Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_7_15_Bor03_iGM)
The rewrite relation of the following TRS is considered.
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
active(p(s(0))) |
→ |
mark(0) |
(3) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
mark(0) |
→ |
active(0) |
(5) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(6) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(7) |
mark(p(X)) |
→ |
active(p(mark(X))) |
(8) |
f(mark(X)) |
→ |
f(X) |
(9) |
f(active(X)) |
→ |
f(X) |
(10) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
s(mark(X)) |
→ |
s(X) |
(15) |
s(active(X)) |
→ |
s(X) |
(16) |
p(mark(X)) |
→ |
p(X) |
(17) |
p(active(X)) |
→ |
p(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#(f(s(0))) |
→ |
f#(p(s(0))) |
(19) |
mark#(f(X)) |
→ |
f#(mark(X)) |
(20) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(21) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(22) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(23) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(24) |
mark#(p(X)) |
→ |
active#(p(mark(X))) |
(25) |
active#(f(0)) |
→ |
s#(0) |
(26) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(27) |
p#(active(X)) |
→ |
p#(X) |
(28) |
mark#(s(X)) |
→ |
mark#(X) |
(29) |
mark#(f(X)) |
→ |
mark#(X) |
(30) |
mark#(p(X)) |
→ |
p#(mark(X)) |
(31) |
active#(f(0)) |
→ |
mark#(cons(0,f(s(0)))) |
(32) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(33) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(34) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(35) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(36) |
active#(f(0)) |
→ |
f#(s(0)) |
(37) |
s#(mark(X)) |
→ |
s#(X) |
(38) |
p#(mark(X)) |
→ |
p#(X) |
(39) |
active#(p(s(0))) |
→ |
mark#(0) |
(40) |
mark#(p(X)) |
→ |
mark#(X) |
(41) |
active#(f(0)) |
→ |
cons#(0,f(s(0))) |
(42) |
mark#(0) |
→ |
active#(0) |
(43) |
active#(f(s(0))) |
→ |
p#(s(0)) |
(44) |
active#(f(s(0))) |
→ |
mark#(f(p(s(0)))) |
(45) |
f#(mark(X)) |
→ |
f#(X) |
(46) |
f#(active(X)) |
→ |
f#(X) |
(47) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(48) |
s#(active(X)) |
→ |
s#(X) |
(49) |
1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(48) |
mark#(f(X)) |
→ |
mark#(X) |
(30) |
mark#(s(X)) |
→ |
mark#(X) |
(29) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(27) |
active#(f(s(0))) |
→ |
mark#(f(p(s(0)))) |
(45) |
mark#(p(X)) |
→ |
active#(p(mark(X))) |
(25) |
mark#(p(X)) |
→ |
mark#(X) |
(41) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(21) |
mark#(f(X)) |
→ |
active#(f(mark(X))) |
(36) |
active#(f(0)) |
→ |
mark#(cons(0,f(s(0)))) |
(32) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
2998 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
29284 |
[p(x1)] |
=
|
29283 |
[mark#(x1)] |
=
|
29284 |
[0] |
=
|
29286 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
29285 |
[f#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
29283 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
p(active(X)) |
→ |
p(X) |
(18) |
s(mark(X)) |
→ |
s(X) |
(15) |
s(active(X)) |
→ |
s(X) |
(16) |
p(mark(X)) |
→ |
p(X) |
(17) |
f(active(X)) |
→ |
f(X) |
(10) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(48) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(27) |
mark#(p(X)) |
→ |
active#(p(mark(X))) |
(25) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(24) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(23) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(34) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(33) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
x1 + 0 |
[s(x1)] |
=
|
x1 + 18006 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
x1 + 48293 |
[p(x1)] |
=
|
45381 |
[mark#(x1)] |
=
|
x1 + 2 |
[0] |
=
|
27376 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[cons(x1, x2)] |
=
|
x1 + 45545 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
p(active(X)) |
→ |
p(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
s(mark(X)) |
→ |
s(X) |
(15) |
mark(p(X)) |
→ |
active(p(mark(X))) |
(8) |
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(p(s(0))) |
→ |
mark(0) |
(3) |
s(active(X)) |
→ |
s(X) |
(16) |
p(mark(X)) |
→ |
p(X) |
(17) |
mark(0) |
→ |
active(0) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(7) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(6) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(23) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(49) |
s#(mark(X)) |
→ |
s#(X) |
(38) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
16335 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
26170 |
[p(x1)] |
=
|
38368 |
[mark#(x1)] |
=
|
x1 + 2 |
[0] |
=
|
0 |
[s#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
1 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
p(active(X)) |
→ |
p(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
s(mark(X)) |
→ |
s(X) |
(15) |
mark(p(X)) |
→ |
active(p(mark(X))) |
(8) |
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(p(s(0))) |
→ |
mark(0) |
(3) |
s(active(X)) |
→ |
s(X) |
(16) |
p(mark(X)) |
→ |
p(X) |
(17) |
mark(0) |
→ |
active(0) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(7) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(6) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
s#(active(X)) |
→ |
s#(X) |
(49) |
s#(mark(X)) |
→ |
s#(X) |
(38) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
p#(active(X)) |
→ |
p#(X) |
(28) |
p#(mark(X)) |
→ |
p#(X) |
(39) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
29273 |
[p#(x1)] |
=
|
x1 + 0 |
[f(x1)] |
=
|
1 |
[p(x1)] |
=
|
39990 |
[mark#(x1)] |
=
|
x1 + 2 |
[0] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
1 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
p(active(X)) |
→ |
p(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
s(mark(X)) |
→ |
s(X) |
(15) |
mark(p(X)) |
→ |
active(p(mark(X))) |
(8) |
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(p(s(0))) |
→ |
mark(0) |
(3) |
s(active(X)) |
→ |
s(X) |
(16) |
p(mark(X)) |
→ |
p(X) |
(17) |
mark(0) |
→ |
active(0) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(7) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(6) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
p#(active(X)) |
→ |
p#(X) |
(28) |
p#(mark(X)) |
→ |
p#(X) |
(39) |
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#(active(X)) |
→ |
f#(X) |
(47) |
f#(mark(X)) |
→ |
f#(X) |
(46) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[cons#(x1, x2)] |
=
|
0 |
[s(x1)] |
=
|
29161 |
[p#(x1)] |
=
|
0 |
[f(x1)] |
=
|
41777 |
[p(x1)] |
=
|
44249 |
[mark#(x1)] |
=
|
x1 + 2 |
[0] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 1 |
[cons(x1, x2)] |
=
|
26815 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
p(active(X)) |
→ |
p(X) |
(18) |
mark(f(X)) |
→ |
active(f(mark(X))) |
(4) |
s(mark(X)) |
→ |
s(X) |
(15) |
mark(p(X)) |
→ |
active(p(mark(X))) |
(8) |
active(f(0)) |
→ |
mark(cons(0,f(s(0)))) |
(1) |
active(p(s(0))) |
→ |
mark(0) |
(3) |
s(active(X)) |
→ |
s(X) |
(16) |
p(mark(X)) |
→ |
p(X) |
(17) |
mark(0) |
→ |
active(0) |
(5) |
f(active(X)) |
→ |
f(X) |
(10) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(7) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(14) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(12) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(11) |
f(mark(X)) |
→ |
f(X) |
(9) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(13) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(6) |
active(f(s(0))) |
→ |
mark(f(p(s(0)))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(active(X)) |
→ |
f#(X) |
(47) |
f#(mark(X)) |
→ |
f#(X) |
(46) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.