Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex15_Luc06_C)
The rewrite relation of the following TRS is considered.
active(f(f(a))) |
→ |
mark(f(g(f(a)))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(2) |
g(mark(X)) |
→ |
mark(g(X)) |
(3) |
proper(f(X)) |
→ |
f(proper(X)) |
(4) |
proper(a) |
→ |
ok(a) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(6) |
f(ok(X)) |
→ |
ok(f(X)) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(8) |
top(mark(X)) |
→ |
top(proper(X)) |
(9) |
top(ok(X)) |
→ |
top(active(X)) |
(10) |
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.
active#(g(X)) |
→ |
active#(X) |
(11) |
top#(mark(X)) |
→ |
proper#(X) |
(12) |
g#(ok(X)) |
→ |
g#(X) |
(13) |
top#(ok(X)) |
→ |
active#(X) |
(14) |
active#(f(f(a))) |
→ |
f#(g(f(a))) |
(15) |
active#(f(f(a))) |
→ |
g#(f(a)) |
(16) |
proper#(f(X)) |
→ |
f#(proper(X)) |
(17) |
f#(ok(X)) |
→ |
f#(X) |
(18) |
g#(mark(X)) |
→ |
g#(X) |
(19) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(20) |
proper#(g(X)) |
→ |
g#(proper(X)) |
(21) |
active#(g(X)) |
→ |
g#(active(X)) |
(22) |
top#(ok(X)) |
→ |
top#(active(X)) |
(23) |
proper#(f(X)) |
→ |
proper#(X) |
(24) |
proper#(g(X)) |
→ |
proper#(X) |
(25) |
1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(23) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(20) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
[a] |
= |
|
[top(x1)] |
= |
|
[top#(x1)] |
= |
· x1 +
|
[f(x1)] |
= |
· x1 +
|
[proper(x1)] |
= |
x1 +
|
[ok(x1)] |
= |
x1 +
|
[mark(x1)] |
= |
· x1 +
|
[f#(x1)] |
= |
|
[g#(x1)] |
= |
|
[proper#(x1)] |
= |
|
[active(x1)] |
= |
· x1 +
|
[active#(x1)] |
= |
|
[g(x1)] |
= |
· x1 +
|
together with the usable
rules
proper(f(X)) |
→ |
f(proper(X)) |
(4) |
g(ok(X)) |
→ |
ok(g(X)) |
(8) |
active(f(f(a))) |
→ |
mark(f(g(f(a)))) |
(1) |
g(mark(X)) |
→ |
mark(g(X)) |
(3) |
proper(a) |
→ |
ok(a) |
(5) |
f(ok(X)) |
→ |
ok(f(X)) |
(7) |
proper(g(X)) |
→ |
g(proper(X)) |
(6) |
active(g(X)) |
→ |
g(active(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
top#(ok(X)) |
→ |
top#(active(X)) |
(23) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(20) |
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) |
(25) |
proper#(f(X)) |
→ |
proper#(X) |
(24) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 591 |
[proper(x1)] |
=
|
4687 |
[ok(x1)] |
=
|
4688 |
[mark(x1)] |
=
|
42845 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
33119 |
[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) |
(25) |
proper#(f(X)) |
→ |
proper#(X) |
(24) |
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
[a] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
12457 |
[ok(x1)] |
=
|
x1 + 12618 |
[mark(x1)] |
=
|
42845 |
[f#(x1)] |
=
|
x1 + 0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
33119 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 0 |
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
active#(g(X)) |
→ |
active#(X) |
(11) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
33041 |
[ok(x1)] |
=
|
x1 + 33041 |
[mark(x1)] |
=
|
53050 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
43324 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
active#(g(X)) |
→ |
active#(X) |
(11) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
g#(ok(X)) |
→ |
g#(X) |
(13) |
g#(mark(X)) |
→ |
g#(X) |
(19) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
11650 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
48683 |
[ok(x1)] |
=
|
x1 + 37034 |
[mark(x1)] |
=
|
x1 + 24609 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
7177 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 24389 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
g#(ok(X)) |
→ |
g#(X) |
(13) |
g#(mark(X)) |
→ |
g#(X) |
(19) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.