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proof of /home/fs5/ayamada/tpdb/relative/Mixed_relative_TRS/ijcar2006.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty
Termination of the given RelTRS could be proven:
(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 196 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 219 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 187 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 1 ms]
(8) YES
----------------------------------------
(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(a, g(y), z) -> f(a, y, g(y))
f(b, g(y), z) -> f(a, y, z)
a -> b
The relative TRS consists of the following S rules:
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, 1], [1, 0]] * x_1 + [[1, 1], [0, 1]] * x_2 + [[1, 0], [0, 0]] * x_3
>>>
<<<
POL(a) = [[0], [1]]
>>>
<<<
POL(g(x_1)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1
>>>
<<<
POL(b) = [[0], [0]]
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
f(a, g(y), z) -> f(a, y, g(y))
Rules from S:
none
----------------------------------------
(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(b, g(y), z) -> f(a, y, z)
a -> b
The relative TRS consists of the following S rules:
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], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2 + [[1, 0], [1, 0]] * x_3
>>>
<<<
POL(b) = [[0], [1]]
>>>
<<<
POL(g(x_1)) = [[0], [1]] + [[1, 0], [1, 1]] * x_1
>>>
<<<
POL(a) = [[1], [1]]
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
a -> b
Rules from S:
none
----------------------------------------
(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(b, g(y), z) -> f(a, y, z)
The relative TRS consists of the following S rules:
f(x, y, z) -> f(x, y, g(z))
----------------------------------------
(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(b) = 1
POL(f(x_1, x_2, x_3)) = x_1 + x_2 + x_3
POL(g(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
f(b, g(y), z) -> f(a, y, z)
Rules from S:
none
----------------------------------------
(6)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
f(x, y, z) -> f(x, y, g(z))
----------------------------------------
(7) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------
(8)
YES