31.01/8.91 YES 32.96/9.46 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 32.96/9.46 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 32.96/9.46 32.96/9.46 32.96/9.46 Termination w.r.t. Q of the given QTRS could be proven: 32.96/9.46 32.96/9.46 (0) QTRS 32.96/9.46 (1) DependencyPairsProof [EQUIVALENT, 33 ms] 32.96/9.46 (2) QDP 32.96/9.46 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 32.96/9.46 (4) AND 32.96/9.46 (5) QDP 32.96/9.46 (6) UsableRulesProof [EQUIVALENT, 0 ms] 32.96/9.46 (7) QDP 32.96/9.46 (8) QDPSizeChangeProof [EQUIVALENT, 2 ms] 32.96/9.46 (9) YES 32.96/9.46 (10) QDP 32.96/9.46 (11) MNOCProof [EQUIVALENT, 0 ms] 32.96/9.46 (12) QDP 32.96/9.46 (13) UsableRulesProof [EQUIVALENT, 0 ms] 32.96/9.46 (14) QDP 32.96/9.46 (15) QDPOrderProof [EQUIVALENT, 447 ms] 32.96/9.46 (16) QDP 32.96/9.46 (17) DependencyGraphProof [EQUIVALENT, 0 ms] 32.96/9.46 (18) TRUE 32.96/9.46 32.96/9.46 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (0) 32.96/9.46 Obligation: 32.96/9.46 Q restricted rewrite system: 32.96/9.46 The TRS R consists of the following rules: 32.96/9.46 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 p(p(s(x1))) -> p(x1) 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 32.96/9.46 Q is empty. 32.96/9.46 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (1) DependencyPairsProof (EQUIVALENT) 32.96/9.46 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (2) 32.96/9.46 Obligation: 32.96/9.46 Q DP problem: 32.96/9.46 The TRS P consists of the following rules: 32.96/9.46 32.96/9.46 FOO(0(x1)) -> P(p(p(s(s(s(p(s(x1)))))))) 32.96/9.46 FOO(0(x1)) -> P(p(s(s(s(p(s(x1))))))) 32.96/9.46 FOO(0(x1)) -> P(s(s(s(p(s(x1)))))) 32.96/9.46 FOO(0(x1)) -> P(s(x1)) 32.96/9.46 FOO(s(x1)) -> P(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 FOO(s(x1)) -> P(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))) 32.96/9.46 FOO(s(x1)) -> P(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1))))))))))))))))))))))) 32.96/9.46 FOO(s(x1)) -> P(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))) 32.96/9.46 FOO(s(x1)) -> P(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1))))))))))))))))))) 32.96/9.46 FOO(s(x1)) -> P(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))) 32.96/9.46 FOO(s(x1)) -> FOO(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))) 32.96/9.46 FOO(s(x1)) -> P(p(s(s(p(s(bar(p(p(s(s(p(s(x1))))))))))))) 32.96/9.46 FOO(s(x1)) -> P(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))) 32.96/9.46 FOO(s(x1)) -> P(s(bar(p(p(s(s(p(s(x1))))))))) 32.96/9.46 FOO(s(x1)) -> BAR(p(p(s(s(p(s(x1))))))) 32.96/9.46 FOO(s(x1)) -> P(p(s(s(p(s(x1)))))) 32.96/9.46 FOO(s(x1)) -> P(s(s(p(s(x1))))) 32.96/9.46 FOO(s(x1)) -> P(s(x1)) 32.96/9.46 BAR(0(x1)) -> P(s(s(s(x1)))) 32.96/9.46 BAR(s(x1)) -> P(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 BAR(s(x1)) -> P(p(s(s(foo(s(p(p(s(s(x1)))))))))) 32.96/9.46 BAR(s(x1)) -> P(s(s(foo(s(p(p(s(s(x1))))))))) 32.96/9.46 BAR(s(x1)) -> FOO(s(p(p(s(s(x1)))))) 32.96/9.46 BAR(s(x1)) -> P(p(s(s(x1)))) 32.96/9.46 BAR(s(x1)) -> P(s(s(x1))) 32.96/9.46 P(p(s(x1))) -> P(x1) 32.96/9.46 32.96/9.46 The TRS R consists of the following rules: 32.96/9.46 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 p(p(s(x1))) -> p(x1) 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 32.96/9.46 Q is empty. 32.96/9.46 We have to consider all minimal (P,Q,R)-chains. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (3) DependencyGraphProof (EQUIVALENT) 32.96/9.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 22 less nodes. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (4) 32.96/9.46 Complex Obligation (AND) 32.96/9.46 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (5) 32.96/9.46 Obligation: 32.96/9.46 Q DP problem: 32.96/9.46 The TRS P consists of the following rules: 32.96/9.46 32.96/9.46 P(p(s(x1))) -> P(x1) 32.96/9.46 32.96/9.46 The TRS R consists of the following rules: 32.96/9.46 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 p(p(s(x1))) -> p(x1) 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 32.96/9.46 Q is empty. 32.96/9.46 We have to consider all minimal (P,Q,R)-chains. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (6) UsableRulesProof (EQUIVALENT) 32.96/9.46 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. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (7) 32.96/9.46 Obligation: 32.96/9.46 Q DP problem: 32.96/9.46 The TRS P consists of the following rules: 32.96/9.46 32.96/9.46 P(p(s(x1))) -> P(x1) 32.96/9.46 32.96/9.46 R is empty. 32.96/9.46 Q is empty. 32.96/9.46 We have to consider all minimal (P,Q,R)-chains. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (8) QDPSizeChangeProof (EQUIVALENT) 32.96/9.46 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. 32.96/9.46 32.96/9.46 From the DPs we obtained the following set of size-change graphs: 32.96/9.46 *P(p(s(x1))) -> P(x1) 32.96/9.46 The graph contains the following edges 1 > 1 32.96/9.46 32.96/9.46 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (9) 32.96/9.46 YES 32.96/9.46 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (10) 32.96/9.46 Obligation: 32.96/9.46 Q DP problem: 32.96/9.46 The TRS P consists of the following rules: 32.96/9.46 32.96/9.46 FOO(s(x1)) -> BAR(p(p(s(s(p(s(x1))))))) 32.96/9.46 BAR(s(x1)) -> FOO(s(p(p(s(s(x1)))))) 32.96/9.46 FOO(s(x1)) -> FOO(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))) 32.96/9.46 32.96/9.46 The TRS R consists of the following rules: 32.96/9.46 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 p(p(s(x1))) -> p(x1) 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 32.96/9.46 Q is empty. 32.96/9.46 We have to consider all minimal (P,Q,R)-chains. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (11) MNOCProof (EQUIVALENT) 32.96/9.46 We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (12) 32.96/9.46 Obligation: 32.96/9.46 Q DP problem: 32.96/9.46 The TRS P consists of the following rules: 32.96/9.46 32.96/9.46 FOO(s(x1)) -> BAR(p(p(s(s(p(s(x1))))))) 32.96/9.46 BAR(s(x1)) -> FOO(s(p(p(s(s(x1)))))) 32.96/9.46 FOO(s(x1)) -> FOO(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))) 32.96/9.46 32.96/9.46 The TRS R consists of the following rules: 32.96/9.46 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 p(p(s(x1))) -> p(x1) 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 32.96/9.46 The set Q consists of the following terms: 32.96/9.46 32.96/9.46 foo(0(x0)) 32.96/9.46 foo(s(x0)) 32.96/9.46 bar(0(x0)) 32.96/9.46 bar(s(x0)) 32.96/9.46 p(s(x0)) 32.96/9.46 p(0(x0)) 32.96/9.46 32.96/9.46 We have to consider all minimal (P,Q,R)-chains. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (13) UsableRulesProof (EQUIVALENT) 32.96/9.46 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. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (14) 32.96/9.46 Obligation: 32.96/9.46 Q DP problem: 32.96/9.46 The TRS P consists of the following rules: 32.96/9.46 32.96/9.46 FOO(s(x1)) -> BAR(p(p(s(s(p(s(x1))))))) 32.96/9.46 BAR(s(x1)) -> FOO(s(p(p(s(s(x1)))))) 32.96/9.46 FOO(s(x1)) -> FOO(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))) 32.96/9.46 32.96/9.46 The TRS R consists of the following rules: 32.96/9.46 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 32.96/9.46 The set Q consists of the following terms: 32.96/9.46 32.96/9.46 foo(0(x0)) 32.96/9.46 foo(s(x0)) 32.96/9.46 bar(0(x0)) 32.96/9.46 bar(s(x0)) 32.96/9.46 p(s(x0)) 32.96/9.46 p(0(x0)) 32.96/9.46 32.96/9.46 We have to consider all minimal (P,Q,R)-chains. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (15) QDPOrderProof (EQUIVALENT) 32.96/9.46 We use the reduction pair processor [LPAR04,JAR06]. 32.96/9.46 32.96/9.46 32.96/9.46 The following pairs can be oriented strictly and are deleted. 32.96/9.46 32.96/9.46 FOO(s(x1)) -> BAR(p(p(s(s(p(s(x1))))))) 32.96/9.46 FOO(s(x1)) -> FOO(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))) 32.96/9.46 The remaining pairs can at least be oriented weakly. 32.96/9.46 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 32.96/9.46 32.96/9.46 POL( BAR_1(x_1) ) = x_1 + 2 32.96/9.46 POL( FOO_1(x_1) ) = x_1 + 2 32.96/9.46 POL( p_1(x_1) ) = max{0, x_1 - 2} 32.96/9.46 POL( s_1(x_1) ) = x_1 + 2 32.96/9.46 POL( 0_1(x_1) ) = max{0, -2} 32.96/9.46 POL( bar_1(x_1) ) = x_1 + 1 32.96/9.46 POL( foo_1(x_1) ) = 2 32.96/9.46 32.96/9.46 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 32.96/9.46 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 32.96/9.46 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (16) 32.96/9.46 Obligation: 32.96/9.46 Q DP problem: 32.96/9.46 The TRS P consists of the following rules: 32.96/9.46 32.96/9.46 BAR(s(x1)) -> FOO(s(p(p(s(s(x1)))))) 32.96/9.46 32.96/9.46 The TRS R consists of the following rules: 32.96/9.46 32.96/9.46 p(s(x1)) -> x1 32.96/9.46 p(0(x1)) -> 0(s(s(s(s(x1))))) 32.96/9.46 bar(0(x1)) -> 0(p(s(s(s(x1))))) 32.96/9.46 bar(s(x1)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x1)))))))))))) 32.96/9.46 foo(s(x1)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x1)))))))))))))))))))))))))) 32.96/9.46 foo(0(x1)) -> 0(s(p(p(p(s(s(s(p(s(x1)))))))))) 32.96/9.46 32.96/9.46 The set Q consists of the following terms: 32.96/9.46 32.96/9.46 foo(0(x0)) 32.96/9.46 foo(s(x0)) 32.96/9.46 bar(0(x0)) 32.96/9.46 bar(s(x0)) 32.96/9.46 p(s(x0)) 32.96/9.46 p(0(x0)) 32.96/9.46 32.96/9.46 We have to consider all minimal (P,Q,R)-chains. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (17) DependencyGraphProof (EQUIVALENT) 32.96/9.46 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 32.96/9.46 ---------------------------------------- 32.96/9.46 32.96/9.46 (18) 32.96/9.46 TRUE 33.38/9.54 EOF