(0) Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
f(x) → x
The relative TRS consists of the following S rules:
rand(x) → rand(s(x))
rand(x) → x
(1) RelTRSRRRProof (EQUIVALENT transformation)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:
POL(c(x1, x2)) = x1 + x2
POL(d(x1)) = x1
POL(f(x1)) = 1 + x1
POL(g(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:
f(x) → x
Rules from S:
rand(x) → x
(2) Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
f(f(x)) → f(d(f(x)))
The relative TRS consists of the following S rules:
rand(x) → rand(s(x))
(3) RelTRSRRRProof (EQUIVALENT transformation)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
| POL(c(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
f(f(x)) → f(d(f(x)))
Rules from S:
none
(4) Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
g(c(x, s(y))) → g(c(s(x), y))
f(c(s(x), y)) → f(c(x, s(y)))
The relative TRS consists of the following S rules:
rand(x) → rand(s(x))
(5) RelTRSRRRProof (EQUIVALENT transformation)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
| POL(c(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
g(c(x, s(y))) → g(c(s(x), y))
Rules from S:
none
(6) Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:
f(c(s(x), y)) → f(c(x, s(y)))
The relative TRS consists of the following S rules:
rand(x) → rand(s(x))
(7) RelTRSRRRProof (EQUIVALENT transformation)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
| POL(c(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
f(c(s(x), y)) → f(c(x, s(y)))
Rules from S:
none
(8) Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:
rand(x) → rand(s(x))
(9) RIsEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(10) YES