(0) Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))
The relative TRS consists of the following S rules:
rand(x) → rand(s(x))
rand(x) → x
(1) RelTRS Reverse (SOUND transformation)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is
f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))
The set of rules S is
rand(x) → rand(s(x))
rand(x) → x
We have obtained the following relative TRS:
The set of rules R is
0'(f(x)) → 0'(s(x))
0'(s(f(x))) → 0'(s(x))
s(s(f(x))) → s(f(f(x)))
The set of rules S is
rand(x) → s(rand(x))
rand(x) → x
(2) Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
0'(f(x)) → 0'(s(x))
0'(s(f(x))) → 0'(s(x))
s(s(f(x))) → s(f(f(x)))
The relative TRS consists of the following S rules:
rand(x) → s(rand(x))
rand(x) → x
(3) RelTRSRRRProof (EQUIVALENT transformation)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:
POL(0'(x1)) = x1
POL(f(x1)) = x1
POL(rand(x1)) = 1 + x1
POL(s(x1)) = x1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:
rand(x) → x
(4) Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
0'(f(x)) → 0'(s(x))
0'(s(f(x))) → 0'(s(x))
s(s(f(x))) → s(f(f(x)))
The relative TRS consists of the following S rules:
rand(x) → s(rand(x))
(5) RelTRSRRRProof (EQUIVALENT transformation)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
0'(f(x)) → 0'(s(x))
0'(s(f(x))) → 0'(s(x))
s(s(f(x))) → s(f(f(x)))
Rules from S:
none
(6) Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
rand(x) → s(rand(x))
(7) RIsEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) YES