Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_4_Luc96b_iGM)
The rewrite relation of the following TRS is considered.
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
mark(f(X1,X2)) |
→ |
active(f(mark(X1),X2)) |
(2) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(3) |
f(mark(X1),X2) |
→ |
f(X1,X2) |
(4) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(5) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(6) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(7) |
g(mark(X)) |
→ |
g(X) |
(8) |
g(active(X)) |
→ |
g(X) |
(9) |
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.
mark#(f(X1,X2)) |
→ |
f#(mark(X1),X2) |
(10) |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(11) |
g#(mark(X)) |
→ |
g#(X) |
(12) |
mark#(g(X)) |
→ |
g#(mark(X)) |
(13) |
active#(f(g(X),Y)) |
→ |
mark#(f(X,f(g(X),Y))) |
(14) |
active#(f(g(X),Y)) |
→ |
f#(X,f(g(X),Y)) |
(15) |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(16) |
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(17) |
mark#(g(X)) |
→ |
mark#(X) |
(18) |
g#(active(X)) |
→ |
g#(X) |
(19) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(20) |
mark#(f(X1,X2)) |
→ |
active#(f(mark(X1),X2)) |
(21) |
mark#(g(X)) |
→ |
active#(g(mark(X))) |
(22) |
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(23) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
mark#(g(X)) |
→ |
active#(g(mark(X))) |
(22) |
mark#(f(X1,X2)) |
→ |
active#(f(mark(X1),X2)) |
(21) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(20) |
active#(f(g(X),Y)) |
→ |
mark#(f(X,f(g(X),Y))) |
(14) |
mark#(g(X)) |
→ |
mark#(X) |
(18) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[f(x1, x2)] |
=
|
x1 + 2438 |
[mark#(x1)] |
=
|
x1 + 30095 |
[mark(x1)] |
=
|
x1 + 30094 |
[f#(x1, x2)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 30096 |
together with the usable
rules
f(mark(X1),X2) |
→ |
f(X1,X2) |
(4) |
g(mark(X)) |
→ |
g(X) |
(8) |
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(3) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(5) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(7) |
g(active(X)) |
→ |
g(X) |
(9) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(6) |
mark(f(X1,X2)) |
→ |
active(f(mark(X1),X2)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(g(X)) |
→ |
active#(g(mark(X))) |
(22) |
mark#(f(X1,X2)) |
→ |
active#(f(mark(X1),X2)) |
(21) |
mark#(f(X1,X2)) |
→ |
mark#(X1) |
(20) |
active#(f(g(X),Y)) |
→ |
mark#(f(X,f(g(X),Y))) |
(14) |
mark#(g(X)) |
→ |
mark#(X) |
(18) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
g#(mark(X)) |
→ |
g#(X) |
(12) |
g#(active(X)) |
→ |
g#(X) |
(19) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[f(x1, x2)] |
=
|
x1 + 8366 |
[mark#(x1)] |
=
|
x1 + 30095 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1, x2)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 0 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
f(mark(X1),X2) |
→ |
f(X1,X2) |
(4) |
g(mark(X)) |
→ |
g(X) |
(8) |
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
mark(g(X)) |
→ |
active(g(mark(X))) |
(3) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(5) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(7) |
g(active(X)) |
→ |
g(X) |
(9) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(6) |
mark(f(X1,X2)) |
→ |
active(f(mark(X1),X2)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
f#(active(X1),X2) |
→ |
f#(X1,X2) |
(23) |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(16) |
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(11) |
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(17) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[f(x1, x2)] |
=
|
x1 + 1324 |
[mark#(x1)] |
=
|
x1 + 30095 |
[mark(x1)] |
=
|
x1 + 1 |
[f#(x1, x2)] |
=
|
x2 + 0 |
[g#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
47339 |
together with the usable
rules
f(mark(X1),X2) |
→ |
f(X1,X2) |
(4) |
g(mark(X)) |
→ |
g(X) |
(8) |
f(X1,mark(X2)) |
→ |
f(X1,X2) |
(5) |
f(X1,active(X2)) |
→ |
f(X1,X2) |
(7) |
g(active(X)) |
→ |
g(X) |
(9) |
f(active(X1),X2) |
→ |
f(X1,X2) |
(6) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
f#(X1,active(X2)) |
→ |
f#(X1,X2) |
(11) |
f#(X1,mark(X2)) |
→ |
f#(X1,X2) |
(17) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 1
component.