Certification Problem
Input (TPDB TRS_Standard/AG01/#3.12)
The rewrite relation of the following TRS is considered.
|
app(nil,y) |
→ |
y |
(1) |
|
app(add(n,x),y) |
→ |
add(n,app(x,y)) |
(2) |
|
reverse(nil) |
→ |
nil |
(3) |
|
reverse(add(n,x)) |
→ |
app(reverse(x),add(n,nil)) |
(4) |
|
shuffle(nil) |
→ |
nil |
(5) |
|
shuffle(add(n,x)) |
→ |
add(n,shuffle(reverse(x))) |
(6) |
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.
|
shuffle#(add(n,x)) |
→ |
shuffle#(reverse(x)) |
(7) |
|
reverse#(add(n,x)) |
→ |
reverse#(x) |
(8) |
|
shuffle#(add(n,x)) |
→ |
reverse#(x) |
(9) |
|
app#(add(n,x),y) |
→ |
app#(x,y) |
(10) |
|
reverse#(add(n,x)) |
→ |
app#(reverse(x),add(n,nil)) |
(11) |
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
shuffle#(add(n,x)) |
→ |
shuffle#(reverse(x)) |
(7) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [reverse#(x1)] |
=
|
0 |
| [reverse(x1)] |
=
|
x1 + 11798 |
| [shuffle(x1)] |
=
|
0 |
| [nil] |
=
|
0 |
| [app#(x1, x2)] |
=
|
0 |
| [shuffle#(x1)] |
=
|
x1 + 0 |
| [add(x1, x2)] |
=
|
x2 + 11799 |
| [app(x1, x2)] |
=
|
x1 + x2 + 0 |
together with the usable
rules
|
reverse(add(n,x)) |
→ |
app(reverse(x),add(n,nil)) |
(4) |
|
app(nil,y) |
→ |
y |
(1) |
|
reverse(nil) |
→ |
nil |
(3) |
|
app(add(n,x),y) |
→ |
add(n,app(x,y)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
shuffle#(add(n,x)) |
→ |
shuffle#(reverse(x)) |
(7) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
reverse#(add(n,x)) |
→ |
reverse#(x) |
(8) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [reverse#(x1)] |
=
|
x1 + 0 |
| [reverse(x1)] |
=
|
x1 + 1 |
| [shuffle(x1)] |
=
|
0 |
| [nil] |
=
|
0 |
| [app#(x1, x2)] |
=
|
0 |
| [shuffle#(x1)] |
=
|
x1 + 0 |
| [add(x1, x2)] |
=
|
x2 + 11799 |
| [app(x1, x2)] |
=
|
x1 + x2 + 0 |
together with the usable
rules
|
reverse(add(n,x)) |
→ |
app(reverse(x),add(n,nil)) |
(4) |
|
app(nil,y) |
→ |
y |
(1) |
|
reverse(nil) |
→ |
nil |
(3) |
|
app(add(n,x),y) |
→ |
add(n,app(x,y)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
reverse#(add(n,x)) |
→ |
reverse#(x) |
(8) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
app#(add(n,x),y) |
→ |
app#(x,y) |
(10) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [reverse#(x1)] |
=
|
0 |
| [reverse(x1)] |
=
|
x1 + 1 |
| [shuffle(x1)] |
=
|
0 |
| [nil] |
=
|
0 |
| [app#(x1, x2)] |
=
|
x1 + 0 |
| [shuffle#(x1)] |
=
|
x1 + 0 |
| [add(x1, x2)] |
=
|
x2 + 11799 |
| [app(x1, x2)] |
=
|
x1 + x2 + 0 |
together with the usable
rules
|
reverse(add(n,x)) |
→ |
app(reverse(x),add(n,nil)) |
(4) |
|
app(nil,y) |
→ |
y |
(1) |
|
reverse(nil) |
→ |
nil |
(3) |
|
app(add(n,x),y) |
→ |
add(n,app(x,y)) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
app#(add(n,x),y) |
→ |
app#(x,y) |
(10) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.