Termination w.r.t. Q proof of /tmp/tmpPJSj0z/PEANO_nokinds_noand

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:

ISNAT(ok(X)) → ISNAT(X)
The graph contains the following edges 1 > 1

↳ 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))

↳ 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)

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

PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2
PLUS(mark(X1), X2) → PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
PLUS(X1, mark(X2)) → PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ 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))

↳ 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)

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

S(ok(X)) → S(X)
The graph contains the following edges 1 > 1
S(mark(X)) → S(X)
The graph contains the following edges 1 > 1

↳ 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))

↳ 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)

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

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
U42¹(ok(X1), ok(X2), ok(X3)) → U42¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3

↳ 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))

↳ 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)

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

U41¹(ok(X1), ok(X2), ok(X3)) → U41¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ 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))

↳ 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)

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

U31¹(mark(X1), X2) → U31¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ 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))

↳ 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)

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

U21¹(ok(X)) → U21¹(X)
The graph contains the following edges 1 > 1
U21¹(mark(X)) → U21¹(X)
The graph contains the following edges 1 > 1

↳ 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))

↳ 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)

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

U12¹(mark(X)) → U12¹(X)
The graph contains the following edges 1 > 1
U12¹(ok(X)) → U12¹(X)
The graph contains the following edges 1 > 1

↳ 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))

↳ 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)

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

U11¹(mark(X1), X2) → U11¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2

↳ 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))

↳ 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)

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

PROPER(isNat(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U42(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2, X3)) → PROPER(X3)
The graph contains the following edges 1 > 1
PROPER(U12(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(plus(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U42(X1, X2, X3)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(U31(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U21(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(U42(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U41(X1, X2, X3)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(plus(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(U11(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1

↳ 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))

↳ 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)

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

ACTIVE(U31(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U21(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U12(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(U11(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1

↳ 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.