/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: succ(P,X1) -> s(X1)
    2: _(X1,X2) -> X1
    3: _(X1,X2) -> X2
Number of strict rules: 3
Direct POLO(bPol) ... removes: 1 3 2
      s 	w: 2 * x1
      _ 	w: 2 * x1 + 2 * x2 + 1
      succ	w: x1 + 2 * x2 + 1
Number of strict rules: 0

...
Input TRS:
    1: succ(P,X1) -> s(X1)
    2: _(X1,X2) -> X1
    3: _(X1,X2) -> X2
Number of strict rules: 3
Direct POLO(bPol) ... removes: 1 3 2
      s 	w: 2 * x1
      _ 	w: 2 * x1 + 2 * x2 + 1
      succ	w: x1 + 2 * x2 + 1
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
 plus : (nat,nat) -> nat
 succ : (nat,nat) -> nat
******** Computation rules ********
 (1)  rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (2)  rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (4)  Y1 + U1 => rec(Y1,U1,succ)
 (5)  xap(H1,W1) => H1[W1]
 (6)  yap(J1,X2) => J1[X2]


******** General Schema criterion ********
Found constructors: 0, s
Checking type order >>OK
Checking positivity of constructors >>OK
Checking function dependency >>OK
Checking  (1) rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (meta X)[is acc in 0,X,%X.%Y.yap(xap(Z,%X),%Y)] [is positive in 0] [is acc in X]  >>True
Checking  (2) rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta I)[is acc in s(U),V,%Z.%U.yap(xap(I,%Z),%U)] [is positive in s(U)] [is positive in yap(xap(I,%Z),%U)]  >>False

Try again using status RL
Checking  (1) rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (meta X)[is acc in 0,X,%X.%Y.yap(xap(Z,%X),%Y)] [is positive in 0] [is acc in X]  >>True
Checking  (2) rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta I)[is acc in s(U),V,%Z.%U.yap(xap(I,%Z),%U)] [is positive in s(U)] [is positive in yap(xap(I,%Z),%U)]  >>False

Try again using status Mul
Checking  (1) rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (meta X)[is acc in 0,X,%X.%Y.yap(xap(Z,%X),%Y)] [is positive in 0] [is acc in X]  >>True
Checking  (2) rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta I)[is acc in s(U),V,%Z.%U.yap(xap(I,%Z),%U)] [is positive in s(U)] [is positive in yap(xap(I,%Z),%U)]  >>False
Found constructors: 0, s, succ
Checking type order >>OK
Checking positivity of constructors >>OK
Checking function dependency >>OK
Checking  (1) rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (meta X)[is acc in 0,X,%X.%Y.yap(xap(Z,%X),%Y)] [is positive in 0] [is acc in X]  >>True
Checking  (2) rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta I)[is acc in s(U),V,%Z.%U.yap(xap(I,%Z),%U)] [is positive in s(U)] [is positive in yap(xap(I,%Z),%U)]  >>False

Try again using status RL
Checking  (1) rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (meta X)[is acc in 0,X,%X.%Y.yap(xap(Z,%X),%Y)] [is positive in 0] [is acc in X]  >>True
Checking  (2) rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta I)[is acc in s(U),V,%Z.%U.yap(xap(I,%Z),%U)] [is positive in s(U)] [is positive in yap(xap(I,%Z),%U)]  >>False

Try again using status Mul
Checking  (1) rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (meta X)[is acc in 0,X,%X.%Y.yap(xap(Z,%X),%Y)] [is positive in 0] [is acc in X]  >>True
Checking  (2) rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (fun rec>yap)
 (fun rec>xap)
 (meta I)[is acc in s(U),V,%Z.%U.yap(xap(I,%Z),%U)] [is positive in s(U)] [is positive in yap(xap(I,%Z),%U)]  >>False
#No idea..
******** Signature ********
 0 : nat
 plus : (nat,nat) -> nat
 rec : (nat,nat,((nat,nat) -> nat)) -> nat
 s : nat -> nat
 succ : (nat,nat) -> nat
 xap : (((nat,nat) -> nat),nat,nat) -> nat
 yap : ((nat -> nat),nat) -> nat
******** Computation Rules ********
 (1)  rec(0,X,%X.%Y.yap(xap(Z,%X),%Y)) => X
 (2)  rec(s(U),V,%Z.%U.yap(xap(I,%Z),%U)) => yap(xap(I,U),rec(U,V,%V.%W.yap(xap(I,%V),%W)))
 (3)  succ(P,X1) => s(X1)
 (4)  Y1 + U1 => rec(Y1,U1,succ)
 (5)  xap(H1,W1) => H1[W1]
 (6)  yap(J1,X2) => J1[X2]

MAYBE