Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z074)
The rewrite relation of the following TRS is considered.
|
r(r(x1)) |
→ |
s(r(x1)) |
(1) |
|
r(s(x1)) |
→ |
s(r(x1)) |
(2) |
|
r(n(x1)) |
→ |
s(r(x1)) |
(3) |
|
r(b(x1)) |
→ |
u(s(b(x1))) |
(4) |
|
r(u(x1)) |
→ |
u(r(x1)) |
(5) |
|
s(u(x1)) |
→ |
u(s(x1)) |
(6) |
|
n(u(x1)) |
→ |
u(n(x1)) |
(7) |
|
t(r(u(x1))) |
→ |
t(c(r(x1))) |
(8) |
|
t(s(u(x1))) |
→ |
t(c(r(x1))) |
(9) |
|
t(n(u(x1))) |
→ |
t(c(r(x1))) |
(10) |
|
c(u(x1)) |
→ |
u(c(x1)) |
(11) |
|
c(s(x1)) |
→ |
s(c(x1)) |
(12) |
|
c(r(x1)) |
→ |
r(c(x1)) |
(13) |
|
c(n(x1)) |
→ |
n(c(x1)) |
(14) |
|
c(n(x1)) |
→ |
n(x1) |
(15) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [n(x1)] |
= |
x1 +
|
| [c(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [u(x1)] |
= |
x1 +
|
| [t(x1)] |
= |
x1 +
|
| [s(x1)] |
= |
x1 +
|
| [r(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
r(r(x1)) |
→ |
s(r(x1)) |
(1) |
|
r(n(x1)) |
→ |
s(r(x1)) |
(3) |
|
t(r(u(x1))) |
→ |
t(c(r(x1))) |
(8) |
|
t(n(u(x1))) |
→ |
t(c(r(x1))) |
(10) |
1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
s(r(x1)) |
→ |
r(s(x1)) |
(16) |
|
b(r(x1)) |
→ |
b(s(u(x1))) |
(17) |
|
u(r(x1)) |
→ |
r(u(x1)) |
(18) |
|
u(s(x1)) |
→ |
s(u(x1)) |
(19) |
|
u(n(x1)) |
→ |
n(u(x1)) |
(20) |
|
u(s(t(x1))) |
→ |
r(c(t(x1))) |
(21) |
|
u(c(x1)) |
→ |
c(u(x1)) |
(22) |
|
s(c(x1)) |
→ |
c(s(x1)) |
(23) |
|
r(c(x1)) |
→ |
c(r(x1)) |
(24) |
|
n(c(x1)) |
→ |
c(n(x1)) |
(25) |
|
n(c(x1)) |
→ |
n(x1) |
(26) |
1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
n#(c(x1)) |
→ |
n#(x1) |
(27) |
|
b#(r(x1)) |
→ |
b#(s(u(x1))) |
(28) |
|
b#(r(x1)) |
→ |
u#(x1) |
(29) |
|
b#(r(x1)) |
→ |
s#(u(x1)) |
(30) |
|
u#(n(x1)) |
→ |
n#(u(x1)) |
(31) |
|
u#(n(x1)) |
→ |
u#(x1) |
(32) |
|
u#(c(x1)) |
→ |
u#(x1) |
(33) |
|
u#(s(x1)) |
→ |
u#(x1) |
(34) |
|
u#(s(x1)) |
→ |
s#(u(x1)) |
(35) |
|
u#(s(t(x1))) |
→ |
r#(c(t(x1))) |
(36) |
|
u#(r(x1)) |
→ |
u#(x1) |
(37) |
|
u#(r(x1)) |
→ |
r#(u(x1)) |
(38) |
|
s#(c(x1)) |
→ |
s#(x1) |
(39) |
|
s#(r(x1)) |
→ |
s#(x1) |
(40) |
|
s#(r(x1)) |
→ |
r#(s(x1)) |
(41) |
|
r#(c(x1)) |
→ |
r#(x1) |
(42) |
1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [n(x1)] |
= |
x1 +
|
| [c(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [u(x1)] |
= |
x1 +
|
| [t(x1)] |
= |
x1 +
|
| [s(x1)] |
= |
x1 +
|
| [r(x1)] |
= |
x1 +
|
| [n#(x1)] |
= |
x1 +
|
| [b#(x1)] |
= |
x1 +
|
| [u#(x1)] |
= |
x1 +
|
| [s#(x1)] |
= |
x1 +
|
| [r#(x1)] |
= |
x1 +
|
together with the usable
rules
|
s(r(x1)) |
→ |
r(s(x1)) |
(16) |
|
b(r(x1)) |
→ |
b(s(u(x1))) |
(17) |
|
u(r(x1)) |
→ |
r(u(x1)) |
(18) |
|
u(s(x1)) |
→ |
s(u(x1)) |
(19) |
|
u(n(x1)) |
→ |
n(u(x1)) |
(20) |
|
u(s(t(x1))) |
→ |
r(c(t(x1))) |
(21) |
|
u(c(x1)) |
→ |
c(u(x1)) |
(22) |
|
s(c(x1)) |
→ |
c(s(x1)) |
(23) |
|
r(c(x1)) |
→ |
c(r(x1)) |
(24) |
|
n(c(x1)) |
→ |
c(n(x1)) |
(25) |
|
n(c(x1)) |
→ |
n(x1) |
(26) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
b#(r(x1)) |
→ |
u#(x1) |
(29) |
|
b#(r(x1)) |
→ |
s#(u(x1)) |
(30) |
|
u#(n(x1)) |
→ |
n#(u(x1)) |
(31) |
|
u#(n(x1)) |
→ |
u#(x1) |
(32) |
|
u#(s(x1)) |
→ |
u#(x1) |
(34) |
|
u#(s(x1)) |
→ |
s#(u(x1)) |
(35) |
|
u#(s(t(x1))) |
→ |
r#(c(t(x1))) |
(36) |
|
u#(r(x1)) |
→ |
u#(x1) |
(37) |
|
u#(r(x1)) |
→ |
r#(u(x1)) |
(38) |
|
s#(r(x1)) |
→ |
s#(x1) |
(40) |
|
s#(r(x1)) |
→ |
r#(s(x1)) |
(41) |
and
no rules
could be deleted.
1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 5
components.
-
The
1st
component contains the
pair
1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c(x1)] |
= |
x1 +
|
| [n#(x1)] |
= |
x1 +
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
and
no rules
could be deleted.
1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
b#(r(x1)) |
→ |
b#(s(u(x1))) |
(28) |
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals
| [n(x1)] |
= |
· x1 +
|
| [c(x1)] |
= |
· x1 +
|
| [u(x1)] |
= |
· x1 +
|
| [t(x1)] |
= |
· x1 +
|
| [s(x1)] |
= |
· x1 +
|
| [r(x1)] |
= |
· x1 +
|
| [b#(x1)] |
= |
· x1 +
|
together with the usable
rules
|
s(r(x1)) |
→ |
r(s(x1)) |
(16) |
|
u(r(x1)) |
→ |
r(u(x1)) |
(18) |
|
u(s(x1)) |
→ |
s(u(x1)) |
(19) |
|
u(n(x1)) |
→ |
n(u(x1)) |
(20) |
|
u(s(t(x1))) |
→ |
r(c(t(x1))) |
(21) |
|
u(c(x1)) |
→ |
c(u(x1)) |
(22) |
|
s(c(x1)) |
→ |
c(s(x1)) |
(23) |
|
r(c(x1)) |
→ |
c(r(x1)) |
(24) |
|
n(c(x1)) |
→ |
c(n(x1)) |
(25) |
|
n(c(x1)) |
→ |
n(x1) |
(26) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
3rd
component contains the
pair
1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c(x1)] |
= |
x1 +
|
| [u#(x1)] |
= |
x1 +
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
and
no rules
could be deleted.
1.1.1.1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
1.1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c(x1)] |
= |
x1 +
|
| [s#(x1)] |
= |
x1 +
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
and
no rules
could be deleted.
1.1.1.1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
5th
component contains the
pair
1.1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [c(x1)] |
= |
x1 +
|
| [r#(x1)] |
= |
x1 +
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
and
no rules
could be deleted.
1.1.1.1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.