YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex25_Luc06_GM.ari

(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:

1, 4, 5, 6, 7, 12, 13, 14, 15, 21, 22, 23, 24, 25, 26

Node 1 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 1 to 5 labelled a__c_1(0), d_1(1), c_1(1)
  • 1 to 4 labelled d_1(0), a__c_1(0), g_1(0), f_1(0), c_1(0), h_1(0), d_1(1), c_1(1)
  • 1 to 1 labelled a__c_1(0), d_1(1), c_1(1)
  • 1 to 12 labelled a__f_1(0), a__h_1(0), f_1(1), h_1(1)
  • 1 to 13 labelled a__c_1(1), d_1(2), c_1(2)
  • 1 to 21 labelled a__c_1(1), d_1(2), c_1(2)
  • 4 to 4 labelled #_1(0)
  • 5 to 6 labelled f_1(0)
  • 6 to 7 labelled g_1(0)
  • 7 to 4 labelled f_1(0)
  • 12 to 4 labelled mark_1(0), a__c_1(1), g_1(1), d_1(1), d_1(2), c_1(2)
  • 12 to 14 labelled a__f_1(1), a__h_1(1), f_1(2), h_1(2)
  • 12 to 15 labelled a__c_1(2), d_1(3), c_1(3)
  • 12 to 24 labelled a__c_1(2), d_1(3), c_1(3)
  • 13 to 12 labelled d_1(1)
  • 14 to 4 labelled mark_1(1), a__c_1(1), g_1(1), d_1(1), d_1(2), c_1(2)
  • 14 to 14 labelled a__f_1(1), a__h_1(1), f_1(2), h_1(2)
  • 14 to 15 labelled a__c_1(2), d_1(3), c_1(3)
  • 14 to 24 labelled a__c_1(2), d_1(3), c_1(3)
  • 15 to 14 labelled d_1(2)
  • 21 to 22 labelled f_1(1)
  • 22 to 23 labelled g_1(1)
  • 23 to 14 labelled f_1(1)
  • 24 to 25 labelled f_1(2)
  • 25 to 26 labelled g_1(2)
  • 26 to 14 labelled f_1(2)

(2) YES