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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 119 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4)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, 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))) 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)) plus(N, 0) -> U11(isNat(N), N) plus(N, s(M)) -> U21(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) 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(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) and/2(YES,YES) isNat/1)YES( n__0/0) n__plus/2(YES,YES) n__isNat/1)YES( n__s/1(YES) 0/0) Quasi precedence: [U21_3, plus_2, n__plus_2] > U11_2 [U21_3, plus_2, n__plus_2] > [s_1, n__s_1] > and_2 [n__0, 0] > tt Status: U11_2: multiset status tt: multiset status U21_3: [3,2,1] s_1: multiset status plus_2: [1,2] and_2: multiset status n__0: multiset status n__plus_2: [1,2] n__s_1: multiset status 0: multiset status 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))) 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)) plus(N, 0) -> U11(isNat(N), N) plus(N, s(M)) -> U21(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) 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(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)) = 2*x_1 POL(n__0) = 1 POL(n__isNat(x_1)) = 2*x_1 POL(n__plus(x_1, x_2)) = 2*x_1 + 2*x_2 POL(n__s(x_1)) = 1 + x_1 POL(plus(x_1, x_2)) = 2*x_1 + 2*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__0) -> 0 activate(n__s(X)) -> s(activate(X)) ---------------------------------------- (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) activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. ---------------------------------------- (5) QTRSRRRProof (EQUIVALENT) Used ordering: 0/0) n__0/0) plus/2(YES,YES) n__plus/2(YES,YES) isNat/1(YES) n__isNat/1)YES( s/1(YES) n__s/1)YES( activate/1(YES) Quasi precedence: 0 > n__0 > n__plus_2 [isNat_1, activate_1] > plus_2 > n__plus_2 s_1 > n__plus_2 Status: 0: multiset status n__0: multiset status plus_2: multiset status n__plus_2: multiset status isNat_1: multiset status s_1: [1] activate_1: multiset status With 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) isNat(X) -> n__isNat(X) s(X) -> n__s(X) activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(X) -> X ---------------------------------------- (6) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: activate(n__isNat(X)) -> isNat(X) Q is empty. ---------------------------------------- (7) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:activate_1 > isNat_1 > n__isNat_1 and weight map: activate_1=1 n__isNat_1=1 isNat_1=2 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: activate(n__isNat(X)) -> isNat(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