL(a24(x_1)) = 4 + x_1 30.62/8.58 POL(a25(x_1)) = 6 + x_1 30.62/8.58 POL(a26(x_1)) = 8 + x_1 30.62/8.58 POL(a34(x_1)) = 1 + x_1 30.62/8.58 POL(a35(x_1)) = 4 + x_1 30.62/8.58 POL(a36(x_1)) = 6 + x_1 30.62/8.58 POL(a45(x_1)) = 1 + x_1 30.62/8.58 POL(a46(x_1)) = 4 + x_1 30.62/8.58 POL(a56(x_1)) = 1 + x_1 30.62/8.58 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 30.62/8.58 30.62/8.58 a12(a12(x1)) -> x1 30.62/8.58 a13(a13(x1)) -> x1 30.62/8.58 a14(a14(x1)) -> x1 30.62/8.58 a15(a15(x1)) -> x1 30.62/8.58 a16(a16(x1)) -> x1 30.62/8.58 a23(a23(x1)) -> x1 30.62/8.58 a24(a24(x1)) -> x1 30.62/8.58 a25(a25(x1)) -> x1 30.62/8.58 a26(a26(x1)) -> x1 30.62/8.58 a34(a34(x1)) -> x1 30.62/8.58 a35(a35(x1)) -> x1 30.62/8.58 a36(a36(x1)) -> x1 30.62/8.58 a45(a45(x1)) -> x1 30.62/8.58 a46(a46(x1)) -> x1 30.62/8.58 a56(a56(x1)) -> x1 30.62/8.58 a13(x1) -> a12(a23(a12(x1))) 30.62/8.58 a14(x1) -> a12(a23(a34(a23(a12(x1))))) 30.62/8.58 a15(x1) -> a12(a23(a34(a45(a34(a23(a12(x1))))))) 30.62/8.58 a16(x1) -> a12(a23(a34(a45(a56(a45(a34(a23(a12(x1))))))))) 30.62/8.58 a24(x1) -> a23(a34(a23(x1))) 30.62/8.58 a25(x1) -> a23(a34(a45(a34(a23(x1))))) 30.62/8.58 a26(x1) -> a23(a34(a45(a56(a45(a34(a23(x1))))))) 30.62/8.58 a35(x1) -> a34(a45(a34(x1))) 30.62/8.58 a36(x1) -> a34(a45(a56(a45(a34(x1))))) 30.62/8.58 a46(x1) -> a45(a56(a45(x1))) 30.62/8.58 a12(a23(a12(a23(a12(a23(x1)))))) -> x1 30.62/8.58 a23(a34(a23(a34(a23(a34(x1)))))) -> x1 30.62/8.58 a34(a45(a34(a45(a34(a45(x1)))))) -> x1 30.62/8.58 a45(a56(a45(a56(a45(a56(x1)))))) -> x1 30.62/8.58 30.62/8.58 30.62/8.58 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (2) 30.62/8.58 Obligation: 30.62/8.58 Q restricted rewrite system: 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 Q is empty. 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (3) DependencyPairsProof (EQUIVALENT) 30.62/8.58 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (4) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A12(a34(x1)) -> A34(a12(x1)) 30.62/8.58 A12(a34(x1)) -> A12(x1) 30.62/8.58 A12(a45(x1)) -> A12(x1) 30.62/8.58 A12(a56(x1)) -> A12(x1) 30.62/8.58 A23(a45(x1)) -> A23(x1) 30.62/8.58 A23(a56(x1)) -> A23(x1) 30.62/8.58 A34(a56(x1)) -> A34(x1) 30.62/8.58 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 Q is empty. 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (5) DependencyGraphProof (EQUIVALENT) 30.62/8.58 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 1 less node. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (6) 30.62/8.58 Complex Obligation (AND) 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (7) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A34(a56(x1)) -> A34(x1) 30.62/8.58 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 Q is empty. 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (8) MNOCProof (EQUIVALENT) 30.62/8.58 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (9) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A34(a56(x1)) -> A34(x1) 30.62/8.58 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 The set Q consists of the following terms: 30.62/8.58 30.62/8.58 a12(a34(x0)) 30.62/8.58 a12(a45(x0)) 30.62/8.58 a12(a56(x0)) 30.62/8.58 a23(a45(x0)) 30.62/8.58 a23(a56(x0)) 30.62/8.58 a34(a56(x0)) 30.62/8.58 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (10) UsableRulesProof (EQUIVALENT) 30.62/8.58 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (11) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A34(a56(x1)) -> A34(x1) 30.62/8.58 30.62/8.58 R is empty. 30.62/8.58 The set Q consists of the following terms: 30.62/8.58 30.62/8.58 a12(a34(x0)) 30.62/8.58 a12(a45(x0)) 30.62/8.58 a12(a56(x0)) 30.62/8.58 a23(a45(x0)) 30.62/8.58 a23(a56(x0)) 30.62/8.58 a34(a56(x0)) 30.62/8.58 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (12) QReductionProof (EQUIVALENT) 30.62/8.58 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 30.62/8.58 30.62/8.58 a12(a34(x0)) 30.62/8.58 a12(a45(x0)) 30.62/8.58 a12(a56(x0)) 30.62/8.58 a23(a45(x0)) 30.62/8.58 a23(a56(x0)) 30.62/8.58 a34(a56(x0)) 30.62/8.58 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (13) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A34(a56(x1)) -> A34(x1) 30.62/8.58 30.62/8.58 R is empty. 30.62/8.58 Q is empty. 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (14) QDPSizeChangeProof (EQUIVALENT) 30.62/8.58 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 30.62/8.58 30.62/8.58 From the DPs we obtained the following set of size-change graphs: 30.62/8.58 *A34(a56(x1)) -> A34(x1) 30.62/8.58 The graph contains the following edges 1 > 1 30.62/8.58 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (15) 30.62/8.58 YES 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (16) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A23(a56(x1)) -> A23(x1) 30.62/8.58 A23(a45(x1)) -> A23(x1) 30.62/8.58 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 Q is empty. 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (17) MNOCProof (EQUIVALENT) 30.62/8.58 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (18) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A23(a56(x1)) -> A23(x1) 30.62/8.58 A23(a45(x1)) -> A23(x1) 30.62/8.58 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 The set Q consists of the following terms: 30.62/8.58 30.62/8.58 a12(a34(x0)) 30.62/8.58 a12(a45(x0)) 30.62/8.58 a12(a56(x0)) 30.62/8.58 a23(a45(x0)) 30.62/8.58 a23(a56(x0)) 30.62/8.58 a34(a56(x0)) 30.62/8.58 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (19) UsableRulesProof (EQUIVALENT) 30.62/8.58 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (20) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A23(a56(x1)) -> A23(x1) 30.62/8.58 A23(a45(x1)) -> A23(x1) 30.62/8.58 30.62/8.58 R is empty. 30.62/8.58 The set Q consists of the following terms: 30.62/8.58 30.62/8.58 a12(a34(x0)) 30.62/8.58 a12(a45(x0)) 30.62/8.58 a12(a56(x0)) 30.62/8.58 a23(a45(x0)) 30.62/8.58 a23(a56(x0)) 30.62/8.58 a34(a56(x0)) 30.62/8.58 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (21) QReductionProof (EQUIVALENT) 30.62/8.58 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 30.62/8.58 30.62/8.58 a12(a34(x0)) 30.62/8.58 a12(a45(x0)) 30.62/8.58 a12(a56(x0)) 30.62/8.58 a23(a45(x0)) 30.62/8.58 a23(a56(x0)) 30.62/8.58 a34(a56(x0)) 30.62/8.58 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (22) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A23(a56(x1)) -> A23(x1) 30.62/8.58 A23(a45(x1)) -> A23(x1) 30.62/8.58 30.62/8.58 R is empty. 30.62/8.58 Q is empty. 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (23) QDPSizeChangeProof (EQUIVALENT) 30.62/8.58 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 30.62/8.58 30.62/8.58 From the DPs we obtained the following set of size-change graphs: 30.62/8.58 *A23(a56(x1)) -> A23(x1) 30.62/8.58 The graph contains the following edges 1 > 1 30.62/8.58 30.62/8.58 30.62/8.58 *A23(a45(x1)) -> A23(x1) 30.62/8.58 The graph contains the following edges 1 > 1 30.62/8.58 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (24) 30.62/8.58 YES 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (25) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A12(a45(x1)) -> A12(x1) 30.62/8.58 A12(a34(x1)) -> A12(x1) 30.62/8.58 A12(a56(x1)) -> A12(x1) 30.62/8.58 30.62/8.58 The TRS R consists of the following rules: 30.62/8.58 30.62/8.58 a12(a34(x1)) -> a34(a12(x1)) 30.62/8.58 a12(a45(x1)) -> a45(a12(x1)) 30.62/8.58 a12(a56(x1)) -> a56(a12(x1)) 30.62/8.58 a23(a45(x1)) -> a45(a23(x1)) 30.62/8.58 a23(a56(x1)) -> a56(a23(x1)) 30.62/8.58 a34(a56(x1)) -> a56(a34(x1)) 30.62/8.58 30.62/8.58 Q is empty. 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (26) UsableRulesProof (EQUIVALENT) 30.62/8.58 We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (27) 30.62/8.58 Obligation: 30.62/8.58 Q DP problem: 30.62/8.58 The TRS P consists of the following rules: 30.62/8.58 30.62/8.58 A12(a45(x1)) -> A12(x1) 30.62/8.58 A12(a34(x1)) -> A12(x1) 30.62/8.58 A12(a56(x1)) -> A12(x1) 30.62/8.58 30.62/8.58 R is empty. 30.62/8.58 Q is empty. 30.62/8.58 We have to consider all minimal (P,Q,R)-chains. 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (28) QDPSizeChangeProof (EQUIVALENT) 30.62/8.58 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 30.62/8.58 30.62/8.58 From the DPs we obtained the following set of size-change graphs: 30.62/8.58 *A12(a45(x1)) -> A12(x1) 30.62/8.58 The graph contains the following edges 1 > 1 30.62/8.58 30.62/8.58 30.62/8.58 *A12(a34(x1)) -> A12(x1) 30.62/8.58 The graph contains the following edges 1 > 1 30.62/8.58 30.62/8.58 30.62/8.58 *A12(a56(x1)) -> A12(x1) 30.62/8.58 The graph contains the following edges 1 > 1 30.62/8.58 30.62/8.58 30.62/8.58 ---------------------------------------- 30.62/8.58 30.62/8.58 (29) 30.62/8.58 YES 30.69/8.64 EOF