/export/starexec/sandbox2/solver/bin/starexec_run_hrs /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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MAYBE

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: xplus(X,0()) -> X
    2: xplus(Y,s(U)) -> s(xplus(Y,U))
    3: _(X1,X2) -> X1
    4: _(X1,X2) -> X2
Number of strict rules: 4
Direct POLO(bPol) ... removes: 4 1 3 2
      s 	w: x1 + 8856
      _ 	w: 2 * x1 + x2 + 1
      0 	w: 1
      xplus	w: 2 * x1 + 2 * x2 + 1237
Number of strict rules: 0

...
Input TRS:
    1: xplus(X,0()) -> X
    2: xplus(Y,s(U)) -> s(xplus(Y,U))
    3: _(X1,X2) -> X1
    4: _(X1,X2) -> X2
Number of strict rules: 4
Direct POLO(bPol) ... removes: 4 1 3 2
      s 	w: x1 + 8856
      _ 	w: 2 * x1 + x2 + 1
      0 	w: 1
      xplus	w: 2 * x1 + 2 * x2 + 1237
Number of strict rules: 0
>>YES

******** Signature ********
 rec : (nat,nat,((nat,nat) -> nat)) -> nat
 0 : nat
 yap : ((nat -> nat),nat) -> nat
 xap : (((nat,nat) -> nat),nat,nat) -> nat
 s : nat -> nat
 xtimes : (nat,nat) -> nat
 xplus : (nat,nat) -> nat
******** Computation rules ********
 (3)  rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (4)  rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (5)  U1 * V1 => rec(V1,0,%G.%F.(U1 + %F))
 (6)  xap(I1,P1) => I1[P1]
 (7)  yap(F2,Y2) => F2[Y2]


******** General Schema criterion ********
Found constructors: 0, s
Checking type order >>OK
Checking positivity of constructors >>OK
Checking function dependency >>OK
Checking  (1) X + 0 => X
 (meta X)[is acc in X,0] [is acc in X]  >>True
Checking  (2) Y + s(U) => s(Y + U)
 (fun xplus>s)
 (fun xplus=xplus) subterm comparison of args w. LR LR
 (meta Y)[is acc in Y,s(U)] [is acc in Y] 
 (meta U)[is acc in Y,s(U)] [is positive in s(U)] [is acc in U]  >>True
Checking  (3) rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (meta V)[is acc in 0,V,%X.%Y.yap(xap(I,%X),%Y)] [is positive in 0] [is acc in V]  >>True
Checking  (4) rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta Z1)[is acc in s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)] [is positive in s(P)] [is positive in yap(xap(Z1,%Z),%U)]  >>False

Try again using status RL
Checking  (1) X + 0 => X
 (meta X)[is acc in X,0] [is acc in X]  >>True
Checking  (2) Y + s(U) => s(Y + U)
 (fun xplus>s)
 (fun xplus=xplus) subterm comparison of args w. RL RL
 (meta Y)[is acc in Y,s(U)] [is acc in Y] 
 (meta U)[is acc in Y,s(U)] [is positive in s(U)] [is acc in U]  >>True
Checking  (3) rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (meta V)[is acc in 0,V,%X.%Y.yap(xap(I,%X),%Y)] [is positive in 0] [is acc in V]  >>True
Checking  (4) rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta Z1)[is acc in s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)] [is positive in s(P)] [is positive in yap(xap(Z1,%Z),%U)]  >>False

Try again using status Mul
Checking  (1) X + 0 => X
 (meta X)[is acc in X,0] [is acc in X]  >>True
Checking  (2) Y + s(U) => s(Y + U)
 (fun xplus>s)
 (fun xplus=xplus) subterm comparison of args w. Mul Mul
 (meta Y)[is acc in Y,s(U)] [is acc in Y] 
 (meta U)[is acc in Y,s(U)] [is positive in s(U)] [is acc in U]  >>True
Checking  (3) rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (meta V)[is acc in 0,V,%X.%Y.yap(xap(I,%X),%Y)] [is positive in 0] [is acc in V]  >>True
Checking  (4) rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta Z1)[is acc in s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)] [is positive in s(P)] [is positive in yap(xap(Z1,%Z),%U)]  >>False
Found constructors: 0, s, xplus
Checking type order >>OK
Checking positivity of constructors >>OK
Checking function dependency >>OK
Checking  (3) rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (meta V)[is acc in 0,V,%X.%Y.yap(xap(I,%X),%Y)] [is positive in 0] [is acc in V]  >>True
Checking  (4) rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta Z1)[is acc in s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)] [is positive in s(P)] [is positive in yap(xap(Z1,%Z),%U)]  >>False

Try again using status RL
Checking  (3) rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (meta V)[is acc in 0,V,%X.%Y.yap(xap(I,%X),%Y)] [is positive in 0] [is acc in V]  >>True
Checking  (4) rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta Z1)[is acc in s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)] [is positive in s(P)] [is positive in yap(xap(Z1,%Z),%U)]  >>False

Try again using status Mul
Checking  (3) rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (meta V)[is acc in 0,V,%X.%Y.yap(xap(I,%X),%Y)] [is positive in 0] [is acc in V]  >>True
Checking  (4) rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta Z1)[is acc in s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)] [is positive in s(P)] [is positive in yap(xap(Z1,%Z),%U)]  >>False
#No idea..
******** Signature ********
 0 : nat
 rec : (nat,nat,((nat,nat) -> nat)) -> nat
 s : nat -> nat
 xap : (((nat,nat) -> nat),nat,nat) -> nat
 xplus : (nat,nat) -> nat
 xtimes : (nat,nat) -> nat
 yap : ((nat -> nat),nat) -> nat
******** Computation Rules ********
 (1)  X + 0 => X
 (2)  Y + s(U) => s(Y + U)
 (3)  rec(0,V,%X.%Y.yap(xap(I,%X),%Y)) => V
 (4)  rec(s(P),X1,%Z.%U.yap(xap(Z1,%Z),%U)) => yap(xap(Z1,P),rec(P,X1,%V.%W.yap(xap(Z1,%V),%W)))
 (5)  U1 * V1 => rec(V1,0,%G.%F.(U1 + %F))
 (6)  xap(I1,P1) => I1[P1]
 (7)  yap(F2,Y2) => F2[Y2]

MAYBE