YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 119 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 3 ms] (6) QTRS (7) RisEmptyProof [EQUIVALENT, 0 ms] (8) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__and(true, X) -> mark(X) a__and(false, Y) -> false a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__first(0, X) -> nil a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) a__from(X) -> cons(X, from(s(X))) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(add(X1, X2)) -> a__add(mark(X1), X2) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(from(X)) -> a__from(X) mark(true) -> true mark(false) -> false mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__and(X1, X2) -> and(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) a__add(X1, X2) -> add(X1, X2) a__first(X1, X2) -> first(X1, X2) a__from(X) -> from(X) The set Q consists of the following terms: a__from(x0) mark(and(x0, x1)) mark(if(x0, x1, x2)) mark(add(x0, x1)) mark(first(x0, x1)) mark(from(x0)) mark(true) mark(false) mark(0) mark(s(x0)) mark(nil) mark(cons(x0, x1)) a__and(x0, x1) a__if(x0, x1, x2) a__add(x0, x1) a__first(x0, x1) ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(a__add(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(a__and(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(a__first(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(a__from(x_1)) = 2 + 2*x_1 POL(a__if(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(add(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(and(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(cons(x_1, x_2)) = 1 + x_1 + x_2 POL(false) = 1 POL(first(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(from(x_1)) = 1 + x_1 POL(if(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(mark(x_1)) = 2*x_1 POL(nil) = 1 POL(s(x_1)) = x_1 POL(true) = 1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__and(true, X) -> mark(X) a__and(false, Y) -> false a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(add(X, Y)) a__first(0, X) -> nil a__first(s(X), cons(Y, Z)) -> cons(Y, first(X, Z)) mark(true) -> true mark(false) -> false mark(0) -> 0 mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) a__and(X1, X2) -> and(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) a__add(X1, X2) -> add(X1, X2) a__first(X1, X2) -> first(X1, X2) a__from(X) -> from(X) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__from(X) -> cons(X, from(s(X))) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(add(X1, X2)) -> a__add(mark(X1), X2) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(from(X)) -> a__from(X) mark(s(X)) -> s(X) The set Q consists of the following terms: a__from(x0) mark(and(x0, x1)) mark(if(x0, x1, x2)) mark(add(x0, x1)) mark(first(x0, x1)) mark(from(x0)) mark(true) mark(false) mark(0) mark(s(x0)) mark(nil) mark(cons(x0, x1)) a__and(x0, x1) a__if(x0, x1, x2) a__add(x0, x1) a__first(x0, x1) ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a__add(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(a__and(x_1, x_2)) = x_1 + 2*x_2 POL(a__first(x_1, x_2)) = x_1 + x_2 POL(a__from(x_1)) = 2 + 2*x_1 POL(a__if(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 POL(add(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(and(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(first(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(from(x_1)) = x_1 POL(if(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + x_3 POL(mark(x_1)) = 2 + 2*x_1 POL(s(x_1)) = 1 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__from(X) -> cons(X, from(s(X))) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(add(X1, X2)) -> a__add(mark(X1), X2) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(s(X)) -> s(X) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: mark(from(X)) -> a__from(X) The set Q consists of the following terms: a__from(x0) mark(and(x0, x1)) mark(if(x0, x1, x2)) mark(add(x0, x1)) mark(first(x0, x1)) mark(from(x0)) mark(true) mark(false) mark(0) mark(s(x0)) mark(nil) mark(cons(x0, x1)) a__and(x0, x1) a__if(x0, x1, x2) a__add(x0, x1) a__first(x0, x1) ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(a__from(x_1)) = 2 + 2*x_1 POL(from(x_1)) = 1 + x_1 POL(mark(x_1)) = 2 + 2*x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: mark(from(X)) -> a__from(X) ---------------------------------------- (6) Obligation: Q restricted rewrite system: R is empty. The set Q consists of the following terms: a__from(x0) mark(and(x0, x1)) mark(if(x0, x1, x2)) mark(add(x0, x1)) mark(first(x0, x1)) mark(from(x0)) mark(true) mark(false) mark(0) mark(s(x0)) mark(nil) mark(cons(x0, x1)) a__and(x0, x1) a__if(x0, x1, x2) a__add(x0, x1) a__first(x0, x1) ---------------------------------------- (7) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (8) YES