Term Rewriting System R:
[x, y]
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(x, h(y)) -> F(h(x), y)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Instantiation Transformation
Dependency Pair:
F(x, h(y)) -> F(h(x), y)
Rules:
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)
Strategy:
innermost
On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule
F(x, h(y)) -> F(h(x), y)
one new Dependency Pair
is created:
F(h(x''), h(y'')) -> F(h(h(x'')), y'')
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳Inst
→DP Problem 2
↳Instantiation Transformation
Dependency Pair:
F(h(x''), h(y'')) -> F(h(h(x'')), y'')
Rules:
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)
Strategy:
innermost
On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule
F(h(x''), h(y'')) -> F(h(h(x'')), y'')
one new Dependency Pair
is created:
F(h(h(x'''')), h(y'''')) -> F(h(h(h(x''''))), y'''')
The transformation is resulting in one new DP problem:
R
↳DPs
→DP Problem 1
↳Inst
→DP Problem 2
↳Inst
...
→DP Problem 3
↳Polynomial Ordering
Dependency Pair:
F(h(h(x'''')), h(y'''')) -> F(h(h(h(x''''))), y'''')
Rules:
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)
Strategy:
innermost
The following dependency pair can be strictly oriented:
F(h(h(x'''')), h(y'''')) -> F(h(h(h(x''''))), y'''')
There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(h(x1)) | = 1 + x1 |
| POL(F(x1, x2)) | = x2 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Inst
→DP Problem 2
↳Inst
...
→DP Problem 4
↳Dependency Graph
Dependency Pair:
Rules:
f(x, g(x)) -> x
f(x, h(y)) -> f(h(x), y)
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes