The
1st
component contains the
pair
mark#(f(X1,X2,X3)) |
→ |
active#(f(X1,mark(X2),X3)) |
(15) |
active#(f(a,X,X)) |
→ |
mark#(f(X,b,b)) |
(12) |
mark#(f(X1,X2,X3)) |
→ |
mark#(X2) |
(17) |
1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1, x2, x3)] |
= |
2 + 1 · x1 + 1 · x2 + 1 · x3
|
[a] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[b] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(f(X1,X2,X3)) |
→ |
mark#(X2) |
(17) |
and
no rules
could be deleted.
1.1.1.1 Instantiation Processor
We instantiate the pair
to the following set of pairs
mark#(f(z0,b,b)) |
→ |
active#(f(z0,mark(b),b)) |
(26) |
1.1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
mark(b) |
→ |
active(b) |
(5) |
f(X1,mark(X2),X3) |
→ |
f(X1,X2,X3) |
(7) |
f(mark(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(6) |
f(X1,X2,mark(X3)) |
→ |
f(X1,X2,X3) |
(8) |
f(active(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(9) |
f(X1,active(X2),X3) |
→ |
f(X1,X2,X3) |
(10) |
f(X1,X2,active(X3)) |
→ |
f(X1,X2,X3) |
(11) |
active(b) |
→ |
mark(a) |
(2) |
mark(a) |
→ |
active(a) |
(4) |
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 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
active(b) |
→ |
mark(a) |
(2) |
f(X1,mark(X2),X3) |
→ |
f(X1,X2,X3) |
(7) |
f(mark(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(6) |
f(X1,X2,mark(X3)) |
→ |
f(X1,X2,X3) |
(8) |
f(active(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(9) |
f(X1,active(X2),X3) |
→ |
f(X1,X2,X3) |
(10) |
f(X1,X2,active(X3)) |
→ |
f(X1,X2,X3) |
(11) |
mark(a) |
→ |
active(a) |
(4) |
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 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
mark(a) |
→ |
active(a) |
(4) |
f(X1,mark(X2),X3) |
→ |
f(X1,X2,X3) |
(7) |
f(mark(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(6) |
f(X1,X2,mark(X3)) |
→ |
f(X1,X2,X3) |
(8) |
f(active(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(9) |
f(X1,active(X2),X3) |
→ |
f(X1,X2,X3) |
(10) |
f(X1,X2,active(X3)) |
→ |
f(X1,X2,X3) |
(11) |
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 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
f(X1,mark(X2),X3) |
→ |
f(X1,X2,X3) |
(7) |
f(mark(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(6) |
f(X1,X2,mark(X3)) |
→ |
f(X1,X2,X3) |
(8) |
f(active(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(9) |
f(X1,active(X2),X3) |
→ |
f(X1,X2,X3) |
(10) |
f(X1,X2,active(X3)) |
→ |
f(X1,X2,X3) |
(11) |
1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1, x2, x3)] |
= |
2 · x1 + 2 · x2 + 2 · x3
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[b] |
= |
0 |
together with the usable
rules
f(X1,mark(X2),X3) |
→ |
f(X1,X2,X3) |
(7) |
f(mark(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(6) |
f(X1,X2,mark(X3)) |
→ |
f(X1,X2,X3) |
(8) |
f(active(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(9) |
f(X1,active(X2),X3) |
→ |
f(X1,X2,X3) |
(10) |
f(X1,X2,active(X3)) |
→ |
f(X1,X2,X3) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rules
f(X1,mark(X2),X3) |
→ |
f(X1,X2,X3) |
(7) |
f(mark(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(6) |
f(X1,X2,mark(X3)) |
→ |
f(X1,X2,X3) |
(8) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(f(a,x0,x0)) |
active(b) |
f(mark(x0),x1,x2) |
f(x0,mark(x1),x2) |
f(x0,x1,mark(x2)) |
f(active(x0),x1,x2) |
f(x0,active(x1),x2) |
f(x0,x1,active(x2)) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[f(x1, x2, x3)] |
= |
2 · x1 + 1 · x2 + 2 · x3
|
[active(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
2 · x1
|
[a] |
= |
1 |
[mark#(x1)] |
= |
2 + 2 · x1
|
[b] |
= |
0 |
the
pair
active#(f(a,X,X)) |
→ |
mark#(f(X,b,b)) |
(12) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f(x1, x2, x3)] |
= |
2 + 2 · x1 + 2 · x2 + 2 · x3
|
[active(x1)] |
= |
1 + 1 · x1
|
[mark#(x1)] |
= |
2 + 2 · x1
|
[b] |
= |
2 |
[active#(x1)] |
= |
1 · x1
|
[a] |
= |
0 |
together with the usable
rules
f(X1,active(X2),X3) |
→ |
f(X1,X2,X3) |
(10) |
f(active(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(9) |
f(X1,X2,active(X3)) |
→ |
f(X1,X2,X3) |
(11) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(f(z0,b,b)) |
→ |
active#(f(z0,active(a),b)) |
(29) |
and
the
rules
f(X1,active(X2),X3) |
→ |
f(X1,X2,X3) |
(10) |
f(active(X1),X2,X3) |
→ |
f(X1,X2,X3) |
(9) |
f(X1,X2,active(X3)) |
→ |
f(X1,X2,X3) |
(11) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.