33.00/9.45 YES 34.10/9.73 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 34.10/9.73 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 34.10/9.73 34.10/9.73 34.10/9.73 Termination w.r.t. Q of the given QTRS could be proven: 34.10/9.73 34.10/9.73 (0) QTRS 34.10/9.73 (1) QTRSRRRProof [EQUIVALENT, 59 ms] 34.10/9.73 (2) QTRS 34.10/9.73 (3) DependencyPairsProof [EQUIVALENT, 39 ms] 34.10/9.73 (4) QDP 34.10/9.73 (5) DependencyGraphProof [EQUIVALENT, 0 ms] 34.10/9.73 (6) AND 34.10/9.73 (7) QDP 34.10/9.73 (8) UsableRulesProof [EQUIVALENT, 0 ms] 34.10/9.73 (9) QDP 34.10/9.73 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 34.10/9.73 (11) YES 34.10/9.73 (12) QDP 34.10/9.73 (13) MNOCProof [EQUIVALENT, 2 ms] 34.10/9.73 (14) QDP 34.10/9.73 (15) UsableRulesProof [EQUIVALENT, 0 ms] 34.10/9.73 (16) QDP 34.10/9.73 (17) QReductionProof [EQUIVALENT, 0 ms] 34.10/9.73 (18) QDP 34.10/9.73 (19) QDPOrderProof [EQUIVALENT, 23 ms] 34.10/9.73 (20) QDP 34.10/9.73 (21) PisEmptyProof [EQUIVALENT, 0 ms] 34.10/9.73 (22) YES 34.10/9.73 (23) QDP 34.10/9.73 (24) MNOCProof [EQUIVALENT, 2 ms] 34.10/9.73 (25) QDP 34.10/9.73 (26) UsableRulesProof [EQUIVALENT, 0 ms] 34.10/9.73 (27) QDP 34.10/9.73 (28) QReductionProof [EQUIVALENT, 0 ms] 34.10/9.73 (29) QDP 34.10/9.73 (30) QDPOrderProof [EQUIVALENT, 39 ms] 34.10/9.73 (31) QDP 34.10/9.73 (32) PisEmptyProof [EQUIVALENT, 0 ms] 34.10/9.73 (33) YES 34.10/9.73 (34) QDP 34.10/9.73 (35) MNOCProof [EQUIVALENT, 1 ms] 34.10/9.73 (36) QDP 34.10/9.73 (37) UsableRulesProof [EQUIVALENT, 0 ms] 34.10/9.73 (38) QDP 34.10/9.73 (39) QReductionProof [EQUIVALENT, 0 ms] 34.10/9.73 (40) QDP 34.10/9.73 (41) QDPOrderProof [EQUIVALENT, 29 ms] 34.10/9.73 (42) QDP 34.10/9.73 (43) PisEmptyProof [EQUIVALENT, 0 ms] 34.10/9.73 (44) YES 34.10/9.73 34.10/9.73 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (0) 34.10/9.73 Obligation: 34.10/9.73 Q restricted rewrite system: 34.10/9.73 The TRS R consists of the following rules: 34.10/9.73 34.10/9.73 tower(0(x1)) -> s(0(p(s(p(s(x1)))))) 34.10/9.73 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.10/9.73 twoto(0(x1)) -> s(0(x1)) 34.10/9.73 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.10/9.73 twice(0(x1)) -> 0(x1) 34.10/9.73 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.10/9.73 p(p(s(x1))) -> p(x1) 34.10/9.73 p(s(x1)) -> x1 34.10/9.73 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.10/9.73 34.10/9.73 Q is empty. 34.10/9.73 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (1) QTRSRRRProof (EQUIVALENT) 34.10/9.73 Used ordering: 34.10/9.73 Polynomial interpretation [POLO]: 34.10/9.73 34.10/9.73 POL(0(x_1)) = x_1 34.10/9.73 POL(p(x_1)) = x_1 34.10/9.73 POL(s(x_1)) = x_1 34.10/9.73 POL(tower(x_1)) = 1 + x_1 34.10/9.73 POL(twice(x_1)) = x_1 34.10/9.73 POL(twoto(x_1)) = x_1 34.10/9.73 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 34.10/9.73 34.10/9.73 tower(0(x1)) -> s(0(p(s(p(s(x1)))))) 34.10/9.73 34.10/9.73 34.10/9.73 34.10/9.73 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (2) 34.10/9.73 Obligation: 34.10/9.73 Q restricted rewrite system: 34.10/9.73 The TRS R consists of the following rules: 34.10/9.73 34.10/9.73 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.10/9.73 twoto(0(x1)) -> s(0(x1)) 34.10/9.73 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.10/9.73 twice(0(x1)) -> 0(x1) 34.10/9.73 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.10/9.73 p(p(s(x1))) -> p(x1) 34.10/9.73 p(s(x1)) -> x1 34.10/9.73 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.10/9.73 34.10/9.73 Q is empty. 34.10/9.73 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (3) DependencyPairsProof (EQUIVALENT) 34.10/9.73 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (4) 34.10/9.73 Obligation: 34.10/9.73 Q DP problem: 34.10/9.73 The TRS P consists of the following rules: 34.10/9.73 34.10/9.73 TOWER(s(x1)) -> P(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.10/9.73 TOWER(s(x1)) -> P(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))) 34.10/9.73 TOWER(s(x1)) -> TWOTO(p(s(p(s(tower(p(s(p(s(x1)))))))))) 34.10/9.73 TOWER(s(x1)) -> P(s(p(s(tower(p(s(p(s(x1))))))))) 34.10/9.73 TOWER(s(x1)) -> P(s(tower(p(s(p(s(x1))))))) 34.10/9.73 TOWER(s(x1)) -> TOWER(p(s(p(s(x1))))) 34.10/9.73 TOWER(s(x1)) -> P(s(p(s(x1)))) 34.10/9.73 TOWER(s(x1)) -> P(s(x1)) 34.10/9.73 TWOTO(s(x1)) -> P(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> TWICE(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(p(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(p(p(s(s(s(twoto(p(s(p(s(x1))))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(p(s(s(s(twoto(p(s(p(s(x1)))))))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(s(s(twoto(p(s(p(s(x1))))))))) 34.10/9.73 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.10/9.73 TWOTO(s(x1)) -> P(s(p(s(x1)))) 34.10/9.73 TWOTO(s(x1)) -> P(s(x1)) 34.10/9.73 TWICE(s(x1)) -> P(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.10/9.73 TWICE(s(x1)) -> P(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1)))))))))))))) 34.10/9.73 TWICE(s(x1)) -> P(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))) 34.10/9.73 TWICE(s(x1)) -> TWICE(p(p(p(s(s(s(x1))))))) 34.10/9.73 TWICE(s(x1)) -> P(p(p(s(s(s(x1)))))) 34.10/9.73 TWICE(s(x1)) -> P(p(s(s(s(x1))))) 34.10/9.73 TWICE(s(x1)) -> P(s(s(s(x1)))) 34.10/9.73 P(p(s(x1))) -> P(x1) 34.10/9.73 34.10/9.73 The TRS R consists of the following rules: 34.10/9.73 34.10/9.73 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.10/9.73 twoto(0(x1)) -> s(0(x1)) 34.10/9.73 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.10/9.73 twice(0(x1)) -> 0(x1) 34.10/9.73 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.10/9.73 p(p(s(x1))) -> p(x1) 34.10/9.73 p(s(x1)) -> x1 34.10/9.73 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.10/9.73 34.10/9.73 Q is empty. 34.10/9.73 We have to consider all minimal (P,Q,R)-chains. 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (5) DependencyGraphProof (EQUIVALENT) 34.10/9.73 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 29 less nodes. 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (6) 34.10/9.73 Complex Obligation (AND) 34.10/9.73 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (7) 34.10/9.73 Obligation: 34.10/9.73 Q DP problem: 34.10/9.73 The TRS P consists of the following rules: 34.10/9.73 34.10/9.73 P(p(s(x1))) -> P(x1) 34.10/9.73 34.10/9.73 The TRS R consists of the following rules: 34.10/9.73 34.10/9.73 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.10/9.73 twoto(0(x1)) -> s(0(x1)) 34.10/9.73 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.10/9.73 twice(0(x1)) -> 0(x1) 34.10/9.73 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.10/9.73 p(p(s(x1))) -> p(x1) 34.10/9.73 p(s(x1)) -> x1 34.10/9.73 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.10/9.73 34.10/9.73 Q is empty. 34.10/9.73 We have to consider all minimal (P,Q,R)-chains. 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (8) UsableRulesProof (EQUIVALENT) 34.10/9.73 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. 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (9) 34.10/9.73 Obligation: 34.10/9.73 Q DP problem: 34.10/9.73 The TRS P consists of the following rules: 34.10/9.73 34.10/9.73 P(p(s(x1))) -> P(x1) 34.10/9.73 34.10/9.73 R is empty. 34.10/9.73 Q is empty. 34.10/9.73 We have to consider all minimal (P,Q,R)-chains. 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (10) QDPSizeChangeProof (EQUIVALENT) 34.10/9.73 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. 34.10/9.73 34.10/9.73 From the DPs we obtained the following set of size-change graphs: 34.10/9.73 *P(p(s(x1))) -> P(x1) 34.10/9.73 The graph contains the following edges 1 > 1 34.10/9.73 34.10/9.73 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (11) 34.10/9.73 YES 34.10/9.73 34.10/9.73 ---------------------------------------- 34.10/9.73 34.10/9.73 (12) 34.10/9.73 Obligation: 34.10/9.73 Q DP problem: 34.10/9.73 The TRS P consists of the following rules: 34.10/9.73 34.10/9.73 TWICE(s(x1)) -> TWICE(p(p(p(s(s(s(x1))))))) 34.10/9.73 34.10/9.73 The TRS R consists of the following rules: 34.10/9.73 34.10/9.73 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.20/9.76 twoto(0(x1)) -> s(0(x1)) 34.20/9.76 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.20/9.76 twice(0(x1)) -> 0(x1) 34.20/9.76 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.20/9.76 p(p(s(x1))) -> p(x1) 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 Q is empty. 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (13) MNOCProof (EQUIVALENT) 34.20/9.76 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (14) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TWICE(s(x1)) -> TWICE(p(p(p(s(s(s(x1))))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.20/9.76 twoto(0(x1)) -> s(0(x1)) 34.20/9.76 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.20/9.76 twice(0(x1)) -> 0(x1) 34.20/9.76 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.20/9.76 p(p(s(x1))) -> p(x1) 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (15) UsableRulesProof (EQUIVALENT) 34.20/9.76 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. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (16) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TWICE(s(x1)) -> TWICE(p(p(p(s(s(s(x1))))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (17) QReductionProof (EQUIVALENT) 34.20/9.76 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (18) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TWICE(s(x1)) -> TWICE(p(p(p(s(s(s(x1))))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (19) QDPOrderProof (EQUIVALENT) 34.20/9.76 We use the reduction pair processor [LPAR04,JAR06]. 34.20/9.76 34.20/9.76 34.20/9.76 The following pairs can be oriented strictly and are deleted. 34.20/9.76 34.20/9.76 TWICE(s(x1)) -> TWICE(p(p(p(s(s(s(x1))))))) 34.20/9.76 The remaining pairs can at least be oriented weakly. 34.20/9.76 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 34.20/9.76 34.20/9.76 POL( TWICE_1(x_1) ) = 2x_1 34.20/9.76 POL( p_1(x_1) ) = max{0, x_1 - 1} 34.20/9.76 POL( s_1(x_1) ) = x_1 + 1 34.20/9.76 POL( 0_1(x_1) ) = 0 34.20/9.76 34.20/9.76 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (20) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 P is empty. 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (21) PisEmptyProof (EQUIVALENT) 34.20/9.76 The TRS P is empty. Hence, there is no (P,Q,R) chain. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (22) 34.20/9.76 YES 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (23) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.20/9.76 twoto(0(x1)) -> s(0(x1)) 34.20/9.76 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.20/9.76 twice(0(x1)) -> 0(x1) 34.20/9.76 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.20/9.76 p(p(s(x1))) -> p(x1) 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 Q is empty. 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (24) MNOCProof (EQUIVALENT) 34.20/9.76 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (25) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.20/9.76 twoto(0(x1)) -> s(0(x1)) 34.20/9.76 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.20/9.76 twice(0(x1)) -> 0(x1) 34.20/9.76 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.20/9.76 p(p(s(x1))) -> p(x1) 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (26) UsableRulesProof (EQUIVALENT) 34.20/9.76 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. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (27) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (28) QReductionProof (EQUIVALENT) 34.20/9.76 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (29) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (30) QDPOrderProof (EQUIVALENT) 34.20/9.76 We use the reduction pair processor [LPAR04,JAR06]. 34.20/9.76 34.20/9.76 34.20/9.76 The following pairs can be oriented strictly and are deleted. 34.20/9.76 34.20/9.76 TWOTO(s(x1)) -> TWOTO(p(s(p(s(x1))))) 34.20/9.76 The remaining pairs can at least be oriented weakly. 34.20/9.76 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 34.20/9.76 34.20/9.76 <<< 34.20/9.76 POL(TWOTO(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 34.20/9.76 >>> 34.20/9.76 34.20/9.76 <<< 34.20/9.76 POL(s(x_1)) = [[1A], [0A], [-I]] + [[1A, -I, -I], [-I, 0A, 0A], [0A, -I, -I]] * x_1 34.20/9.76 >>> 34.20/9.76 34.20/9.76 <<< 34.20/9.76 POL(p(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, 0A], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 34.20/9.76 >>> 34.20/9.76 34.20/9.76 34.20/9.76 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (31) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 P is empty. 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (32) PisEmptyProof (EQUIVALENT) 34.20/9.76 The TRS P is empty. Hence, there is no (P,Q,R) chain. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (33) 34.20/9.76 YES 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (34) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TOWER(s(x1)) -> TOWER(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.20/9.76 twoto(0(x1)) -> s(0(x1)) 34.20/9.76 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.20/9.76 twice(0(x1)) -> 0(x1) 34.20/9.76 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.20/9.76 p(p(s(x1))) -> p(x1) 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 Q is empty. 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (35) MNOCProof (EQUIVALENT) 34.20/9.76 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (36) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TOWER(s(x1)) -> TOWER(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 tower(s(x1)) -> p(s(p(s(twoto(p(s(p(s(tower(p(s(p(s(x1)))))))))))))) 34.20/9.76 twoto(0(x1)) -> s(0(x1)) 34.20/9.76 twoto(s(x1)) -> p(p(s(p(p(p(s(s(p(s(s(p(s(s(p(s(twice(p(s(p(s(p(p(p(s(s(s(twoto(p(s(p(s(x1)))))))))))))))))))))))))))))))) 34.20/9.76 twice(0(x1)) -> 0(x1) 34.20/9.76 twice(s(x1)) -> p(p(p(s(s(s(s(s(twice(p(p(p(s(s(s(x1))))))))))))))) 34.20/9.76 p(p(s(x1))) -> p(x1) 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 p(0(x1)) -> 0(s(s(s(s(s(s(s(s(x1))))))))) 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (37) UsableRulesProof (EQUIVALENT) 34.20/9.76 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. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (38) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TOWER(s(x1)) -> TOWER(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (39) QReductionProof (EQUIVALENT) 34.20/9.76 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 34.20/9.76 34.20/9.76 tower(s(x0)) 34.20/9.76 twoto(0(x0)) 34.20/9.76 twoto(s(x0)) 34.20/9.76 twice(0(x0)) 34.20/9.76 twice(s(x0)) 34.20/9.76 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (40) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 The TRS P consists of the following rules: 34.20/9.76 34.20/9.76 TOWER(s(x1)) -> TOWER(p(s(p(s(x1))))) 34.20/9.76 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (41) QDPOrderProof (EQUIVALENT) 34.20/9.76 We use the reduction pair processor [LPAR04,JAR06]. 34.20/9.76 34.20/9.76 34.20/9.76 The following pairs can be oriented strictly and are deleted. 34.20/9.76 34.20/9.76 TOWER(s(x1)) -> TOWER(p(s(p(s(x1))))) 34.20/9.76 The remaining pairs can at least be oriented weakly. 34.20/9.76 Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: 34.20/9.76 34.20/9.76 <<< 34.20/9.76 POL(TOWER(x_1)) = [[-I]] + [[0A, -I, -I]] * x_1 34.20/9.76 >>> 34.20/9.76 34.20/9.76 <<< 34.20/9.76 POL(s(x_1)) = [[1A], [0A], [-I]] + [[1A, -I, -I], [-I, 0A, 0A], [0A, -I, -I]] * x_1 34.20/9.76 >>> 34.20/9.76 34.20/9.76 <<< 34.20/9.76 POL(p(x_1)) = [[0A], [-I], [-I]] + [[-I, -I, 0A], [-I, 0A, -I], [-I, 0A, 0A]] * x_1 34.20/9.76 >>> 34.20/9.76 34.20/9.76 34.20/9.76 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (42) 34.20/9.76 Obligation: 34.20/9.76 Q DP problem: 34.20/9.76 P is empty. 34.20/9.76 The TRS R consists of the following rules: 34.20/9.76 34.20/9.76 p(s(x1)) -> x1 34.20/9.76 34.20/9.76 The set Q consists of the following terms: 34.20/9.76 34.20/9.76 p(s(x0)) 34.20/9.76 p(0(x0)) 34.20/9.76 34.20/9.76 We have to consider all minimal (P,Q,R)-chains. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (43) PisEmptyProof (EQUIVALENT) 34.20/9.76 The TRS P is empty. Hence, there is no (P,Q,R) chain. 34.20/9.76 ---------------------------------------- 34.20/9.76 34.20/9.76 (44) 34.20/9.76 YES 34.41/9.85 EOF