/export/starexec/sandbox/solver/bin/starexec_run_hrs /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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 ********
 add : (nat,nat) -> nat
 s : nat -> nat
 0 : nat
 rec : (((nat,nat) -> nat),nat,nat) -> nat
 yap : ((nat -> nat),nat) -> nat
 xap : (((nat,nat) -> nat),nat,nat) -> nat
 succ : (nat,nat) -> nat
******** Computation rules ********
 (6)  add(W1,s(P1)) => s(add(W1,P1))
 (5)  add(V1,0) => V1
 (1)  rec(%X.%Y.yap(xap(F,%X),%Y),Y,0) => Y
 (2)  rec(%Z.%U.yap(xap(G,%Z),%U),V,s(W)) => yap(xap(G,W),rec(%V.%W.yap(xap(G,%V),%W),V,W))
 (4)  add(Y1,U1) => rec(%F.%G.yap(xap(succ,%F),%G),Y1,U1)
 (7)  xap(F2,Y2) => F2[Y2]
 (8)  yap(G2,V2) => G2[V2]


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

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

Try again using status Mul
Checking  (1) rec(%X.%Y.yap(xap(F,%X),%Y),Y,0) => Y
 (meta Y)[is acc in %X.%Y.yap(xap(F,%X),%Y),Y,0] [is positive in yap(xap(F,%X),%Y)] [is acc in Y]  >>True
Checking  (2) rec(%Z.%U.yap(xap(G,%Z),%U),V,s(W)) => yap(xap(G,W),rec(%V.%W.yap(xap(G,%V),%W),V,W))
 (fun rec>yap)
 (fun rec>xap)
 (meta G)[is acc in %Z.%U.yap(xap(G,%Z),%U),V,s(W)] [is positive in yap(xap(G,%Z),%U)] [is positive in s(W)]  >>False
Found constructors: s, 0, succ
Checking type order >>OK
Checking positivity of constructors >>OK
Checking function dependency >>OK
Checking  (6) add(W1,s(P1)) => s(add(W1,P1))
 (fun add>s)
 (fun add=add) subterm comparison of args w. LR LR
 (meta W1)[is acc in W1,s(P1)] [is acc in W1] 
 (meta P1)[is acc in W1,s(P1)] [is positive in s(P1)] [is acc in P1]  >>True
Checking  (5) add(V1,0) => V1
 (meta V1)[is acc in V1,0] [is acc in V1]  >>True
Checking  (1) rec(%X.%Y.yap(xap(F,%X),%Y),Y,0) => Y
 (meta Y)[is acc in %X.%Y.yap(xap(F,%X),%Y),Y,0] [is positive in yap(xap(F,%X),%Y)] [is acc in Y]  >>True
Checking  (2) rec(%Z.%U.yap(xap(G,%Z),%U),V,s(W)) => yap(xap(G,W),rec(%V.%W.yap(xap(G,%V),%W),V,W))
 (fun rec>yap)
 (fun rec>xap)
 (meta G)[is acc in %Z.%U.yap(xap(G,%Z),%U),V,s(W)] [is positive in yap(xap(G,%Z),%U)] [is positive in s(W)]  >>False

Try again using status RL
Checking  (6) add(W1,s(P1)) => s(add(W1,P1))
 (fun add>s)
 (fun add=add) subterm comparison of args w. RL RL
 (meta W1)[is acc in W1,s(P1)] [is acc in W1] 
 (meta P1)[is acc in W1,s(P1)] [is positive in s(P1)] [is acc in P1]  >>True
Checking  (5) add(V1,0) => V1
 (meta V1)[is acc in V1,0] [is acc in V1]  >>True
Checking  (1) rec(%X.%Y.yap(xap(F,%X),%Y),Y,0) => Y
 (meta Y)[is acc in %X.%Y.yap(xap(F,%X),%Y),Y,0] [is positive in yap(xap(F,%X),%Y)] [is acc in Y]  >>True
Checking  (2) rec(%Z.%U.yap(xap(G,%Z),%U),V,s(W)) => yap(xap(G,W),rec(%V.%W.yap(xap(G,%V),%W),V,W))
 (fun rec>yap)
 (fun rec>xap)
 (meta G)[is acc in %Z.%U.yap(xap(G,%Z),%U),V,s(W)] [is positive in yap(xap(G,%Z),%U)] [is positive in s(W)]  >>False

Try again using status Mul
Checking  (6) add(W1,s(P1)) => s(add(W1,P1))
 (fun add>s)
 (fun add=add) subterm comparison of args w. Mul Mul
 (meta W1)[is acc in W1,s(P1)] [is acc in W1] 
 (meta P1)[is acc in W1,s(P1)] [is positive in s(P1)] [is acc in P1]  >>True
Checking  (5) add(V1,0) => V1
 (meta V1)[is acc in V1,0] [is acc in V1]  >>True
Checking  (1) rec(%X.%Y.yap(xap(F,%X),%Y),Y,0) => Y
 (meta Y)[is acc in %X.%Y.yap(xap(F,%X),%Y),Y,0] [is positive in yap(xap(F,%X),%Y)] [is acc in Y]  >>True
Checking  (2) rec(%Z.%U.yap(xap(G,%Z),%U),V,s(W)) => yap(xap(G,W),rec(%V.%W.yap(xap(G,%V),%W),V,W))
 (fun rec>yap)
 (fun rec>xap)
 (meta G)[is acc in %Z.%U.yap(xap(G,%Z),%U),V,s(W)] [is positive in yap(xap(G,%Z),%U)] [is positive in s(W)]  >>False
#No idea..
******** Signature ********
 0 : nat
 add : (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(%X.%Y.yap(xap(F,%X),%Y),Y,0) => Y
 (2)  rec(%Z.%U.yap(xap(G,%Z),%U),V,s(W)) => yap(xap(G,W),rec(%V.%W.yap(xap(G,%V),%W),V,W))
 (3)  succ(P,X1) => s(X1)
 (4)  add(Y1,U1) => rec(%F.%G.yap(xap(succ,%F),%G),Y1,U1)
 (5)  add(V1,0) => V1
 (6)  add(W1,s(P1)) => s(add(W1,P1))
 (7)  xap(F2,Y2) => F2[Y2]
 (8)  yap(G2,V2) => G2[V2]

MAYBE