/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES We split firstr-order part and higher-order part, and do modular checking by a general modularity. ******** FO SN check ******** Check SN using NaTT (Nagoya Termination Tool) Input TRS: 1: quot(0(),s(X),s(Y)) -> 0() 2: quot(s(U),s(V),W) -> quot(U,V,W) 3: quot(P,0(),s(X1)) -> s(quot(P,s(X1),s(X1))) 4: _(X1,X2) -> X1 5: _(X1,X2) -> X2 Number of strict rules: 5 Direct POLO(bPol) ... failed. Uncurrying ... failed. Dependency Pairs: #1: #quot(s(U),s(V),W) -> #quot(U,V,W) #2: #quot(P,0(),s(X1)) -> #quot(P,s(X1),s(X1)) Number of SCCs: 1, DPs: 2 SCC { #1 #2 } POLO(Sum)... succeeded. s w: x1 + 1 _ w: 0 0 w: 1 quot w: 0 #_ w: 0 #quot w: x1 USABLE RULES: { } Removed DPs: #1 Number of SCCs: 0, DPs: 0 ... Input TRS: 1: quot(0(),s(X),s(Y)) -> 0() 2: quot(s(U),s(V),W) -> quot(U,V,W) 3: quot(P,0(),s(X1)) -> s(quot(P,s(X1),s(X1))) 4: _(X1,X2) -> X1 5: _(X1,X2) -> X2 Number of strict rules: 5 Direct POLO(bPol) ... failed. Uncurrying ... failed. Dependency Pairs: #1: #quot(s(U),s(V),W) -> #quot(U,V,W) #2: #quot(P,0(),s(X1)) -> #quot(P,s(X1),s(X1)) Number of SCCs: 1, DPs: 2 SCC { #1 #2 } POLO(Sum)... succeeded. s w: x1 + 1 _ w: 0 0 w: 1 quot w: 0 #_ w: 0 #quot w: x1 USABLE RULES: { } Removed DPs: #1 Number of SCCs: 0, DPs: 0 >>YES ******** Signature ******** map : ((c -> c),d) -> d nil : d cons : (c,d) -> d filter : ((c -> b),d) -> d filter2 : (b,(c -> b),c,d) -> d true : b false : b ******** Computation rules ******** (4) map(Z1,nil) => nil (5) map(G1,cons(V1,W1)) => cons(G1[V1],map(G1,W1)) (6) filter(J1,nil) => nil (7) filter(F2,cons(Y2,U2)) => filter2(F2[Y2],F2,Y2,U2) (8) filter2(true,H2,W2,P2) => cons(W2,filter(H2,P2)) (9) filter2(false,F3,Y3,U3) => filter(F3,U3) ******** General Schema criterion ******** Found constructors: 0, cons, false, nil, s, true Checking type order >>OK Checking positivity of constructors >>OK Checking function dependency >>Regared as equal: filter2, filter Checking (1) quot(0,s(X),s(Y)) => 0 (fun quot>0) >>True Checking (2) quot(s(U),s(V),W) => quot(U,V,W) (fun quot=quot) subterm comparison of args w. LR LR (meta U)[is acc in s(U),s(V),W] [is positive in s(U)] [is acc in U] (meta V)[is acc in s(U),s(V),W] [is positive in s(U)] [is positive in s(V)] [is acc in V] (meta W)[is acc in s(U),s(V),W] [is positive in s(U)] [is positive in s(V)] [is acc in W] >>True Checking (3) quot(P,0,s(X1)) => s(quot(P,s(X1),s(X1))) (fun quot>s) (fun quot=quot) subterm comparison of args w. LR LR >>False Try again using status RL Checking (1) quot(0,s(X),s(Y)) => 0 (fun quot>0) >>True Checking (2) quot(s(U),s(V),W) => quot(U,V,W) (fun quot=quot) subterm comparison of args w. RL RL (meta U)[is acc in s(U),s(V),W] [is positive in s(U)] [is acc in U] (meta V)[is acc in s(U),s(V),W] [is positive in s(U)] [is positive in s(V)] [is acc in V] (meta W)[is acc in s(U),s(V),W] [is positive in s(U)] [is positive in s(V)] [is acc in W] >>True Checking (3) quot(P,0,s(X1)) => s(quot(P,s(X1),s(X1))) (fun quot>s) (fun quot=quot) subterm comparison of args w. RL RL >>False Try again using status Mul Checking (1) quot(0,s(X),s(Y)) => 0 (fun quot>0) >>True Checking (2) quot(s(U),s(V),W) => quot(U,V,W) (fun quot=quot) subterm comparison of args w. Mul Mul (meta U)[is acc in s(U),s(V),W] [is positive in s(U)] [is acc in U] (meta V)[is acc in s(U),s(V),W] [is positive in s(U)] [is positive in s(V)] [is acc in V] (meta W)[is acc in s(U),s(V),W] [is positive in s(U)] [is positive in s(V)] [is acc in W] >>True Checking (3) quot(P,0,s(X1)) => s(quot(P,s(X1),s(X1))) (fun quot>s) (fun quot=quot) subterm comparison of args w. Mul Mul >>False Found constructors: nil, cons, true, false Checking type order >>OK Checking positivity of constructors >>OK Checking function dependency >>Regared as equal: filter2, filter Checking (4) map(Z1,nil) => nil (fun map>nil) >>True Checking (5) map(G1,cons(V1,W1)) => cons(G1[V1],map(G1,W1)) (fun map>cons) (meta G1)[is acc in G1,cons(V1,W1)] [is acc in G1] (meta V1)[is acc in G1,cons(V1,W1)] [is positive in cons(V1,W1)] [is acc in V1] (fun map=map) subterm comparison of args w. LR LR (meta G1)[is acc in G1,cons(V1,W1)] [is acc in G1] (meta W1)[is acc in G1,cons(V1,W1)] [is positive in cons(V1,W1)] [is acc in W1] >>True Checking (6) filter(J1,nil) => nil (fun filter>nil) >>True Checking (7) filter(F2,cons(Y2,U2)) => filter2(F2[Y2],F2,Y2,U2) (fun filter=filter2) subterm comparison of args w. Arg [1,2] Arg [2,4,3,1] (meta F2)[is acc in F2,cons(Y2,U2)] [is acc in F2] (meta Y2)[is acc in F2,cons(Y2,U2)] [is positive in cons(Y2,U2)] [is acc in Y2] (meta F2)[is acc in F2,cons(Y2,U2)] [is acc in F2] (meta Y2)[is acc in F2,cons(Y2,U2)] [is positive in cons(Y2,U2)] [is acc in Y2] (meta U2)[is acc in F2,cons(Y2,U2)] [is positive in cons(Y2,U2)] [is acc in U2] >>True Checking (8) filter2(true,H2,W2,P2) => cons(W2,filter(H2,P2)) (fun filter2>cons) (meta W2)[is acc in true,H2,W2,P2] [is positive in true] [is acc in W2] (fun filter2=filter) subterm comparison of args w. Arg [2,4,3,1] Arg [1,2] (meta H2)[is acc in true,H2,W2,P2] [is positive in true] [is acc in H2] (meta P2)[is acc in true,H2,W2,P2] [is positive in true] [is acc in P2] >>True Checking (9) filter2(false,F3,Y3,U3) => filter(F3,U3) (fun filter2=filter) subterm comparison of args w. Arg [2,4,3,1] Arg [1,2] (meta F3)[is acc in false,F3,Y3,U3] [is positive in false] [is acc in F3] (meta U3)[is acc in false,F3,Y3,U3] [is positive in false] [is acc in U3] >>True #SN! ******** Signature ******** 0 : a cons : (c,d) -> d false : b filter : ((c -> b),d) -> d filter2 : (b,(c -> b),c,d) -> d map : ((c -> c),d) -> d nil : d quot : (a,a,a) -> a s : a -> a true : b ******** Computation Rules ******** (1) quot(0,s(X),s(Y)) => 0 (2) quot(s(U),s(V),W) => quot(U,V,W) (3) quot(P,0,s(X1)) => s(quot(P,s(X1),s(X1))) (4) map(Z1,nil) => nil (5) map(G1,cons(V1,W1)) => cons(G1[V1],map(G1,W1)) (6) filter(J1,nil) => nil (7) filter(F2,cons(Y2,U2)) => filter2(F2[Y2],F2,Y2,U2) (8) filter2(true,H2,W2,P2) => cons(W2,filter(H2,P2)) (9) filter2(false,F3,Y3,U3) => filter(F3,U3) YES