Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex23_Luc06_iGM)
The rewrite relation of the following TRS is considered.
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(2) |
mark(a) |
→ |
active(a) |
(3) |
mark(c(X)) |
→ |
active(c(X)) |
(4) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(5) |
f(mark(X)) |
→ |
f(X) |
(6) |
f(active(X)) |
→ |
f(X) |
(7) |
c(mark(X)) |
→ |
c(X) |
(8) |
c(active(X)) |
→ |
c(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(10) |
g(active(X)) |
→ |
g(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.
mark#(f(X)) |
→ |
f#(mark(X)) |
(12) |
g#(active(X)) |
→ |
g#(X) |
(13) |
active#(f(f(a))) |
→ |
mark#(c(f(g(f(a))))) |
(14) |
mark#(g(X)) |
→ |
active#(g(mark(X))) |
(15) |
mark#(g(X)) |
→ |
mark#(X) |
(16) |
mark#(a) |
→ |
active#(a) |
(17) |
active#(f(f(a))) |
→ |
c#(f(g(f(a)))) |
(18) |
f#(active(X)) |
→ |
f#(X) |
(19) |
mark#(g(X)) |
→ |
g#(mark(X)) |
(20) |
c#(active(X)) |
→ |
c#(X) |
(21) |
c#(mark(X)) |
→ |
c#(X) |
(22) |
active#(f(f(a))) |
→ |
g#(f(a)) |
(23) |
mark#(f(X)) |
→ |
mark#(X) |
(24) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(25) |
g#(mark(X)) |
→ |
g#(X) |
(26) |
active#(f(f(a))) |
→ |
f#(g(f(a))) |
(27) |
f#(mark(X)) |
→ |
f#(X) |
(28) |
mark#(c(X)) |
→ |
active#(c(X)) |
(29) |
1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
mark#(f(X)) |
→ |
mark#(X) |
(24) |
mark#(g(X)) |
→ |
mark#(X) |
(16) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
0 |
[c(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 1 |
[mark#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
0 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[c#(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
mark#(f(X)) |
→ |
mark#(X) |
(24) |
mark#(g(X)) |
→ |
mark#(X) |
(16) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
mark#(c(X)) |
→ |
active#(c(X)) |
(29) |
active#(f(f(a))) |
→ |
mark#(c(f(g(f(a))))) |
(14) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
3118 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 8683 |
[mark#(x1)] |
=
|
x1 + 11798 |
[mark(x1)] |
=
|
x1 + 0 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
c(mark(X)) |
→ |
c(X) |
(8) |
g(mark(X)) |
→ |
g(X) |
(10) |
f(active(X)) |
→ |
f(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(11) |
c(active(X)) |
→ |
c(X) |
(9) |
f(mark(X)) |
→ |
f(X) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(c(X)) |
→ |
active#(c(X)) |
(29) |
active#(f(f(a))) |
→ |
mark#(c(f(g(f(a))))) |
(14) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
c#(mark(X)) |
→ |
c#(X) |
(22) |
c#(active(X)) |
→ |
c#(X) |
(21) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
3118 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 8683 |
[mark#(x1)] |
=
|
x1 + 11798 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
c(mark(X)) |
→ |
c(X) |
(8) |
g(mark(X)) |
→ |
g(X) |
(10) |
f(active(X)) |
→ |
f(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(11) |
c(active(X)) |
→ |
c(X) |
(9) |
f(mark(X)) |
→ |
f(X) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
c#(mark(X)) |
→ |
c#(X) |
(22) |
c#(active(X)) |
→ |
c#(X) |
(21) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
g#(mark(X)) |
→ |
g#(X) |
(26) |
g#(active(X)) |
→ |
g#(X) |
(13) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
3118 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 8683 |
[mark#(x1)] |
=
|
x1 + 11798 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
c(mark(X)) |
→ |
c(X) |
(8) |
g(mark(X)) |
→ |
g(X) |
(10) |
f(active(X)) |
→ |
f(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(11) |
c(active(X)) |
→ |
c(X) |
(9) |
f(mark(X)) |
→ |
f(X) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
g#(mark(X)) |
→ |
g#(X) |
(26) |
g#(active(X)) |
→ |
g#(X) |
(13) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
f#(mark(X)) |
→ |
f#(X) |
(28) |
f#(active(X)) |
→ |
f#(X) |
(19) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
3118 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 8683 |
[mark#(x1)] |
=
|
x1 + 11798 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
1 |
together with the usable
rules
c(mark(X)) |
→ |
c(X) |
(8) |
g(mark(X)) |
→ |
g(X) |
(10) |
f(active(X)) |
→ |
f(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(11) |
c(active(X)) |
→ |
c(X) |
(9) |
f(mark(X)) |
→ |
f(X) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(mark(X)) |
→ |
f#(X) |
(28) |
f#(active(X)) |
→ |
f#(X) |
(19) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.