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(b(x1)) → b(a(a(x1)))
b(x1) → c(a(c(x1)))
a(a(x1)) → a(c(a(x1)))
Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x1)) → b(a(a(x1)))
b(x1) → c(a(c(x1)))
a(a(x1)) → a(c(a(x1)))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(b(x1)) → b(a(a(x1)))
b(x1) → c(a(c(x1)))
a(a(x1)) → a(c(a(x1)))
The set Q is empty.
We have obtained the following QTRS:
b(a(x)) → a(a(b(x)))
b(x) → c(a(c(x)))
a(a(x)) → a(c(a(x)))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ RFCMatchBoundsTRSProof
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(x)) → a(a(b(x)))
b(x) → c(a(c(x)))
a(a(x)) → a(c(a(x)))
Q is empty.
Termination of the TRS R could be shown with a Match Bound [6,7] of 3. This implies Q-termination of R.
The following rules were used to construct the certificate:
b(a(x)) → a(a(b(x)))
b(x) → c(a(c(x)))
a(a(x)) → a(c(a(x)))
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
147, 148, 150, 149, 152, 151, 153, 154, 156, 155, 158, 157, 159, 160, 161, 162, 164, 163, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174
Node 147 is start node and node 148 is final node.
Those nodes are connect through the following edges:
- 147 to 149 labelled a_1(0)
- 147 to 151 labelled c_1(0)
- 147 to 153 labelled a_1(0)
- 147 to 161 labelled a_1(1)
- 148 to 148 labelled #_1(0)
- 150 to 148 labelled b_1(0)
- 150 to 155 labelled c_1(1)
- 150 to 157 labelled a_1(1)
- 150 to 165 labelled a_1(2)
- 149 to 150 labelled a_1(0)
- 149 to 167 labelled a_1(1)
- 152 to 148 labelled c_1(0)
- 151 to 152 labelled a_1(0)
- 153 to 154 labelled c_1(0)
- 154 to 148 labelled a_1(0)
- 154 to 159 labelled a_1(1)
- 156 to 148 labelled c_1(1)
- 155 to 156 labelled a_1(1)
- 158 to 148 labelled b_1(1)
- 158 to 163 labelled c_1(2)
- 158 to 157 labelled a_1(1)
- 158 to 165 labelled a_1(2)
- 157 to 158 labelled a_1(1)
- 157 to 169 labelled a_1(2)
- 159 to 160 labelled c_1(1)
- 160 to 148 labelled a_1(1)
- 160 to 159 labelled a_1(1)
- 161 to 162 labelled c_1(1)
- 162 to 150 labelled a_1(1)
- 162 to 169 labelled a_1(2)
- 162 to 167 labelled a_1(1)
- 164 to 148 labelled c_1(2)
- 163 to 164 labelled a_1(2)
- 165 to 166 labelled c_1(2)
- 166 to 158 labelled a_1(2)
- 166 to 169 labelled a_1(2)
- 166 to 171 labelled a_1(3)
- 167 to 168 labelled c_1(1)
- 168 to 157 labelled a_1(1)
- 168 to 165 labelled a_1(2), a_1(1)
- 169 to 170 labelled c_1(2)
- 170 to 157 labelled a_1(2)
- 170 to 165 labelled a_1(2)
- 170 to 173 labelled a_1(3)
- 171 to 172 labelled c_1(3)
- 172 to 165 labelled a_1(3)
- 173 to 174 labelled c_1(3)
- 174 to 169 labelled a_1(3)
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x1)) → b(a(a(x1)))
b(x1) → c(a(c(x1)))
a(a(x1)) → a(c(a(x1)))
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
b(x1) → c(a(c(x1)))
Used ordering:
Polynomial interpretation [25]:
POL(a(x1)) = x1
POL(b(x1)) = 1 + x1
POL(c(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ RRRPoloQTRSProof
↳ QTRS
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(x1)) → b(a(a(x1)))
a(a(x1)) → a(c(a(x1)))
Q is empty.