YES
proof of MNZ_10_0.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) TransformationProof [EQUIVALENT, 0 ms]
(6) QDP
(7) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) QDP
(9) QDPOrderProof [EQUIVALENT, 69 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) AND
    (13) QDP
        (14) UsableRulesProof [EQUIVALENT, 0 ms]
        (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) QDP
        (19) TransformationProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) DependencyGraphProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) TransformationProof [EQUIVALENT, 0 ms]
        (24) QDP
        (25) DependencyGraphProof [EQUIVALENT, 0 ms]
        (26) QDP
        (27) TransformationProof [EQUIVALENT, 0 ms]
        (28) QDP
        (29) DependencyGraphProof [EQUIVALENT, 0 ms]
        (30) QDP
        (31) TransformationProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) DependencyGraphProof [EQUIVALENT, 0 ms]
        (34) QDP
        (35) TransformationProof [EQUIVALENT, 0 ms]
        (36) QDP
        (37) DependencyGraphProof [EQUIVALENT, 0 ms]
        (38) QDP
        (39) TransformationProof [EQUIVALENT, 0 ms]
        (40) QDP
        (41) DependencyGraphProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QDPOrderProof [EQUIVALENT, 0 ms]
        (44) QDP
        (45) QDPOrderProof [EQUIVALENT, 967 ms]
        (46) QDP
        (47) QDPOrderProof [EQUIVALENT, 935 ms]
        (48) QDP
        (49) QDPOrderProof [EQUIVALENT, 576 ms]
        (50) QDP
        (51) QDPOrderProof [EQUIVALENT, 765 ms]
        (52) QDP
        (53) QDPOrderProof [EQUIVALENT, 783 ms]
        (54) QDP
        (55) QDPOrderProof [EQUIVALENT, 588 ms]
        (56) QDP
        (57) QDPOrderProof [EQUIVALENT, 783 ms]
        (58) QDP
        (59) QDPOrderProof [EQUIVALENT, 1842 ms]
        (60) QDP
        (61) DependencyGraphProof [EQUIVALENT, 0 ms]
        (62) TRUE


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(0)) -> F(0)
   S(s(s(0))) -> F(s(0))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(s(s(0))))))
   F(s(s(0))) -> S(s(s(s(s(0)))))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   G(x) -> H(x, x)
   S(x) -> H(x, 0)
   S(x) -> H(0, x)
   F(g(x)) -> G(g(f(x)))
   F(g(x)) -> G(f(x))
   F(g(x)) -> F(x)
   G(s(x)) -> S(s(g(x)))
   G(s(x)) -> S(g(x))
   G(s(x)) -> G(x)
   H(f(x), g(x)) -> F(s(x))
   H(f(x), g(x)) -> S(x)

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(0))) -> F(s(0))
   F(s(s(0))) -> S(s(s(s(s(s(0))))))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(s(0)))))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   F(g(x)) -> G(g(f(x)))
   G(x) -> H(x, x)
   H(f(x), g(x)) -> F(s(x))
   F(g(x)) -> G(f(x))
   G(s(x)) -> S(s(g(x)))
   G(s(x)) -> S(g(x))
   G(s(x)) -> G(x)
   F(g(x)) -> F(x)
   H(f(x), g(x)) -> S(x)

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule S(s(s(0))) -> F(s(0)) at position [0] we obtained the following new rules [LPAR04]:

   (S(s(s(0))) -> F(h(0, 0)),S(s(s(0))) -> F(h(0, 0)))


----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(s(s(s(s(0))))))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(s(0)))))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   F(g(x)) -> G(g(f(x)))
   G(x) -> H(x, x)
   H(f(x), g(x)) -> F(s(x))
   F(g(x)) -> G(f(x))
   G(s(x)) -> S(s(g(x)))
   G(s(x)) -> S(g(x))
   G(s(x)) -> G(x)
   F(g(x)) -> F(x)
   H(f(x), g(x)) -> S(x)
   S(s(s(0))) -> F(h(0, 0))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(s(s(0))))))
   F(s(s(0))) -> S(s(s(s(s(0)))))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   F(g(x)) -> G(g(f(x)))
   G(x) -> H(x, x)
   H(f(x), g(x)) -> F(s(x))
   F(g(x)) -> G(f(x))
   G(s(x)) -> S(s(g(x)))
   G(s(x)) -> S(g(x))
   G(s(x)) -> G(x)
   F(g(x)) -> F(x)
   H(f(x), g(x)) -> S(x)

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   G(x) -> H(x, x)
   G(s(x)) -> S(s(g(x)))
   G(s(x)) -> S(g(x))
   F(g(x)) -> F(x)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( F_1(x_1) ) = x_1
POL( G_1(x_1) ) = 1
POL( S_1(x_1) ) = 0
POL( s_1(x_1) ) = max{0, -1}
POL( h_2(x_1, x_2) ) = 0
POL( 0 ) = 0
POL( f_1(x_1) ) = 2x_1
POL( g_1(x_1) ) = x_1 + 1
POL( H_2(x_1, x_2) ) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(s(s(0))))))
   F(s(s(0))) -> S(s(s(s(s(0)))))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   F(g(x)) -> G(g(f(x)))
   H(f(x), g(x)) -> F(s(x))
   F(g(x)) -> G(f(x))
   G(s(x)) -> G(x)
   H(f(x), g(x)) -> S(x)

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.
----------------------------------------

(12)
Complex Obligation (AND)

----------------------------------------

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   G(s(x)) -> G(x)

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(14) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   G(s(x)) -> G(x)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(16) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*G(s(x)) -> G(x)
The graph contains the following edges 1 > 1


----------------------------------------

(17)
YES

----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(s(s(s(s(0))))))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(s(0)))))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule F(s(s(0))) -> S(s(s(s(s(s(0)))))) at position [0] we obtained the following new rules [LPAR04]:

   (F(s(s(0))) -> S(h(s(s(s(s(0)))), 0)),F(s(s(0))) -> S(h(s(s(s(s(0)))), 0)))
   (F(s(s(0))) -> S(h(0, s(s(s(s(0)))))),F(s(s(0))) -> S(h(0, s(s(s(s(0)))))))
   (F(s(s(0))) -> S(s(h(s(s(s(0))), 0))),F(s(s(0))) -> S(s(h(s(s(s(0))), 0))))
   (F(s(s(0))) -> S(s(h(0, s(s(s(0)))))),F(s(s(0))) -> S(s(h(0, s(s(s(0)))))))
   (F(s(s(0))) -> S(s(s(f(s(0))))),F(s(s(0))) -> S(s(s(f(s(0))))))
   (F(s(s(0))) -> S(s(s(h(s(s(0)), 0)))),F(s(s(0))) -> S(s(s(h(s(s(0)), 0)))))
   (F(s(s(0))) -> S(s(s(h(0, s(s(0)))))),F(s(s(0))) -> S(s(s(h(0, s(s(0)))))))
   (F(s(s(0))) -> S(s(s(s(f(0))))),F(s(s(0))) -> S(s(s(s(f(0))))))
   (F(s(s(0))) -> S(s(s(s(h(s(0), 0))))),F(s(s(0))) -> S(s(s(s(h(s(0), 0))))))
   (F(s(s(0))) -> S(s(s(s(h(0, s(0)))))),F(s(s(0))) -> S(s(s(s(h(0, s(0)))))))
   (F(s(s(0))) -> S(s(s(s(s(h(0, 0)))))),F(s(s(0))) -> S(s(s(s(s(h(0, 0)))))))


----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(s(0)))))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   F(s(s(0))) -> S(h(s(s(s(s(0)))), 0))
   F(s(s(0))) -> S(h(0, s(s(s(s(0))))))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(21) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(s(s(s(0)))))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(23) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule F(s(s(0))) -> S(s(s(s(s(0))))) at position [0] we obtained the following new rules [LPAR04]:

   (F(s(s(0))) -> S(h(s(s(s(0))), 0)),F(s(s(0))) -> S(h(s(s(s(0))), 0)))
   (F(s(s(0))) -> S(h(0, s(s(s(0))))),F(s(s(0))) -> S(h(0, s(s(s(0))))))
   (F(s(s(0))) -> S(s(f(s(0)))),F(s(s(0))) -> S(s(f(s(0)))))
   (F(s(s(0))) -> S(s(h(s(s(0)), 0))),F(s(s(0))) -> S(s(h(s(s(0)), 0))))
   (F(s(s(0))) -> S(s(h(0, s(s(0))))),F(s(s(0))) -> S(s(h(0, s(s(0))))))
   (F(s(s(0))) -> S(s(s(f(0)))),F(s(s(0))) -> S(s(s(f(0)))))
   (F(s(s(0))) -> S(s(s(h(s(0), 0)))),F(s(s(0))) -> S(s(s(h(s(0), 0)))))
   (F(s(s(0))) -> S(s(s(h(0, s(0))))),F(s(s(0))) -> S(s(s(h(0, s(0))))))
   (F(s(s(0))) -> S(s(s(s(h(0, 0))))),F(s(s(0))) -> S(s(s(s(h(0, 0))))))


----------------------------------------

(24)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(s(0))))
   F(s(s(0))) -> S(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(h(s(s(s(0))), 0))
   F(s(s(0))) -> S(h(0, s(s(s(0)))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(25) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(26)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(s(s(0))))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(27) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule F(s(s(0))) -> S(s(s(s(0)))) at position [0] we obtained the following new rules [LPAR04]:

   (F(s(s(0))) -> S(f(s(0))),F(s(s(0))) -> S(f(s(0))))
   (F(s(s(0))) -> S(h(s(s(0)), 0)),F(s(s(0))) -> S(h(s(s(0)), 0)))
   (F(s(s(0))) -> S(h(0, s(s(0)))),F(s(s(0))) -> S(h(0, s(s(0)))))
   (F(s(s(0))) -> S(s(f(0))),F(s(s(0))) -> S(s(f(0))))
   (F(s(s(0))) -> S(s(h(s(0), 0))),F(s(s(0))) -> S(s(h(s(0), 0))))
   (F(s(s(0))) -> S(s(h(0, s(0)))),F(s(s(0))) -> S(s(h(0, s(0)))))
   (F(s(s(0))) -> S(s(s(h(0, 0)))),F(s(s(0))) -> S(s(s(h(0, 0)))))


----------------------------------------

(28)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(h(s(s(0)), 0))
   F(s(s(0))) -> S(h(0, s(s(0))))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(29) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(s(0)))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(31) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule F(s(s(0))) -> S(s(s(0))) at position [0] we obtained the following new rules [LPAR04]:

   (F(s(s(0))) -> S(f(0)),F(s(s(0))) -> S(f(0)))
   (F(s(s(0))) -> S(h(s(0), 0)),F(s(s(0))) -> S(h(s(0), 0)))
   (F(s(s(0))) -> S(h(0, s(0))),F(s(s(0))) -> S(h(0, s(0))))
   (F(s(s(0))) -> S(s(h(0, 0))),F(s(s(0))) -> S(s(h(0, 0))))


----------------------------------------

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))
   F(s(s(0))) -> S(f(0))
   F(s(s(0))) -> S(h(s(0), 0))
   F(s(s(0))) -> S(h(0, s(0)))
   F(s(s(0))) -> S(s(h(0, 0)))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(33) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))
   F(s(s(0))) -> S(f(0))
   F(s(s(0))) -> S(s(h(0, 0)))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(35) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule F(s(s(0))) -> S(f(0)) at position [0] we obtained the following new rules [LPAR04]:

   (F(s(s(0))) -> S(0),F(s(s(0))) -> S(0))


----------------------------------------

(36)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))
   F(s(s(0))) -> S(s(h(0, 0)))
   F(s(s(0))) -> S(0)

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(37) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(38)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))
   F(s(s(0))) -> S(s(h(0, 0)))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(39) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule F(s(s(0))) -> S(s(h(0, 0))) at position [0] we obtained the following new rules [LPAR04]:

   (F(s(s(0))) -> S(h(h(0, 0), 0)),F(s(s(0))) -> S(h(h(0, 0), 0)))
   (F(s(s(0))) -> S(h(0, h(0, 0))),F(s(s(0))) -> S(h(0, h(0, 0))))


----------------------------------------

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))
   F(s(s(0))) -> S(h(h(0, 0), 0))
   F(s(s(0))) -> S(h(0, h(0, 0)))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(41) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(f(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(43) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(s(f(0)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(F(x_1)) =  	[[0]] 	 +  	[[0, 1]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[1], [0]] 	 +  	[[0, 0], [1, 0]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(S(x_1)) =  	[[0]] 	 +  	[[0, 1]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[1], [0]] 	 +  	[[0, 0], [0, 0]] 	* 	x_1 	 +  	[[0, 0], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(f(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
>>>

   <<<
 POL(g(x_1)) =  	[[1], [0]] 	 +  	[[0, 0], [1, 0]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(44)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(h(0, s(0))))
   F(s(s(0))) -> S(s(s(h(0, 0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(45) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(s(h(0, s(0))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(F(x_1)) =  	[[1A]] 	 +  	[[2A, 3A, 0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0A], [3A], [-I]] 	 +  	[[-I, 0A, 1A], [0A, 0A, 2A], [-I, -I, 1A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[3A], [-I], [-I]]
>>>

   <<<
 POL(S(x_1)) =  	[[3A]] 	 +  	[[3A, 0A, 2A]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, -I], [-I, 0A, -I], [-I, -I, -I]] 	* 	x_1 	 +  	[[-I, 0A, -I], [0A, -I, 2A], [-I, -I, -I]] 	* 	x_2
>>>

   <<<
 POL(f(x_1)) =  	[[3A], [3A], [-I]] 	 +  	[[0A, -I, 2A], [0A, -I, 2A], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(g(x_1)) =  	[[0A], [3A], [-I]] 	 +  	[[0A, 0A, 3A], [0A, 0A, 3A], [-I, -I, -I]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
   F(s(s(0))) -> S(s(s(h(0, 0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(47) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(s(h(s(s(s(0))), 0)))
   F(s(s(0))) -> S(s(h(s(s(0)), 0)))
   F(s(s(0))) -> S(s(h(s(0), 0)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(F(x_1)) =  	[[0A]] 	 +  	[[1A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[3A], [3A], [-I]] 	 +  	[[0A, -I, -I], [3A, 0A, 3A], [-I, -I, 3A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A], [2A], [-I]]
>>>

   <<<
 POL(S(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[0A], [3A], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1 	 +  	[[0A, -I, -I], [3A, -I, 0A], [-I, -I, -I]] 	* 	x_2
>>>

   <<<
 POL(f(x_1)) =  	[[-I], [3A], [-I]] 	 +  	[[0A, -I, -I], [3A, -I, -I], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(g(x_1)) =  	[[3A], [3A], [-I]] 	 +  	[[0A, -I, -I], [3A, -I, 0A], [-I, -I, -I]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(48)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(s(h(0, 0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(49) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(s(h(0, s(s(s(0))))))
   F(s(s(0))) -> S(s(h(0, s(s(0)))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[3A]] 	 +  	[[-I, -I, 3A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[3A], [3A], [3A]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [2A, 3A, 0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[1A], [0A], [0A]]
>>>

   <<<
 POL(F(x_1)) =  	[[1A]] 	 +  	[[0A, -I, 3A]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[3A], [1A], [-I]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [-I, 3A, -I]] 	* 	x_1 	 +  	[[-I, -I, -I], [-I, -I, -I], [2A, 0A, -I]] 	* 	x_2
>>>

   <<<
 POL(f(x_1)) =  	[[3A], [3A], [0A]] 	 +  	[[0A, -I, -I], [0A, 0A, -I], [3A, 3A, -I]] 	* 	x_1
>>>

   <<<
 POL(g(x_1)) =  	[[3A], [3A], [3A]] 	 +  	[[0A, 0A, -I], [0A, -I, -I], [3A, 3A, 0A]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(f(s(0)))
   F(s(s(0))) -> S(s(s(h(0, 0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(51) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(s(s(h(s(s(0)), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(s(0))))))
   F(s(s(0))) -> S(s(s(s(h(s(0), 0)))))
   F(s(s(0))) -> S(s(s(s(h(0, s(0))))))
   F(s(s(0))) -> S(s(s(s(s(h(0, 0))))))
   F(s(s(0))) -> S(s(s(h(s(0), 0))))
   F(s(s(0))) -> S(s(s(h(0, s(0)))))
   F(s(s(0))) -> S(s(s(s(h(0, 0)))))
   F(s(s(0))) -> S(s(s(h(0, 0))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[0A]] 	 +  	[[1A, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, -I], [-I, 0A, -I], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[1A], [-I], [-I]]
>>>

   <<<
 POL(F(x_1)) =  	[[1A]] 	 +  	[[1A, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(f(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, -I, -I], [-I, 0A, -I], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[0A], [-I], [-I]] 	 +  	[[-I, 0A, -I], [-I, 0A, -I], [-I, -I, -I]] 	* 	x_1 	 +  	[[-I, 0A, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_2
>>>

   <<<
 POL(g(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [0A, 0A, -I], [-I, -I, 2A]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(52)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(s(f(0))))
   F(s(s(0))) -> S(f(s(0)))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(53) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(f(s(0)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[0A]] 	 +  	[[-I, 3A, -I]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0A], [1A], [0A]] 	 +  	[[-I, -I, -I], [0A, 0A, 3A], [0A, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A], [0A], [-I]]
>>>

   <<<
 POL(F(x_1)) =  	[[2A]] 	 +  	[[1A, 3A, -I]] 	* 	x_1
>>>

   <<<
 POL(f(x_1)) =  	[[0A], [2A], [0A]] 	 +  	[[0A, -I, -I], [1A, 0A, -I], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, -I], [-I, 0A, -I], [-I, -I, -I]] 	* 	x_1 	 +  	[[-I, -I, -I], [0A, -I, 3A], [-I, -I, -I]] 	* 	x_2
>>>

   <<<
 POL(g(x_1)) =  	[[0A], [3A], [0A]] 	 +  	[[-I, -I, 0A], [2A, 0A, 3A], [-I, -I, 0A]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(54)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(f(s(0))))
   F(s(s(0))) -> S(s(s(f(0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(55) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(s(f(s(0))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[1A]] 	 +  	[[1A, -I, 2A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[2A], [2A], [-I]] 	 +  	[[-I, -I, 2A], [-I, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[1A], [-I], [-I]]
>>>

   <<<
 POL(F(x_1)) =  	[[1A]] 	 +  	[[-I, 3A, -I]] 	* 	x_1
>>>

   <<<
 POL(f(x_1)) =  	[[0A], [2A], [0A]] 	 +  	[[0A, 2A, 2A], [-I, 0A, 0A], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[2A], [0A], [-I]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1 	 +  	[[-I, -I, 2A], [-I, -I, 0A], [-I, 0A, 0A]] 	* 	x_2
>>>

   <<<
 POL(g(x_1)) =  	[[3A], [2A], [2A]] 	 +  	[[0A, 2A, 2A], [-I, -I, 0A], [-I, 0A, 0A]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(56)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(s(f(0)))))
   F(s(s(0))) -> S(s(s(f(0))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(57) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F(s(s(0))) -> S(s(s(f(0))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[2A]] 	 +  	[[3A, 2A, 0A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[1A], [-I], [-I]] 	 +  	[[0A, -I, -I], [-I, 0A, 2A], [0A, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A], [-I], [-I]]
>>>

   <<<
 POL(F(x_1)) =  	[[1A]] 	 +  	[[0A, 3A, -I]] 	* 	x_1
>>>

   <<<
 POL(f(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, 0A], [-I, 1A, -I], [0A, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[1A], [-I], [-I]] 	 +  	[[-I, -I, -I], [-I, 0A, 2A], [-I, -I, -I]] 	* 	x_1 	 +  	[[0A, -I, -I], [-I, 0A, 2A], [-I, -I, 0A]] 	* 	x_2
>>>

   <<<
 POL(g(x_1)) =  	[[1A], [3A], [1A]] 	 +  	[[0A, -I, 0A], [3A, 0A, 3A], [0A, -I, 0A]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(58)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(s(f(0)))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(59) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   S(s(s(s(s(s(s(s(0)))))))) -> F(s(s(0)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(S(x_1)) =  	[[0A]] 	 +  	[[-I, 3A, 2A]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [0A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A], [2A], [-I]]
>>>

   <<<
 POL(F(x_1)) =  	[[1A]] 	 +  	[[1A, 2A, 2A]] 	* 	x_1
>>>

   <<<
 POL(f(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, -I, -I], [0A, 0A, -I], [0A, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(h(x_1, x_2)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [-I, -I, 0A], [0A, -I, -I]] 	* 	x_1 	 +  	[[0A, -I, -I], [-I, -I, -I], [0A, -I, -I]] 	* 	x_2
>>>

   <<<
 POL(g(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   s(s(0)) -> f(0)
   s(s(s(0))) -> f(s(0))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(g(x)) -> g(g(f(x)))
   g(x) -> h(x, x)
   h(f(x), g(x)) -> f(s(x))
   g(s(x)) -> s(s(g(x)))
   f(0) -> 0
   f(s(0)) -> s(0)


----------------------------------------

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F(s(s(0))) -> S(s(s(f(s(0)))))
   F(s(s(0))) -> S(s(s(s(f(0)))))

The TRS R consists of the following rules:

   s(s(0)) -> f(0)
   f(0) -> 0
   s(s(s(0))) -> f(s(0))
   f(s(0)) -> s(0)
   s(s(s(s(s(s(s(s(0)))))))) -> f(s(s(0)))
   f(s(s(0))) -> s(s(s(s(s(s(0))))))
   g(x) -> h(x, x)
   s(x) -> h(x, 0)
   s(x) -> h(0, x)
   f(g(x)) -> g(g(f(x)))
   g(s(x)) -> s(s(g(x)))
   h(f(x), g(x)) -> f(s(x))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(61) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(62)
TRUE