Term Rewriting System R:
[y, x]
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
+'(s(x), y) -> +'(x, y)
-'(s(x), s(y)) -> -'(x, y)
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
→DP Problem 2
↳AFS
Dependency Pair:
+'(s(x), y) -> +'(x, y)
Rules:
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
Strategy:
innermost
The following dependency pair can be strictly oriented:
+'(s(x), y) -> +'(x, y)
There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System: +'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳AFS
Dependency Pair:
Rules:
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Argument Filtering and Ordering
Dependency Pair:
-'(s(x), s(y)) -> -'(x, y)
Rules:
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
Strategy:
innermost
The following dependency pair can be strictly oriented:
-'(s(x), s(y)) -> -'(x, y)
There are no usable rules for innermost that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System: -'(x1, x2) -> -'(x1, x2)
s(x1) -> s(x1)
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳AFS
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes