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proof of /home/fs5/ayamada/tpdb/relative/Relative_05/rt3-9.trs
# AProVE Commit ID: 2b684c7cda508b1711f707cb42f127e64fe1df88 ffrohn 20140415 dirty
Termination of the given RelTRS could be proven:
(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 223 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 215 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 226 ms]
(6) RelTRS
(7) RIsEmptyProof [EQUIVALENT, 0 ms]
(8) YES
----------------------------------------
(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
l(m(x)) -> m(l(x))
m(r(x)) -> r(m(x))
f(m(x), y) -> f(x, m(y))
The relative TRS consists of the following S rules:
b -> l(b)
f(x, y) -> f(x, r(y))
----------------------------------------
(1) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(l(x_1)) = [[0], [0]] + [[1, 1], [1, 1]] * x_1
>>>
<<<
POL(m(x_1)) = [[0], [1]] + [[1, 0], [0, 1]] * x_1
>>>
<<<
POL(r(x_1)) = [[0], [0]] + [[1, 0], [0, 1]] * x_1
>>>
<<<
POL(f(x_1, x_2)) = [[1], [1]] + [[1, 1], [1, 1]] * x_1 + [[1, 1], [1, 1]] * x_2
>>>
<<<
POL(b) = [[0], [0]]
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
l(m(x)) -> m(l(x))
Rules from S:
none
----------------------------------------
(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
m(r(x)) -> r(m(x))
f(m(x), y) -> f(x, m(y))
The relative TRS consists of the following S rules:
b -> l(b)
f(x, y) -> f(x, r(y))
----------------------------------------
(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
m/1(YES)
r/1)YES(
f/2(YES,YES)
b/0)
l/1)YES(
Quasi precedence:
f_2 > m_1
b > m_1
Status:
m_1: [1]
f_2: [1,2]
b: multiset status
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
f(m(x), y) -> f(x, m(y))
Rules from S:
none
----------------------------------------
(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
m(r(x)) -> r(m(x))
The relative TRS consists of the following S rules:
b -> l(b)
f(x, y) -> f(x, r(y))
----------------------------------------
(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(m(x_1)) = [[1], [0]] + [[1, 1], [1, 1]] * x_1
>>>
<<<
POL(r(x_1)) = [[0], [1]] + [[1, 0], [0, 1]] * x_1
>>>
<<<
POL(b) = [[1], [1]]
>>>
<<<
POL(l(x_1)) = [[0], [0]] + [[1, 0], [1, 0]] * x_1
>>>
<<<
POL(f(x_1, x_2)) = [[1], [1]] + [[1, 0], [0, 0]] * x_1 + [[1, 0], [1, 0]] * x_2
>>>
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
m(r(x)) -> r(m(x))
Rules from S:
none
----------------------------------------
(6)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
b -> l(b)
f(x, y) -> f(x, r(y))
----------------------------------------
(7) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------
(8)
YES