(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(0) = 2
POL(U11(x1, x2, x3)) = x1 + x2 + x3
POL(U12(x1, x2, x3)) = 2·x1 + x2 + x3
POL(activate(x1)) = x1
POL(plus(x1, x2)) = x1 + x2
POL(s(x1)) = x1
POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
plus(N, 0) → N
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1, x2, x3)) = x1 + 2·x2 + 2·x3
POL(U12(x1, x2, x3)) = x1 + 2·x2 + 2·x3
POL(activate(x1)) = x1
POL(plus(x1, x2)) = 2·x1 + 2·x2
POL(s(x1)) = 1 + x1
POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
plus(N, s(M)) → U11(tt, M, N)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(U11(x1, x2, x3)) = 3 + x1 + x2 + x3
POL(U12(x1, x2, x3)) = x1 + x2 + x3
POL(activate(x1)) = 1 + x1
POL(plus(x1, x2)) = x1 + x2
POL(s(x1)) = x1
POL(tt) = 3
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
activate(X) → X
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) TRUE
(9) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) TRUE
(11) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(12) TRUE