3.52/1.73 YES 3.52/1.74 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 3.52/1.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.52/1.74 3.52/1.74 3.52/1.74 Termination w.r.t. Q of the given QTRS could be proven: 3.52/1.74 3.52/1.74 (0) QTRS 3.52/1.74 (1) QTRSToCSRProof [SOUND, 0 ms] 3.52/1.74 (2) CSR 3.52/1.74 (3) CSRRRRProof [EQUIVALENT, 28 ms] 3.52/1.74 (4) CSR 3.52/1.74 (5) CSRRRRProof [EQUIVALENT, 0 ms] 3.52/1.74 (6) CSR 3.52/1.74 (7) RisEmptyProof [EQUIVALENT, 0 ms] 3.52/1.74 (8) YES 3.52/1.74 3.52/1.74 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (0) 3.52/1.74 Obligation: 3.52/1.74 Q restricted rewrite system: 3.52/1.74 The TRS R consists of the following rules: 3.52/1.74 3.52/1.74 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 3.52/1.74 active(__(X, nil)) -> mark(X) 3.52/1.74 active(__(nil, X)) -> mark(X) 3.52/1.74 active(and(tt, X)) -> mark(X) 3.52/1.74 active(isNePal(__(I, __(P, I)))) -> mark(tt) 3.52/1.74 active(__(X1, X2)) -> __(active(X1), X2) 3.52/1.74 active(__(X1, X2)) -> __(X1, active(X2)) 3.52/1.74 active(and(X1, X2)) -> and(active(X1), X2) 3.52/1.74 active(isNePal(X)) -> isNePal(active(X)) 3.52/1.74 __(mark(X1), X2) -> mark(__(X1, X2)) 3.52/1.74 __(X1, mark(X2)) -> mark(__(X1, X2)) 3.52/1.74 and(mark(X1), X2) -> mark(and(X1, X2)) 3.52/1.74 isNePal(mark(X)) -> mark(isNePal(X)) 3.52/1.74 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 3.52/1.74 proper(nil) -> ok(nil) 3.52/1.74 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 3.52/1.74 proper(tt) -> ok(tt) 3.52/1.74 proper(isNePal(X)) -> isNePal(proper(X)) 3.52/1.74 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 3.52/1.74 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 3.52/1.74 isNePal(ok(X)) -> ok(isNePal(X)) 3.52/1.74 top(mark(X)) -> top(proper(X)) 3.52/1.74 top(ok(X)) -> top(active(X)) 3.52/1.74 3.52/1.74 The set Q consists of the following terms: 3.52/1.74 3.52/1.74 active(__(x0, x1)) 3.52/1.74 active(and(x0, x1)) 3.52/1.74 active(isNePal(x0)) 3.52/1.74 __(mark(x0), x1) 3.52/1.74 __(x0, mark(x1)) 3.52/1.74 and(mark(x0), x1) 3.52/1.74 isNePal(mark(x0)) 3.52/1.74 proper(__(x0, x1)) 3.52/1.74 proper(nil) 3.52/1.74 proper(and(x0, x1)) 3.52/1.74 proper(tt) 3.52/1.74 proper(isNePal(x0)) 3.52/1.74 __(ok(x0), ok(x1)) 3.52/1.74 and(ok(x0), ok(x1)) 3.52/1.74 isNePal(ok(x0)) 3.52/1.74 top(mark(x0)) 3.52/1.74 top(ok(x0)) 3.52/1.74 3.52/1.74 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (1) QTRSToCSRProof (SOUND) 3.52/1.74 The following Q TRS is given: Q restricted rewrite system: 3.52/1.74 The TRS R consists of the following rules: 3.52/1.74 3.52/1.74 active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z))) 3.52/1.74 active(__(X, nil)) -> mark(X) 3.52/1.74 active(__(nil, X)) -> mark(X) 3.52/1.74 active(and(tt, X)) -> mark(X) 3.52/1.74 active(isNePal(__(I, __(P, I)))) -> mark(tt) 3.52/1.74 active(__(X1, X2)) -> __(active(X1), X2) 3.52/1.74 active(__(X1, X2)) -> __(X1, active(X2)) 3.52/1.74 active(and(X1, X2)) -> and(active(X1), X2) 3.52/1.74 active(isNePal(X)) -> isNePal(active(X)) 3.52/1.74 __(mark(X1), X2) -> mark(__(X1, X2)) 3.52/1.74 __(X1, mark(X2)) -> mark(__(X1, X2)) 3.52/1.74 and(mark(X1), X2) -> mark(and(X1, X2)) 3.52/1.74 isNePal(mark(X)) -> mark(isNePal(X)) 3.52/1.74 proper(__(X1, X2)) -> __(proper(X1), proper(X2)) 3.52/1.74 proper(nil) -> ok(nil) 3.52/1.74 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 3.52/1.74 proper(tt) -> ok(tt) 3.52/1.74 proper(isNePal(X)) -> isNePal(proper(X)) 3.52/1.74 __(ok(X1), ok(X2)) -> ok(__(X1, X2)) 3.52/1.74 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 3.52/1.74 isNePal(ok(X)) -> ok(isNePal(X)) 3.52/1.74 top(mark(X)) -> top(proper(X)) 3.52/1.74 top(ok(X)) -> top(active(X)) 3.52/1.74 3.52/1.74 The set Q consists of the following terms: 3.52/1.74 3.52/1.74 active(__(x0, x1)) 3.52/1.74 active(and(x0, x1)) 3.52/1.74 active(isNePal(x0)) 3.52/1.74 __(mark(x0), x1) 3.52/1.74 __(x0, mark(x1)) 3.52/1.74 and(mark(x0), x1) 3.52/1.74 isNePal(mark(x0)) 3.52/1.74 proper(__(x0, x1)) 3.52/1.74 proper(nil) 3.52/1.74 proper(and(x0, x1)) 3.52/1.74 proper(tt) 3.52/1.74 proper(isNePal(x0)) 3.52/1.74 __(ok(x0), ok(x1)) 3.52/1.74 and(ok(x0), ok(x1)) 3.52/1.74 isNePal(ok(x0)) 3.52/1.74 top(mark(x0)) 3.52/1.74 top(ok(x0)) 3.52/1.74 3.52/1.74 Special symbols used for the transformation (see [GM04]): 3.52/1.74 top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1 3.52/1.74 The replacement map contains the following entries: 3.52/1.74 3.52/1.74 __: {1, 2} 3.52/1.74 nil: empty set 3.52/1.74 and: {1} 3.52/1.74 tt: empty set 3.52/1.74 isNePal: {1} 3.52/1.74 The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete. 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (2) 3.52/1.74 Obligation: 3.52/1.74 Context-sensitive rewrite system: 3.52/1.74 The TRS R consists of the following rules: 3.52/1.74 3.52/1.74 __(__(X, Y), Z) -> __(X, __(Y, Z)) 3.52/1.74 __(X, nil) -> X 3.52/1.74 __(nil, X) -> X 3.52/1.74 and(tt, X) -> X 3.52/1.74 isNePal(__(I, __(P, I))) -> tt 3.52/1.74 3.52/1.74 The replacement map contains the following entries: 3.52/1.74 3.52/1.74 __: {1, 2} 3.52/1.74 nil: empty set 3.52/1.74 and: {1} 3.52/1.74 tt: empty set 3.52/1.74 isNePal: {1} 3.52/1.74 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (3) CSRRRRProof (EQUIVALENT) 3.52/1.74 The following CSR is given: Context-sensitive rewrite system: 3.52/1.74 The TRS R consists of the following rules: 3.52/1.74 3.52/1.74 __(__(X, Y), Z) -> __(X, __(Y, Z)) 3.52/1.74 __(X, nil) -> X 3.52/1.74 __(nil, X) -> X 3.52/1.74 and(tt, X) -> X 3.52/1.74 isNePal(__(I, __(P, I))) -> tt 3.52/1.74 3.52/1.74 The replacement map contains the following entries: 3.52/1.74 3.52/1.74 __: {1, 2} 3.52/1.74 nil: empty set 3.52/1.74 and: {1} 3.52/1.74 tt: empty set 3.52/1.74 isNePal: {1} 3.52/1.74 Used ordering: 3.52/1.74 Polynomial interpretation [POLO]: 3.52/1.74 3.52/1.74 POL(__(x_1, x_2)) = 1 + 2*x_1 + x_2 3.52/1.74 POL(and(x_1, x_2)) = 2*x_1 + x_2 3.52/1.74 POL(isNePal(x_1)) = 1 + 2*x_1 3.52/1.74 POL(nil) = 1 3.52/1.74 POL(tt) = 0 3.52/1.74 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.52/1.74 3.52/1.74 __(__(X, Y), Z) -> __(X, __(Y, Z)) 3.52/1.74 __(X, nil) -> X 3.52/1.74 __(nil, X) -> X 3.52/1.74 isNePal(__(I, __(P, I))) -> tt 3.52/1.74 3.52/1.74 3.52/1.74 3.52/1.74 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (4) 3.52/1.74 Obligation: 3.52/1.74 Context-sensitive rewrite system: 3.52/1.74 The TRS R consists of the following rules: 3.52/1.74 3.52/1.74 and(tt, X) -> X 3.52/1.74 3.52/1.74 The replacement map contains the following entries: 3.52/1.74 3.52/1.74 and: {1} 3.52/1.74 tt: empty set 3.52/1.74 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (5) CSRRRRProof (EQUIVALENT) 3.52/1.74 The following CSR is given: Context-sensitive rewrite system: 3.52/1.74 The TRS R consists of the following rules: 3.52/1.74 3.52/1.74 and(tt, X) -> X 3.52/1.74 3.52/1.74 The replacement map contains the following entries: 3.52/1.74 3.52/1.74 and: {1} 3.52/1.74 tt: empty set 3.52/1.74 Used ordering: 3.52/1.74 Polynomial interpretation [POLO]: 3.52/1.74 3.52/1.74 POL(and(x_1, x_2)) = x_1 + x_2 3.52/1.74 POL(tt) = 1 3.52/1.74 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 3.52/1.74 3.52/1.74 and(tt, X) -> X 3.52/1.74 3.52/1.74 3.52/1.74 3.52/1.74 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (6) 3.52/1.74 Obligation: 3.52/1.74 Context-sensitive rewrite system: 3.52/1.74 R is empty. 3.52/1.74 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (7) RisEmptyProof (EQUIVALENT) 3.52/1.74 The CSR R is empty. Hence, termination is trivially proven. 3.52/1.74 ---------------------------------------- 3.52/1.74 3.52/1.74 (8) 3.52/1.74 YES 3.52/1.75 EOF