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 NaTT @ termCOMP 2023)
1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
active#(c(X)) |
→ |
d#(X) |
(19) |
c#(mark(X)) |
→ |
c#(X) |
(20) |
d#(active(X)) |
→ |
d#(X) |
(21) |
mark#(d(X)) |
→ |
active#(d(X)) |
(22) |
mark#(c(X)) |
→ |
active#(c(X)) |
(23) |
d#(mark(X)) |
→ |
d#(X) |
(24) |
mark#(h(X)) |
→ |
mark#(X) |
(25) |
h#(mark(X)) |
→ |
h#(X) |
(26) |
active#(h(X)) |
→ |
mark#(c(d(X))) |
(27) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(28) |
active#(f(f(X))) |
→ |
g#(f(X)) |
(29) |
c#(active(X)) |
→ |
c#(X) |
(30) |
f#(active(X)) |
→ |
f#(X) |
(31) |
f#(mark(X)) |
→ |
f#(X) |
(32) |
h#(active(X)) |
→ |
h#(X) |
(33) |
mark#(h(X)) |
→ |
h#(mark(X)) |
(34) |
mark#(f(X)) |
→ |
mark#(X) |
(35) |
active#(f(f(X))) |
→ |
mark#(c(f(g(f(X))))) |
(36) |
active#(f(f(X))) |
→ |
f#(g(f(X))) |
(37) |
mark#(f(X)) |
→ |
f#(mark(X)) |
(38) |
mark#(h(X)) |
→ |
active#(h(mark(X))) |
(39) |
active#(h(X)) |
→ |
d#(X) |
(40) |
mark#(g(X)) |
→ |
active#(g(X)) |
(41) |
active#(c(X)) |
→ |
mark#(d(X)) |
(42) |
g#(active(X)) |
→ |
g#(X) |
(43) |
active#(h(X)) |
→ |
c#(d(X)) |
(44) |
g#(mark(X)) |
→ |
g#(X) |
(45) |
active#(f(f(X))) |
→ |
c#(f(g(f(X)))) |
(46) |
1.1 Dependency Graph Processor
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(28) |
active#(h(X)) |
→ |
mark#(c(d(X))) |
(27) |
active#(c(X)) |
→ |
mark#(d(X)) |
(42) |
mark#(h(X)) |
→ |
mark#(X) |
(25) |
mark#(g(X)) |
→ |
active#(g(X)) |
(41) |
mark#(h(X)) |
→ |
active#(h(mark(X))) |
(39) |
mark#(c(X)) |
→ |
active#(c(X)) |
(23) |
mark#(d(X)) |
→ |
active#(d(X)) |
(22) |
active#(f(f(X))) |
→ |
mark#(c(f(g(f(X))))) |
(36) |
mark#(f(X)) |
→ |
mark#(X) |
(35) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
x1 + 60194 |
[d(x1)] |
=
|
1 |
[c(x1)] |
=
|
x1 + 30097 |
[f(x1)] |
=
|
x1 + 74428 |
[mark#(x1)] |
=
|
x1 + 30095 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 30094 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
14235 |
together with the usable
rules
h(active(X)) |
→ |
h(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
d(mark(X)) |
→ |
d(X) |
(15) |
mark(h(X)) |
→ |
active(h(mark(X))) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(active(X)) |
→ |
d(X) |
(16) |
h(mark(X)) |
→ |
h(X) |
(17) |
mark(c(X)) |
→ |
active(c(X)) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(d(X)) |
→ |
active(d(X)) |
(7) |
g(active(X)) |
→ |
g(X) |
(14) |
c(active(X)) |
→ |
c(X) |
(12) |
c(mark(X)) |
→ |
c(X) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(13) |
mark(g(X)) |
→ |
active(g(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(28) |
active#(h(X)) |
→ |
mark#(c(d(X))) |
(27) |
active#(c(X)) |
→ |
mark#(d(X)) |
(42) |
mark#(h(X)) |
→ |
mark#(X) |
(25) |
mark#(g(X)) |
→ |
active#(g(X)) |
(41) |
mark#(h(X)) |
→ |
active#(h(mark(X))) |
(39) |
mark#(c(X)) |
→ |
active#(c(X)) |
(23) |
mark#(d(X)) |
→ |
active#(d(X)) |
(22) |
active#(f(f(X))) |
→ |
mark#(c(f(g(f(X))))) |
(36) |
mark#(f(X)) |
→ |
mark#(X) |
(35) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
h#(mark(X)) |
→ |
h#(X) |
(26) |
h#(active(X)) |
→ |
h#(X) |
(33) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
118388 |
[d(x1)] |
=
|
1143 |
[c(x1)] |
=
|
1143 |
[f(x1)] |
=
|
16304 |
[mark#(x1)] |
=
|
x1 + 30095 |
[h#(x1)] |
=
|
x1 + 0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
30894 |
together with the usable
rules
h(active(X)) |
→ |
h(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
d(mark(X)) |
→ |
d(X) |
(15) |
mark(h(X)) |
→ |
active(h(mark(X))) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(active(X)) |
→ |
d(X) |
(16) |
h(mark(X)) |
→ |
h(X) |
(17) |
mark(c(X)) |
→ |
active(c(X)) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(d(X)) |
→ |
active(d(X)) |
(7) |
g(active(X)) |
→ |
g(X) |
(14) |
c(active(X)) |
→ |
c(X) |
(12) |
c(mark(X)) |
→ |
c(X) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(13) |
mark(g(X)) |
→ |
active(g(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
h#(mark(X)) |
→ |
h#(X) |
(26) |
h#(active(X)) |
→ |
h#(X) |
(33) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
f#(mark(X)) |
→ |
f#(X) |
(32) |
f#(active(X)) |
→ |
f#(X) |
(31) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
17678 |
[d(x1)] |
=
|
5054 |
[c(x1)] |
=
|
5054 |
[f(x1)] |
=
|
8946 |
[mark#(x1)] |
=
|
x1 + 30095 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
47941 |
together with the usable
rules
h(active(X)) |
→ |
h(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
d(mark(X)) |
→ |
d(X) |
(15) |
mark(h(X)) |
→ |
active(h(mark(X))) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(active(X)) |
→ |
d(X) |
(16) |
h(mark(X)) |
→ |
h(X) |
(17) |
mark(c(X)) |
→ |
active(c(X)) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(d(X)) |
→ |
active(d(X)) |
(7) |
g(active(X)) |
→ |
g(X) |
(14) |
c(active(X)) |
→ |
c(X) |
(12) |
c(mark(X)) |
→ |
c(X) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(13) |
mark(g(X)) |
→ |
active(g(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(mark(X)) |
→ |
f#(X) |
(32) |
f#(active(X)) |
→ |
f#(X) |
(31) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
c#(mark(X)) |
→ |
c#(X) |
(20) |
c#(active(X)) |
→ |
c#(X) |
(30) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
1 |
[d(x1)] |
=
|
1 |
[c(x1)] |
=
|
1 |
[f(x1)] |
=
|
28882 |
[mark#(x1)] |
=
|
x1 + 30095 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
47941 |
together with the usable
rules
h(active(X)) |
→ |
h(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
d(mark(X)) |
→ |
d(X) |
(15) |
mark(h(X)) |
→ |
active(h(mark(X))) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(active(X)) |
→ |
d(X) |
(16) |
h(mark(X)) |
→ |
h(X) |
(17) |
mark(c(X)) |
→ |
active(c(X)) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(d(X)) |
→ |
active(d(X)) |
(7) |
g(active(X)) |
→ |
g(X) |
(14) |
c(active(X)) |
→ |
c(X) |
(12) |
c(mark(X)) |
→ |
c(X) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(13) |
mark(g(X)) |
→ |
active(g(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
c#(mark(X)) |
→ |
c#(X) |
(20) |
c#(active(X)) |
→ |
c#(X) |
(30) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
d#(mark(X)) |
→ |
d#(X) |
(24) |
d#(active(X)) |
→ |
d#(X) |
(21) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
17748 |
[d(x1)] |
=
|
6704 |
[c(x1)] |
=
|
16908 |
[f(x1)] |
=
|
32869 |
[mark#(x1)] |
=
|
x1 + 30095 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
15064 |
together with the usable
rules
h(active(X)) |
→ |
h(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
d(mark(X)) |
→ |
d(X) |
(15) |
mark(h(X)) |
→ |
active(h(mark(X))) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(active(X)) |
→ |
d(X) |
(16) |
h(mark(X)) |
→ |
h(X) |
(17) |
mark(c(X)) |
→ |
active(c(X)) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(d(X)) |
→ |
active(d(X)) |
(7) |
g(active(X)) |
→ |
g(X) |
(14) |
c(active(X)) |
→ |
c(X) |
(12) |
c(mark(X)) |
→ |
c(X) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(13) |
mark(g(X)) |
→ |
active(g(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
d#(mark(X)) |
→ |
d#(X) |
(24) |
d#(active(X)) |
→ |
d#(X) |
(21) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
g#(mark(X)) |
→ |
g#(X) |
(45) |
g#(active(X)) |
→ |
g#(X) |
(43) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[h(x1)] |
=
|
52849 |
[d(x1)] |
=
|
1 |
[c(x1)] |
=
|
1 |
[f(x1)] |
=
|
1 |
[mark#(x1)] |
=
|
x1 + 30095 |
[h#(x1)] |
=
|
0 |
[d#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
47421 |
together with the usable
rules
h(active(X)) |
→ |
h(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
d(mark(X)) |
→ |
d(X) |
(15) |
mark(h(X)) |
→ |
active(h(mark(X))) |
(8) |
active(f(f(X))) |
→ |
mark(c(f(g(f(X))))) |
(1) |
active(h(X)) |
→ |
mark(c(d(X))) |
(3) |
d(active(X)) |
→ |
d(X) |
(16) |
h(mark(X)) |
→ |
h(X) |
(17) |
mark(c(X)) |
→ |
active(c(X)) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(d(X)) |
→ |
active(d(X)) |
(7) |
g(active(X)) |
→ |
g(X) |
(14) |
c(active(X)) |
→ |
c(X) |
(12) |
c(mark(X)) |
→ |
c(X) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
g(mark(X)) |
→ |
g(X) |
(13) |
mark(g(X)) |
→ |
active(g(X)) |
(6) |
active(c(X)) |
→ |
mark(d(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
g#(mark(X)) |
→ |
g#(X) |
(45) |
g#(active(X)) |
→ |
g#(X) |
(43) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.