/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 213 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4)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: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U31(tt) -> 0 a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: a__U11/2(YES,YES) tt/0) mark/1)YES( a__U21/3(YES,YES,YES) s/1(YES) a__plus/2(YES,YES) a__U31/1)YES( 0/0) a__U41/3(YES,YES,YES) a__x/2(YES,YES) a__and/2(YES,YES) a__isNat/1)YES( plus/2(YES,YES) isNat/1)YES( x/2(YES,YES) U11/2(YES,YES) U21/3(YES,YES,YES) U31/1)YES( U41/3(YES,YES,YES) and/2(YES,YES) Quasi precedence: [tt, 0] > [a__U21_3, a__plus_2, plus_2, U21_3] > [a__U11_2, U11_2] [tt, 0] > [a__U21_3, a__plus_2, plus_2, U21_3] > [s_1, a__and_2, and_2] [a__U41_3, a__x_2, x_2, U41_3] > [a__U21_3, a__plus_2, plus_2, U21_3] > [a__U11_2, U11_2] [a__U41_3, a__x_2, x_2, U41_3] > [a__U21_3, a__plus_2, plus_2, U21_3] > [s_1, a__and_2, and_2] Status: a__U11_2: multiset status tt: multiset status a__U21_3: [2,3,1] s_1: multiset status a__plus_2: [2,1] 0: multiset status a__U41_3: [2,3,1] a__x_2: [2,1] a__and_2: multiset status plus_2: [2,1] x_2: [2,1] U11_2: multiset status U21_3: [2,3,1] U41_3: [2,3,1] and_2: multiset status With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: a__U11(tt, N) -> mark(N) a__U21(tt, M, N) -> s(a__plus(mark(N), mark(M))) a__U41(tt, M, N) -> a__plus(a__x(mark(N), mark(M)), mark(N)) a__and(tt, X) -> mark(X) a__isNat(plus(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__isNat(s(V1)) -> a__isNat(V1) a__isNat(x(V1, V2)) -> a__and(a__isNat(V1), isNat(V2)) a__plus(N, 0) -> a__U11(a__isNat(N), N) a__plus(N, s(M)) -> a__U21(a__and(a__isNat(M), isNat(N)), M, N) a__x(N, 0) -> a__U31(a__isNat(N)) a__x(N, s(M)) -> a__U41(a__and(a__isNat(M), isNat(N)), M, N) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__U31(tt) -> 0 a__isNat(0) -> tt mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:mark_1 > s_1 > a__and_2 > and_2 > a__plus_2 > plus_2 > a__U31_1 > a__U41_3 > U41_3 > a__isNat_1 > U31_1 > isNat_1 > a__x_2 > x_2 > tt > a__U21_3 > U21_3 > a__U11_2 > U11_2 > 0 and weight map: tt=2 0=1 a__U31_1=1 a__isNat_1=1 mark_1=0 U31_1=1 isNat_1=1 s_1=1 U11_2=0 a__U11_2=0 U21_3=0 a__U21_3=0 plus_2=0 a__plus_2=0 U41_3=0 a__U41_3=0 x_2=0 a__x_2=0 and_2=0 a__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: a__U31(tt) -> 0 a__isNat(0) -> tt mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U21(X1, X2, X3)) -> a__U21(mark(X1), X2, X3) mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) mark(U31(X)) -> a__U31(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(x(X1, X2)) -> a__x(mark(X1), mark(X2)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__U11(X1, X2) -> U11(X1, X2) a__U21(X1, X2, X3) -> U21(X1, X2, X3) a__plus(X1, X2) -> plus(X1, X2) a__U31(X) -> U31(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__x(X1, X2) -> x(X1, X2) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(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