32.66/9.39 YES 34.48/9.84 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 34.48/9.84 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 34.48/9.84 34.48/9.84 34.48/9.84 Termination w.r.t. Q of the given QTRS could be proven: 34.48/9.84 34.48/9.84 (0) QTRS 34.48/9.84 (1) QTRSRRRProof [EQUIVALENT, 50 ms] 34.48/9.84 (2) QTRS 34.48/9.84 (3) DependencyPairsProof [EQUIVALENT, 57 ms] 34.48/9.84 (4) QDP 34.48/9.84 (5) DependencyGraphProof [EQUIVALENT, 7 ms] 34.48/9.84 (6) AND 34.48/9.84 (7) QDP 34.48/9.84 (8) UsableRulesProof [EQUIVALENT, 0 ms] 34.48/9.84 (9) QDP 34.48/9.84 (10) QDPSizeChangeProof [EQUIVALENT, 2 ms] 34.48/9.84 (11) YES 34.48/9.84 (12) QDP 34.48/9.84 (13) MNOCProof [EQUIVALENT, 0 ms] 34.48/9.84 (14) QDP 34.48/9.84 (15) UsableRulesProof [EQUIVALENT, 0 ms] 34.48/9.84 (16) QDP 34.48/9.84 (17) QReductionProof [EQUIVALENT, 0 ms] 34.48/9.84 (18) QDP 34.48/9.84 (19) QDPOrderProof [EQUIVALENT, 18 ms] 34.48/9.84 (20) QDP 34.48/9.84 (21) PisEmptyProof [EQUIVALENT, 0 ms] 34.48/9.84 (22) YES 34.48/9.84 (23) QDP 34.48/9.84 (24) MNOCProof [EQUIVALENT, 0 ms] 34.48/9.84 (25) QDP 34.48/9.84 (26) UsableRulesProof [EQUIVALENT, 0 ms] 34.48/9.84 (27) QDP 34.48/9.84 (28) QReductionProof [EQUIVALENT, 1 ms] 34.48/9.84 (29) QDP 34.48/9.84 (30) QDPOrderProof [EQUIVALENT, 11 ms] 34.48/9.84 (31) QDP 34.48/9.84 (32) PisEmptyProof [EQUIVALENT, 0 ms] 34.48/9.84 (33) YES 34.48/9.84 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (0) 34.48/9.84 Obligation: 34.48/9.84 Q restricted rewrite system: 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(0(x1)) -> p(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1)))))))))))))))) 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 0(x1) -> x1 34.48/9.84 34.48/9.84 Q is empty. 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (1) QTRSRRRProof (EQUIVALENT) 34.48/9.84 Used ordering: 34.48/9.84 Polynomial interpretation [POLO]: 34.48/9.84 34.48/9.84 POL(0(x_1)) = 1 + x_1 34.48/9.84 POL(p(x_1)) = x_1 34.48/9.84 POL(s(x_1)) = x_1 34.48/9.84 POL(twice(x_1)) = x_1 34.48/9.84 POL(twoto(x_1)) = 1 + x_1 34.48/9.84 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 34.48/9.84 34.48/9.84 twoto(0(x1)) -> p(p(s(s(s(p(p(p(s(s(s(0(p(p(s(s(x1)))))))))))))))) 34.48/9.84 0(x1) -> x1 34.48/9.84 34.48/9.84 34.48/9.84 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (2) 34.48/9.84 Obligation: 34.48/9.84 Q restricted rewrite system: 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 34.48/9.84 Q is empty. 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (3) DependencyPairsProof (EQUIVALENT) 34.48/9.84 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (4) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWOTO(s(x1)) -> P(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> TWICE(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(p(s(s(s(twoto(p(s(p(s(x1)))))))))) 34.48/9.84 TWOTO(s(x1)) -> P(s(s(s(twoto(p(s(p(s(x1))))))))) 34.48/9.84 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.48/9.84 TWOTO(s(x1)) -> P(s(p(s(x1)))) 34.48/9.84 TWOTO(s(x1)) -> P(s(x1)) 34.48/9.84 TWICE(0(x1)) -> P(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 TWICE(0(x1)) -> P(s(0(s(p(s(s(s(s(p(s(x1))))))))))) 34.48/9.84 TWICE(0(x1)) -> P(s(s(s(s(p(s(x1))))))) 34.48/9.84 TWICE(0(x1)) -> P(s(x1)) 34.48/9.84 TWICE(s(x1)) -> P(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))))) 34.48/9.84 TWICE(s(x1)) -> P(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))) 34.48/9.84 TWICE(s(x1)) -> P(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1)))))))))))))))) 34.48/9.84 TWICE(s(x1)) -> P(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))) 34.48/9.84 TWICE(s(x1)) -> TWICE(p(s(p(s(p(s(p(s(x1))))))))) 34.48/9.84 TWICE(s(x1)) -> P(s(p(s(p(s(p(s(x1)))))))) 34.48/9.84 TWICE(s(x1)) -> P(s(p(s(p(s(x1)))))) 34.48/9.84 TWICE(s(x1)) -> P(s(p(s(x1)))) 34.48/9.84 TWICE(s(x1)) -> P(s(x1)) 34.48/9.84 P(p(s(x1))) -> P(x1) 34.48/9.84 P(0(x1)) -> P(s(x1)) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 34.48/9.84 Q is empty. 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (5) DependencyGraphProof (EQUIVALENT) 34.48/9.84 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 26 less nodes. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (6) 34.48/9.84 Complex Obligation (AND) 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (7) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 P(p(s(x1))) -> P(x1) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 34.48/9.84 Q is empty. 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (8) UsableRulesProof (EQUIVALENT) 34.48/9.84 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. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (9) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 P(p(s(x1))) -> P(x1) 34.48/9.84 34.48/9.84 R is empty. 34.48/9.84 Q is empty. 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (10) QDPSizeChangeProof (EQUIVALENT) 34.48/9.84 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. 34.48/9.84 34.48/9.84 From the DPs we obtained the following set of size-change graphs: 34.48/9.84 *P(p(s(x1))) -> P(x1) 34.48/9.84 The graph contains the following edges 1 > 1 34.48/9.84 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (11) 34.48/9.84 YES 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (12) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWICE(s(x1)) -> TWICE(p(s(p(s(p(s(p(s(x1))))))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 34.48/9.84 Q is empty. 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (13) MNOCProof (EQUIVALENT) 34.48/9.84 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (14) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWICE(s(x1)) -> TWICE(p(s(p(s(p(s(p(s(x1))))))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 twoto(s(x0)) 34.48/9.84 twice(0(x0)) 34.48/9.84 twice(s(x0)) 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (15) UsableRulesProof (EQUIVALENT) 34.48/9.84 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. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (16) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWICE(s(x1)) -> TWICE(p(s(p(s(p(s(p(s(x1))))))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 twoto(s(x0)) 34.48/9.84 twice(0(x0)) 34.48/9.84 twice(s(x0)) 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (17) QReductionProof (EQUIVALENT) 34.48/9.84 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 34.48/9.84 34.48/9.84 twoto(s(x0)) 34.48/9.84 twice(0(x0)) 34.48/9.84 twice(s(x0)) 34.48/9.84 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (18) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWICE(s(x1)) -> TWICE(p(s(p(s(p(s(p(s(x1))))))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (19) QDPOrderProof (EQUIVALENT) 34.48/9.84 We use the reduction pair processor [LPAR04,JAR06]. 34.48/9.84 34.48/9.84 34.48/9.84 The following pairs can be oriented strictly and are deleted. 34.48/9.84 34.48/9.84 TWICE(s(x1)) -> TWICE(p(s(p(s(p(s(p(s(x1))))))))) 34.48/9.84 The remaining pairs can at least be oriented weakly. 34.48/9.84 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 34.48/9.84 34.48/9.84 POL( TWICE_1(x_1) ) = max{0, x_1 - 1} 34.48/9.84 POL( p_1(x_1) ) = max{0, x_1 - 2} 34.48/9.84 POL( s_1(x_1) ) = x_1 + 2 34.48/9.84 34.48/9.84 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (20) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 P is empty. 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (21) PisEmptyProof (EQUIVALENT) 34.48/9.84 The TRS P is empty. Hence, there is no (P,Q,R) chain. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (22) 34.48/9.84 YES 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (23) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 34.48/9.84 Q is empty. 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (24) MNOCProof (EQUIVALENT) 34.48/9.84 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (25) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 twoto(s(x1)) -> p(p(s(s(p(p(p(s(s(s(twice(p(p(s(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))) 34.48/9.84 twice(0(x1)) -> p(s(p(s(0(s(p(s(s(s(s(p(s(x1))))))))))))) 34.48/9.84 twice(s(x1)) -> s(p(p(p(p(s(s(s(s(s(twice(p(s(p(s(p(s(p(s(x1))))))))))))))))))) 34.48/9.84 p(p(s(x1))) -> p(x1) 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 p(0(x1)) -> 0(s(s(s(s(p(s(x1))))))) 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 twoto(s(x0)) 34.48/9.84 twice(0(x0)) 34.48/9.84 twice(s(x0)) 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (26) UsableRulesProof (EQUIVALENT) 34.48/9.84 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. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (27) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 twoto(s(x0)) 34.48/9.84 twice(0(x0)) 34.48/9.84 twice(s(x0)) 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (28) QReductionProof (EQUIVALENT) 34.48/9.84 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 34.48/9.84 34.48/9.84 twoto(s(x0)) 34.48/9.84 twice(0(x0)) 34.48/9.84 twice(s(x0)) 34.48/9.84 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (29) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 The TRS P consists of the following rules: 34.48/9.84 34.48/9.84 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.48/9.84 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (30) QDPOrderProof (EQUIVALENT) 34.48/9.84 We use the reduction pair processor [LPAR04,JAR06]. 34.48/9.84 34.48/9.84 34.48/9.84 The following pairs can be oriented strictly and are deleted. 34.48/9.84 34.48/9.84 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.48/9.84 The remaining pairs can at least be oriented weakly. 34.48/9.84 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 34.48/9.84 34.48/9.84 POL( TWOTO_1(x_1) ) = x_1 + 2 34.48/9.84 POL( p_1(x_1) ) = max{0, x_1 - 1} 34.48/9.84 POL( s_1(x_1) ) = x_1 + 1 34.48/9.84 34.48/9.84 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (31) 34.48/9.84 Obligation: 34.48/9.84 Q DP problem: 34.48/9.84 P is empty. 34.48/9.84 The TRS R consists of the following rules: 34.48/9.84 34.48/9.84 p(s(x1)) -> x1 34.48/9.84 34.48/9.84 The set Q consists of the following terms: 34.48/9.84 34.48/9.84 p(s(x0)) 34.48/9.84 p(0(x0)) 34.48/9.84 34.48/9.84 We have to consider all minimal (P,Q,R)-chains. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (32) PisEmptyProof (EQUIVALENT) 34.48/9.84 The TRS P is empty. Hence, there is no (P,Q,R) chain. 34.48/9.84 ---------------------------------------- 34.48/9.84 34.48/9.84 (33) 34.48/9.84 YES 34.56/9.90 EOF