/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 141 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4) QTRS (5) RisEmptyProof [EQUIVALENT, 0 ms] (6) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__and(tt, X) -> mark(X) a__isList(V) -> a__isNeList(V) a__isList(nil) -> tt a__isList(__(V1, V2)) -> a__and(a__isList(V1), isList(V2)) a__isNeList(V) -> a__isQid(V) a__isNeList(__(V1, V2)) -> a__and(a__isList(V1), isNeList(V2)) a__isNeList(__(V1, V2)) -> a__and(a__isNeList(V1), isList(V2)) a__isNePal(V) -> a__isQid(V) a__isNePal(__(I, __(P, I))) -> a__and(a__isQid(I), isPal(P)) a__isPal(V) -> a__isNePal(V) a__isPal(nil) -> tt a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isList(X)) -> a__isList(X) mark(isNeList(X)) -> a__isNeList(X) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isList(X) -> isList(X) a__isNeList(X) -> isNeList(X) a__isQid(X) -> isQid(X) a__isNePal(X) -> isNePal(X) a__isPal(X) -> isPal(X) The set Q consists of the following terms: a__isList(x0) a__isNeList(x0) a__isNePal(x0) a__isPal(x0) mark(__(x0, x1)) mark(and(x0, x1)) mark(isList(x0)) mark(isNeList(x0)) mark(isQid(x0)) mark(isNePal(x0)) mark(isPal(x0)) mark(nil) mark(tt) mark(a) mark(e) mark(i) mark(o) mark(u) a____(x0, x1) a__and(x0, x1) a__isQid(x0) ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(__(x_1, x_2)) = x_1 + x_2 POL(a) = 2 POL(a____(x_1, x_2)) = x_1 + x_2 POL(a__and(x_1, x_2)) = x_1 + x_2 POL(a__isList(x_1)) = x_1 POL(a__isNeList(x_1)) = x_1 POL(a__isNePal(x_1)) = x_1 POL(a__isPal(x_1)) = x_1 POL(a__isQid(x_1)) = x_1 POL(and(x_1, x_2)) = x_1 + x_2 POL(e) = 2 POL(i) = 1 POL(isList(x_1)) = x_1 POL(isNeList(x_1)) = x_1 POL(isNePal(x_1)) = x_1 POL(isPal(x_1)) = x_1 POL(isQid(x_1)) = x_1 POL(mark(x_1)) = x_1 POL(nil) = 2 POL(o) = 1 POL(tt) = 0 POL(u) = 0 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a____(X, nil) -> mark(X) a____(nil, X) -> mark(X) a__isList(nil) -> tt a__isPal(nil) -> tt a__isQid(a) -> tt a__isQid(e) -> tt a__isQid(i) -> tt a__isQid(o) -> tt ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a__and(tt, X) -> mark(X) a__isList(V) -> a__isNeList(V) a__isList(__(V1, V2)) -> a__and(a__isList(V1), isList(V2)) a__isNeList(V) -> a__isQid(V) a__isNeList(__(V1, V2)) -> a__and(a__isList(V1), isNeList(V2)) a__isNeList(__(V1, V2)) -> a__and(a__isNeList(V1), isList(V2)) a__isNePal(V) -> a__isQid(V) a__isNePal(__(I, __(P, I))) -> a__and(a__isQid(I), isPal(P)) a__isPal(V) -> a__isNePal(V) a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isList(X)) -> a__isList(X) mark(isNeList(X)) -> a__isNeList(X) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isList(X) -> isList(X) a__isNeList(X) -> isNeList(X) a__isQid(X) -> isQid(X) a__isNePal(X) -> isNePal(X) a__isPal(X) -> isPal(X) The set Q consists of the following terms: a__isList(x0) a__isNeList(x0) a__isNePal(x0) a__isPal(x0) mark(__(x0, x1)) mark(and(x0, x1)) mark(isList(x0)) mark(isNeList(x0)) mark(isQid(x0)) mark(isNePal(x0)) mark(isPal(x0)) mark(nil) mark(tt) mark(a) mark(e) mark(i) mark(o) mark(u) a____(x0, x1) a__and(x0, x1) a__isQid(x0) ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:mark_1 > o > i > a > a__and_2 > e > nil > a__isPal_1 > a__isNePal_1 > isNePal_1 > a__isQid_1 > isQid_1 > and_2 > u > a__isNeList_1 > isNeList_1 > a__isList_1 > tt > a_____2 > ___2 > isPal_1 > isList_1 and weight map: tt=2 u=1 nil=1 a=1 e=1 i=1 o=1 mark_1=0 a__isList_1=3 a__isNeList_1=2 isList_1=3 a__isQid_1=1 isNeList_1=2 a__isNePal_1=1 isPal_1=1 a__isPal_1=1 isQid_1=1 isNePal_1=1 ___2=4 a_____2=4 a__and_2=0 and_2=0 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a____(__(X, Y), Z) -> a____(mark(X), a____(mark(Y), mark(Z))) a__and(tt, X) -> mark(X) a__isList(V) -> a__isNeList(V) a__isList(__(V1, V2)) -> a__and(a__isList(V1), isList(V2)) a__isNeList(V) -> a__isQid(V) a__isNeList(__(V1, V2)) -> a__and(a__isList(V1), isNeList(V2)) a__isNeList(__(V1, V2)) -> a__and(a__isNeList(V1), isList(V2)) a__isNePal(V) -> a__isQid(V) a__isNePal(__(I, __(P, I))) -> a__and(a__isQid(I), isPal(P)) a__isPal(V) -> a__isNePal(V) a__isQid(u) -> tt mark(__(X1, X2)) -> a____(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isList(X)) -> a__isList(X) mark(isNeList(X)) -> a__isNeList(X) mark(isQid(X)) -> a__isQid(X) mark(isNePal(X)) -> a__isNePal(X) mark(isPal(X)) -> a__isPal(X) mark(nil) -> nil mark(tt) -> tt mark(a) -> a mark(e) -> e mark(i) -> i mark(o) -> o mark(u) -> u a____(X1, X2) -> __(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isList(X) -> isList(X) a__isNeList(X) -> isNeList(X) a__isQid(X) -> isQid(X) a__isNePal(X) -> isNePal(X) a__isPal(X) -> isPal(X) ---------------------------------------- (4) Obligation: Q restricted rewrite system: R is empty. The set Q consists of the following terms: a__isList(x0) a__isNeList(x0) a__isNePal(x0) a__isPal(x0) mark(__(x0, x1)) mark(and(x0, x1)) mark(isList(x0)) mark(isNeList(x0)) mark(isQid(x0)) mark(isNePal(x0)) mark(isPal(x0)) mark(nil) mark(tt) mark(a) mark(e) mark(i) mark(o) mark(u) a____(x0, x1) a__and(x0, x1) a__isQid(x0) ---------------------------------------- (5) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (6) YES