The rewrite relation of the following TRS is considered.
|
Tucp(SRlbeta(x)) |
→ |
W43(k,x) |
|
Tucp(SRlbeta(x)) |
→ |
W48(k,x) |
|
Tucp(SRcp(x)) |
→ |
W83(k,x) |
|
Tucp(SRcp(x)) |
→ |
W88(k,x) |
|
Tucp(SRlll(x)) |
→ |
W108(k,x) |
|
Tucp(W109(x)) |
→ |
W113(k,x) |
|
Tgc(SRlbeta(x)) |
→ |
W166(k,x) |
|
Tgc(SRcp(x)) |
→ |
W188(k,x) |
|
Tgc(SRlll(x)) |
→ |
W203(k,x) |
|
Tucp(SRlbeta(x)) |
→ |
SRlbeta(Tucp(x)) |
|
Tucp(SRlbeta(x)) |
→ |
SRcp(SRlbeta(Tgc(x))) |
|
Tucp(SRcp(x)) |
→ |
SRcp(Tucp(x)) |
|
Tucp(SRcp(x)) |
→ |
SRcp(Tgc(x)) |
|
Tucp(SRlll(x)) |
→ |
SRlll(Tucp(x)) |
|
Tucp(W22(x)) |
→ |
SRlll(Tucp(x)) |
|
Tucp(SRlll(x)) |
→ |
Tucp(W22(x)) |
|
Tucp(W22(SRlll(x))) |
→ |
Tucp(W22(x)) |
|
Tucp(SRlll(SRlll(x))) |
→ |
SRlll(Tucp(x)) |
|
Tucp(Answer) |
→ |
Answer |
|
Tucp(Answer) |
→ |
SRcp(Answer) |
|
W43(s(k),x) |
→ |
SRlll(W43(k,x)) |
|
W43(s(k),x) |
→ |
SRlll(SRcp(SRlbeta(Tgc(x)))) |
|
W48(s(k),x) |
→ |
SRlll(W48(k,x)) |
|
W48(s(k),x) |
→ |
SRlll(SRlbeta(Tucp(x))) |
|
Tucp(SRlbeta(x)) |
→ |
SRlll(SRcp(SRlbeta(SRlll(Tgc(x))))) |
|
Tucp(SRlbeta(x)) |
→ |
SRlbeta(SRlll(Tucp(x))) |
|
Tucp(SRlbeta(x)) |
→ |
SRlll(SRlll(SRcp(SRlbeta(Tgc(x))))) |
|
Tucp(SRlbeta(x)) |
→ |
SRlll(SRlbeta(Tucp(x))) |
|
Tucp(SRlbeta(x)) |
→ |
SRlll(SRcp(SRlbeta(Tgc(x)))) |
|
W83(s(k),x) |
→ |
SRlll(W83(k,x)) |
|
W83(s(k),x) |
→ |
SRlll(SRcp(Tgc(x))) |
|
W88(s(k),x) |
→ |
SRlll(W88(k,x)) |
|
W88(s(k),x) |
→ |
SRlll(SRcp(Tucp(x))) |
|
Tucp(SRcp(x)) |
→ |
SRlll(SRcp(Tgc(x))) |
|
Tucp(SRcp(x)) |
→ |
SRlll(SRcp(Tucp(x))) |
|
Tucp(SRlll(x)) |
→ |
SRlll(SRlll(Tucp(x))) |
|
W108(s(k),x) |
→ |
SRlll(W108(k,x)) |
|
W108(s(k),x) |
→ |
SRlll(SRlll(Tucp(x))) |
|
Tucp(SRlll(x)) |
→ |
Tucp(W109(x)) |
|
Tucp(W109(SRlll(x))) |
→ |
Tucp(W109(x)) |
|
W113(s(k),x) |
→ |
SRlll(W113(k,x)) |
|
W113(s(k),x) |
→ |
SRlll(SRlll(Tucp(x))) |
|
Tucp(SRlll(x)) |
→ |
SRlll(SRlll(SRlll(Tucp(x)))) |
|
Tucp(SRlll(SRlll(x))) |
→ |
SRlll(SRlll(SRlll(Tucp(x)))) |
|
Tucp(W127(x)) |
→ |
SRlll(SRlll(Tucp(x))) |
|
Tucp(SRlll(x)) |
→ |
Tucp(W127(x)) |
|
Tucp(W127(SRlll(x))) |
→ |
Tucp(W127(x)) |
|
Tucp(SRlll(SRlll(x))) |
→ |
SRlll(SRlll(Tucp(x))) |
|
Tucp(Answer) |
→ |
SRlll(SRcp(Answer)) |
|
Tucp(Answer) |
→ |
SRlll(Answer) |
|
Tgc(SRlbeta(x)) |
→ |
SRlbeta(Tgc(x)) |
|
Tgc(SRcp(x)) |
→ |
SRcp(Tgc(x)) |
|
Tgc(SRlll(x)) |
→ |
SRlll(Tgc(x)) |
|
Tgc(Answer) |
→ |
Answer |
|
W166(s(k),x) |
→ |
SRlll(W166(k,x)) |
|
W166(s(k),x) |
→ |
SRlll(SRlbeta(Tgc(x))) |
|
Tgc(SRlbeta(x)) |
→ |
SRlll(SRlbeta(SRlll(Tgc(x)))) |
|
Tgc(SRlbeta(x)) |
→ |
SRlll(SRlll(SRlbeta(Tgc(x)))) |
|
Tgc(SRlbeta(x)) |
→ |
SRlll(SRlbeta(Tgc(x))) |
|
W188(s(k),x) |
→ |
SRlll(W188(k,x)) |
|
W188(s(k),x) |
→ |
SRlll(SRcp(Tgc(x))) |
|
Tgc(SRcp(x)) |
→ |
SRlll(SRcp(Tgc(x))) |
|
Tgc(SRlll(x)) |
→ |
SRlll(SRlll(Tgc(x))) |
|
W203(s(k),x) |
→ |
SRlll(W203(k,x)) |
|
W203(s(k),x) |
→ |
SRlll(SRlll(Tgc(x))) |
|
Tgc(SRlll(x)) |
→ |
SRlll(SRlll(SRlll(Tgc(x)))) |
|
Tgc(Answer) |
→ |
SRlll(Answer) |
The evaluation strategy is innermost.There are no rules.
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pairs
|
Tucp#(SRlbeta(x)) |
→ |
W48#(k,x) |
|
W48#(s(k),x) |
→ |
W48#(k,x) |
|
W48#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRcp(x)) |
→ |
W88#(k,x) |
|
W88#(s(k),x) |
→ |
W88#(k,x) |
|
W88#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(W109(x)) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlbeta(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRcp(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(x) |
|
Tucp#(W22(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W22(x)) |
|
Tucp#(W22(SRlll(x))) |
→ |
Tucp#(W22(x)) |
|
Tucp#(SRlll(SRlll(x))) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W109(SRlll(x))) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W127(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W127(x)) |
|
Tucp#(W127(SRlll(x))) |
→ |
Tucp#(W127(x)) |
1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the naturals
| [Tucp#(x1)] |
= |
+ ·
x1
|
| [SRlbeta(x1)] |
= |
+ ·
x1
|
| [W48#(x1, x2)] |
= |
+ ·
x1 + ·
x2
|
| [s(x1)] |
= |
+ ·
x1
|
| [SRcp(x1)] |
= |
+ ·
x1
|
| [W88#(x1, x2)] |
= |
+ ·
x1 + ·
x2
|
| [SRlll(x1)] |
= |
+ ·
x1
|
| [W108#(x1, x2)] |
= |
+ ·
x1 + ·
x2
|
| [W109(x1)] |
= |
+ ·
x1
|
| [W113#(x1, x2)] |
= |
+ ·
x1 + ·
x2
|
| [W22(x1)] |
= |
+ ·
x1
|
| [W127(x1)] |
= |
+ ·
x1
|
the
pairs
|
Tucp#(SRlbeta(x)) |
→ |
W48#(k,x) |
|
W48#(s(k),x) |
→ |
W48#(k,x) |
|
W48#(s(k),x) |
→ |
Tucp#(x) |
|
W88#(s(k),x) |
→ |
W88#(k,x) |
|
W88#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(W109(x)) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlbeta(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(x) |
|
Tucp#(W22(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W22(x)) |
|
Tucp#(W22(SRlll(x))) |
→ |
Tucp#(W22(x)) |
|
Tucp#(SRlll(SRlll(x))) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W109(SRlll(x))) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W127(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W127(x)) |
|
Tucp#(W127(SRlll(x))) |
→ |
Tucp#(W127(x)) |
remain.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pairs
|
W48#(s(k),x) |
→ |
W48#(k,x) |
|
W48#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlbeta(x)) |
→ |
W48#(k,x) |
|
Tucp#(SRlll(x)) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(W109(x)) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlbeta(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(x) |
|
Tucp#(W22(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W22(x)) |
|
Tucp#(W22(SRlll(x))) |
→ |
Tucp#(W22(x)) |
|
Tucp#(SRlll(SRlll(x))) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W109(SRlll(x))) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W127(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W127(x)) |
|
Tucp#(W127(SRlll(x))) |
→ |
Tucp#(W127(x)) |
1.1.1.1.1.1.1.2 Reduction Pair Processor
Using the matrix interpretations of dimension 3 with strict dimension 1 over the naturals
| [W48#(x1, x2)] |
= |
+ ·
x1 + ·
x2
|
| [s(x1)] |
= |
+ ·
x1
|
| [Tucp#(x1)] |
= |
+ ·
x1
|
| [SRlbeta(x1)] |
= |
+ ·
x1
|
| [SRlll(x1)] |
= |
+ ·
x1
|
| [W108#(x1, x2)] |
= |
+ ·
x1 + ·
x2
|
| [W109(x1)] |
= |
+ ·
x1
|
| [W113#(x1, x2)] |
= |
+ ·
x1 + ·
x2
|
| [W22(x1)] |
= |
+ ·
x1
|
| [W127(x1)] |
= |
+ ·
x1
|
the
pairs
|
W48#(s(k),x) |
→ |
W48#(k,x) |
|
W48#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(W109(x)) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(x) |
|
Tucp#(W22(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W22(x)) |
|
Tucp#(W22(SRlll(x))) |
→ |
Tucp#(W22(x)) |
|
Tucp#(SRlll(SRlll(x))) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W109(SRlll(x))) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W127(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W127(x)) |
|
Tucp#(W127(SRlll(x))) |
→ |
Tucp#(W127(x)) |
remain.
1.1.1.1.1.1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
1.1.1.1.1.1.1.2.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pairs
|
W108#(s(k),x) |
→ |
W108#(k,x) |
|
W108#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
W108#(k,x) |
|
Tucp#(W109(x)) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(x) |
|
Tucp#(W22(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W22(x)) |
|
Tucp#(W22(SRlll(x))) |
→ |
Tucp#(W22(x)) |
|
Tucp#(SRlll(SRlll(x))) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W109(SRlll(x))) |
→ |
Tucp#(W109(x)) |
|
Tucp#(W127(x)) |
→ |
Tucp#(x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W127(x)) |
|
Tucp#(W127(SRlll(x))) |
→ |
Tucp#(W127(x)) |
1.1.1.1.1.1.1.2.1.2 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
| [W108#(x1, x2)] |
= |
1 + 1 ·
x2
|
| [s(x1)] |
= |
0 |
| [Tucp#(x1)] |
= |
1 ·
x1
|
| [SRlll(x1)] |
= |
1 + 1 ·
x1
|
| [W109(x1)] |
= |
1 + 1 ·
x1
|
| [W113#(x1, x2)] |
= |
1 + 1 ·
x2
|
| [W22(x1)] |
= |
1 + 1 ·
x1
|
| [W127(x1)] |
= |
1 + 1 ·
x1
|
the
pairs
|
W108#(s(k),x) |
→ |
W108#(k,x) |
|
Tucp#(SRlll(x)) |
→ |
W108#(k,x) |
|
Tucp#(W109(x)) |
→ |
W113#(k,x) |
|
W113#(s(k),x) |
→ |
W113#(k,x) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W22(x)) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W109(x)) |
|
Tucp#(SRlll(x)) |
→ |
Tucp#(W127(x)) |
remain.
1.1.1.1.1.1.1.2.1.2.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
W108#(s(k),x) |
→ |
W108#(k,x) |
1.1.1.1.1.1.1.2.1.2.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
|
W113#(s(k),x) |
→ |
W113#(k,x) |
1.1.1.1.1.1.1.2.1.2.1.2 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
1.1.1.2 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.2.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.1.2.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
1.1.1.3 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.3.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.1.3.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pairs
|
W166#(s(k),x) |
→ |
W166#(k,x) |
|
W166#(s(k),x) |
→ |
Tgc#(x) |
|
Tgc#(SRlbeta(x)) |
→ |
W166#(k,x) |
|
Tgc#(SRcp(x)) |
→ |
W188#(k,x) |
|
W188#(s(k),x) |
→ |
W188#(k,x) |
|
W188#(s(k),x) |
→ |
Tgc#(x) |
|
Tgc#(SRlll(x)) |
→ |
W203#(k,x) |
|
W203#(s(k),x) |
→ |
W203#(k,x) |
|
W203#(s(k),x) |
→ |
Tgc#(x) |
|
Tgc#(SRlbeta(x)) |
→ |
Tgc#(x) |
|
Tgc#(SRcp(x)) |
→ |
Tgc#(x) |
|
Tgc#(SRlll(x)) |
→ |
Tgc#(x) |
1.1.1.4 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.4.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
1.1.1.4.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
-
W166#(s(k),x) → W166#(k,x):
1>1, 2>=2
-
W166#(s(k),x) → Tgc#(x):
2>=1
-
Tgc#(SRlbeta(x)) → W166#(k,x):
1>2
-
W188#(s(k),x) → Tgc#(x):
2>=1
-
W203#(s(k),x) → Tgc#(x):
2>=1
-
W188#(s(k),x) → W188#(k,x):
1>1, 2>=2
-
W203#(s(k),x) → W203#(k,x):
1>1, 2>=2
-
Tgc#(SRcp(x)) → W188#(k,x):
1>2
-
Tgc#(SRlll(x)) → W203#(k,x):
1>2
-
Tgc#(SRlbeta(x)) → Tgc#(x):
1>1
-
Tgc#(SRcp(x)) → Tgc#(x):
1>1
-
Tgc#(SRlll(x)) → Tgc#(x):
1>1
As there is no critical graph in the transitive closure, there are no infinite chains.