/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 357 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4)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: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U13(tt) -> tt U21(tt, V1) -> U22(isNat(activate(V1))) U22(tt) -> tt U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) U32(tt, V2) -> U33(isNat(activate(V2))) U33(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0 U71(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)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) and(X1, X2) -> n__and(X1, X2) isNat(X) -> n__isNat(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(n__and(X1, X2)) -> and(activate(X1), X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: U11/3(YES,YES,YES) tt/0) U12/2(YES,YES) isNat/1)YES( activate/1)YES( U13/1(YES) U21/2(YES,YES) U22/1)YES( U31/3(YES,YES,YES) U32/2(YES,YES) U33/1(YES) U41/2(YES,YES) U51/3(YES,YES,YES) s/1(YES) plus/2(YES,YES) U61/1(YES) 0/0) U71/3(YES,YES,YES) x/2(YES,YES) and/2(YES,YES) n__0/0) n__plus/2(YES,YES) isNatKind/1)YES( n__isNatKind/1)YES( n__s/1(YES) n__x/2(YES,YES) n__and/2(YES,YES) n__isNat/1)YES( Quasi precedence: [U71_3, x_2, n__x_2] > U31_3 > U32_2 > U33_1 [U71_3, x_2, n__x_2] > [U51_3, plus_2, n__plus_2] > U11_3 > [U12_2, U13_1] > tt > U33_1 [U71_3, x_2, n__x_2] > [U51_3, plus_2, n__plus_2] > U11_3 > [U12_2, U13_1] > tt > [s_1, n__s_1] > U21_2 [U71_3, x_2, n__x_2] > [U51_3, plus_2, n__plus_2] > U41_2 [U71_3, x_2, n__x_2] > [U51_3, plus_2, n__plus_2] > [and_2, n__and_2] [U71_3, x_2, n__x_2] > U61_1 > [0, n__0] > tt > U33_1 [U71_3, x_2, n__x_2] > U61_1 > [0, n__0] > tt > [s_1, n__s_1] > U21_2 [U71_3, x_2, n__x_2] > U61_1 > [0, n__0] > U41_2 [U71_3, x_2, n__x_2] > U61_1 > [0, n__0] > [and_2, n__and_2] Status: U11_3: multiset status tt: multiset status U12_2: multiset status U13_1: multiset status U21_2: multiset status U31_3: [3,2,1] U32_2: [1,2] U33_1: [1] U41_2: multiset status U51_3: [3,2,1] s_1: multiset status plus_2: [1,2] U61_1: multiset status 0: multiset status U71_3: [3,2,1] x_2: [1,2] and_2: multiset status n__0: multiset status n__plus_2: [1,2] n__s_1: multiset status n__x_2: [1,2] n__and_2: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U13(tt) -> tt U21(tt, V1) -> U22(isNat(activate(V1))) U31(tt, V1, V2) -> U32(isNat(activate(V1)), activate(V2)) U32(tt, V2) -> U33(isNat(activate(V2))) U33(tt) -> tt U41(tt, N) -> activate(N) U51(tt, M, N) -> s(plus(activate(N), activate(M))) U61(tt) -> 0 U71(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)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNat(n__x(V1, V2)) -> U31(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) isNatKind(n__x(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) plus(N, 0) -> U41(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U51(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) x(N, 0) -> U61(and(isNat(N), n__isNatKind(N))) x(N, s(M)) -> U71(and(and(isNat(M), n__isNatKind(M)), n__and(n__isNat(N), n__isNatKind(N))), M, N) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U22(tt) -> tt 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) and(X1, X2) -> n__and(X1, X2) isNat(X) -> n__isNat(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(n__and(X1, X2)) -> and(activate(X1), X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:activate_1 > and_2 > n__and_2 > 0 > x_2 > n__x_2 > isNatKind_1 > n__isNatKind_1 > s_1 > n__s_1 > n__0 > isNat_1 > n__isNat_1 > plus_2 > n__plus_2 > tt > U22_1 and weight map: tt=1 0=1 n__0=1 U22_1=1 isNatKind_1=1 n__isNatKind_1=1 s_1=1 n__s_1=1 isNat_1=1 n__isNat_1=1 activate_1=0 plus_2=0 n__plus_2=0 x_2=0 n__x_2=0 and_2=0 n__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: U22(tt) -> tt 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) x(X1, X2) -> n__x(X1, X2) and(X1, X2) -> n__and(X1, X2) isNat(X) -> n__isNat(X) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(activate(X1), activate(X2)) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(activate(X)) activate(n__x(X1, X2)) -> x(activate(X1), activate(X2)) activate(n__and(X1, X2)) -> and(activate(X1), X2) activate(n__isNat(X)) -> isNat(X) activate(X) -> X ---------------------------------------- (4) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (5) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (6) YES