The rewrite relation of the following TRS is considered.
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(40) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(38) |
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[proper(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[c] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[true] |
= |
0 |
[false] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
proper(f(X)) |
→ |
f(proper(X)) |
(10) |
proper(if(X1,X2,X3)) |
→ |
if(proper(X1),proper(X2),proper(X3)) |
(11) |
proper(c) |
→ |
ok(c) |
(12) |
proper(true) |
→ |
ok(true) |
(13) |
proper(false) |
→ |
ok(false) |
(14) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
f(mark(X)) |
→ |
mark(f(X)) |
(7) |
f(ok(X)) |
→ |
ok(f(X)) |
(15) |
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(5) |
active(if(X1,X2,X3)) |
→ |
if(X1,active(X2),X3) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[top#(x1)] |
= |
2 + x1
|
[active(x1)] |
= |
x1 |
[f(x1)] |
= |
2 + 2 · x1
|
[mark(x1)] |
= |
1 + x1
|
[if(x1, x2, x3)] |
= |
x1 + x2
|
[c] |
= |
0 |
[true] |
= |
2 |
[proper(x1)] |
= |
x1 |
[ok(x1)] |
= |
x1 |
[false] |
= |
0 |
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(38) |
could be deleted.
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[c] |
= |
0 |
[true] |
= |
0 |
[ok(x1)] |
= |
1 · x1
|
[top#(x1)] |
= |
1 · x1
|
together with the usable
rules
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(5) |
active(if(X1,X2,X3)) |
→ |
if(X1,active(X2),X3) |
(6) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
f(mark(X)) |
→ |
mark(f(X)) |
(7) |
f(ok(X)) |
→ |
ok(f(X)) |
(15) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(c) |
= |
1 |
|
weight(c) |
= |
1 |
|
|
|
prec(true) |
= |
3 |
|
weight(true) |
= |
1 |
|
|
|
prec(active) |
= |
7 |
|
weight(active) |
= |
4 |
|
|
|
prec(f) |
= |
5 |
|
weight(f) |
= |
6 |
|
|
|
prec(mark) |
= |
0 |
|
weight(mark) |
= |
1 |
|
|
|
prec(ok) |
= |
4 |
|
weight(ok) |
= |
5 |
|
|
|
prec(top#) |
= |
6 |
|
weight(top#) |
= |
1 |
|
|
|
prec(if) |
= |
2 |
|
weight(if) |
= |
0 |
|
|
|
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(40) |
and
the
rules
active(f(X)) |
→ |
mark(if(X,c,f(true))) |
(1) |
active(if(true,X,Y)) |
→ |
mark(X) |
(2) |
active(f(X)) |
→ |
f(active(X)) |
(4) |
active(if(X1,X2,X3)) |
→ |
if(active(X1),X2,X3) |
(5) |
active(if(X1,X2,X3)) |
→ |
if(X1,active(X2),X3) |
(6) |
if(mark(X1),X2,X3) |
→ |
mark(if(X1,X2,X3)) |
(8) |
if(X1,mark(X2),X3) |
→ |
mark(if(X1,X2,X3)) |
(9) |
if(ok(X1),ok(X2),ok(X3)) |
→ |
ok(if(X1,X2,X3)) |
(16) |
f(mark(X)) |
→ |
mark(f(X)) |
(7) |
f(ok(X)) |
→ |
ok(f(X)) |
(15) |
could be deleted.
1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(24) |
active#(f(X)) |
→ |
active#(X) |
(22) |
active#(if(X1,X2,X3)) |
→ |
active#(X2) |
(26) |
1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[f(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
active#(if(X1,X2,X3)) |
→ |
active#(X1) |
(24) |
|
1 |
> |
1 |
active#(f(X)) |
→ |
active#(X) |
(22) |
|
1 |
> |
1 |
active#(if(X1,X2,X3)) |
→ |
active#(X2) |
(26) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
proper#(if(X1,X2,X3)) |
→ |
proper#(X1) |
(33) |
proper#(f(X)) |
→ |
proper#(X) |
(31) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X2) |
(34) |
proper#(if(X1,X2,X3)) |
→ |
proper#(X3) |
(35) |
1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[if(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[f(x1)] |
= |
1 · x1
|
[proper#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
proper#(if(X1,X2,X3)) |
→ |
proper#(X1) |
(33) |
|
1 |
> |
1 |
proper#(f(X)) |
→ |
proper#(X) |
(31) |
|
1 |
> |
1 |
proper#(if(X1,X2,X3)) |
→ |
proper#(X2) |
(34) |
|
1 |
> |
1 |
proper#(if(X1,X2,X3)) |
→ |
proper#(X3) |
(35) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
f#(ok(X)) |
→ |
f#(X) |
(36) |
f#(mark(X)) |
→ |
f#(X) |
(27) |
1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[ok(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.
1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
f#(ok(X)) |
→ |
f#(X) |
(36) |
|
1 |
> |
1 |
f#(mark(X)) |
→ |
f#(X) |
(27) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
if#(X1,mark(X2),X3) |
→ |
if#(X1,X2,X3) |
(29) |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(28) |
if#(ok(X1),ok(X2),ok(X3)) |
→ |
if#(X1,X2,X3) |
(37) |
1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[ok(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.
1.1.1.5.1 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) |
(29) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
if#(mark(X1),X2,X3) |
→ |
if#(X1,X2,X3) |
(28) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
if#(ok(X1),ok(X2),ok(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.