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/rt3-1.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty
Termination of the given RelTRS could be proven:
(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 457 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 195 ms]
(4) RelTRS
(5) RIsEmptyProof [EQUIVALENT, 27 ms]
(6) YES
----------------------------------------
(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(g(x), y, z) -> f(x, y, g(z))
The relative TRS consists of the following S rules:
f(x, y, g(z)) -> f(x, g(y), z)
f(x, a, z) -> f(x, g(a), z)
f(x, y, z) -> f(x, y, g(z))
----------------------------------------
(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(f(x_1, x_2, x_3)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 0]] * x_2 + [[1, 0], [1, 0]] * x_3
>>>
<<<
POL(g(x_1)) = [[0], [0]] + [[1, 0], [0, 0]] * x_1
>>>
<<<
POL(a) = [[0], [1]]
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:
f(x, a, z) -> f(x, g(a), z)
----------------------------------------
(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(g(x), y, z) -> f(x, y, g(z))
The relative TRS consists of the following S rules:
f(x, y, g(z)) -> f(x, g(y), z)
f(x, y, z) -> f(x, y, g(z))
----------------------------------------
(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(f(x_1, x_2, x_3)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1 + [[1, 0], [1, 0]] * x_2 + [[1, 0], [1, 0]] * x_3
>>>
<<<
POL(g(x_1)) = [[0], [1]] + [[1, 0], [0, 1]] * x_1
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
f(g(x), y, z) -> f(x, y, g(z))
Rules from S:
none
----------------------------------------
(4)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
f(x, y, g(z)) -> f(x, g(y), z)
f(x, y, z) -> f(x, y, g(z))
----------------------------------------
(5) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------
(6)
YES