Platform claims not to support CPU time, falling back to wall time.
Cannot find out cpu time on external processes, falling back to wall time!
proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt1-1.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty
Termination of the given RelTRS could be proven:
(0) RelTRS
(1) RelTRS Reverse [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 339 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 0 ms]
(6) YES
----------------------------------------
(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(x) -> x
The relative TRS consists of the following S rules:
a -> g(a)
----------------------------------------
(1) RelTRS Reverse (EQUIVALENT)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is
f(x) -> x
The set of rules S is
a -> g(a)
We have obtained the following relative TRS:
The set of rules R is
f(x) -> x
The set of rules S is
a'(x) -> a'(g(x))
----------------------------------------
(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(x) -> x
The relative TRS consists of the following S rules:
a'(x) -> a'(g(x))
----------------------------------------
(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(f(x_1)) = [[1], [0]] + [[1, 0], [0, 1]] * x_1
>>>
<<<
POL(a'(x_1)) = [[2], [2]] + [[1, 0], [0, 0]] * x_1
>>>
<<<
POL(g(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
f(x) -> x
Rules from S:
none
----------------------------------------
(4)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
a'(x) -> a'(g(x))
----------------------------------------
(5) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------
(6)
YES