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__f(f(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(X)
Q is empty.
↳ QTRS
↳ RFCMatchBoundsTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(f(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(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:
a__f(f(X)) → a__c(f(g(f(X))))
a__c(X) → d(X)
a__h(X) → a__c(d(X))
mark(f(X)) → a__f(mark(X))
mark(c(X)) → a__c(X)
mark(h(X)) → a__h(mark(X))
mark(g(X)) → g(X)
mark(d(X)) → d(X)
a__f(X) → f(X)
a__c(X) → c(X)
a__h(X) → h(X)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
1, 2, 3, 5, 6, 4, 7, 8, 9, 10, 11, 12, 14, 15, 13
Node 1 is start node and node 2 is final node.
Those nodes are connect through the following edges:
- 1 to 3 labelled a__h_1(0), a__f_1(0), f_1(1), h_1(1)
- 1 to 2 labelled h_1(0), c_1(0), d_1(0), g_1(0), a__c_1(0), f_1(0), c_1(1), d_1(1)
- 1 to 1 labelled a__c_1(0), c_1(1), d_1(1)
- 1 to 4 labelled a__c_1(0), c_1(1), d_1(1)
- 1 to 7 labelled a__c_1(1), c_1(2), d_1(2)
- 1 to 10 labelled a__c_1(1), c_1(2), d_1(2)
- 2 to 2 labelled #_1(0)
- 3 to 2 labelled mark_1(0), g_1(1), a__c_1(1), d_1(1), c_1(2), d_1(2)
- 3 to 8 labelled a__h_1(1), a__f_1(1), f_1(2), h_1(2)
- 3 to 9 labelled a__c_1(2), c_1(3), d_1(3)
- 3 to 13 labelled a__c_1(2), c_1(3), d_1(3)
- 5 to 6 labelled g_1(0)
- 6 to 2 labelled f_1(0)
- 4 to 5 labelled f_1(0)
- 7 to 3 labelled d_1(1)
- 8 to 2 labelled mark_1(1), g_1(1), a__c_1(1), d_1(1), c_1(2), d_1(2)
- 8 to 8 labelled a__h_1(1), a__f_1(1), f_1(2), h_1(2)
- 8 to 9 labelled a__c_1(2), c_1(3), d_1(3)
- 8 to 13 labelled a__c_1(2), c_1(3), d_1(3)
- 9 to 8 labelled d_1(2)
- 10 to 11 labelled f_1(1)
- 11 to 12 labelled g_1(1)
- 12 to 8 labelled f_1(1)
- 14 to 15 labelled g_1(2)
- 15 to 8 labelled f_1(2)
- 13 to 14 labelled f_1(2)