Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

U521(mark(X1), X2, X3) → U521(X1, X2, X3)
U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(x(N, 0)) → U611(isNat(N))
ACTIVE(U71(X1, X2, X3)) → U711(active(X1), X2, X3)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(plus(N, s(M))) → ISNAT(M)
U521(ok(X1), ok(X2), ok(X3)) → U521(X1, X2, X3)
ACTIVE(U52(tt, M, N)) → PLUS(N, M)
X(X1, mark(X2)) → X(X1, X2)
ACTIVE(plus(X1, X2)) → PLUS(active(X1), X2)
ACTIVE(U72(tt, M, N)) → PLUS(x(N, M), N)
PROPER(U72(X1, X2, X3)) → PROPER(X3)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(tt, V2)) → U121(isNat(V2))
U321(ok(X)) → U321(X)
U311(mark(X1), X2) → U311(X1, X2)
U411(ok(X1), ok(X2)) → U411(X1, X2)
PROPER(U11(X1, X2)) → PROPER(X2)
ACTIVE(isNat(x(V1, V2))) → U311(isNat(V1), V2)
PROPER(plus(X1, X2)) → PROPER(X1)
ACTIVE(U41(X1, X2)) → U411(active(X1), X2)
PROPER(U32(X)) → U321(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
U611(ok(X)) → U611(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(x(X1, X2)) → ACTIVE(X2)
ACTIVE(plus(N, s(M))) → U511(isNat(M), M, N)
U321(mark(X)) → U321(X)
PROPER(U72(X1, X2, X3)) → U721(proper(X1), proper(X2), proper(X3))
ACTIVE(isNat(x(V1, V2))) → ISNAT(V1)
S(ok(X)) → S(X)
ACTIVE(U12(X)) → U121(active(X))
ACTIVE(U72(tt, M, N)) → X(N, M)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
PROPER(U71(X1, X2, X3)) → U711(proper(X1), proper(X2), proper(X3))
ACTIVE(plus(X1, X2)) → PLUS(X1, active(X2))
TOP(mark(X)) → PROPER(X)
PROPER(U52(X1, X2, X3)) → PROPER(X1)
U121(ok(X)) → U121(X)
ACTIVE(U31(tt, V2)) → ISNAT(V2)
ACTIVE(plus(N, 0)) → ISNAT(N)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
X(mark(X1), X2) → X(X1, X2)
ACTIVE(isNat(s(V1))) → U211(isNat(V1))
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
PROPER(U21(X)) → U211(proper(X))
U721(ok(X1), ok(X2), ok(X3)) → U721(X1, X2, X3)
U311(ok(X1), ok(X2)) → U311(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
ACTIVE(U31(tt, V2)) → U321(isNat(V2))
PROPER(U52(X1, X2, X3)) → PROPER(X2)
S(mark(X)) → S(X)
PROPER(U41(X1, X2)) → PROPER(X2)
ACTIVE(isNat(plus(V1, V2))) → ISNAT(V1)
ACTIVE(x(N, s(M))) → U711(isNat(M), M, N)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U11(tt, V2)) → ISNAT(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U21(X)) → PROPER(X)
U611(mark(X)) → U611(X)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
ACTIVE(x(X1, X2)) → ACTIVE(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
ACTIVE(U51(tt, M, N)) → U521(isNat(N), M, N)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
U121(mark(X)) → U121(X)
ACTIVE(U12(X)) → ACTIVE(X)
PROPER(U12(X)) → U121(proper(X))
U411(mark(X1), X2) → U411(X1, X2)
U211(ok(X)) → U211(X)
PROPER(plus(X1, X2)) → PLUS(proper(X1), proper(X2))
ACTIVE(U61(X)) → U611(active(X))
PROPER(U72(X1, X2, X3)) → PROPER(X1)
ACTIVE(plus(N, 0)) → U411(isNat(N), N)
PROPER(U52(X1, X2, X3)) → U521(proper(X1), proper(X2), proper(X3))
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
U721(mark(X1), X2, X3) → U721(X1, X2, X3)
PROPER(U61(X)) → U611(proper(X))
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
ACTIVE(U51(tt, M, N)) → ISNAT(N)
ACTIVE(x(X1, X2)) → X(active(X1), X2)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → X(proper(X1), proper(X2))
PROPER(s(X)) → S(proper(X))
U211(mark(X)) → U211(X)
U111(ok(X1), ok(X2)) → U111(X1, X2)
PROPER(U72(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
U111(mark(X1), X2) → U111(X1, X2)
PROPER(U61(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
ACTIVE(U71(tt, M, N)) → ISNAT(N)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)
PROPER(U12(X)) → PROPER(X)
ACTIVE(U51(X1, X2, X3)) → U511(active(X1), X2, X3)
ACTIVE(x(X1, X2)) → X(X1, active(X2))
PROPER(U41(X1, X2)) → U411(proper(X1), proper(X2))
PROPER(s(X)) → PROPER(X)
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
ACTIVE(x(N, s(M))) → ISNAT(M)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(U32(X)) → U321(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → U721(active(X1), X2, X3)
ACTIVE(x(N, 0)) → ISNAT(N)
PROPER(U32(X)) → PROPER(X)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(U52(tt, M, N)) → S(plus(N, M))
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
ACTIVE(isNat(plus(V1, V2))) → U111(isNat(V1), V2)
X(ok(X1), ok(X2)) → X(X1, X2)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
ACTIVE(U21(X)) → U211(active(X))
ACTIVE(U71(tt, M, N)) → U721(isNat(N), M, N)
TOP(mark(X)) → TOP(proper(X))
PROPER(U51(X1, X2, X3)) → U511(proper(X1), proper(X2), proper(X3))
PROPER(plus(X1, X2)) → PROPER(X2)
ACTIVE(U52(X1, X2, X3)) → U521(active(X1), X2, X3)
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

U521(mark(X1), X2, X3) → U521(X1, X2, X3)
U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(x(N, 0)) → U611(isNat(N))
ACTIVE(U71(X1, X2, X3)) → U711(active(X1), X2, X3)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(plus(N, s(M))) → ISNAT(M)
U521(ok(X1), ok(X2), ok(X3)) → U521(X1, X2, X3)
ACTIVE(U52(tt, M, N)) → PLUS(N, M)
X(X1, mark(X2)) → X(X1, X2)
ACTIVE(plus(X1, X2)) → PLUS(active(X1), X2)
ACTIVE(U72(tt, M, N)) → PLUS(x(N, M), N)
PROPER(U72(X1, X2, X3)) → PROPER(X3)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U11(tt, V2)) → U121(isNat(V2))
U321(ok(X)) → U321(X)
U311(mark(X1), X2) → U311(X1, X2)
U411(ok(X1), ok(X2)) → U411(X1, X2)
PROPER(U11(X1, X2)) → PROPER(X2)
ACTIVE(isNat(x(V1, V2))) → U311(isNat(V1), V2)
PROPER(plus(X1, X2)) → PROPER(X1)
ACTIVE(U41(X1, X2)) → U411(active(X1), X2)
PROPER(U32(X)) → U321(proper(X))
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
U611(ok(X)) → U611(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(x(X1, X2)) → ACTIVE(X2)
ACTIVE(plus(N, s(M))) → U511(isNat(M), M, N)
U321(mark(X)) → U321(X)
PROPER(U72(X1, X2, X3)) → U721(proper(X1), proper(X2), proper(X3))
ACTIVE(isNat(x(V1, V2))) → ISNAT(V1)
S(ok(X)) → S(X)
ACTIVE(U12(X)) → U121(active(X))
ACTIVE(U72(tt, M, N)) → X(N, M)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
PROPER(U71(X1, X2, X3)) → U711(proper(X1), proper(X2), proper(X3))
ACTIVE(plus(X1, X2)) → PLUS(X1, active(X2))
TOP(mark(X)) → PROPER(X)
PROPER(U52(X1, X2, X3)) → PROPER(X1)
U121(ok(X)) → U121(X)
ACTIVE(U31(tt, V2)) → ISNAT(V2)
ACTIVE(plus(N, 0)) → ISNAT(N)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
X(mark(X1), X2) → X(X1, X2)
ACTIVE(isNat(s(V1))) → U211(isNat(V1))
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
PROPER(U21(X)) → U211(proper(X))
U721(ok(X1), ok(X2), ok(X3)) → U721(X1, X2, X3)
U311(ok(X1), ok(X2)) → U311(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
ACTIVE(U31(tt, V2)) → U321(isNat(V2))
PROPER(U52(X1, X2, X3)) → PROPER(X2)
S(mark(X)) → S(X)
PROPER(U41(X1, X2)) → PROPER(X2)
ACTIVE(isNat(plus(V1, V2))) → ISNAT(V1)
ACTIVE(x(N, s(M))) → U711(isNat(M), M, N)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U11(tt, V2)) → ISNAT(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
U711(mark(X1), X2, X3) → U711(X1, X2, X3)
PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U21(X)) → PROPER(X)
U611(mark(X)) → U611(X)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
ACTIVE(x(X1, X2)) → ACTIVE(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)
ACTIVE(U51(tt, M, N)) → U521(isNat(N), M, N)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
U121(mark(X)) → U121(X)
ACTIVE(U12(X)) → ACTIVE(X)
PROPER(U12(X)) → U121(proper(X))
U411(mark(X1), X2) → U411(X1, X2)
U211(ok(X)) → U211(X)
PROPER(plus(X1, X2)) → PLUS(proper(X1), proper(X2))
ACTIVE(U61(X)) → U611(active(X))
PROPER(U72(X1, X2, X3)) → PROPER(X1)
ACTIVE(plus(N, 0)) → U411(isNat(N), N)
PROPER(U52(X1, X2, X3)) → U521(proper(X1), proper(X2), proper(X3))
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
U721(mark(X1), X2, X3) → U721(X1, X2, X3)
PROPER(U61(X)) → U611(proper(X))
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
ACTIVE(U51(tt, M, N)) → ISNAT(N)
ACTIVE(x(X1, X2)) → X(active(X1), X2)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(x(X1, X2)) → X(proper(X1), proper(X2))
PROPER(s(X)) → S(proper(X))
U211(mark(X)) → U211(X)
U111(ok(X1), ok(X2)) → U111(X1, X2)
PROPER(U72(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
U111(mark(X1), X2) → U111(X1, X2)
PROPER(U61(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
ACTIVE(U71(tt, M, N)) → ISNAT(N)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)
PROPER(U12(X)) → PROPER(X)
ACTIVE(U51(X1, X2, X3)) → U511(active(X1), X2, X3)
ACTIVE(x(X1, X2)) → X(X1, active(X2))
PROPER(U41(X1, X2)) → U411(proper(X1), proper(X2))
PROPER(s(X)) → PROPER(X)
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
ACTIVE(x(N, s(M))) → ISNAT(M)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
TOP(ok(X)) → TOP(active(X))
ACTIVE(U32(X)) → U321(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → U721(active(X1), X2, X3)
ACTIVE(x(N, 0)) → ISNAT(N)
PROPER(U32(X)) → PROPER(X)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(U52(tt, M, N)) → S(plus(N, M))
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
ACTIVE(isNat(plus(V1, V2))) → U111(isNat(V1), V2)
X(ok(X1), ok(X2)) → X(X1, X2)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
ACTIVE(U21(X)) → U211(active(X))
ACTIVE(U71(tt, M, N)) → U721(isNat(N), M, N)
TOP(mark(X)) → TOP(proper(X))
PROPER(U51(X1, X2, X3)) → U511(proper(X1), proper(X2), proper(X3))
PROPER(plus(X1, X2)) → PROPER(X2)
ACTIVE(U52(X1, X2, X3)) → U521(active(X1), X2, X3)
PROPER(U11(X1, X2)) → PROPER(X1)
ACTIVE(s(X)) → S(active(X))

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 18 SCCs with 59 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNAT(ok(X)) → ISNAT(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ISNAT(ok(X)) → ISNAT(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

X(ok(X1), ok(X2)) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

X(ok(X1), ok(X2)) → X(X1, X2)
X(X1, mark(X2)) → X(X1, X2)
X(mark(X1), X2) → X(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U721(mark(X1), X2, X3) → U721(X1, X2, X3)
U721(ok(X1), ok(X2), ok(X3)) → U721(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U721(mark(X1), X2, X3) → U721(X1, X2, X3)
U721(ok(X1), ok(X2), ok(X3)) → U721(X1, X2, X3)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U711(mark(X1), X2, X3) → U711(X1, X2, X3)
U711(ok(X1), ok(X2), ok(X3)) → U711(X1, X2, X3)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U611(ok(X)) → U611(X)
U611(mark(X)) → U611(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U611(ok(X)) → U611(X)
U611(mark(X)) → U611(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U521(mark(X1), X2, X3) → U521(X1, X2, X3)
U521(ok(X1), ok(X2), ok(X3)) → U521(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U521(mark(X1), X2, X3) → U521(X1, X2, X3)
U521(ok(X1), ok(X2), ok(X3)) → U521(X1, X2, X3)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(ok(X1), ok(X2), ok(X3)) → U511(X1, X2, X3)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U411(ok(X1), ok(X2)) → U411(X1, X2)
U411(mark(X1), X2) → U411(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U411(ok(X1), ok(X2)) → U411(X1, X2)
U411(mark(X1), X2) → U411(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U321(ok(X)) → U321(X)
U321(mark(X)) → U321(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U321(ok(X)) → U321(X)
U321(mark(X)) → U321(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U311(mark(X1), X2) → U311(X1, X2)
U311(ok(X1), ok(X2)) → U311(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U311(mark(X1), X2) → U311(X1, X2)
U311(ok(X1), ok(X2)) → U311(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U211(ok(X)) → U211(X)
U211(mark(X)) → U211(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U211(ok(X)) → U211(X)
U211(mark(X)) → U211(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U121(mark(X)) → U121(X)
U121(ok(X)) → U121(X)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U121(mark(X)) → U121(X)
U121(ok(X)) → U121(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U111(mark(X1), X2) → U111(X1, X2)
U111(ok(X1), ok(X2)) → U111(X1, X2)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

U111(ok(X1), ok(X2)) → U111(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(U61(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U72(X1, X2, X3)) → PROPER(X1)
PROPER(U72(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(U32(X)) → PROPER(X)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U21(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER(U61(X)) → PROPER(X)
PROPER(isNat(X)) → PROPER(X)
PROPER(U52(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2, X3)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X1)
PROPER(U12(X)) → PROPER(X)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U72(X1, X2, X3)) → PROPER(X1)
PROPER(U72(X1, X2, X3)) → PROPER(X3)
PROPER(U52(X1, X2, X3)) → PROPER(X2)
PROPER(U32(X)) → PROPER(X)
PROPER(U41(X1, X2)) → PROPER(X2)
PROPER(x(X1, X2)) → PROPER(X1)
PROPER(U41(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(U72(X1, X2, X3)) → PROPER(X2)
PROPER(U51(X1, X2, X3)) → PROPER(X3)
PROPER(U71(X1, X2, X3)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X3)
PROPER(U21(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U71(X1, X2, X3)) → PROPER(X1)
PROPER(x(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2, X3)) → PROPER(X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X2)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X1)

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(U52(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U71(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U32(X)) → ACTIVE(X)
ACTIVE(U72(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U61(X)) → ACTIVE(X)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(x(X1, X2)) → ACTIVE(X2)
ACTIVE(U41(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U51(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(x(X1, X2)) → ACTIVE(X1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = 2·x1   
POL(U11(x1, x2)) = x1 + 2·x2   
POL(U12(x1)) = 2·x1   
POL(U21(x1)) = 2·x1   
POL(U31(x1, x2)) = x1 + 2·x2   
POL(U32(x1)) = 2·x1   
POL(U41(x1, x2)) = 2·x1 + 2·x2   
POL(U51(x1, x2, x3)) = 2·x1 + x2 + x3   
POL(U52(x1, x2, x3)) = x1 + 2·x2 + x3   
POL(U61(x1)) = 2·x1   
POL(U71(x1, x2, x3)) = x1 + x2 + 2·x3   
POL(U72(x1, x2, x3)) = 2·x1 + x2 + 2·x3   
POL(active(x1)) = 2·x1   
POL(isNat(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = 2·x1   
POL(plus(x1, x2)) = 2·x1 + 2·x2   
POL(proper(x1)) = x1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(x(x1, x2)) = x1 + x2   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(X)) → TOP(proper(X)) at position [0] we obtained the following new rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(X)) → TOP(active(X))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(x(x0, 0))) → TOP(mark(U61(isNat(x0))))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U61(tt))) → TOP(mark(0))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U21(tt))) → TOP(mark(tt))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNat(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(ok(isNat(x(x0, x1)))) → TOP(mark(U31(isNat(x0), x1)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(U32(tt))) → TOP(mark(tt))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(tt)) → TOP(ok(tt))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(x(x0, 0))) → TOP(mark(U61(isNat(x0))))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U61(tt))) → TOP(mark(0))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(tt))) → TOP(mark(tt))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNat(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(isNat(x(x0, x1)))) → TOP(mark(U31(isNat(x0), x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(x(x0, 0))) → TOP(mark(U61(isNat(x0))))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNat(x0))))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(isNat(x(x0, x1)))) → TOP(mark(U31(isNat(x0), x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(ok(x(x0, 0))) → TOP(mark(U61(isNat(x0))))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNat(x0))))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(isNat(x(x0, x1)))) → TOP(mark(U31(isNat(x0), x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1, x2)) = x2   
POL(U51(x1, x2, x3)) = 0   
POL(U52(x1, x2, x3)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2, x3)) = 1   
POL(U72(x1, x2, x3)) = 1   
POL(active(x1)) = x1   
POL(isNat(x1)) = 0   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = x1   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(x(x1, x2)) = 1   

The following usable rules [17] were oriented:

active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
active(U12(X)) → U12(active(X))
proper(tt) → ok(tt)
active(U21(X)) → U21(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(X1, active(X2))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(0) → ok(0)
proper(U61(X)) → U61(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U12(X)) → U12(proper(X))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
active(U12(tt)) → mark(tt)
active(U11(tt, V2)) → mark(U12(isNat(V2)))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNat(x0))))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(ok(isNat(x(x0, x1)))) → TOP(mark(U31(isNat(x0), x1)))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNat(x0))))
TOP(ok(isNat(x(x0, x1)))) → TOP(mark(U31(isNat(x0), x1)))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 1   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1, x2)) = x2   
POL(U51(x1, x2, x3)) = 0   
POL(U52(x1, x2, x3)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2, x3)) = 0   
POL(U72(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = x1   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(x(x1, x2)) = 0   

The following usable rules [17] were oriented:

active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
active(U12(X)) → U12(active(X))
proper(tt) → ok(tt)
active(U21(X)) → U21(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(X1, active(X2))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(0) → ok(0)
proper(U61(X)) → U61(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U12(X)) → U12(proper(X))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
active(U12(tt)) → mark(tt)
active(U11(tt, V2)) → mark(U12(isNat(V2)))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(ok(U31(tt, x0))) → TOP(mark(U32(isNat(x0))))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 1   
POL(U32(x1)) = 0   
POL(U41(x1, x2)) = x2   
POL(U51(x1, x2, x3)) = 0   
POL(U52(x1, x2, x3)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2, x3)) = 1   
POL(U72(x1, x2, x3)) = 1   
POL(active(x1)) = x1   
POL(isNat(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = x1   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(x(x1, x2)) = 1   

The following usable rules [17] were oriented:

active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
active(U12(X)) → U12(active(X))
proper(tt) → ok(tt)
active(U21(X)) → U21(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(X1, active(X2))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(0) → ok(0)
proper(U61(X)) → U61(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U12(X)) → U12(proper(X))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
active(U12(tt)) → mark(tt)
active(U11(tt, V2)) → mark(U12(isNat(V2)))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1, x2)) = x2   
POL(U51(x1, x2, x3)) = 0   
POL(U52(x1, x2, x3)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2, x3)) = 0   
POL(U72(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = 1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = x1   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(x(x1, x2)) = 0   

The following usable rules [17] were oriented:

active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
active(U12(X)) → U12(active(X))
proper(tt) → ok(tt)
active(U21(X)) → U21(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(X1, active(X2))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(0) → ok(0)
proper(U61(X)) → U61(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U12(X)) → U12(proper(X))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
active(U12(tt)) → mark(tt)
active(U11(tt, V2)) → mark(U12(isNat(V2)))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 1   
POL(U12(x1)) = 1   
POL(U21(x1)) = 1   
POL(U31(x1, x2)) = 1   
POL(U32(x1)) = 1   
POL(U41(x1, x2)) = 1   
POL(U51(x1, x2, x3)) = 1   
POL(U52(x1, x2, x3)) = 1   
POL(U61(x1)) = 1   
POL(U71(x1, x2, x3)) = 1   
POL(U72(x1, x2, x3)) = 1   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(mark(x1)) = 1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = 1   
POL(proper(x1)) = 0   
POL(s(x1)) = 1   
POL(tt) = 0   
POL(x(x1, x2)) = 1   

The following usable rules [17] were oriented:

U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(ok(x(x0, s(x1)))) → TOP(mark(U71(isNat(x1), x1, x0)))
TOP(ok(U52(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U72(tt, x0, x1))) → TOP(mark(plus(x(x1, x0), x1)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U51(isNat(x1), x1, x0)))
TOP(ok(plus(x0, 0))) → TOP(mark(U41(isNat(x0), x0)))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
Used ordering: Combined order from the following AFS and order.
TOP(x1)  =  TOP(x1)
mark(x1)  =  x1
x(x1, x2)  =  x(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
s(x1)  =  s(x1)
U71(x1, x2, x3)  =  U71(x1, x2, x3)
isNat(x1)  =  isNat
plus(x1, x2)  =  plus(x1, x2)
active(x1)  =  x1
U72(x1, x2, x3)  =  U72(x1, x2, x3)
U52(x1, x2, x3)  =  U52(x1, x2, x3)
tt  =  tt
U51(x1, x2, x3)  =  U51(x1, x2, x3)
U11(x1, x2)  =  U11
U41(x1, x2)  =  x2
U12(x1)  =  U12
U32(x1)  =  x1
U61(x1)  =  U61(x1)
U31(x1, x2)  =  U31
U21(x1)  =  x1
0  =  0

Recursive path order with status [2].
Quasi-Precedence:
TOP1 > [x2, U713, U723] > [plus2, U523, U513] > [isNat, tt, U11, U12, U611, U31, 0] > s1

Status:
plus2: [1,2]
U611: multiset
U31: []
U12: []
U11: []
x2: [2,1]
0: multiset
U523: [3,2,1]
isNat: []
tt: multiset
U723: [2,3,1]
s1: multiset
TOP1: [1]
U513: [3,2,1]
U713: [2,3,1]


The following usable rules [17] were oriented:

active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
active(U12(X)) → U12(active(X))
proper(tt) → ok(tt)
active(U21(X)) → U21(active(X))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(x(X1, X2)) → x(X1, active(X2))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(0) → ok(0)
proper(U61(X)) → U61(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U12(X)) → U12(proper(X))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
active(U12(tt)) → mark(tt)
active(U11(tt, V2)) → mark(U12(isNat(V2)))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(ok(U41(tt, x0))) → TOP(mark(x0))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(ok(U41(tt, x0))) → TOP(mark(x0))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1, x2)) = 1 + x2   
POL(U51(x1, x2, x3)) = 0   
POL(U52(x1, x2, x3)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2, x3)) = x3   
POL(U72(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(x(x1, x2)) = 0   

The following usable rules [17] were oriented:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(0) → ok(0)
proper(U61(X)) → U61(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U21(X)) → U21(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U12(X)) → U12(proper(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


TOP(ok(U71(tt, x0, x1))) → TOP(mark(U72(isNat(x1), x0, x1)))
TOP(ok(U51(tt, x0, x1))) → TOP(mark(U52(isNat(x1), x0, x1)))
The remaining pairs can at least be oriented weakly.

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U32(x1)) = 0   
POL(U41(x1, x2)) = 0   
POL(U51(x1, x2, x3)) = 1   
POL(U52(x1, x2, x3)) = 0   
POL(U61(x1)) = 0   
POL(U71(x1, x2, x3)) = 1   
POL(U72(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 0   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = 0   
POL(proper(x1)) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   
POL(x(x1, x2)) = 0   

The following usable rules [17] were oriented:

U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
x(mark(X1), X2) → mark(x(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U21(ok(X)) → ok(U21(X))
U12(ok(X)) → ok(U12(X))
U12(mark(X)) → mark(U12(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U21(mark(X)) → mark(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U32(mark(X)) → mark(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ UsableRulesReductionPairsProof
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

TOP(mark(x(x0, x1))) → TOP(x(proper(x0), proper(x1)))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(U72(x0, x1, x2))) → TOP(U72(proper(x0), proper(x1), proper(x2)))
TOP(mark(U51(x0, x1, x2))) → TOP(U51(proper(x0), proper(x1), proper(x2)))
TOP(mark(U71(x0, x1, x2))) → TOP(U71(proper(x0), proper(x1), proper(x2)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U52(x0, x1, x2))) → TOP(U52(proper(x0), proper(x1), proper(x2)))
TOP(ok(U71(x0, x1, x2))) → TOP(U71(active(x0), x1, x2))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(U32(x0))) → TOP(U32(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U41(x0, x1))) → TOP(U41(proper(x0), proper(x1)))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U61(x0))) → TOP(U61(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U52(x0, x1, x2))) → TOP(U52(active(x0), x1, x2))
TOP(ok(U41(x0, x1))) → TOP(U41(active(x0), x1))
TOP(ok(x(x0, x1))) → TOP(x(active(x0), x1))
TOP(ok(U32(x0))) → TOP(U32(active(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(x(x0, x1))) → TOP(x(x0, active(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U51(x0, x1, x2))) → TOP(U51(active(x0), x1, x2))
TOP(ok(U61(x0))) → TOP(U61(active(x0)))
TOP(ok(U72(x0, x1, x2))) → TOP(U72(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))

The TRS R consists of the following rules:

proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(X1, X2, X3))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U61(mark(X)) → mark(U61(X))
U61(ok(X)) → ok(U61(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
U52(mark(X1), X2, X3) → mark(U52(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U41(mark(X1), X2) → mark(U41(X1, X2))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U32(mark(X)) → mark(U32(X))
U32(ok(X)) → ok(U32(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(X))
isNat(ok(X)) → ok(isNat(X))
U12(mark(X)) → mark(U12(X))
U12(ok(X)) → ok(U12(X))
U11(mark(X1), X2) → mark(U11(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.