Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

U11¹(tt, V2) → U12¹(isNat(activate(V2)))
U11¹(tt, V2) → ISNAT(activate(V2))
U11¹(tt, V2) → ACTIVATE(V2)
U31¹(tt, N) → ACTIVATE(N)
U41¹(tt, M, N) → U42¹(isNat(activate(N)), activate(M), activate(N))
U41¹(tt, M, N) → ISNAT(activate(N))
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)
U42¹(tt, M, N) → S(plus(activate(N), activate(M)))
U42¹(tt, M, N) → PLUS(activate(N), activate(M))
U42¹(tt, M, N) → ACTIVATE(N)
U42¹(tt, M, N) → ACTIVATE(M)
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → U21¹(isNat(activate(V1)))
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, 0) → U31¹(isNat(N), N)
PLUS(N, 0) → ISNAT(N)
PLUS(N, s(M)) → U41¹(isNat(M), M, N)
PLUS(N, s(M)) → ISNAT(M)
ACTIVATE(n__0) → 0¹
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
ACTIVATE(n__s(X)) → S(X)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(4) Obligation:

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

U11¹(tt, V2) → ISNAT(activate(V2))
ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2))
U11¹(tt, V2) → ACTIVATE(V2)
ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2)
PLUS(N, 0) → U31¹(isNat(N), N)
U31¹(tt, N) → ACTIVATE(N)
PLUS(N, 0) → ISNAT(N)
ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1))
ISNAT(n__plus(V1, V2)) → ACTIVATE(V1)
ISNAT(n__plus(V1, V2)) → ACTIVATE(V2)
ISNAT(n__s(V1)) → ISNAT(activate(V1))
ISNAT(n__s(V1)) → ACTIVATE(V1)
PLUS(N, s(M)) → U41¹(isNat(M), M, N)
U41¹(tt, M, N) → U42¹(isNat(activate(N)), activate(M), activate(N))
U42¹(tt, M, N) → PLUS(activate(N), activate(M))
PLUS(N, s(M)) → ISNAT(M)
U42¹(tt, M, N) → ACTIVATE(N)
U42¹(tt, M, N) → ACTIVATE(M)
U41¹(tt, M, N) → ISNAT(activate(N))
U41¹(tt, M, N) → ACTIVATE(N)
U41¹(tt, M, N) → ACTIVATE(M)

The TRS R consists of the following rules:

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(X) → X

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

(5) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Polynomial interpretation [POLO]:

POL(0) = 1   
POL(U11(x₁, x₂)) = x₁   
POL(U12(x₁)) = x₁   
POL(U21(x₁)) = 1   
POL(U31(x₁, x₂)) = x₂   
POL(U41(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(U42(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(activate(x₁)) = x₁   
POL(isNat(x₁)) = 1   
POL(n__0) = 1   
POL(n__plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(n__s(x₁)) = 1 + x₁   
POL(plus(x₁, x₂)) = 1 + x₁ + x₂   
POL(s(x₁)) = 1 + x₁   
POL(tt) = 1   

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

• ISNAT(n__plus(V1, V2)) → U11¹(isNat(activate(V1)), activate(V2)) (allowed arguments on rhs = {2})
  The graph contains the following edges 1 > 2

• U11¹(tt, V2) → ISNAT(activate(V2)) (allowed arguments on rhs = {1})
  The graph contains the following edges 2 >= 1

• U11¹(tt, V2) → ACTIVATE(V2) (allowed arguments on rhs = {1})
  The graph contains the following edges 2 >= 1

• ACTIVATE(n__plus(X1, X2)) → PLUS(X1, X2) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 1 > 1, 1 > 2

• U41¹(tt, M, N) → ISNAT(activate(N)) (allowed arguments on rhs = {1})
  The graph contains the following edges 3 >= 1

• U31¹(tt, N) → ACTIVATE(N) (allowed arguments on rhs = {1})
  The graph contains the following edges 2 >= 1

• U42¹(tt, M, N) → PLUS(activate(N), activate(M)) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 3 >= 1, 2 >= 2

• U41¹(tt, M, N) → U42¹(isNat(activate(N)), activate(M), activate(N)) (allowed arguments on rhs = {2, 3})
  The graph contains the following edges 2 >= 2, 3 >= 3

• PLUS(N, 0) → U31¹(isNat(N), N) (allowed arguments on rhs = {2})
  The graph contains the following edges 1 >= 2

• PLUS(N, s(M)) → U41¹(isNat(M), M, N) (allowed arguments on rhs = {2, 3})
  The graph contains the following edges 2 > 2, 1 >= 3

• ISNAT(n__plus(V1, V2)) → ISNAT(activate(V1)) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• ISNAT(n__s(V1)) → ISNAT(activate(V1)) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• ISNAT(n__plus(V1, V2)) → ACTIVATE(V1) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• ISNAT(n__plus(V1, V2)) → ACTIVATE(V2) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• ISNAT(n__s(V1)) → ACTIVATE(V1) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• PLUS(N, 0) → ISNAT(N) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 >= 1

• PLUS(N, s(M)) → ISNAT(M) (allowed arguments on rhs = {1})
  The graph contains the following edges 2 > 1

• U41¹(tt, M, N) → ACTIVATE(N) (allowed arguments on rhs = {1})
  The graph contains the following edges 3 >= 1

• U41¹(tt, M, N) → ACTIVATE(M) (allowed arguments on rhs = {1})
  The graph contains the following edges 2 >= 1

• U42¹(tt, M, N) → ACTIVATE(N) (allowed arguments on rhs = {1})
  The graph contains the following edges 3 >= 1

• U42¹(tt, M, N) → ACTIVATE(M) (allowed arguments on rhs = {1})
  The graph contains the following edges 2 >= 1

We oriented the following set of usable rules [AAECC05,FROCOS05].

s(X) → n__s(X)
plus(X1, X2) → n__plus(X1, X2)
plus(N, s(M)) → U41(isNat(M), M, N)
plus(N, 0) → U31(isNat(N), N)
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__0) → tt
activate(X) → X
activate(n__s(X)) → s(X)
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__0) → 0
U42(tt, M, N) → s(plus(activate(N), activate(M)))
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U31(tt, N) → activate(N)
U21(tt) → tt
U12(tt) → tt
U11(tt, V2) → U12(isNat(activate(V2)))
0 → n__0

(6) TRUE