The rewrite relation of the following TRS is considered.
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(21) |
active#(f(X)) |
→ |
mark#(if(X,c,f(true))) |
(17) |
mark#(f(X)) |
→ |
mark#(X) |
(23) |
mark#(if(X1,X2,X3)) |
→ |
active#(if(mark(X1),mark(X2),X3)) |
(24) |
active#(if(true,X,Y)) |
→ |
mark#(X) |
(20) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(26) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X2) |
(27) |
1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 + 2 · x1
|
[mark(x1)] |
= |
1 · x1
|
[if(x1, x2, x3)] |
= |
1 · x1 + 2 · x2 + 1 · x3
|
[c] |
= |
0 |
[true] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(f(X)) |
→ |
mark#(X) |
(23) |
and
no rules
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(f) |
= |
2 |
|
weight(f) |
= |
4 |
|
|
|
prec(if) |
= |
3 |
|
weight(if) |
= |
1 |
|
|
|
prec(c) |
= |
1 |
|
weight(c) |
= |
2 |
|
|
|
prec(true) |
= |
0 |
|
weight(true) |
= |
1 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(f) |
= |
[1] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(if) |
= |
[1,2] |
π(c) |
= |
[] |
π(true) |
= |
[] |
π(active) |
= |
1 |
the
pairs
active#(f(X)) |
→ |
mark#(if(X,c,f(true))) |
(17) |
active#(if(true,X,Y)) |
→ |
mark#(X) |
(20) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X1) |
(26) |
mark#(if(X1,X2,X3)) |
→ |
mark#(X2) |
(27) |
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
f#(active(X)) |
→ |
f#(X) |
(31) |
f#(mark(X)) |
→ |
f#(X) |
(30) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[f#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(active(X)) |
→ |
f#(X) |
(31) |
|
1 |
> |
1 |
f#(mark(X)) |
→ |
f#(X) |
(30) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(33) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(32) |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(34) |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(35) |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(36) |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(37) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[if#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(33) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(32) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
if#(X1,X2,mark(X3)) |
→ |
if#(X1,X2,X3) |
(34) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
if#(active(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(35) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
if#(X1,active(X2),X3) |
→ |
if#(X1,X2,X3) |
(36) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
if#(X1,X2,active(X3)) |
→ |
if#(X1,X2,X3) |
(37) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.