The rewrite relation of the following TRS is considered.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
mark#(g(X1,X2)) |
→ |
active#(g(mark(X1),X2)) |
(30) |
active#(h(X)) |
→ |
mark#(g(X,X)) |
(20) |
mark#(g(X1,X2)) |
→ |
mark#(X1) |
(32) |
mark#(h(X)) |
→ |
active#(h(mark(X))) |
(27) |
active#(g(a,X)) |
→ |
mark#(f(b,X)) |
(22) |
mark#(f(X1,X2)) |
→ |
active#(f(mark(X1),X2)) |
(34) |
active#(f(X,X)) |
→ |
mark#(h(a)) |
(24) |
mark#(h(X)) |
→ |
mark#(X) |
(29) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(36) |
1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[h(x1)] |
= |
2 + 2 · x1
|
[mark(x1)] |
= |
1 · x1
|
[g(x1, x2)] |
= |
2 + 1 · x1 + 1 · x2
|
[a] |
= |
0 |
[f(x1, x2)] |
= |
2 + 2 · x1 + 1 · x2
|
[b] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pairs
mark#(g(X1,X2)) |
→ |
mark#(X1) |
(32) |
mark#(h(X)) |
→ |
mark#(X) |
(29) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(36) |
and
no rules
could be deleted.
1.1.1.1 Instantiation Processor
We instantiate the pair
to the following set of pairs
mark#(g(z0,z0)) |
→ |
active#(g(mark(z0),z0)) |
(48) |
1.1.1.1.1 Instantiation Processor
We instantiate the pair
to the following set of pairs
mark#(h(a)) |
→ |
active#(h(mark(a))) |
(49) |
1.1.1.1.1.1 Rewriting Processor
We rewrite the right hand side of the pair
resulting in
1.1.1.1.1.1.1 Rewriting Processor
We rewrite the right hand side of the pair
resulting in
1.1.1.1.1.1.1.1 Rewriting Processor
We rewrite the right hand side of the pair
resulting in
1.1.1.1.1.1.1.1.1 Rewriting Processor
We rewrite the right hand side of the pair
resulting in
1.1.1.1.1.1.1.1.1.1 Instantiation Processor
We instantiate the pair
to the following set of pairs
mark#(f(b,z0)) |
→ |
active#(f(mark(b),z0)) |
(54) |
1.1.1.1.1.1.1.1.1.1.1 Rewriting Processor
We rewrite the right hand side of the pair
resulting in
1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + x1
|
[g(x1, x2)] |
= |
1 + x1
|
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[h(x1)] |
= |
1 + x1
|
[a] |
= |
1 |
[f(x1, x2)] |
= |
2 |
[b] |
= |
0 |
[mark#(x1)] |
= |
-1 + x1
|
the
pair
mark#(h(a)) |
→ |
active#(h(b)) |
(53) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
h#(active(X)) |
→ |
h#(X) |
(39) |
h#(mark(X)) |
→ |
h#(X) |
(38) |
1.1.2 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.2.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(h(x0)) |
active(g(a,x0)) |
active(f(x0,x0)) |
active(a) |
mark(h(x0)) |
mark(g(x0,x1)) |
mark(a) |
mark(f(x0,x1)) |
mark(b) |
1.1.2.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
h#(active(X)) |
→ |
h#(X) |
(39) |
|
1 |
> |
1 |
h#(mark(X)) |
→ |
h#(X) |
(38) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
g#(X1,mark(X2)) |
→ |
g#(X1,X2) |
(41) |
g#(mark(X1),X2) |
→ |
g#(X1,X2) |
(40) |
g#(active(X1),X2) |
→ |
g#(X1,X2) |
(42) |
g#(X1,active(X2)) |
→ |
g#(X1,X2) |
(43) |
1.1.3 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.3.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(h(x0)) |
active(g(a,x0)) |
active(f(x0,x0)) |
active(a) |
mark(h(x0)) |
mark(g(x0,x1)) |
mark(a) |
mark(f(x0,x1)) |
mark(b) |
1.1.3.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
g#(X1,mark(X2)) |
→ |
g#(X1,X2) |
(41) |
|
1 |
≥ |
1 |
2 |
> |
2 |
g#(mark(X1),X2) |
→ |
g#(X1,X2) |
(40) |
|
1 |
> |
1 |
2 |
≥ |
2 |
g#(active(X1),X2) |
→ |
g#(X1,X2) |
(42) |
|
1 |
> |
1 |
2 |
≥ |
2 |
g#(X1,active(X2)) |
→ |
g#(X1,X2) |
(43) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(45) |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(44) |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(46) |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(47) |
1.1.4 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.4.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(h(x0)) |
active(g(a,x0)) |
active(f(x0,x0)) |
active(a) |
mark(h(x0)) |
mark(g(x0,x1)) |
mark(a) |
mark(f(x0,x1)) |
mark(b) |
1.1.4.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(45) |
|
1 |
≥ |
1 |
2 |
> |
2 |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(44) |
|
1 |
> |
1 |
2 |
≥ |
2 |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(46) |
|
1 |
> |
1 |
2 |
≥ |
2 |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(47) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.