Termination w.r.t. Q of the following Term Rewriting System could be proven:
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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
Q is empty.
The TRS is overlay and locally confluent. By [19] we can switch to innermost.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
The set Q consists of the following terms:
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
U111(tt, M, N) → ACTIVATE(N)
U221(tt, M, N) → ACTIVATE(N)
U211(tt, M, N) → U221(tt, activate(M), activate(N))
U221(tt, M, N) → ACTIVATE(M)
U121(tt, M, N) → PLUS(activate(N), activate(M))
U111(tt, M, N) → ACTIVATE(M)
U221(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U211(tt, M, N)
U111(tt, M, N) → U121(tt, activate(M), activate(N))
U211(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U111(tt, M, N)
U221(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U211(tt, M, N) → ACTIVATE(N)
U121(tt, M, N) → ACTIVATE(N)
U121(tt, M, N) → ACTIVATE(M)
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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
The set Q consists of the following terms:
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U111(tt, M, N) → ACTIVATE(N)
U221(tt, M, N) → ACTIVATE(N)
U211(tt, M, N) → U221(tt, activate(M), activate(N))
U221(tt, M, N) → ACTIVATE(M)
U121(tt, M, N) → PLUS(activate(N), activate(M))
U111(tt, M, N) → ACTIVATE(M)
U221(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U211(tt, M, N)
U111(tt, M, N) → U121(tt, activate(M), activate(N))
U211(tt, M, N) → ACTIVATE(M)
PLUS(N, s(M)) → U111(tt, M, N)
U221(tt, M, N) → PLUS(x(activate(N), activate(M)), activate(N))
U211(tt, M, N) → ACTIVATE(N)
U121(tt, M, N) → ACTIVATE(N)
U121(tt, M, N) → ACTIVATE(M)
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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
The set Q consists of the following terms:
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 9 less nodes.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS(N, s(M)) → U111(tt, M, N)
U121(tt, M, N) → PLUS(activate(N), activate(M))
U111(tt, M, N) → U121(tt, activate(M), activate(N))
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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
The set Q consists of the following terms:
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS(N, s(M)) → U111(tt, M, N)
U121(tt, M, N) → PLUS(activate(N), activate(M))
U111(tt, M, N) → U121(tt, activate(M), activate(N))
The TRS R consists of the following rules:
activate(X) → X
The set Q consists of the following terms:
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
PLUS(N, s(M)) → U111(tt, M, N)
U121(tt, M, N) → PLUS(activate(N), activate(M))
U111(tt, M, N) → U121(tt, activate(M), activate(N))
The TRS R consists of the following rules:
activate(X) → X
The set Q consists of the following terms:
activate(x0)
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.
The following dependency pairs can be deleted:
PLUS(N, s(M)) → U111(tt, M, N)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [25]:
POL(PLUS(x1, x2)) = x1 + x2
POL(U111(x1, x2, x3)) = x1 + 2·x2 + x3
POL(U121(x1, x2, x3)) = x1 + x2 + x3
POL(activate(x1)) = x1
POL(s(x1)) = 1 + 2·x1
POL(tt) = 0
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
U121(tt, M, N) → PLUS(activate(N), activate(M))
U111(tt, M, N) → U121(tt, activate(M), activate(N))
The TRS R consists of the following rules:
activate(X) → X
The set Q consists of the following terms:
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
U211(tt, M, N) → U221(tt, activate(M), activate(N))
U221(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U211(tt, M, N)
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)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
The set Q consists of the following terms:
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
U211(tt, M, N) → U221(tt, activate(M), activate(N))
U221(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U211(tt, M, N)
The TRS R consists of the following rules:
activate(X) → X
The set Q consists of the following terms:
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
U11(tt, x0, x1)
U12(tt, x0, x1)
U21(tt, x0, x1)
U22(tt, x0, x1)
plus(x0, 0)
plus(x0, s(x1))
x(x0, 0)
x(x0, s(x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
Q DP problem:
The TRS P consists of the following rules:
U211(tt, M, N) → U221(tt, activate(M), activate(N))
U221(tt, M, N) → X(activate(N), activate(M))
X(N, s(M)) → U211(tt, M, N)
The TRS R consists of the following rules:
activate(X) → X
The set Q consists of the following terms:
activate(x0)
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.
The following dependency pairs can be deleted:
X(N, s(M)) → U211(tt, M, N)
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [25]:
POL(U211(x1, x2, x3)) = x1 + 2·x2 + x3
POL(U221(x1, x2, x3)) = x1 + 2·x2 + x3
POL(X(x1, x2)) = 1 + x1 + 2·x2
POL(activate(x1)) = x1
POL(s(x1)) = 2 + 2·x1
POL(tt) = 1
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
U211(tt, M, N) → U221(tt, activate(M), activate(N))
U221(tt, M, N) → X(activate(N), activate(M))
The TRS R consists of the following rules:
activate(X) → X
The set Q consists of the following terms:
activate(x0)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.