/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) Overlay + Local Confluence [EQUIVALENT, 0 ms]
(2) QTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) QDP
(5) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(6) YES


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

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

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

Q is empty.

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

(1) Overlay + Local Confluence (EQUIVALENT)
The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

The set Q consists of the following terms:

   f(a, g(x0))
   f(g(x0), a)
   f(g(x0), g(x1))
   h(g(x0), x1, x2)
   h(a, x0, x1)


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

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

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

   F(g(x), a) -> F(x, g(a))
   F(g(x), g(y)) -> H(g(y), x, g(y))
   H(g(x), y, z) -> F(y, h(x, y, z))
   H(g(x), y, z) -> H(x, y, z)

The TRS R consists of the following rules:

   f(a, g(y)) -> g(g(y))
   f(g(x), a) -> f(x, g(a))
   f(g(x), g(y)) -> h(g(y), x, g(y))
   h(g(x), y, z) -> f(y, h(x, y, z))
   h(a, y, z) -> z

The set Q consists of the following terms:

   f(a, g(x0))
   f(g(x0), a)
   f(g(x0), g(x1))
   h(g(x0), x1, x2)
   h(a, x0, x1)

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

(5) 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:
*F(g(x), g(y)) -> H(g(y), x, g(y))
The graph contains the following edges 2 >= 1, 1 > 2, 2 >= 3


*H(g(x), y, z) -> F(y, h(x, y, z))
The graph contains the following edges 2 >= 1


*H(g(x), y, z) -> H(x, y, z)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


*F(g(x), a) -> F(x, g(a))
The graph contains the following edges 1 > 1


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

(6)
YES