Certification Problem
Input (TPDB TRS_Innermost/Transformed_CSR_innermost_04/Ex23_Luc06_GM)
The rewrite relation of the following TRS is considered.
|
a__f(f(a)) |
→ |
c(f(g(f(a)))) |
(1) |
|
mark(f(X)) |
→ |
a__f(mark(X)) |
(2) |
|
mark(a) |
→ |
a |
(3) |
|
mark(c(X)) |
→ |
c(X) |
(4) |
|
mark(g(X)) |
→ |
g(mark(X)) |
(5) |
|
a__f(X) |
→ |
f(X) |
(6) |
The evaluation strategy is innermost.Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Constant to Unary
Every constant is turned into a unary function symbol to obtain the TRS
|
a__f(f(a'(x))) |
→ |
c(f(g(f(a'(x))))) |
(7) |
|
mark(f(X)) |
→ |
a__f(mark(X)) |
(2) |
|
mark(a'(x)) |
→ |
a'(x) |
(8) |
|
mark(c(X)) |
→ |
c(X) |
(4) |
|
mark(g(X)) |
→ |
g(mark(X)) |
(5) |
|
a__f(X) |
→ |
f(X) |
(6) |
1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a'(f(a__f(x))) |
→ |
a'(f(g(f(c(x))))) |
(9) |
|
f(mark(X)) |
→ |
mark(a__f(X)) |
(10) |
|
a'(mark(x)) |
→ |
a'(x) |
(11) |
|
c(mark(X)) |
→ |
c(X) |
(12) |
|
g(mark(X)) |
→ |
mark(g(X)) |
(13) |
|
a__f(X) |
→ |
f(X) |
(6) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [a'(x1)] |
= |
1 · x1
|
| [f(x1)] |
= |
1 · x1
|
| [a__f(x1)] |
= |
1 · x1
|
| [g(x1)] |
= |
1 · x1
|
| [c(x1)] |
= |
1 · x1
|
| [mark(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
|
a'(mark(x)) |
→ |
a'(x) |
(11) |
|
c(mark(X)) |
→ |
c(X) |
(12) |
1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
a'#(f(a__f(x))) |
→ |
a'#(f(g(f(c(x))))) |
(14) |
|
a'#(f(a__f(x))) |
→ |
f#(g(f(c(x)))) |
(15) |
|
a'#(f(a__f(x))) |
→ |
g#(f(c(x))) |
(16) |
|
a'#(f(a__f(x))) |
→ |
f#(c(x)) |
(17) |
|
f#(mark(X)) |
→ |
a__f#(X) |
(18) |
|
g#(mark(X)) |
→ |
g#(X) |
(19) |
|
a__f#(X) |
→ |
f#(X) |
(20) |
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 2
components.
-
The
1st
component contains the
pair
|
a__f#(X) |
→ |
f#(X) |
(20) |
|
f#(mark(X)) |
→ |
a__f#(X) |
(18) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [mark(x1)] |
= |
1 · x1
|
| [f#(x1)] |
= |
1 · x1
|
| [a__f#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
f#(mark(X)) |
→ |
a__f#(X) |
(18) |
|
| 1 |
> |
1 |
|
a__f#(X) |
→ |
f#(X) |
(20) |
|
| 1 |
≥ |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
| [mark(x1)] |
= |
1 · x1
|
| [g#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
g#(mark(X)) |
→ |
g#(X) |
(19) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.