Term Rewriting System R:
[x, y]
minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

MINUS(+(x, y)) -> MINUS(minus(minus(x)))
MINUS(+(x, y)) -> MINUS(minus(x))
MINUS(+(x, y)) -> MINUS(x)
MINUS(+(x, y)) -> MINUS(minus(minus(y)))
MINUS(+(x, y)) -> MINUS(minus(y))
MINUS(+(x, y)) -> MINUS(y)
MINUS(*(x, y)) -> MINUS(minus(minus(x)))
MINUS(*(x, y)) -> MINUS(minus(x))
MINUS(*(x, y)) -> MINUS(x)
MINUS(*(x, y)) -> MINUS(minus(minus(y)))
MINUS(*(x, y)) -> MINUS(minus(y))
MINUS(*(x, y)) -> MINUS(y)
F(minus(x)) -> MINUS(minus(minus(f(x))))
F(minus(x)) -> MINUS(minus(f(x)))
F(minus(x)) -> MINUS(f(x))
F(minus(x)) -> F(x)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Argument Filtering and Ordering

Dependency Pairs:

MINUS(*(x, y)) -> MINUS(y)
MINUS(*(x, y)) -> MINUS(minus(y))
MINUS(*(x, y)) -> MINUS(minus(minus(y)))
MINUS(*(x, y)) -> MINUS(x)
MINUS(*(x, y)) -> MINUS(minus(x))
MINUS(*(x, y)) -> MINUS(minus(minus(x)))
MINUS(+(x, y)) -> MINUS(y)
MINUS(+(x, y)) -> MINUS(minus(y))
MINUS(+(x, y)) -> MINUS(minus(minus(y)))
MINUS(+(x, y)) -> MINUS(x)
MINUS(+(x, y)) -> MINUS(minus(x))
MINUS(+(x, y)) -> MINUS(minus(minus(x)))

Rules:

minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

MINUS(*(x, y)) -> MINUS(y)
MINUS(*(x, y)) -> MINUS(minus(y))
MINUS(*(x, y)) -> MINUS(minus(minus(y)))
MINUS(*(x, y)) -> MINUS(x)
MINUS(*(x, y)) -> MINUS(minus(x))
MINUS(*(x, y)) -> MINUS(minus(minus(x)))
MINUS(+(x, y)) -> MINUS(y)
MINUS(+(x, y)) -> MINUS(minus(y))
MINUS(+(x, y)) -> MINUS(minus(minus(y)))
MINUS(+(x, y)) -> MINUS(x)
MINUS(+(x, y)) -> MINUS(minus(x))
MINUS(+(x, y)) -> MINUS(minus(minus(x)))

The following usable rules for innermost can be oriented:

minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{+, *}

resulting in one new DP problem.
Used Argument Filtering System:
MINUS(x1) -> MINUS(x1)
+(x1, x2) -> +(x1, x2)
minus(x1) -> x1
*(x1, x2) -> *(x1, x2)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Dependency Graph

Dependency Pair:

Rules:

minus(minus(x)) -> x
minus(+(x, y)) -> *(minus(minus(minus(x))), minus(minus(minus(y))))
minus(*(x, y)) -> +(minus(minus(minus(x))), minus(minus(minus(y))))
f(minus(x)) -> minus(minus(minus(f(x))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes