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/rt2-7.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) RootLabelingProof [EQUIVALENT, 2 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 679 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 432 ms]
(8) RelTRS
(9) RIsEmptyProof [EQUIVALENT, 0 ms]
(10) YES
----------------------------------------
(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(g(f(x))) -> f(g(g(g(f(x)))))
The relative TRS consists of the following S rules:
g(x) -> g(g(x))
----------------------------------------
(1) RelTRS Reverse (EQUIVALENT)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is
f(g(f(x))) -> f(g(g(g(f(x)))))
The set of rules S is
g(x) -> g(g(x))
We have obtained the following relative TRS:
The set of rules R is
f(g(f(x))) -> f(g(g(g(f(x)))))
The set of rules S is
g(x) -> g(g(x))
----------------------------------------
(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(g(f(x))) -> f(g(g(g(f(x)))))
The relative TRS consists of the following S rules:
g(x) -> g(g(x))
----------------------------------------
(3) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
----------------------------------------
(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{f_1}(x)))))
f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
The relative TRS consists of the following S rules:
g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))
----------------------------------------
(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(f_{g_1}(x_1)) = [[0], [0]] + [[2, 2], [2, 2]] * x_1
>>>
<<<
POL(g_{f_1}(x_1)) = [[0], [0]] + [[2, 2], [2, 2]] * x_1
>>>
<<<
POL(f_{f_1}(x_1)) = [[2], [2]] + [[1, 0], [0, 0]] * x_1
>>>
<<<
POL(g_{g_1}(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_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{f_1}(x)))))
Rules from S:
none
----------------------------------------
(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
The relative TRS consists of the following S rules:
g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))
----------------------------------------
(7) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
<<<
POL(f_{g_1}(x_1)) = [[0], [0]] + [[2, 2], [0, 0]] * x_1
>>>
<<<
POL(g_{f_1}(x_1)) = [[2], [2]] + [[1, 0], [0, 0]] * x_1
>>>
<<<
POL(g_{g_1}(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_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
Rules from S:
none
----------------------------------------
(8)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))
----------------------------------------
(9) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------
(10)
YES