31.67/9.09 YES 32.38/9.25 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 32.38/9.25 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 32.38/9.25 32.38/9.25 32.38/9.25 Termination w.r.t. Q of the given QTRS could be proven: 32.38/9.25 32.38/9.25 (0) QTRS 32.38/9.25 (1) QTRSRRRProof [EQUIVALENT, 71 ms] 32.38/9.25 (2) QTRS 32.38/9.25 (3) DependencyPairsProof [EQUIVALENT, 28 ms] 32.38/9.25 (4) QDP 32.38/9.25 (5) DependencyGraphProof [EQUIVALENT, 3 ms] 32.38/9.25 (6) AND 32.38/9.25 (7) QDP 32.38/9.25 (8) UsableRulesProof [EQUIVALENT, 0 ms] 32.38/9.25 (9) QDP 32.38/9.25 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 32.38/9.25 (11) YES 32.38/9.25 (12) QDP 32.38/9.25 (13) MNOCProof [EQUIVALENT, 0 ms] 32.38/9.25 (14) QDP 32.38/9.25 (15) UsableRulesProof [EQUIVALENT, 0 ms] 32.38/9.25 (16) QDP 32.38/9.25 (17) QReductionProof [EQUIVALENT, 0 ms] 32.38/9.25 (18) QDP 32.38/9.25 (19) QDPOrderProof [EQUIVALENT, 18 ms] 32.38/9.25 (20) QDP 32.38/9.25 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 32.38/9.25 (22) TRUE 32.38/9.25 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (0) 32.38/9.25 Obligation: 32.38/9.25 Q restricted rewrite system: 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) 32.38/9.25 v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) 32.38/9.25 w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) 32.38/9.25 w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) 32.38/9.25 p(p(s(x1))) -> p(x1) 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 Q is empty. 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (1) QTRSRRRProof (EQUIVALENT) 32.38/9.25 Used ordering: 32.38/9.25 Polynomial interpretation [POLO]: 32.38/9.25 32.38/9.25 POL(0(x_1)) = x_1 32.38/9.25 POL(p(x_1)) = x_1 32.38/9.25 POL(s(x_1)) = x_1 32.38/9.25 POL(v(x_1)) = 1 + x_1 32.38/9.25 POL(w(x_1)) = 1 + x_1 32.38/9.25 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 32.38/9.25 32.38/9.25 v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) 32.38/9.25 w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) 32.38/9.25 32.38/9.25 32.38/9.25 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (2) 32.38/9.25 Obligation: 32.38/9.25 Q restricted rewrite system: 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) 32.38/9.25 w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) 32.38/9.25 p(p(s(x1))) -> p(x1) 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 Q is empty. 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (3) DependencyPairsProof (EQUIVALENT) 32.38/9.25 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (4) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 V(s(x1)) -> P(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1))))))))))))))))))) 32.38/9.25 V(s(x1)) -> P(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))) 32.38/9.25 V(s(x1)) -> W(p(p(s(s(p(s(p(s(x1))))))))) 32.38/9.25 V(s(x1)) -> P(p(s(s(p(s(p(s(x1)))))))) 32.38/9.25 V(s(x1)) -> P(s(s(p(s(p(s(x1))))))) 32.38/9.25 V(s(x1)) -> P(s(p(s(x1)))) 32.38/9.25 V(s(x1)) -> P(s(x1)) 32.38/9.25 W(s(x1)) -> P(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))) 32.38/9.25 W(s(x1)) -> P(s(s(v(p(p(s(s(s(p(p(s(s(x1))))))))))))) 32.38/9.25 W(s(x1)) -> V(p(p(s(s(s(p(p(s(s(x1)))))))))) 32.38/9.25 W(s(x1)) -> P(p(s(s(s(p(p(s(s(x1))))))))) 32.38/9.25 W(s(x1)) -> P(s(s(s(p(p(s(s(x1)))))))) 32.38/9.25 W(s(x1)) -> P(p(s(s(x1)))) 32.38/9.25 W(s(x1)) -> P(s(s(x1))) 32.38/9.25 P(p(s(x1))) -> P(x1) 32.38/9.25 P(0(x1)) -> P(s(x1)) 32.38/9.25 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) 32.38/9.25 w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) 32.38/9.25 p(p(s(x1))) -> p(x1) 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 Q is empty. 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (5) DependencyGraphProof (EQUIVALENT) 32.38/9.25 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 13 less nodes. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (6) 32.38/9.25 Complex Obligation (AND) 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (7) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 P(p(s(x1))) -> P(x1) 32.38/9.25 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) 32.38/9.25 w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) 32.38/9.25 p(p(s(x1))) -> p(x1) 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 Q is empty. 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (8) UsableRulesProof (EQUIVALENT) 32.38/9.25 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. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (9) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 P(p(s(x1))) -> P(x1) 32.38/9.25 32.38/9.25 R is empty. 32.38/9.25 Q is empty. 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (10) QDPSizeChangeProof (EQUIVALENT) 32.38/9.25 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. 32.38/9.25 32.38/9.25 From the DPs we obtained the following set of size-change graphs: 32.38/9.25 *P(p(s(x1))) -> P(x1) 32.38/9.25 The graph contains the following edges 1 > 1 32.38/9.25 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (11) 32.38/9.25 YES 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (12) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 V(s(x1)) -> W(p(p(s(s(p(s(p(s(x1))))))))) 32.38/9.25 W(s(x1)) -> V(p(p(s(s(s(p(p(s(s(x1)))))))))) 32.38/9.25 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) 32.38/9.25 w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) 32.38/9.25 p(p(s(x1))) -> p(x1) 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 Q is empty. 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (13) MNOCProof (EQUIVALENT) 32.38/9.25 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (14) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 V(s(x1)) -> W(p(p(s(s(p(s(p(s(x1))))))))) 32.38/9.25 W(s(x1)) -> V(p(p(s(s(s(p(p(s(s(x1)))))))))) 32.38/9.25 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) 32.38/9.25 w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) 32.38/9.25 p(p(s(x1))) -> p(x1) 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 The set Q consists of the following terms: 32.38/9.25 32.38/9.25 v(s(x0)) 32.38/9.25 w(s(x0)) 32.38/9.25 p(s(x0)) 32.38/9.25 p(0(x0)) 32.38/9.25 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (15) UsableRulesProof (EQUIVALENT) 32.38/9.25 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. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (16) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 V(s(x1)) -> W(p(p(s(s(p(s(p(s(x1))))))))) 32.38/9.25 W(s(x1)) -> V(p(p(s(s(s(p(p(s(s(x1)))))))))) 32.38/9.25 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 The set Q consists of the following terms: 32.38/9.25 32.38/9.25 v(s(x0)) 32.38/9.25 w(s(x0)) 32.38/9.25 p(s(x0)) 32.38/9.25 p(0(x0)) 32.38/9.25 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (17) QReductionProof (EQUIVALENT) 32.38/9.25 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 32.38/9.25 32.38/9.25 v(s(x0)) 32.38/9.25 w(s(x0)) 32.38/9.25 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (18) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 V(s(x1)) -> W(p(p(s(s(p(s(p(s(x1))))))))) 32.38/9.25 W(s(x1)) -> V(p(p(s(s(s(p(p(s(s(x1)))))))))) 32.38/9.25 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 The set Q consists of the following terms: 32.38/9.25 32.38/9.25 p(s(x0)) 32.38/9.25 p(0(x0)) 32.38/9.25 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (19) QDPOrderProof (EQUIVALENT) 32.38/9.25 We use the reduction pair processor [LPAR04,JAR06]. 32.38/9.25 32.38/9.25 32.38/9.25 The following pairs can be oriented strictly and are deleted. 32.38/9.25 32.38/9.25 W(s(x1)) -> V(p(p(s(s(s(p(p(s(s(x1)))))))))) 32.38/9.25 The remaining pairs can at least be oriented weakly. 32.38/9.25 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 32.38/9.25 32.38/9.25 POL( V_1(x_1) ) = max{0, x_1 - 2} 32.38/9.25 POL( W_1(x_1) ) = x_1 32.38/9.25 POL( p_1(x_1) ) = max{0, x_1 - 2} 32.38/9.25 POL( s_1(x_1) ) = x_1 + 2 32.38/9.25 POL( 0_1(x_1) ) = 0 32.38/9.25 32.38/9.25 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 32.38/9.25 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (20) 32.38/9.25 Obligation: 32.38/9.25 Q DP problem: 32.38/9.25 The TRS P consists of the following rules: 32.38/9.25 32.38/9.25 V(s(x1)) -> W(p(p(s(s(p(s(p(s(x1))))))))) 32.38/9.25 32.38/9.25 The TRS R consists of the following rules: 32.38/9.25 32.38/9.25 p(s(x1)) -> x1 32.38/9.25 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) 32.38/9.25 32.38/9.25 The set Q consists of the following terms: 32.38/9.25 32.38/9.25 p(s(x0)) 32.38/9.25 p(0(x0)) 32.38/9.25 32.38/9.25 We have to consider all minimal (P,Q,R)-chains. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (21) DependencyGraphProof (EQUIVALENT) 32.38/9.25 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 32.38/9.25 ---------------------------------------- 32.38/9.25 32.38/9.25 (22) 32.38/9.25 TRUE 32.57/9.41 EOF