3.34/1.55 YES 3.34/1.56 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.34/1.56 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.34/1.56 3.34/1.56 3.34/1.56 Termination w.r.t. Q of the given QTRS could be proven: 3.34/1.56 3.34/1.56 (0) QTRS 3.34/1.56 (1) QTRSRRRProof [EQUIVALENT, 0 ms] 3.34/1.56 (2) QTRS 3.34/1.56 (3) RisEmptyProof [EQUIVALENT, 0 ms] 3.34/1.56 (4) YES 3.34/1.56 3.34/1.56 3.34/1.56 ---------------------------------------- 3.34/1.56 3.34/1.56 (0) 3.34/1.56 Obligation: 3.34/1.56 Q restricted rewrite system: 3.34/1.56 The TRS R consists of the following rules: 3.34/1.56 3.34/1.56 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 3.34/1.56 a____(X, nil) -> mark(X) 3.34/1.56 a____(nil, X) -> mark(X) 3.34/1.56 a__and(tt, X) -> mark(X) 3.34/1.56 a__isNePal(__(I, __(P, I))) -> tt 3.34/1.56 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 3.34/1.56 mark(and(X1, X2)) -> a__and(mark(X1), X2) 3.34/1.56 mark(isNePal(X)) -> a__isNePal(mark(X)) 3.34/1.56 mark(nil) -> nil 3.34/1.56 mark(tt) -> tt 3.34/1.56 a____(X1, X2) -> __(X1, X2) 3.34/1.56 a__and(X1, X2) -> and(X1, X2) 3.34/1.56 a__isNePal(X) -> isNePal(X) 3.34/1.56 3.34/1.56 The set Q consists of the following terms: 3.34/1.56 3.34/1.56 mark(__(x0, x1)) 3.34/1.56 mark(and(x0, x1)) 3.34/1.56 mark(isNePal(x0)) 3.34/1.56 mark(nil) 3.34/1.56 mark(tt) 3.34/1.56 a____(x0, x1) 3.34/1.56 a__and(x0, x1) 3.34/1.56 a__isNePal(x0) 3.34/1.56 3.34/1.56 3.34/1.56 ---------------------------------------- 3.34/1.56 3.34/1.56 (1) QTRSRRRProof (EQUIVALENT) 3.34/1.56 Used ordering: 3.34/1.56 Knuth-Bendix order [KBO] with precedence:mark_1 > a__isNePal_1 > tt > a__and_2 > nil > a_____2 > isNePal_1 > and_2 > ___2 3.34/1.56 3.34/1.56 and weight map: 3.34/1.56 3.34/1.56 nil=1 3.34/1.56 tt=4 3.34/1.56 mark_1=0 3.34/1.56 a__isNePal_1=1 3.34/1.56 isNePal_1=1 3.34/1.56 ___2=0 3.34/1.56 a_____2=0 3.34/1.56 a__and_2=0 3.34/1.56 and_2=0 3.34/1.56 3.34/1.56 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.34/1.56 3.34/1.56 a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) 3.34/1.56 a____(X, nil) -> mark(X) 3.34/1.56 a____(nil, X) -> mark(X) 3.34/1.56 a__and(tt, X) -> mark(X) 3.34/1.56 a__isNePal(__(I, __(P, I))) -> tt 3.34/1.56 mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) 3.34/1.56 mark(and(X1, X2)) -> a__and(mark(X1), X2) 3.34/1.56 mark(isNePal(X)) -> a__isNePal(mark(X)) 3.34/1.56 mark(nil) -> nil 3.34/1.56 mark(tt) -> tt 3.34/1.56 a____(X1, X2) -> __(X1, X2) 3.34/1.56 a__and(X1, X2) -> and(X1, X2) 3.34/1.56 a__isNePal(X) -> isNePal(X) 3.34/1.56 3.34/1.56 3.34/1.56 3.34/1.56 3.34/1.56 ---------------------------------------- 3.34/1.56 3.34/1.56 (2) 3.34/1.56 Obligation: 3.34/1.56 Q restricted rewrite system: 3.34/1.56 R is empty. 3.34/1.56 The set Q consists of the following terms: 3.34/1.56 3.34/1.56 mark(__(x0, x1)) 3.34/1.56 mark(and(x0, x1)) 3.34/1.56 mark(isNePal(x0)) 3.34/1.56 mark(nil) 3.34/1.56 mark(tt) 3.34/1.56 a____(x0, x1) 3.34/1.56 a__and(x0, x1) 3.34/1.56 a__isNePal(x0) 3.34/1.56 3.34/1.56 3.34/1.56 ---------------------------------------- 3.34/1.56 3.34/1.56 (3) RisEmptyProof (EQUIVALENT) 3.34/1.56 The TRS R is empty. Hence, termination is trivially proven. 3.34/1.56 ---------------------------------------- 3.34/1.56 3.34/1.56 (4) 3.34/1.56 YES 3.34/1.58 EOF