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


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


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(4) YES


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

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

   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   log(s(0)) -> 0
   log(s(s(x))) -> s(log(s(half(x))))

Q is empty.

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

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

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

   0'(half(x)) -> 0'(x)
   s(s(half(x))) -> half(s(x))
   0'(s(log(x))) -> 0'(x)
   s(s(log(x))) -> half(s(log(s(x))))

Q is empty.

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

(3) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

   0'(half(x)) -> 0'(x)
   s(s(half(x))) -> half(s(x))
   0'(s(log(x))) -> 0'(x)
   s(s(log(x))) -> half(s(log(s(x))))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
23, 24, 37, 38, 39, 40, 41, 42, 43, 44

Node 23 is start node and node 24 is final node.

Those nodes are connected through the following edges:

* 23 to 24 labelled 0'_1(0), 0'_1(1)* 23 to 37 labelled half_1(0)* 23 to 38 labelled half_1(0)* 24 to 24 labelled #_1(0)* 37 to 24 labelled s_1(0)* 37 to 41 labelled half_1(1)* 37 to 42 labelled half_1(1)* 38 to 39 labelled s_1(0)* 39 to 40 labelled log_1(0)* 40 to 24 labelled s_1(0)* 40 to 41 labelled half_1(1)* 40 to 42 labelled half_1(1)* 41 to 24 labelled s_1(1)* 41 to 41 labelled half_1(1)* 41 to 42 labelled half_1(1)* 42 to 43 labelled s_1(1)* 43 to 44 labelled log_1(1)* 44 to 24 labelled s_1(1)* 44 to 41 labelled half_1(1)* 44 to 42 labelled half_1(1)


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

(4)
YES