Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_Zan97_C)
The rewrite relation of the following TRS is considered.
active(g(X)) |
→ |
mark(h(X)) |
(1) |
active(c) |
→ |
mark(d) |
(2) |
active(h(d)) |
→ |
mark(g(c)) |
(3) |
proper(g(X)) |
→ |
g(proper(X)) |
(4) |
proper(h(X)) |
→ |
h(proper(X)) |
(5) |
proper(c) |
→ |
ok(c) |
(6) |
proper(d) |
→ |
ok(d) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(8) |
h(ok(X)) |
→ |
ok(h(X)) |
(9) |
top(mark(X)) |
→ |
top(proper(X)) |
(10) |
top(ok(X)) |
→ |
top(active(X)) |
(11) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by NaTT @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
top#(ok(X)) |
→ |
top#(active(X)) |
(12) |
proper#(h(X)) |
→ |
h#(proper(X)) |
(13) |
active#(g(X)) |
→ |
h#(X) |
(14) |
proper#(g(X)) |
→ |
g#(proper(X)) |
(15) |
proper#(g(X)) |
→ |
proper#(X) |
(16) |
proper#(h(X)) |
→ |
proper#(X) |
(17) |
g#(ok(X)) |
→ |
g#(X) |
(18) |
top#(mark(X)) |
→ |
proper#(X) |
(19) |
top#(ok(X)) |
→ |
active#(X) |
(20) |
h#(ok(X)) |
→ |
h#(X) |
(21) |
active#(h(d)) |
→ |
g#(c) |
(22) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(23) |
1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(23) |
top#(ok(X)) |
→ |
top#(active(X)) |
(12) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
[h(x1)] |
= |
· x1 +
|
[d] |
= |
|
[top(x1)] |
= |
|
[top#(x1)] |
= |
· x1 +
|
[c] |
= |
|
[proper(x1)] |
= |
· x1 +
|
[ok(x1)] |
= |
· x1 +
|
[h#(x1)] |
= |
|
[mark(x1)] |
= |
· x1 +
|
[g#(x1)] |
= |
|
[proper#(x1)] |
= |
|
[active(x1)] |
= |
· x1 +
|
[active#(x1)] |
= |
|
[g(x1)] |
= |
· x1 +
|
together with the usable
rules
proper(g(X)) |
→ |
g(proper(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(8) |
active(g(X)) |
→ |
mark(h(X)) |
(1) |
active(h(d)) |
→ |
mark(g(c)) |
(3) |
proper(h(X)) |
→ |
h(proper(X)) |
(5) |
proper(d) |
→ |
ok(d) |
(7) |
h(ok(X)) |
→ |
ok(h(X)) |
(9) |
proper(c) |
→ |
ok(c) |
(6) |
active(c) |
→ |
mark(d) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
top#(mark(X)) |
→ |
top#(proper(X)) |
(23) |
top#(ok(X)) |
→ |
top#(active(X)) |
(12) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
proper#(g(X)) |
→ |
proper#(X) |
(16) |
proper#(h(X)) |
→ |
proper#(X) |
(17) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
42004 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
2 |
[h#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
proper#(g(X)) |
→ |
proper#(X) |
(16) |
proper#(h(X)) |
→ |
proper#(X) |
(17) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d] |
=
|
23684 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
1 |
[proper(x1)] |
=
|
267 |
[ok(x1)] |
=
|
x1 + 591 |
[h#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
1 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 17748 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 1 |
[d] |
=
|
19496 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
12457 |
[proper(x1)] |
=
|
3440 |
[ok(x1)] |
=
|
x1 + 591 |
[h#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
1 |
[g#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.