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:
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
The set Q is empty.
We have obtained the following QTRS:
a(x) → x
b(b(x)) → a(x)
c(c(b(x))) → b(c(b(c(c(x)))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
b(b(x)) → a(x)
c(c(b(x))) → b(c(b(c(c(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:
B(c(c(x1))) → B(c(b(x1)))
B(b(x1)) → A(x1)
B(c(c(x1))) → B(x1)
The TRS R consists of the following rules:
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(c(c(x1))) → B(c(b(x1)))
B(b(x1)) → A(x1)
B(c(c(x1))) → B(x1)
The TRS R consists of the following rules:
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
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 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(c(c(x1))) → B(c(b(x1)))
B(c(c(x1))) → B(x1)
The TRS R consists of the following rules:
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(c(c(x1))) → B(c(b(x1))) at position [0,0] we obtained the following new rules:
B(c(c(c(c(x0))))) → B(c(c(c(b(c(b(x0)))))))
B(c(c(b(x0)))) → B(c(a(x0)))
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(c(c(x1))) → B(x1)
B(c(c(b(x0)))) → B(c(a(x0)))
B(c(c(c(c(x0))))) → B(c(c(c(b(c(b(x0)))))))
The TRS R consists of the following rules:
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(c(c(b(x0)))) → B(c(a(x0))) at position [0,0] we obtained the following new rules:
B(c(c(b(x0)))) → B(c(x0))
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(c(c(x1))) → B(x1)
B(c(c(b(x0)))) → B(c(x0))
B(c(c(c(c(x0))))) → B(c(c(c(b(c(b(x0)))))))
The TRS R consists of the following rules:
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R.
Interpretation over the domain with elements from 0 to 1.c: 1 + x0
B: 0
a: x0
b: 1 + x0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
B.1(c.0(c.1(b.0(x0)))) → B.1(c.0(x0))
B.1(c.0(c.1(x1))) → B.1(x1)
B.0(c.1(c.0(b.1(x0)))) → B.0(c.1(x0))
B.1(c.0(c.1(c.0(c.1(x0))))) → B.1(c.0(c.1(c.0(b.1(c.0(b.1(x0)))))))
B.0(c.1(c.0(x1))) → B.0(x1)
B.0(c.1(c.0(c.1(c.0(x0))))) → B.0(c.1(c.0(c.1(b.0(c.1(b.0(x0)))))))
The TRS R consists of the following rules:
b.0(c.1(c.0(x1))) → c.0(c.1(b.0(c.1(b.0(x1)))))
b.1(c.0(c.1(x1))) → c.1(c.0(b.1(c.0(b.1(x1)))))
a.1(x1) → x1
b.0(b.1(x1)) → a.1(x1)
b.1(b.0(x1)) → a.0(x1)
a.0(x1) → x1
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.1(c.0(c.1(b.0(x0)))) → B.1(c.0(x0))
B.1(c.0(c.1(x1))) → B.1(x1)
B.0(c.1(c.0(b.1(x0)))) → B.0(c.1(x0))
B.1(c.0(c.1(c.0(c.1(x0))))) → B.1(c.0(c.1(c.0(b.1(c.0(b.1(x0)))))))
B.0(c.1(c.0(x1))) → B.0(x1)
B.0(c.1(c.0(c.1(c.0(x0))))) → B.0(c.1(c.0(c.1(b.0(c.1(b.0(x0)))))))
The TRS R consists of the following rules:
b.0(c.1(c.0(x1))) → c.0(c.1(b.0(c.1(b.0(x1)))))
b.1(c.0(c.1(x1))) → c.1(c.0(b.1(c.0(b.1(x1)))))
a.1(x1) → x1
b.0(b.1(x1)) → a.1(x1)
b.1(b.0(x1)) → a.0(x1)
a.0(x1) → x1
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.1(c.0(c.1(b.0(x0)))) → B.1(c.0(x0))
B.1(c.0(c.1(x1))) → B.1(x1)
B.1(c.0(c.1(c.0(c.1(x0))))) → B.1(c.0(c.1(c.0(b.1(c.0(b.1(x0)))))))
The TRS R consists of the following rules:
b.0(c.1(c.0(x1))) → c.0(c.1(b.0(c.1(b.0(x1)))))
b.1(c.0(c.1(x1))) → c.1(c.0(b.1(c.0(b.1(x1)))))
a.1(x1) → x1
b.0(b.1(x1)) → a.1(x1)
b.1(b.0(x1)) → a.0(x1)
a.0(x1) → x1
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.
The following dependency pairs can be deleted:
B.1(c.0(c.1(b.0(x0)))) → B.1(c.0(x0))
The following rules are removed from R:
b.1(b.0(x1)) → a.0(x1)
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.1(x1)) = x1
POL(a.0(x1)) = x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.1(c.0(c.1(x1))) → B.1(x1)
B.1(c.0(c.1(c.0(c.1(x0))))) → B.1(c.0(c.1(c.0(b.1(c.0(b.1(x0)))))))
The TRS R consists of the following rules:
b.1(c.0(c.1(x1))) → c.1(c.0(b.1(c.0(b.1(x1)))))
a.0(x1) → x1
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(B.1(x1)) = x1
POL(b.1(x1)) = x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.1(c.0(c.1(x1))) → B.1(x1)
B.1(c.0(c.1(c.0(c.1(x0))))) → B.1(c.0(c.1(c.0(b.1(c.0(b.1(x0)))))))
The TRS R consists of the following rules:
b.1(c.0(c.1(x1))) → c.1(c.0(b.1(c.0(b.1(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
B.1(c.0(c.1(x1))) → B.1(x1)
B.1(c.0(c.1(c.0(c.1(x0))))) → B.1(c.0(c.1(c.0(b.1(c.0(b.1(x0)))))))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.1(x1)) = x1
POL(b.1(x1)) = x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = 1 + x1
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QTRS Reverse
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b.1(c.0(c.1(x1))) → c.1(c.0(b.1(c.0(b.1(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.0(c.1(c.0(b.1(x0)))) → B.0(c.1(x0))
B.0(c.1(c.0(x1))) → B.0(x1)
B.0(c.1(c.0(c.1(c.0(x0))))) → B.0(c.1(c.0(c.1(b.0(c.1(b.0(x0)))))))
The TRS R consists of the following rules:
b.0(c.1(c.0(x1))) → c.0(c.1(b.0(c.1(b.0(x1)))))
b.1(c.0(c.1(x1))) → c.1(c.0(b.1(c.0(b.1(x1)))))
a.1(x1) → x1
b.0(b.1(x1)) → a.1(x1)
b.1(b.0(x1)) → a.0(x1)
a.0(x1) → x1
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.
The following dependency pairs can be deleted:
B.0(c.1(c.0(b.1(x0)))) → B.0(c.1(x0))
The following rules are removed from R:
b.0(b.1(x1)) → a.1(x1)
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = x1
POL(a.1(x1)) = x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.0(c.1(c.0(x1))) → B.0(x1)
B.0(c.1(c.0(c.1(c.0(x0))))) → B.0(c.1(c.0(c.1(b.0(c.1(b.0(x0)))))))
The TRS R consists of the following rules:
b.0(c.1(c.0(x1))) → c.0(c.1(b.0(c.1(b.0(x1)))))
a.1(x1) → x1
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(B.0(x1)) = x1
POL(b.0(x1)) = x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.0(c.1(c.0(x1))) → B.0(x1)
B.0(c.1(c.0(c.1(c.0(x0))))) → B.0(c.1(c.0(c.1(b.0(c.1(b.0(x0)))))))
The TRS R consists of the following rules:
b.0(c.1(c.0(x1))) → c.0(c.1(b.0(c.1(b.0(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
B.0(c.1(c.0(x1))) → B.0(x1)
B.0(c.1(c.0(c.1(c.0(x0))))) → B.0(c.1(c.0(c.1(b.0(c.1(b.0(x0)))))))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = x1
POL(b.0(x1)) = x1
POL(c.0(x1)) = 1 + x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ PisEmptyProof
↳ QTRS Reverse
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b.0(c.1(c.0(x1))) → c.0(c.1(b.0(c.1(b.0(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.
We have reversed the following QTRS:
The set of rules R is
a(x1) → x1
b(b(x1)) → a(x1)
b(c(c(x1))) → c(c(b(c(b(x1)))))
The set Q is empty.
We have obtained the following QTRS:
a(x) → x
b(b(x)) → a(x)
c(c(b(x))) → b(c(b(c(c(x)))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
b(b(x)) → a(x)
c(c(b(x))) → b(c(b(c(c(x)))))
Q is empty.