30.90/8.88 YES 31.76/9.08 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 31.76/9.08 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 31.76/9.08 31.76/9.08 31.76/9.08 Termination w.r.t. Q of the given QTRS could be proven: 31.76/9.08 31.76/9.08 (0) QTRS 31.76/9.08 (1) QTRSRRRProof [EQUIVALENT, 70 ms] 31.76/9.08 (2) QTRS 31.76/9.08 (3) DependencyPairsProof [EQUIVALENT, 45 ms] 31.76/9.08 (4) QDP 31.76/9.08 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 31.76/9.08 (6) AND 31.76/9.08 (7) QDP 31.76/9.08 (8) UsableRulesProof [EQUIVALENT, 0 ms] 31.76/9.08 (9) QDP 31.76/9.08 (10) QDPSizeChangeProof [EQUIVALENT, 1 ms] 31.76/9.08 (11) YES 31.76/9.08 (12) QDP 31.76/9.08 (13) MNOCProof [EQUIVALENT, 6 ms] 31.76/9.08 (14) QDP 31.76/9.08 (15) UsableRulesProof [EQUIVALENT, 0 ms] 31.76/9.08 (16) QDP 31.76/9.08 (17) QReductionProof [EQUIVALENT, 0 ms] 31.76/9.08 (18) QDP 31.76/9.08 (19) QDPOrderProof [EQUIVALENT, 0 ms] 31.76/9.08 (20) QDP 31.76/9.08 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 31.76/9.08 (22) TRUE 31.76/9.08 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (0) 31.76/9.08 Obligation: 31.76/9.08 Q restricted rewrite system: 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 i(0(x1)) -> p(s(p(s(0(p(s(p(s(x1))))))))) 31.76/9.08 i(s(x1)) -> p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))))) 31.76/9.08 j(0(x1)) -> p(s(p(p(s(s(0(p(s(p(s(x1))))))))))) 31.76/9.08 j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1))))))))))))) 31.76/9.08 p(p(s(x1))) -> p(x1) 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 Q is empty. 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (1) QTRSRRRProof (EQUIVALENT) 31.76/9.08 Used ordering: 31.76/9.08 Polynomial interpretation [POLO]: 31.76/9.08 31.76/9.08 POL(0(x_1)) = x_1 31.76/9.08 POL(i(x_1)) = 1 + x_1 31.76/9.08 POL(j(x_1)) = 1 + x_1 31.76/9.08 POL(p(x_1)) = x_1 31.76/9.08 POL(s(x_1)) = x_1 31.76/9.08 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 31.76/9.08 31.76/9.08 i(0(x1)) -> p(s(p(s(0(p(s(p(s(x1))))))))) 31.76/9.08 j(0(x1)) -> p(s(p(p(s(s(0(p(s(p(s(x1))))))))))) 31.76/9.08 31.76/9.08 31.76/9.08 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (2) 31.76/9.08 Obligation: 31.76/9.08 Q restricted rewrite system: 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 i(s(x1)) -> p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))))) 31.76/9.08 j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1))))))))))))) 31.76/9.08 p(p(s(x1))) -> p(x1) 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 Q is empty. 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (3) DependencyPairsProof (EQUIVALENT) 31.76/9.08 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (4) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 I(s(x1)) -> P(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))))) 31.76/9.08 I(s(x1)) -> P(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))) 31.76/9.08 I(s(x1)) -> J(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))) 31.76/9.08 I(s(x1)) -> P(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))) 31.76/9.08 I(s(x1)) -> P(s(p(p(p(p(s(s(s(s(x1)))))))))) 31.76/9.08 I(s(x1)) -> P(p(p(p(s(s(s(s(x1)))))))) 31.76/9.08 I(s(x1)) -> P(p(p(s(s(s(s(x1))))))) 31.76/9.08 I(s(x1)) -> P(p(s(s(s(s(x1)))))) 31.76/9.08 I(s(x1)) -> P(s(s(s(s(x1))))) 31.76/9.08 J(s(x1)) -> P(p(s(s(i(p(s(p(s(x1))))))))) 31.76/9.08 J(s(x1)) -> P(s(s(i(p(s(p(s(x1)))))))) 31.76/9.08 J(s(x1)) -> I(p(s(p(s(x1))))) 31.76/9.08 J(s(x1)) -> P(s(p(s(x1)))) 31.76/9.08 J(s(x1)) -> P(s(x1)) 31.76/9.08 P(p(s(x1))) -> P(x1) 31.76/9.08 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 i(s(x1)) -> p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))))) 31.76/9.08 j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1))))))))))))) 31.76/9.08 p(p(s(x1))) -> p(x1) 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 Q is empty. 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (5) DependencyGraphProof (EQUIVALENT) 31.76/9.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (6) 31.76/9.08 Complex Obligation (AND) 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (7) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 P(p(s(x1))) -> P(x1) 31.76/9.08 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 i(s(x1)) -> p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))))) 31.76/9.08 j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1))))))))))))) 31.76/9.08 p(p(s(x1))) -> p(x1) 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 Q is empty. 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (8) UsableRulesProof (EQUIVALENT) 31.76/9.08 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. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (9) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 P(p(s(x1))) -> P(x1) 31.76/9.08 31.76/9.08 R is empty. 31.76/9.08 Q is empty. 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (10) QDPSizeChangeProof (EQUIVALENT) 31.76/9.08 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. 31.76/9.08 31.76/9.08 From the DPs we obtained the following set of size-change graphs: 31.76/9.08 *P(p(s(x1))) -> P(x1) 31.76/9.08 The graph contains the following edges 1 > 1 31.76/9.08 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (11) 31.76/9.08 YES 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (12) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 I(s(x1)) -> J(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))) 31.76/9.08 J(s(x1)) -> I(p(s(p(s(x1))))) 31.76/9.08 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 i(s(x1)) -> p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))))) 31.76/9.08 j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1))))))))))))) 31.76/9.08 p(p(s(x1))) -> p(x1) 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 Q is empty. 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (13) MNOCProof (EQUIVALENT) 31.76/9.08 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (14) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 I(s(x1)) -> J(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))) 31.76/9.08 J(s(x1)) -> I(p(s(p(s(x1))))) 31.76/9.08 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 i(s(x1)) -> p(s(p(s(s(j(p(s(p(s(p(p(p(p(s(s(s(s(x1)))))))))))))))))) 31.76/9.08 j(s(x1)) -> s(s(s(s(p(p(s(s(i(p(s(p(s(x1))))))))))))) 31.76/9.08 p(p(s(x1))) -> p(x1) 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 The set Q consists of the following terms: 31.76/9.08 31.76/9.08 i(s(x0)) 31.76/9.08 j(s(x0)) 31.76/9.08 p(s(x0)) 31.76/9.08 p(0(x0)) 31.76/9.08 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (15) UsableRulesProof (EQUIVALENT) 31.76/9.08 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. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (16) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 I(s(x1)) -> J(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))) 31.76/9.08 J(s(x1)) -> I(p(s(p(s(x1))))) 31.76/9.08 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 The set Q consists of the following terms: 31.76/9.08 31.76/9.08 i(s(x0)) 31.76/9.08 j(s(x0)) 31.76/9.08 p(s(x0)) 31.76/9.08 p(0(x0)) 31.76/9.08 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (17) QReductionProof (EQUIVALENT) 31.76/9.08 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 31.76/9.08 31.76/9.08 i(s(x0)) 31.76/9.08 j(s(x0)) 31.76/9.08 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (18) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 I(s(x1)) -> J(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))) 31.76/9.08 J(s(x1)) -> I(p(s(p(s(x1))))) 31.76/9.08 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 The set Q consists of the following terms: 31.76/9.08 31.76/9.08 p(s(x0)) 31.76/9.08 p(0(x0)) 31.76/9.08 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (19) QDPOrderProof (EQUIVALENT) 31.76/9.08 We use the reduction pair processor [LPAR04,JAR06]. 31.76/9.08 31.76/9.08 31.76/9.08 The following pairs can be oriented strictly and are deleted. 31.76/9.08 31.76/9.08 J(s(x1)) -> I(p(s(p(s(x1))))) 31.76/9.08 The remaining pairs can at least be oriented weakly. 31.76/9.08 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 31.76/9.08 31.76/9.08 POL( I_1(x_1) ) = max{0, x_1 - 2} 31.76/9.08 POL( J_1(x_1) ) = max{0, x_1 - 1} 31.76/9.08 POL( p_1(x_1) ) = max{0, x_1 - 2} 31.76/9.08 POL( s_1(x_1) ) = x_1 + 2 31.76/9.08 POL( 0_1(x_1) ) = max{0, -2} 31.76/9.08 31.76/9.08 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 31.76/9.08 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (20) 31.76/9.08 Obligation: 31.76/9.08 Q DP problem: 31.76/9.08 The TRS P consists of the following rules: 31.76/9.08 31.76/9.08 I(s(x1)) -> J(p(s(p(s(p(p(p(p(s(s(s(s(x1))))))))))))) 31.76/9.08 31.76/9.08 The TRS R consists of the following rules: 31.76/9.08 31.76/9.08 p(s(x1)) -> x1 31.76/9.08 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 31.76/9.08 31.76/9.08 The set Q consists of the following terms: 31.76/9.08 31.76/9.08 p(s(x0)) 31.76/9.08 p(0(x0)) 31.76/9.08 31.76/9.08 We have to consider all minimal (P,Q,R)-chains. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (21) DependencyGraphProof (EQUIVALENT) 31.76/9.08 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 31.76/9.08 ---------------------------------------- 31.76/9.08 31.76/9.08 (22) 31.76/9.08 TRUE 31.98/9.18 EOF