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:

f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0))) → g(d(1))
g(c(1)) → g(d(h(0)))
g(h(x)) → g(x)

Q is empty.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(h(0))) → g(d(1))
g(c(1)) → g(d(h(0)))
g(h(x)) → g(x)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

7539900, 7539901, 7539902, 7539904, 7539903, 7539906, 7539905, 7539907, 7539908, 7539909, 7539910, 7539911, 7539912, 7539913, 7539914, 7539915

Node 7539900 is start node and node 7539901 is final node.

Those nodes are connect through the following edges:

  • 7539900 to 7539901 labelled f_1(0), h_1(0), 1(0), c_1(0), g_1(0), 0(0), d_1(0), f_1(1), h_1(1), 1(1), c_1(1), g_1(1), 0(1), d_1(1), 1(2)
  • 7539900 to 7539902 labelled g_1(0)
  • 7539900 to 7539905 labelled f_1(0)
  • 7539900 to 7539907 labelled f_1(0)
  • 7539900 to 7539909 labelled f_1(1)
  • 7539900 to 7539911 labelled f_1(1)
  • 7539900 to 7539913 labelled g_1(1)
  • 7539900 to 7539904 labelled h_1(1)
  • 7539900 to 7539915 labelled h_1(2)
  • 7539901 to 7539901 labelled #_1(0)
  • 7539902 to 7539903 labelled d_1(0)
  • 7539904 to 7539901 labelled 0(0)
  • 7539903 to 7539904 labelled h_1(0)
  • 7539903 to 7539901 labelled 1(0)
  • 7539906 to 7539901 labelled f_1(0)
  • 7539906 to 7539909 labelled f_1(1)
  • 7539906 to 7539911 labelled f_1(1)
  • 7539905 to 7539906 labelled c_1(0)
  • 7539907 to 7539908 labelled d_1(0)
  • 7539908 to 7539901 labelled f_1(0)
  • 7539908 to 7539909 labelled f_1(1)
  • 7539908 to 7539911 labelled f_1(1)
  • 7539909 to 7539910 labelled c_1(1)
  • 7539910 to 7539901 labelled f_1(1)
  • 7539910 to 7539909 labelled f_1(1)
  • 7539910 to 7539911 labelled f_1(1)
  • 7539911 to 7539912 labelled d_1(1)
  • 7539912 to 7539901 labelled f_1(1)
  • 7539912 to 7539909 labelled f_1(1)
  • 7539912 to 7539911 labelled f_1(1)
  • 7539913 to 7539914 labelled d_1(1)
  • 7539914 to 7539915 labelled h_1(1)
  • 7539914 to 7539901 labelled 1(1)
  • 7539915 to 7539901 labelled 0(1)

(2) TRUE