/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 260 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 20 ms] (4) QTRS (5) QTRSRRRProof [EQUIVALENT, 0 ms] (6) QTRS (7) QTRSRRRProof [EQUIVALENT, 0 ms] (8) QTRS (9) RisEmptyProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U11(tt, N) -> activate(N) U21(tt, M, N) -> s(plus(activate(N), activate(M))) U31(tt) -> 0 U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) plus(N, 0) -> U11(isNat(N), N) plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) x(N, 0) -> U31(isNat(N)) x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNat(X) -> n__isNat(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(X) -> X Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: U11/2(YES,YES) tt/0) activate/1)YES( U21/3(YES,YES,YES) s/1(YES) plus/2(YES,YES) U31/1(YES) 0/0) U41/3(YES,YES,YES) x/2(YES,YES) and/2(YES,YES) isNat/1(YES) n__0/0) n__plus/2(YES,YES) n__isNat/1(YES) n__s/1(YES) n__x/2(YES,YES) Quasi precedence: [U41_3, x_2, n__x_2] > [U21_3, plus_2, n__plus_2] > [U11_2, tt, s_1, and_2, isNat_1, n__isNat_1, n__s_1] [U41_3, x_2, n__x_2] > [U31_1, 0, n__0] > [U11_2, tt, s_1, and_2, isNat_1, n__isNat_1, n__s_1] Status: U11_2: multiset status tt: multiset status U21_3: [2,3,1] s_1: multiset status plus_2: [2,1] U31_1: multiset status 0: multiset status U41_3: [2,3,1] x_2: [2,1] and_2: multiset status isNat_1: multiset status n__0: multiset status n__plus_2: [2,1] n__isNat_1: multiset status n__s_1: multiset status n__x_2: [2,1] With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U11(tt, N) -> activate(N) U21(tt, M, N) -> s(plus(activate(N), activate(M))) U31(tt) -> 0 U41(tt, M, N) -> plus(x(activate(N), activate(M)), activate(N)) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) isNat(n__s(V1)) -> isNat(activate(V1)) isNat(n__x(V1, V2)) -> and(isNat(activate(V1)), n__isNat(activate(V2))) plus(N, 0) -> U11(isNat(N), N) plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) x(N, 0) -> U31(isNat(N)) x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNat(X) -> n__isNat(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(X) -> X Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 1 POL(activate(x_1)) = 2*x_1 POL(isNat(x_1)) = 1 + 2*x_1 POL(n__0) = 1 POL(n__isNat(x_1)) = 1 + x_1 POL(n__plus(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(n__s(x_1)) = x_1 POL(n__x(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(plus(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(s(x_1)) = x_1 POL(x(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) ---------------------------------------- (4) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNat(X) -> n__isNat(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: Polynomial interpretation [POLO]: POL(0) = 2 POL(activate(x_1)) = x_1 POL(isNat(x_1)) = 1 + 2*x_1 POL(n__0) = 2 POL(n__isNat(x_1)) = 2*x_1 POL(n__plus(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(n__s(x_1)) = 2*x_1 POL(n__x(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(plus(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(s(x_1)) = 2*x_1 POL(x(x_1, x_2)) = 2 + x_1 + 2*x_2 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: isNat(X) -> n__isNat(X) ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__s(X)) -> s(activate(X)) activate(X) -> X Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:0 > x_2 > activate_1 > plus_2 > n__plus_2 > n__x_2 > s_1 > n__s_1 > n__0 and weight map: 0=1 n__0=1 s_1=1 n__s_1=1 activate_1=2 plus_2=0 n__plus_2=0 x_2=1 n__x_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: 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) activate(n__s(X)) -> s(activate(X)) activate(X) -> X ---------------------------------------- (8) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (9) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (10) YES