Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/Ex4_4_Luc96b_C)
The rewrite relation of the following TRS is considered.
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
active(f(X1,X2)) |
→ |
f(active(X1),X2) |
(2) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
f(mark(X1),X2) |
→ |
mark(f(X1,X2)) |
(4) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
proper(f(X1,X2)) |
→ |
f(proper(X1),proper(X2)) |
(6) |
proper(g(X)) |
→ |
g(proper(X)) |
(7) |
f(ok(X1),ok(X2)) |
→ |
ok(f(X1,X2)) |
(8) |
g(ok(X)) |
→ |
ok(g(X)) |
(9) |
top(mark(X)) |
→ |
top(proper(X)) |
(10) |
top(ok(X)) |
→ |
top(active(X)) |
(11) |
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(X1,X2)) |
→ |
active#(X1) |
(12) |
g#(ok(X)) |
→ |
g#(X) |
(13) |
g#(mark(X)) |
→ |
g#(X) |
(14) |
proper#(g(X)) |
→ |
g#(proper(X)) |
(15) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(16) |
top#(mark(X)) |
→ |
proper#(X) |
(17) |
active#(g(X)) |
→ |
g#(active(X)) |
(18) |
top#(ok(X)) |
→ |
top#(active(X)) |
(19) |
proper#(g(X)) |
→ |
proper#(X) |
(20) |
proper#(f(X1,X2)) |
→ |
proper#(X2) |
(21) |
f#(ok(X1),ok(X2)) |
→ |
f#(X1,X2) |
(22) |
active#(f(g(X),Y)) |
→ |
f#(X,f(g(X),Y)) |
(23) |
proper#(f(X1,X2)) |
→ |
f#(proper(X1),proper(X2)) |
(24) |
active#(f(X1,X2)) |
→ |
f#(active(X1),X2) |
(25) |
top#(ok(X)) |
→ |
active#(X) |
(26) |
active#(g(X)) |
→ |
active#(X) |
(27) |
proper#(f(X1,X2)) |
→ |
proper#(X1) |
(28) |
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(29) |
1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(16) |
top#(ok(X)) |
→ |
top#(active(X)) |
(19) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[f(x1, x2)] |
=
|
x1 + 0 |
[proper(x1)] |
=
|
x1 + 1 |
[ok(x1)] |
=
|
x1 + 2 |
[mark(x1)] |
=
|
x1 + 2 |
[f#(x1, x2)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 1 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
f(mark(X1),X2) |
→ |
mark(f(X1,X2)) |
(4) |
f(ok(X1),ok(X2)) |
→ |
ok(f(X1,X2)) |
(8) |
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(9) |
proper(f(X1,X2)) |
→ |
f(proper(X1),proper(X2)) |
(6) |
active(f(X1,X2)) |
→ |
f(active(X1),X2) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
top#(mark(X)) |
→ |
top#(proper(X)) |
(16) |
top#(ok(X)) |
→ |
top#(active(X)) |
(19) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
proper#(f(X1,X2)) |
→ |
proper#(X1) |
(28) |
proper#(f(X1,X2)) |
→ |
proper#(X2) |
(21) |
proper#(g(X)) |
→ |
proper#(X) |
(20) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[f(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
6295 |
[f#(x1, x2)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
x1 + 0 |
[active(x1)] |
=
|
x1 + 46572 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
f(mark(X1),X2) |
→ |
mark(f(X1,X2)) |
(4) |
f(ok(X1),ok(X2)) |
→ |
ok(f(X1,X2)) |
(8) |
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(9) |
proper(f(X1,X2)) |
→ |
f(proper(X1),proper(X2)) |
(6) |
active(f(X1,X2)) |
→ |
f(active(X1),X2) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
proper#(g(X)) |
→ |
proper#(X) |
(20) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
active#(g(X)) |
→ |
active#(X) |
(27) |
active#(f(X1,X2)) |
→ |
active#(X1) |
(12) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
x1 + 0 |
[f(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
x1 + 0 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
16908 |
[f#(x1, x2)] |
=
|
0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
x1 + 39825 |
[active#(x1)] |
=
|
x1 + 0 |
[g(x1)] |
=
|
x1 + 1 |
together with the usable
rules
f(mark(X1),X2) |
→ |
mark(f(X1,X2)) |
(4) |
f(ok(X1),ok(X2)) |
→ |
ok(f(X1,X2)) |
(8) |
active(f(g(X),Y)) |
→ |
mark(f(X,f(g(X),Y))) |
(1) |
active(g(X)) |
→ |
g(active(X)) |
(3) |
g(mark(X)) |
→ |
mark(g(X)) |
(5) |
proper(g(X)) |
→ |
g(proper(X)) |
(7) |
g(ok(X)) |
→ |
ok(g(X)) |
(9) |
proper(f(X1,X2)) |
→ |
f(proper(X1),proper(X2)) |
(6) |
active(f(X1,X2)) |
→ |
f(active(X1),X2) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(g(X)) |
→ |
active#(X) |
(27) |
active#(f(X1,X2)) |
→ |
active#(X1) |
(12) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
g#(mark(X)) |
→ |
g#(X) |
(14) |
g#(ok(X)) |
→ |
g#(X) |
(13) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[f(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 16908 |
[f#(x1, x2)] |
=
|
0 |
[g#(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
981 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
g#(mark(X)) |
→ |
g#(X) |
(14) |
g#(ok(X)) |
→ |
g#(X) |
(13) |
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#(mark(X1),X2) |
→ |
f#(X1,X2) |
(29) |
f#(ok(X1),ok(X2)) |
→ |
f#(X1,X2) |
(22) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[top(x1)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[f(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
x1 + 1 |
[mark(x1)] |
=
|
x1 + 16908 |
[f#(x1, x2)] |
=
|
x1 + x2 + 0 |
[g#(x1)] |
=
|
0 |
[proper#(x1)] |
=
|
0 |
[active(x1)] |
=
|
12213 |
[active#(x1)] |
=
|
0 |
[g(x1)] |
=
|
x1 + 0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
f#(mark(X1),X2) |
→ |
f#(X1,X2) |
(29) |
f#(ok(X1),ok(X2)) |
→ |
f#(X1,X2) |
(22) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.