YES proof of Transformed_CSR_04_Ex2_Luc03b_Z.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 67 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 10 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 0 ms] (8) QTRS (9) QTRSRRRProof [EQUIVALENT, 0 ms] (10) QTRS (11) RisEmptyProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: fst(0, Z) -> nil fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z))) from(X) -> cons(X, n__from(s(X))) add(0, X) -> X add(s(X), Y) -> s(n__add(activate(X), Y)) len(nil) -> 0 len(cons(X, Z)) -> s(n__len(activate(Z))) fst(X1, X2) -> n__fst(X1, X2) from(X) -> n__from(X) add(X1, X2) -> n__add(X1, X2) len(X) -> n__len(X) activate(n__fst(X1, X2)) -> fst(X1, X2) activate(n__from(X)) -> from(X) activate(n__add(X1, X2)) -> add(X1, X2) activate(n__len(X)) -> len(X) activate(X) -> X Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(activate(x_1)) = 2 + 2*x_1 POL(add(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(cons(x_1, x_2)) = 2 + x_1 + x_2 POL(from(x_1)) = 2 + 2*x_1 POL(fst(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(len(x_1)) = 1 + 2*x_1 POL(n__add(x_1, x_2)) = x_1 + x_2 POL(n__from(x_1)) = x_1 POL(n__fst(x_1, x_2)) = x_1 + x_2 POL(n__len(x_1)) = 1 + x_1 POL(nil) = 1 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: fst(0, Z) -> nil add(0, X) -> X len(nil) -> 0 len(cons(X, Z)) -> s(n__len(activate(Z))) fst(X1, X2) -> n__fst(X1, X2) from(X) -> n__from(X) add(X1, X2) -> n__add(X1, X2) activate(n__len(X)) -> len(X) activate(X) -> X ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z))) from(X) -> cons(X, n__from(s(X))) add(s(X), Y) -> s(n__add(activate(X), Y)) len(X) -> n__len(X) activate(n__fst(X1, X2)) -> fst(X1, X2) activate(n__from(X)) -> from(X) activate(n__add(X1, X2)) -> add(X1, X2) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(activate(x_1)) = 2*x_1 POL(add(x_1, x_2)) = 2*x_1 + 2*x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(from(x_1)) = 2*x_1 POL(fst(x_1, x_2)) = 2*x_1 + 2*x_2 POL(len(x_1)) = 2 + 2*x_1 POL(n__add(x_1, x_2)) = x_1 + 2*x_2 POL(n__from(x_1)) = x_1 POL(n__fst(x_1, x_2)) = x_1 + x_2 POL(n__len(x_1)) = 1 + x_1 POL(s(x_1)) = x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: len(X) -> n__len(X) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z))) from(X) -> cons(X, n__from(s(X))) add(s(X), Y) -> s(n__add(activate(X), Y)) activate(n__fst(X1, X2)) -> fst(X1, X2) activate(n__from(X)) -> from(X) activate(n__add(X1, X2)) -> add(X1, X2) Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(activate(x_1)) = 2 + 2*x_1 POL(add(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(cons(x_1, x_2)) = x_1 + x_2 POL(from(x_1)) = 2 + 2*x_1 POL(fst(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(n__add(x_1, x_2)) = x_1 + x_2 POL(n__from(x_1)) = x_1 POL(n__fst(x_1, x_2)) = 1 + x_1 + x_2 POL(s(x_1)) = 2 + x_1 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: add(s(X), Y) -> s(n__add(activate(X), Y)) activate(n__fst(X1, X2)) -> fst(X1, X2) activate(n__add(X1, X2)) -> add(X1, X2) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z))) from(X) -> cons(X, n__from(s(X))) activate(n__from(X)) -> from(X) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(activate(x_1)) = 2 + 2*x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(from(x_1)) = 1 + 2*x_1 POL(fst(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(n__from(x_1)) = x_1 POL(n__fst(x_1, x_2)) = x_1 + x_2 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: activate(n__from(X)) -> from(X) ---------------------------------------- (8) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z))) from(X) -> cons(X, n__from(s(X))) Q is empty. ---------------------------------------- (9) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(activate(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(from(x_1)) = 2 + 2*x_1 POL(fst(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(n__from(x_1)) = x_1 POL(n__fst(x_1, x_2)) = x_1 + x_2 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: fst(s(X), cons(Y, Z)) -> cons(Y, n__fst(activate(X), activate(Z))) from(X) -> cons(X, n__from(s(X))) ---------------------------------------- (10) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (11) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (12) YES