Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 RFCMatchBoundsTRSProof (⇔)
↳2 TRUE

(0) Obligation:

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.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] 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:

8867181, 8867182, 8867183, 8867186, 8867185, 8867184, 8867187, 8867188, 8867189, 8867190, 8867191, 8867192, 8867193, 8867195, 8867194

Node 8867181 is start node and node 8867182 is final node.

Those nodes are connect through the following edges:

  • 8867181 to 8867183 labelled a__h_1(0), a__f_1(0), f_1(1), h_1(1)
  • 8867181 to 8867182 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)
  • 8867181 to 8867181 labelled a__c_1(0), c_1(1), d_1(1)
  • 8867181 to 8867184 labelled a__c_1(0), c_1(1), d_1(1)
  • 8867181 to 8867187 labelled a__c_1(1), c_1(2), d_1(2)
  • 8867181 to 8867190 labelled a__c_1(1), c_1(2), d_1(2)
  • 8867182 to 8867182 labelled #_1(0)
  • 8867183 to 8867182 labelled mark_1(0), g_1(1), a__c_1(1), d_1(1), c_1(2), d_1(2)
  • 8867183 to 8867188 labelled a__h_1(1), a__f_1(1), f_1(2), h_1(2)
  • 8867183 to 8867189 labelled a__c_1(2), c_1(3), d_1(3)
  • 8867183 to 8867193 labelled a__c_1(2), c_1(3), d_1(3)
  • 8867186 to 8867182 labelled f_1(0)
  • 8867185 to 8867186 labelled g_1(0)
  • 8867184 to 8867185 labelled f_1(0)
  • 8867187 to 8867183 labelled d_1(1)
  • 8867188 to 8867182 labelled mark_1(1), g_1(1), a__c_1(1), d_1(1), c_1(2), d_1(2)
  • 8867188 to 8867188 labelled a__h_1(1), a__f_1(1), h_1(2), f_1(2)
  • 8867188 to 8867189 labelled a__c_1(2), c_1(3), d_1(3)
  • 8867188 to 8867193 labelled a__c_1(2), c_1(3), d_1(3)
  • 8867189 to 8867188 labelled d_1(2)
  • 8867190 to 8867191 labelled f_1(1)
  • 8867191 to 8867192 labelled g_1(1)
  • 8867192 to 8867188 labelled f_1(1)
  • 8867193 to 8867194 labelled f_1(2)
  • 8867195 to 8867188 labelled f_1(2)
  • 8867194 to 8867195 labelled g_1(2)

(2) TRUE