Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex24_Luc06_C)
The rewrite relation of the following TRS is considered.
active(f(b,X,c)) |
→ |
mark(f(X,c,X)) |
(1) |
active(c) |
→ |
mark(b) |
(2) |
active(f(X1,X2,X3)) |
→ |
f(X1,active(X2),X3) |
(3) |
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
proper(f(X1,X2,X3)) |
→ |
f(proper(X1),proper(X2),proper(X3)) |
(5) |
proper(b) |
→ |
ok(b) |
(6) |
proper(c) |
→ |
ok(c) |
(7) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
top(mark(X)) |
→ |
top(proper(X)) |
(9) |
top(ok(X)) |
→ |
top(active(X)) |
(10) |
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.
top#(mark(X)) |
→ |
proper#(X) |
(11) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X1) |
(12) |
f#(X1,mark(X2),X3) |
→ |
f#(X1,X2,X3) |
(13) |
active#(f(X1,X2,X3)) |
→ |
f#(X1,active(X2),X3) |
(14) |
active#(f(b,X,c)) |
→ |
f#(X,c,X) |
(15) |
f#(ok(X1),ok(X2),ok(X3)) |
→ |
f#(X1,X2,X3) |
(16) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X2) |
(17) |
active#(f(X1,X2,X3)) |
→ |
active#(X2) |
(18) |
proper#(f(X1,X2,X3)) |
→ |
f#(proper(X1),proper(X2),proper(X3)) |
(19) |
top#(ok(X)) |
→ |
top#(active(X)) |
(20) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X3) |
(22) |
top#(ok(X)) |
→ |
active#(X) |
(23) |
1.1 Dependency Graph Processor
The dependency pairs are split into 4
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
top#(ok(X)) |
→ |
top#(active(X)) |
(20) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
[b] |
= |
|
[top(x1)] |
= |
|
[top#(x1)] |
= |
· x1 +
|
[c] |
= |
|
[f(x1, x2, x3)] |
= |
· x1 + · x2 + · x3 +
|
[proper(x1)] |
= |
x1 +
|
[ok(x1)] |
= |
x1 +
|
[mark(x1)] |
= |
x1 +
|
[f#(x1, x2, x3)] |
= |
|
[proper#(x1)] |
= |
|
[active(x1)] |
= |
· x1 +
|
[active#(x1)] |
= |
|
together with the usable
rules
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
active(f(b,X,c)) |
→ |
mark(f(X,c,X)) |
(1) |
active(f(X1,X2,X3)) |
→ |
f(X1,active(X2),X3) |
(3) |
proper(f(X1,X2,X3)) |
→ |
f(proper(X1),proper(X2),proper(X3)) |
(5) |
proper(c) |
→ |
ok(c) |
(7) |
proper(b) |
→ |
ok(b) |
(6) |
active(c) |
→ |
mark(b) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(21) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
active#(f(X1,X2,X3)) |
→ |
active#(X2) |
(18) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[b] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
1 |
[f(x1, x2, x3)] |
=
|
x2 + 1 |
[proper(x1)] |
=
|
50763 |
[ok(x1)] |
=
|
50763 |
[mark(x1)] |
=
|
20585 |
[f#(x1, x2, x3)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
33855 |
[active#(x1)] |
=
|
x1 + 0 |
together with the usable
rules
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
active(f(b,X,c)) |
→ |
mark(f(X,c,X)) |
(1) |
proper(c) |
→ |
ok(c) |
(7) |
proper(b) |
→ |
ok(b) |
(6) |
active(c) |
→ |
mark(b) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(f(X1,X2,X3)) |
→ |
active#(X2) |
(18) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
proper#(f(X1,X2,X3)) |
→ |
proper#(X3) |
(22) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X1) |
(12) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X2) |
(17) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[b] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
43391 |
[f(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
36983 |
[mark(x1)] |
=
|
7630 |
[f#(x1, x2, x3)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
33855 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
active(f(b,X,c)) |
→ |
mark(f(X,c,X)) |
(1) |
active(c) |
→ |
mark(b) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(f(X1,X2,X3)) |
→ |
proper#(X3) |
(22) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X1) |
(12) |
proper#(f(X1,X2,X3)) |
→ |
proper#(X2) |
(17) |
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(X1),ok(X2),ok(X3)) |
→ |
f#(X1,X2,X3) |
(16) |
f#(X1,mark(X2),X3) |
→ |
f#(X1,X2,X3) |
(13) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[b] |
=
|
1 |
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[c] |
=
|
14661 |
[f(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 7177 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 36983 |
[mark(x1)] |
=
|
1 |
[f#(x1, x2, x3)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[active#(x1)] |
=
|
0 |
together with the usable
rules
f(X1,mark(X2),X3) |
→ |
mark(f(X1,X2,X3)) |
(4) |
f(ok(X1),ok(X2),ok(X3)) |
→ |
ok(f(X1,X2,X3)) |
(8) |
active(f(b,X,c)) |
→ |
mark(f(X,c,X)) |
(1) |
active(c) |
→ |
mark(b) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
f#(ok(X1),ok(X2),ok(X3)) |
→ |
f#(X1,X2,X3) |
(16) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.