Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex15_Luc06_iGM)
The rewrite relation of the following TRS is considered.
active(f(f(a))) |
→ |
mark(f(g(f(a)))) |
(1) |
mark(f(X)) |
→ |
active(f(X)) |
(2) |
mark(a) |
→ |
active(a) |
(3) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(4) |
f(mark(X)) |
→ |
f(X) |
(5) |
f(active(X)) |
→ |
f(X) |
(6) |
g(mark(X)) |
→ |
g(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(8) |
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
active(f(f(a'(x)))) |
→ |
mark(f(g(f(a'(x))))) |
(9) |
mark(f(X)) |
→ |
active(f(X)) |
(2) |
mark(a'(x)) |
→ |
active(a'(x)) |
(10) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(4) |
f(mark(X)) |
→ |
f(X) |
(5) |
f(active(X)) |
→ |
f(X) |
(6) |
g(mark(X)) |
→ |
g(X) |
(7) |
g(active(X)) |
→ |
g(X) |
(8) |
1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
a'(f(f(active(x)))) |
→ |
a'(f(g(f(mark(x))))) |
(11) |
f(mark(X)) |
→ |
f(active(X)) |
(12) |
a'(mark(x)) |
→ |
a'(active(x)) |
(13) |
g(mark(X)) |
→ |
mark(g(active(X))) |
(14) |
mark(f(X)) |
→ |
f(X) |
(15) |
active(f(X)) |
→ |
f(X) |
(16) |
mark(g(X)) |
→ |
g(X) |
(17) |
active(g(X)) |
→ |
g(X) |
(18) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
a'#(f(f(active(x)))) |
→ |
a'#(f(g(f(mark(x))))) |
(19) |
a'#(f(f(active(x)))) |
→ |
f#(g(f(mark(x)))) |
(20) |
a'#(f(f(active(x)))) |
→ |
g#(f(mark(x))) |
(21) |
a'#(f(f(active(x)))) |
→ |
f#(mark(x)) |
(22) |
a'#(f(f(active(x)))) |
→ |
mark#(x) |
(23) |
f#(mark(X)) |
→ |
f#(active(X)) |
(24) |
f#(mark(X)) |
→ |
active#(X) |
(25) |
a'#(mark(x)) |
→ |
a'#(active(x)) |
(26) |
a'#(mark(x)) |
→ |
active#(x) |
(27) |
g#(mark(X)) |
→ |
mark#(g(active(X))) |
(28) |
g#(mark(X)) |
→ |
g#(active(X)) |
(29) |
g#(mark(X)) |
→ |
active#(X) |
(30) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
a'#(mark(x)) |
→ |
a'#(active(x)) |
(26) |
a'#(f(f(active(x)))) |
→ |
a'#(f(g(f(mark(x))))) |
(19) |
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
|
[g(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[a'#(x1)] |
= |
1 · x1
|
together with the usable
rules
mark(f(X)) |
→ |
f(X) |
(15) |
mark(g(X)) |
→ |
g(X) |
(17) |
g(mark(X)) |
→ |
mark(g(active(X))) |
(14) |
f(mark(X)) |
→ |
f(active(X)) |
(12) |
active(f(X)) |
→ |
f(X) |
(16) |
active(g(X)) |
→ |
g(X) |
(18) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[a'#(x1)] |
= |
2 + 2 · x1
|
[active(x1)] |
= |
x1 |
[f(x1)] |
= |
1 + 2 · x1
|
[g(x1)] |
= |
0 |
[mark(x1)] |
= |
x1 |
the
pair
a'#(f(f(active(x)))) |
→ |
a'#(f(g(f(mark(x))))) |
(19) |
could be deleted.
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 + 1 · x1
|
[f(x1)] |
= |
3 · x1
|
[g(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
1 · x1
|
[a'#(x1)] |
= |
3 · x1
|
the
pair
a'#(mark(x)) |
→ |
a'#(active(x)) |
(26) |
and
the
rules
mark(f(X)) |
→ |
f(X) |
(15) |
mark(g(X)) |
→ |
g(X) |
(17) |
g(mark(X)) |
→ |
mark(g(active(X))) |
(14) |
f(mark(X)) |
→ |
f(active(X)) |
(12) |
could be deleted.
1.1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
f#(mark(X)) |
→ |
f#(active(X)) |
(24) |
1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[f#(x1)] |
= |
1 · x1
|
together with the usable
rules
active(f(X)) |
→ |
f(X) |
(16) |
active(g(X)) |
→ |
g(X) |
(18) |
mark(g(X)) |
→ |
g(X) |
(17) |
g(mark(X)) |
→ |
mark(g(active(X))) |
(14) |
mark(f(X)) |
→ |
f(X) |
(15) |
f(mark(X)) |
→ |
f(active(X)) |
(12) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.2.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
3 + 3 · x1
|
[g(x1)] |
= |
1 + 2 · x1
|
[mark(x1)] |
= |
1 + 1 · x1
|
[f#(x1)] |
= |
3 · x1
|
the
pair
f#(mark(X)) |
→ |
f#(active(X)) |
(24) |
and
the
rules
mark(g(X)) |
→ |
g(X) |
(17) |
g(mark(X)) |
→ |
mark(g(active(X))) |
(14) |
mark(f(X)) |
→ |
f(X) |
(15) |
f(mark(X)) |
→ |
f(active(X)) |
(12) |
could be deleted.
1.1.1.1.2.1.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
g#(mark(X)) |
→ |
g#(active(X)) |
(29) |
1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[g#(x1)] |
= |
1 · x1
|
together with the usable
rules
active(f(X)) |
→ |
f(X) |
(16) |
active(g(X)) |
→ |
g(X) |
(18) |
mark(g(X)) |
→ |
g(X) |
(17) |
g(mark(X)) |
→ |
mark(g(active(X))) |
(14) |
mark(f(X)) |
→ |
f(X) |
(15) |
f(mark(X)) |
→ |
f(active(X)) |
(12) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.3.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
3 + 3 · x1
|
[g(x1)] |
= |
1 + 2 · x1
|
[mark(x1)] |
= |
1 + 1 · x1
|
[g#(x1)] |
= |
3 · x1
|
the
pair
g#(mark(X)) |
→ |
g#(active(X)) |
(29) |
and
the
rules
mark(g(X)) |
→ |
g(X) |
(17) |
g(mark(X)) |
→ |
mark(g(active(X))) |
(14) |
mark(f(X)) |
→ |
f(X) |
(15) |
f(mark(X)) |
→ |
f(active(X)) |
(12) |
could be deleted.
1.1.1.1.3.1.1 P is empty
There are no pairs anymore.