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proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt2-3.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty
Termination of the given RelTRS could be proven:
(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 101 ms]
(2) RelTRS
(3) RIsEmptyProof [EQUIVALENT, 0 ms]
(4) YES
----------------------------------------
(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
R(x, B2) -> B2
W(x, B2) -> B2
The relative TRS consists of the following S rules:
B1 -> R(T, B1)
B1 -> W(T, B1)
----------------------------------------
(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(R(x_1, x_2)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 1]] * x_2
>>>
<<<
POL(B2) = [[1], [1]]
>>>
<<<
POL(W(x_1, x_2)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1 + [[1, 1], [0, 1]] * x_2
>>>
<<<
POL(B1) = [[1], [0]]
>>>
<<<
POL(T) = [[0], [1]]
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
R(x, B2) -> B2
W(x, B2) -> B2
Rules from S:
none
----------------------------------------
(2)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
B1 -> R(T, B1)
B1 -> W(T, B1)
----------------------------------------
(3) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------
(4)
YES