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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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:

U121(mark(X)) → U121(X)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(plus(N, s(M))) → U411(isNat(M), M, N)
PROPER(U12(X)) → U121(proper(X))
U421(mark(X1), X2, X3) → U421(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
U211(ok(X)) → U211(X)
PROPER(plus(X1, X2)) → PLUS(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(U42(X1, X2, X3)) → U421(active(X1), X2, X3)
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(plus(X1, X2)) → PLUS(active(X1), X2)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
ACTIVE(U11(tt, V2)) → U121(isNat(V2))
U311(mark(X1), X2) → U311(X1, X2)
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → S(proper(X))
U211(mark(X)) → U211(X)
PROPER(plus(X1, X2)) → PROPER(X1)
ACTIVE(U42(tt, M, N)) → PLUS(N, M)
U111(ok(X1), ok(X2)) → U111(X1, X2)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
U421(ok(X1), ok(X2), ok(X3)) → U421(X1, X2, X3)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
ACTIVE(U41(tt, M, N)) → U421(isNat(N), M, N)
U111(mark(X1), X2) → U111(X1, X2)
ACTIVE(U42(tt, M, N)) → S(plus(N, M))
S(ok(X)) → S(X)
PROPER(isNat(X)) → PROPER(X)
ACTIVE(U12(X)) → U121(active(X))
ACTIVE(U41(tt, M, N)) → ISNAT(N)
ACTIVE(plus(X1, X2)) → PLUS(X1, active(X2))
TOP(mark(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → U411(proper(X1), proper(X2), proper(X3))
U121(ok(X)) → U121(X)
ACTIVE(plus(N, 0)) → ISNAT(N)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PROPER(U12(X)) → PROPER(X)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → U411(active(X1), X2, X3)
ACTIVE(isNat(s(V1))) → U211(isNat(V1))
ACTIVE(plus(N, 0)) → U311(isNat(N), N)
PROPER(s(X)) → PROPER(X)
PROPER(U21(X)) → U211(proper(X))
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
U311(ok(X1), ok(X2)) → U311(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
TOP(ok(X)) → TOP(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
S(mark(X)) → S(X)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
ACTIVE(isNat(plus(V1, V2))) → ISNAT(V1)
ACTIVE(isNat(plus(V1, V2))) → U111(isNat(V1), V2)
PROPER(U42(X1, X2, X3)) → PROPER(X1)
ACTIVE(U11(tt, V2)) → ISNAT(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
ACTIVE(U21(X)) → U211(active(X))
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
PROPER(U21(X)) → PROPER(X)
TOP(mark(X)) → TOP(proper(X))
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U42(X1, X2, X3)) → U421(proper(X1), proper(X2), proper(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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:

U121(mark(X)) → U121(X)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(plus(N, s(M))) → U411(isNat(M), M, N)
PROPER(U12(X)) → U121(proper(X))
U421(mark(X1), X2, X3) → U421(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
PROPER(U11(X1, X2)) → U111(proper(X1), proper(X2))
U211(ok(X)) → U211(X)
PROPER(plus(X1, X2)) → PLUS(proper(X1), proper(X2))
PROPER(U31(X1, X2)) → PROPER(X1)
ACTIVE(U42(X1, X2, X3)) → U421(active(X1), X2, X3)
ACTIVE(plus(N, s(M))) → ISNAT(M)
ACTIVE(plus(X1, X2)) → PLUS(active(X1), X2)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
ACTIVE(U11(tt, V2)) → U121(isNat(V2))
U311(mark(X1), X2) → U311(X1, X2)
ACTIVE(U31(X1, X2)) → U311(active(X1), X2)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(s(X)) → S(proper(X))
U211(mark(X)) → U211(X)
PROPER(plus(X1, X2)) → PROPER(X1)
ACTIVE(U42(tt, M, N)) → PLUS(N, M)
U111(ok(X1), ok(X2)) → U111(X1, X2)
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
U421(ok(X1), ok(X2), ok(X3)) → U421(X1, X2, X3)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
ACTIVE(U41(tt, M, N)) → U421(isNat(N), M, N)
U111(mark(X1), X2) → U111(X1, X2)
ACTIVE(U42(tt, M, N)) → S(plus(N, M))
S(ok(X)) → S(X)
PROPER(isNat(X)) → PROPER(X)
ACTIVE(U12(X)) → U121(active(X))
ACTIVE(U41(tt, M, N)) → ISNAT(N)
ACTIVE(plus(X1, X2)) → PLUS(X1, active(X2))
TOP(mark(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → U411(proper(X1), proper(X2), proper(X3))
U121(ok(X)) → U121(X)
ACTIVE(plus(N, 0)) → ISNAT(N)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PROPER(U12(X)) → PROPER(X)
PROPER(U31(X1, X2)) → U311(proper(X1), proper(X2))
TOP(ok(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → U411(active(X1), X2, X3)
ACTIVE(isNat(s(V1))) → U211(isNat(V1))
ACTIVE(plus(N, 0)) → U311(isNat(N), N)
PROPER(s(X)) → PROPER(X)
PROPER(U21(X)) → U211(proper(X))
ACTIVE(U11(X1, X2)) → U111(active(X1), X2)
U311(ok(X1), ok(X2)) → U311(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
TOP(ok(X)) → TOP(active(X))
PROPER(isNat(X)) → ISNAT(proper(X))
ACTIVE(U21(X)) → ACTIVE(X)
S(mark(X)) → S(X)
ISNAT(ok(X)) → ISNAT(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
ACTIVE(isNat(plus(V1, V2))) → ISNAT(V1)
ACTIVE(isNat(plus(V1, V2))) → U111(isNat(V1), V2)
PROPER(U42(X1, X2, X3)) → PROPER(X1)
ACTIVE(U11(tt, V2)) → ISNAT(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
ACTIVE(U21(X)) → U211(active(X))
U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
PROPER(U21(X)) → PROPER(X)
TOP(mark(X)) → TOP(proper(X))
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U42(X1, X2, X3)) → U421(proper(X1), proper(X2), proper(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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 12 SCCs with 34 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ UsableRulesProof
          ↳ 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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

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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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
            ↳ UsableRulesProof
          ↳ 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U421(mark(X1), X2, X3) → U421(X1, X2, X3)
U421(ok(X1), ok(X2), ok(X3)) → U421(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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

U421(mark(X1), X2, X3) → U421(X1, X2, X3)
U421(ok(X1), ok(X2), ok(X3)) → U421(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

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

U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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

U411(ok(X1), ok(X2), ok(X3)) → U411(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(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
            ↳ 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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
            ↳ 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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
            ↳ 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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
            ↳ 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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
            ↳ UsableRulesProof
          ↳ QDP
          ↳ QDP

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

PROPER(isNat(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U12(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(plus(X1, X2)) → 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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

PROPER(isNat(X)) → PROPER(X)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U12(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(U21(X)) → PROPER(X)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(U11(X1, X2)) → PROPER(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
            ↳ UsableRulesProof
          ↳ QDP

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

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U11(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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

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

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X)) → ACTIVE(X)
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X)) → ACTIVE(X)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(U11(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
            ↳ 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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)) = x1   
POL(U11(x1, x2)) = x1 + x2   
POL(U12(x1)) = x1   
POL(U21(x1)) = 2·x1   
POL(U31(x1, x2)) = 2·x1 + 2·x2   
POL(U41(x1, x2, x3)) = x1 + x2 + x3   
POL(U42(x1, x2, x3)) = x1 + 2·x2 + 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 + x2   
POL(proper(x1)) = x1   
POL(s(x1)) = x1   
POL(tt) = 0   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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(tt)) → TOP(ok(tt))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(X)) → TOP(active(X))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U41(isNat(x1), x1, x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(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(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U31(isNat(x0), x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(active(x0), x1, x2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(mark(tt)) → TOP(ok(tt))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U41(isNat(x1), x1, x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(isNat(0))) → TOP(mark(tt))
TOP(ok(U12(tt))) → TOP(mark(tt))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U21(x0))) → TOP(U21(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(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(plus(x0, 0))) → TOP(mark(U31(isNat(x0), x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U41(isNat(x1), x1, x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
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(plus(x0, 0))) → TOP(mark(U31(isNat(x0), x0)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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(plus(x0, 0))) → TOP(mark(U31(isNat(x0), x0)))
The remaining pairs can at least be oriented weakly.

TOP(ok(isNat(s(x0)))) → TOP(mark(U21(isNat(x0))))
TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U41(isNat(x1), x1, x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
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(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(active(x0), x1, x2))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = x2   
POL(U41(x1, x2, x3)) = 0   
POL(U42(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(isNat(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = x1 + x2   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

proper(s(X)) → s(proper(X))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
proper(0) → ok(0)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(U11(X1, X2)) → U11(active(X1), X2)
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U21(X)) → U21(active(X))
active(U12(X)) → U12(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U31(X1, X2)) → U31(active(X1), X2)
active(s(X)) → s(active(X))
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(plus(X1, X2)) → plus(X1, active(X2))
active(plus(X1, X2)) → plus(active(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(plus(x0, s(x1)))) → TOP(mark(U41(isNat(x1), x1, x0)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
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(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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(plus(x0, s(x1)))) → TOP(mark(U41(isNat(x1), x1, x0)))
TOP(ok(isNat(plus(x0, x1)))) → TOP(mark(U11(isNat(x0), x1)))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(isNat(x0))))
The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = x2   
POL(U41(x1, x2, x3)) = 0   
POL(U42(x1, x2, x3)) = 0   
POL(active(x1)) = 0   
POL(isNat(x1)) = 1 + x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = 1   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 0   

The following usable rules [17] were oriented:

proper(s(X)) → s(proper(X))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
proper(0) → ok(0)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
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))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(tt, x0, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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, x1))) → TOP(mark(U42(isNat(x1), x0, x1)))
The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(active(x0), x1, x2))
Used ordering: Polynomial interpretation with max and min functions [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)) = x2   
POL(U41(x1, x2, x3)) = 1 + x3   
POL(U42(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   

The following usable rules [17] were oriented:

proper(s(X)) → s(proper(X))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
proper(0) → ok(0)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
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))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(isNat(x0))) → TOP(isNat(proper(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(x1, x2, x3)) = 1   
POL(U42(x1, x2, x3)) = 1   
POL(active(x1)) = 1 + x1   
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   

The following usable rules [17] were oriented:

plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
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))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U31(tt, x0))) → TOP(mark(x0))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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(x0))
The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(active(x0), x1, x2))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1   
POL(TOP(x1)) = x1   
POL(U11(x1, x2)) = x1   
POL(U12(x1)) = x1   
POL(U21(x1)) = 1   
POL(U31(x1, x2)) = x1 + x2   
POL(U41(x1, x2, x3)) = x3   
POL(U42(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 + x2   
POL(proper(x1)) = x1   
POL(s(x1)) = 0   
POL(tt) = 1   

The following usable rules [17] were oriented:

proper(s(X)) → s(proper(X))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
proper(0) → ok(0)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(U11(X1, X2)) → U11(active(X1), X2)
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U21(X)) → U21(active(X))
active(U12(X)) → U12(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U31(X1, X2)) → U31(active(X1), X2)
active(s(X)) → s(active(X))
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(plus(X1, X2)) → plus(X1, active(X2))
active(plus(X1, X2)) → plus(active(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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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(U42(tt, x0, x1))) → TOP(mark(s(plus(x1, x0))))
The remaining pairs can at least be oriented weakly.

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(x1, x2, x3)) = 0   
POL(U42(x1, x2, x3)) = 1   
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   

The following usable rules [17] were oriented:

plus(X1, mark(X2)) → mark(plus(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
s(mark(X)) → mark(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U21(mark(X)) → mark(U21(X))
U21(ok(X)) → ok(U21(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))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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

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

TOP(mark(plus(x0, x1))) → TOP(plus(proper(x0), proper(x1)))
TOP(mark(U12(x0))) → TOP(U12(proper(x0)))
TOP(ok(plus(x0, x1))) → TOP(plus(x0, active(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(U42(x0, x1, x2))) → TOP(U42(proper(x0), proper(x1), proper(x2)))
TOP(ok(U41(x0, x1, x2))) → TOP(U41(active(x0), x1, x2))
TOP(ok(U12(x0))) → TOP(U12(active(x0)))
TOP(mark(U21(x0))) → TOP(U21(proper(x0)))
TOP(mark(U31(x0, x1))) → TOP(U31(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(plus(x0, x1))) → TOP(plus(active(x0), x1))
TOP(ok(U21(x0))) → TOP(U21(active(x0)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U31(x0, x1))) → TOP(U31(active(x0), x1))
TOP(mark(U41(x0, x1, x2))) → TOP(U41(proper(x0), proper(x1), proper(x2)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U42(x0, x1, x2))) → TOP(U42(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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))

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