Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex25_Luc06_iGM)
The rewrite relation of the following TRS is considered.
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
mark(c(X)) |
→ |
active(c(X)) |
(5) |
mark(g(X)) |
→ |
active(g(X)) |
(6) |
mark(d(X)) |
→ |
active(d(X)) |
(7) |
mark(h(X)) |
→ |
active(h(mark(X))) |
(8) |
f(mark(X)) |
→ |
f(X) |
(9) |
f(active(X)) |
→ |
f(X) |
(10) |
c(mark(X)) |
→ |
c(X) |
(11) |
c(active(X)) |
→ |
c(X) |
(12) |
g(mark(X)) |
→ |
g(X) |
(13) |
g(active(X)) |
→ |
g(X) |
(14) |
d(mark(X)) |
→ |
d(X) |
(15) |
d(active(X)) |
→ |
d(X) |
(16) |
h(mark(X)) |
→ |
h(X) |
(17) |
h(active(X)) |
→ |
h(X) |
(18) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
f(f(active(X))) |
→ |
f(g(f(c(mark(X))))) |
(19) |
c(active(X)) |
→ |
d(mark(X)) |
(20) |
h(active(X)) |
→ |
d(c(mark(X))) |
(21) |
f(mark(X)) |
→ |
mark(f(active(X))) |
(22) |
c(mark(X)) |
→ |
c(active(X)) |
(23) |
g(mark(X)) |
→ |
g(active(X)) |
(24) |
d(mark(X)) |
→ |
d(active(X)) |
(25) |
h(mark(X)) |
→ |
mark(h(active(X))) |
(26) |
mark(f(X)) |
→ |
f(X) |
(27) |
active(f(X)) |
→ |
f(X) |
(28) |
mark(c(X)) |
→ |
c(X) |
(29) |
active(c(X)) |
→ |
c(X) |
(30) |
mark(g(X)) |
→ |
g(X) |
(31) |
active(g(X)) |
→ |
g(X) |
(32) |
mark(d(X)) |
→ |
d(X) |
(33) |
active(d(X)) |
→ |
d(X) |
(34) |
mark(h(X)) |
→ |
h(X) |
(35) |
active(h(X)) |
→ |
h(X) |
(36) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
[f(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[g(x1)] |
= |
1 · x1
|
[c(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[d(x1)] |
= |
1 · x1
|
[h(x1)] |
= |
1 · x1 + 1 |
all of the following rules can be deleted.
h(active(X)) |
→ |
d(c(mark(X))) |
(21) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
f#(f(active(X))) |
→ |
f#(g(f(c(mark(X))))) |
(37) |
f#(f(active(X))) |
→ |
g#(f(c(mark(X)))) |
(38) |
f#(f(active(X))) |
→ |
f#(c(mark(X))) |
(39) |
f#(f(active(X))) |
→ |
c#(mark(X)) |
(40) |
f#(f(active(X))) |
→ |
mark#(X) |
(41) |
c#(active(X)) |
→ |
d#(mark(X)) |
(42) |
c#(active(X)) |
→ |
mark#(X) |
(43) |
f#(mark(X)) |
→ |
mark#(f(active(X))) |
(44) |
f#(mark(X)) |
→ |
f#(active(X)) |
(45) |
f#(mark(X)) |
→ |
active#(X) |
(46) |
c#(mark(X)) |
→ |
c#(active(X)) |
(47) |
c#(mark(X)) |
→ |
active#(X) |
(48) |
g#(mark(X)) |
→ |
g#(active(X)) |
(49) |
g#(mark(X)) |
→ |
active#(X) |
(50) |
d#(mark(X)) |
→ |
d#(active(X)) |
(51) |
d#(mark(X)) |
→ |
active#(X) |
(52) |
h#(mark(X)) |
→ |
mark#(h(active(X))) |
(53) |
h#(mark(X)) |
→ |
h#(active(X)) |
(54) |
h#(mark(X)) |
→ |
active#(X) |
(55) |
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
f#(f(active(X))) |
→ |
f#(c(mark(X))) |
(39) |
f#(f(active(X))) |
→ |
f#(g(f(c(mark(X))))) |
(37) |
f#(mark(X)) |
→ |
f#(active(X)) |
(45) |
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[f#(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[c(x1)] |
= |
0 |
[mark(x1)] |
= |
1 + 1 · x1
|
[g(x1)] |
= |
0 |
[h(x1)] |
= |
1 + 1 · x1
|
[d(x1)] |
= |
0 |
the
pair
f#(mark(X)) |
→ |
f#(active(X)) |
(45) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[f#(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 |
[active(x1)] |
= |
0 |
[c(x1)] |
= |
0 |
[mark(x1)] |
= |
0 |
[g(x1)] |
= |
0 |
[h(x1)] |
= |
0 |
[d(x1)] |
= |
0 |
together with the usable
rules
c(active(X)) |
→ |
d(mark(X)) |
(20) |
c(mark(X)) |
→ |
c(active(X)) |
(23) |
g(mark(X)) |
→ |
g(active(X)) |
(24) |
d(mark(X)) |
→ |
d(active(X)) |
(25) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(f(active(X))) |
→ |
f#(c(mark(X))) |
(39) |
f#(f(active(X))) |
→ |
f#(g(f(c(mark(X))))) |
(37) |
could be deleted.
1.1.1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
c#(mark(X)) |
→ |
c#(active(X)) |
(47) |
1.1.1.1.2 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[c#(x1)] |
= |
1 + x1
|
[active(x1)] |
= |
x1 |
[f(x1)] |
= |
2 · x1
|
[c(x1)] |
= |
-2 |
[g(x1)] |
= |
x1 |
[d(x1)] |
= |
-2 |
[h(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
1 + 2 · x1
|
the
pair
c#(mark(X)) |
→ |
c#(active(X)) |
(47) |
could be deleted.
1.1.1.1.2.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
g#(mark(X)) |
→ |
g#(active(X)) |
(49) |
1.1.1.1.3 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[g#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 + 1 · x1
|
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 + 1 · x1
|
[c(x1)] |
= |
0 |
[g(x1)] |
= |
1 · x1
|
[d(x1)] |
= |
0 |
[h(x1)] |
= |
1 + 1 · x1
|
the
pair
g#(mark(X)) |
→ |
g#(active(X)) |
(49) |
could be deleted.
1.1.1.1.3.1 P is empty
There are no pairs anymore.
-
The
4th
component contains the
pair
d#(mark(X)) |
→ |
d#(active(X)) |
(51) |
1.1.1.1.4 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[d#(x1)] |
= |
1 + x1
|
[active(x1)] |
= |
x1 |
[f(x1)] |
= |
2 · x1
|
[c(x1)] |
= |
-2 |
[g(x1)] |
= |
x1 |
[d(x1)] |
= |
-2 |
[h(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
1 + 2 · x1
|
the
pair
d#(mark(X)) |
→ |
d#(active(X)) |
(51) |
could be deleted.
1.1.1.1.4.1 P is empty
There are no pairs anymore.
-
The
5th
component contains the
pair
h#(mark(X)) |
→ |
h#(active(X)) |
(54) |
1.1.1.1.5 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[h#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 + 1 · x1
|
[active(x1)] |
= |
1 · x1
|
[f(x1)] |
= |
1 + 1 · x1
|
[c(x1)] |
= |
0 |
[g(x1)] |
= |
1 · x1
|
[d(x1)] |
= |
0 |
[h(x1)] |
= |
1 + 1 · x1
|
the
pair
h#(mark(X)) |
→ |
h#(active(X)) |
(54) |
could be deleted.
1.1.1.1.5.1 P is empty
There are no pairs anymore.