Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex23_Luc06_C)
The rewrite relation of the following TRS is considered.
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
proper(a) |
→ |
ok(a) |
(7) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
top(mark(X)) |
→ |
top(proper(X)) |
(13) |
top(ok(X)) |
→ |
top(active(X)) |
(14) |
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(X)) |
→ |
active#(X) |
(15) |
top#(mark(X)) |
→ |
proper#(X) |
(16) |
f#(ok(X)) |
→ |
f#(X) |
(17) |
c#(ok(X)) |
→ |
c#(X) |
(18) |
top#(ok(X)) |
→ |
top#(active(X)) |
(19) |
top#(ok(X)) |
→ |
active#(X) |
(20) |
g#(mark(X)) |
→ |
g#(X) |
(21) |
proper#(c(X)) |
→ |
c#(proper(X)) |
(22) |
proper#(g(X)) |
→ |
g#(proper(X)) |
(23) |
g#(ok(X)) |
→ |
g#(X) |
(24) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(25) |
f#(mark(X)) |
→ |
f#(X) |
(26) |
active#(f(f(a))) |
→ |
c#(f(g(f(a)))) |
(27) |
active#(f(f(a))) |
→ |
g#(f(a)) |
(28) |
proper#(c(X)) |
→ |
proper#(X) |
(29) |
active#(g(X)) |
→ |
active#(X) |
(30) |
proper#(f(X)) |
→ |
f#(proper(X)) |
(31) |
active#(f(X)) |
→ |
f#(active(X)) |
(32) |
proper#(g(X)) |
→ |
proper#(X) |
(33) |
active#(g(X)) |
→ |
g#(active(X)) |
(34) |
proper#(f(X)) |
→ |
proper#(X) |
(35) |
active#(f(f(a))) |
→ |
f#(g(f(a))) |
(36) |
1.1 Dependency Graph Processor
The dependency pairs are split into 6
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(19) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(25) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
7579 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
51 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 2 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
proper(a) |
→ |
ok(a) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(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
proper#(f(X)) |
→ |
proper#(X) |
(35) |
proper#(g(X)) |
→ |
proper#(X) |
(33) |
proper#(c(X)) |
→ |
proper#(X) |
(29) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 24773 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 52085 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 27311 |
together with the usable
rules
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
proper(a) |
→ |
ok(a) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(f(X)) |
→ |
proper#(X) |
(35) |
proper#(g(X)) |
→ |
proper#(X) |
(33) |
proper#(c(X)) |
→ |
proper#(X) |
(29) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
active#(g(X)) |
→ |
active#(X) |
(30) |
active#(f(X)) |
→ |
active#(X) |
(15) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 44049 |
[f(x1)] |
=
|
x1 + 46876 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 44051 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
proper(a) |
→ |
ok(a) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(g(X)) |
→ |
active#(X) |
(30) |
active#(f(X)) |
→ |
active#(X) |
(15) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
f#(ok(X)) |
→ |
f#(X) |
(17) |
f#(mark(X)) |
→ |
f#(X) |
(26) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
2805 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 33124 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
x1 + 0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 33126 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
proper(a) |
→ |
ok(a) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(ok(X)) |
→ |
f#(X) |
(17) |
f#(mark(X)) |
→ |
f#(X) |
(26) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
g#(mark(X)) |
→ |
g#(X) |
(21) |
g#(ok(X)) |
→ |
g#(X) |
(24) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
11575 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 35355 |
[f(x1)] |
=
|
x1 + 1 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 77360 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 42004 |
together with the usable
rules
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
proper(a) |
→ |
ok(a) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
g#(mark(X)) |
→ |
g#(X) |
(21) |
g#(ok(X)) |
→ |
g#(X) |
(24) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[a] |
=
|
11576 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[c(x1)] |
=
|
x1 + 1 |
[f(x1)] |
=
|
x1 + 22313 |
[proper(x1)] |
=
|
x1 + 25192 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[c#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 26172 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 26170 |
together with the usable
rules
f(mark(X)) |
→ |
mark(f(X)) |
(4) |
proper(c(X)) |
→ |
c(proper(X)) |
(8) |
active(f(f(a))) |
→ |
mark(c(f(g(f(a))))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
f(ok(X)) |
→ |
ok(f(X)) |
(10) |
proper(a) |
→ |
ok(a) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(12) |
c(ok(X)) |
→ |
ok(c(X)) |
(11) |
proper(g(X)) |
→ |
g(proper(X)) |
(9) |
proper(f(X)) |
→ |
f(proper(X)) |
(6) |
active(f(X)) |
→ |
f(active(X)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 0
components.