Certification Problem
Input (TPDB SRS_Standard/ICFP_2010/214320)
The rewrite relation of the following TRS is considered.
|
0(1(0(x1))) |
→ |
0(2(1(0(x1)))) |
(1) |
|
0(1(0(x1))) |
→ |
0(0(2(1(0(x1))))) |
(2) |
|
0(1(0(x1))) |
→ |
0(0(2(1(2(x1))))) |
(3) |
|
0(1(0(x1))) |
→ |
0(2(1(0(2(x1))))) |
(4) |
|
0(1(0(x1))) |
→ |
0(3(2(1(0(x1))))) |
(5) |
|
0(1(0(x1))) |
→ |
1(0(0(0(2(x1))))) |
(6) |
|
0(1(0(x1))) |
→ |
1(0(0(2(0(x1))))) |
(7) |
|
0(1(0(x1))) |
→ |
1(0(4(2(0(x1))))) |
(8) |
|
0(1(0(x1))) |
→ |
1(4(0(4(0(x1))))) |
(9) |
|
0(1(0(x1))) |
→ |
4(0(0(2(1(x1))))) |
(10) |
|
0(1(0(x1))) |
→ |
5(0(0(4(1(x1))))) |
(11) |
|
0(1(0(x1))) |
→ |
5(1(0(4(0(x1))))) |
(12) |
|
0(1(0(x1))) |
→ |
0(2(1(0(3(2(x1)))))) |
(13) |
|
0(1(0(x1))) |
→ |
0(4(0(4(1(3(x1)))))) |
(14) |
|
0(1(0(x1))) |
→ |
0(4(2(2(1(0(x1)))))) |
(15) |
|
0(1(0(x1))) |
→ |
0(5(2(1(2(0(x1)))))) |
(16) |
|
0(1(0(x1))) |
→ |
0(5(2(5(1(0(x1)))))) |
(17) |
|
0(1(0(x1))) |
→ |
1(0(0(5(4(4(x1)))))) |
(18) |
|
0(1(0(x1))) |
→ |
1(0(4(4(4(0(x1)))))) |
(19) |
|
0(1(0(x1))) |
→ |
1(5(0(0(4(2(x1)))))) |
(20) |
|
0(1(0(x1))) |
→ |
3(0(0(4(1(4(x1)))))) |
(21) |
|
0(1(0(x1))) |
→ |
4(5(1(0(2(0(x1)))))) |
(22) |
|
0(1(0(x1))) |
→ |
5(5(1(0(0(2(x1)))))) |
(23) |
|
0(0(1(0(x1)))) |
→ |
1(0(0(2(0(x1))))) |
(24) |
|
0(0(1(0(x1)))) |
→ |
0(1(5(0(0(2(x1)))))) |
(25) |
|
0(1(0(3(x1)))) |
→ |
1(0(3(3(0(2(x1)))))) |
(26) |
|
0(1(0(3(x1)))) |
→ |
1(0(5(3(2(0(x1)))))) |
(27) |
|
0(1(1(0(x1)))) |
→ |
0(4(4(1(1(0(x1)))))) |
(28) |
|
0(1(1(3(x1)))) |
→ |
3(4(5(1(1(0(x1)))))) |
(29) |
|
0(1(2(0(x1)))) |
→ |
1(1(0(2(0(x1))))) |
(30) |
|
0(1(2(0(x1)))) |
→ |
3(0(2(1(0(x1))))) |
(31) |
|
0(1(2(0(x1)))) |
→ |
4(1(0(0(2(x1))))) |
(32) |
|
0(1(2(0(x1)))) |
→ |
0(0(4(2(5(1(x1)))))) |
(33) |
|
0(1(2(0(x1)))) |
→ |
1(1(2(0(4(0(x1)))))) |
(34) |
|
0(1(2(0(x1)))) |
→ |
3(0(2(1(0(4(x1)))))) |
(35) |
|
0(1(3(0(x1)))) |
→ |
1(0(3(0(2(x1))))) |
(36) |
|
0(1(4(0(x1)))) |
→ |
0(3(4(2(1(0(x1)))))) |
(37) |
|
0(1(5(0(x1)))) |
→ |
0(5(1(4(0(x1))))) |
(38) |
|
0(1(5(0(x1)))) |
→ |
1(5(3(0(2(0(x1)))))) |
(39) |
|
0(3(1(0(x1)))) |
→ |
1(2(3(0(5(0(x1)))))) |
(40) |
|
5(0(1(0(x1)))) |
→ |
1(4(0(0(5(1(x1)))))) |
(41) |
|
5(0(1(0(x1)))) |
→ |
2(1(0(0(4(5(x1)))))) |
(42) |
|
0(1(0(0(0(x1))))) |
→ |
0(0(5(1(0(0(x1)))))) |
(43) |
|
0(1(2(4(0(x1))))) |
→ |
0(0(5(4(2(1(x1)))))) |
(44) |
|
0(1(2(5(0(x1))))) |
→ |
1(0(2(0(5(4(x1)))))) |
(45) |
|
0(1(4(0(0(x1))))) |
→ |
0(0(0(4(1(0(x1)))))) |
(46) |
|
0(1(4(5(0(x1))))) |
→ |
1(5(0(0(4(2(x1)))))) |
(47) |
|
0(3(0(1(0(x1))))) |
→ |
0(3(0(0(2(1(x1)))))) |
(48) |
|
3(0(3(1(0(x1))))) |
→ |
0(1(3(2(3(0(x1)))))) |
(49) |
|
5(0(1(2(0(x1))))) |
→ |
0(0(5(2(1(0(x1)))))) |
(50) |
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.
There are 125 ruless (increase limit for explicit display).
1.1 Dependency Graph Processor
The dependency pairs are split into 3
components.
-
The
1st
component contains the
pair
|
0#(1(1(3(x1)))) |
→ |
0#(x1) |
(161) |
|
0#(1(0(3(x1)))) |
→ |
0#(x1) |
(96) |
|
0#(3(1(0(x1)))) |
→ |
0#(5(0(x1))) |
(153) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
| [0#(x1)] |
= |
· x1 +
|
| [1(x1)] |
= |
· x1 +
|
| [4(x1)] |
= |
|
| [5(x1)] |
= |
· x1 +
|
| [3(x1)] |
= |
· x1 +
|
| [0(x1)] |
= |
· x1 +
|
| [3#(x1)] |
= |
|
| [5#(x1)] |
= |
|
| [2(x1)] |
= |
|
together with the usable
rules
|
0(1(0(x1))) |
→ |
1(0(0(5(4(4(x1)))))) |
(18) |
|
5(0(1(2(0(x1))))) |
→ |
0(0(5(2(1(0(x1)))))) |
(50) |
|
0(1(0(x1))) |
→ |
0(2(1(0(2(x1))))) |
(4) |
|
0(1(0(x1))) |
→ |
0(4(2(2(1(0(x1)))))) |
(15) |
|
0(1(0(x1))) |
→ |
1(0(4(2(0(x1))))) |
(8) |
|
0(1(0(x1))) |
→ |
0(2(1(0(x1)))) |
(1) |
|
0(1(0(x1))) |
→ |
0(0(2(1(2(x1))))) |
(3) |
|
0(1(0(x1))) |
→ |
0(5(2(1(2(0(x1)))))) |
(16) |
|
0(1(0(x1))) |
→ |
3(0(0(4(1(4(x1)))))) |
(21) |
|
0(1(3(0(x1)))) |
→ |
1(0(3(0(2(x1))))) |
(36) |
|
0(1(0(3(x1)))) |
→ |
1(0(3(3(0(2(x1)))))) |
(26) |
|
0(1(0(x1))) |
→ |
1(0(4(4(4(0(x1)))))) |
(19) |
|
0(1(2(0(x1)))) |
→ |
4(1(0(0(2(x1))))) |
(32) |
|
0(1(0(x1))) |
→ |
0(5(2(5(1(0(x1)))))) |
(17) |
|
0(1(0(3(x1)))) |
→ |
1(0(5(3(2(0(x1)))))) |
(27) |
|
0(1(2(0(x1)))) |
→ |
1(1(2(0(4(0(x1)))))) |
(34) |
|
0(1(0(x1))) |
→ |
4(5(1(0(2(0(x1)))))) |
(22) |
|
0(1(1(0(x1)))) |
→ |
0(4(4(1(1(0(x1)))))) |
(28) |
|
0(1(2(4(0(x1))))) |
→ |
0(0(5(4(2(1(x1)))))) |
(44) |
|
0(1(0(x1))) |
→ |
0(3(2(1(0(x1))))) |
(5) |
|
0(1(2(0(x1)))) |
→ |
0(0(4(2(5(1(x1)))))) |
(33) |
|
0(1(0(x1))) |
→ |
4(0(0(2(1(x1))))) |
(10) |
|
0(1(5(0(x1)))) |
→ |
1(5(3(0(2(0(x1)))))) |
(39) |
|
0(1(0(x1))) |
→ |
1(0(0(2(0(x1))))) |
(7) |
|
0(1(0(x1))) |
→ |
1(5(0(0(4(2(x1)))))) |
(20) |
|
0(0(1(0(x1)))) |
→ |
0(1(5(0(0(2(x1)))))) |
(25) |
|
3(0(3(1(0(x1))))) |
→ |
0(1(3(2(3(0(x1)))))) |
(49) |
|
0(1(2(0(x1)))) |
→ |
1(1(0(2(0(x1))))) |
(30) |
|
0(1(0(x1))) |
→ |
0(4(0(4(1(3(x1)))))) |
(14) |
|
0(1(2(0(x1)))) |
→ |
3(0(2(1(0(x1))))) |
(31) |
|
0(1(0(x1))) |
→ |
5(1(0(4(0(x1))))) |
(12) |
|
0(1(2(5(0(x1))))) |
→ |
1(0(2(0(5(4(x1)))))) |
(45) |
|
0(1(0(x1))) |
→ |
5(5(1(0(0(2(x1)))))) |
(23) |
|
0(0(1(0(x1)))) |
→ |
1(0(0(2(0(x1))))) |
(24) |
|
0(1(0(x1))) |
→ |
5(0(0(4(1(x1))))) |
(11) |
|
0(1(0(x1))) |
→ |
1(4(0(4(0(x1))))) |
(9) |
|
0(1(0(x1))) |
→ |
0(2(1(0(3(2(x1)))))) |
(13) |
|
0(3(1(0(x1)))) |
→ |
1(2(3(0(5(0(x1)))))) |
(40) |
|
0(1(0(x1))) |
→ |
1(0(0(0(2(x1))))) |
(6) |
|
0(1(5(0(x1)))) |
→ |
0(5(1(4(0(x1))))) |
(38) |
|
0(3(0(1(0(x1))))) |
→ |
0(3(0(0(2(1(x1)))))) |
(48) |
|
0(1(4(5(0(x1))))) |
→ |
1(5(0(0(4(2(x1)))))) |
(47) |
|
0(1(4(0(x1)))) |
→ |
0(3(4(2(1(0(x1)))))) |
(37) |
|
5(0(1(0(x1)))) |
→ |
1(4(0(0(5(1(x1)))))) |
(41) |
|
5(0(1(0(x1)))) |
→ |
2(1(0(0(4(5(x1)))))) |
(42) |
|
0(1(4(0(0(x1))))) |
→ |
0(0(0(4(1(0(x1)))))) |
(46) |
|
0(1(2(0(x1)))) |
→ |
3(0(2(1(0(4(x1)))))) |
(35) |
|
0(1(1(3(x1)))) |
→ |
3(4(5(1(1(0(x1)))))) |
(29) |
|
0(1(0(0(0(x1))))) |
→ |
0(0(5(1(0(0(x1)))))) |
(43) |
|
0(1(0(x1))) |
→ |
0(0(2(1(0(x1))))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
0#(1(1(3(x1)))) |
→ |
0#(x1) |
(161) |
|
0#(1(0(3(x1)))) |
→ |
0#(x1) |
(96) |
|
0#(3(1(0(x1)))) |
→ |
0#(5(0(x1))) |
(153) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
2nd
component contains the
pair
|
5#(0(1(0(x1)))) |
→ |
5#(x1) |
(142) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
| [0#(x1)] |
=
|
0 |
| [1(x1)] |
=
|
x1 + 1 |
| [4(x1)] |
=
|
x1 + 2 |
| [5(x1)] |
=
|
x1 + 2 |
| [3(x1)] |
=
|
28876 |
| [0(x1)] |
=
|
x1 + 1 |
| [3#(x1)] |
=
|
0 |
| [5#(x1)] |
=
|
x1 + 0 |
| [2(x1)] |
=
|
x1 + 1 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
|
5#(0(1(0(x1)))) |
→ |
5#(x1) |
(142) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
|
3#(0(3(1(0(x1))))) |
→ |
3#(0(x1)) |
(129) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 2 with strict dimension 1 over the naturals
| [0#(x1)] |
= |
· x1 +
|
| [1(x1)] |
= |
· x1 +
|
| [4(x1)] |
= |
|
| [5(x1)] |
= |
· x1 +
|
| [3(x1)] |
= |
x1 +
|
| [0(x1)] |
= |
· x1 +
|
| [3#(x1)] |
= |
· x1 +
|
| [5#(x1)] |
= |
|
| [2(x1)] |
= |
|
together with the usable
rules
|
0(1(0(x1))) |
→ |
1(0(0(5(4(4(x1)))))) |
(18) |
|
5(0(1(2(0(x1))))) |
→ |
0(0(5(2(1(0(x1)))))) |
(50) |
|
0(1(0(x1))) |
→ |
0(2(1(0(2(x1))))) |
(4) |
|
0(1(0(x1))) |
→ |
0(4(2(2(1(0(x1)))))) |
(15) |
|
0(1(0(x1))) |
→ |
1(0(4(2(0(x1))))) |
(8) |
|
0(1(0(x1))) |
→ |
0(2(1(0(x1)))) |
(1) |
|
0(1(0(x1))) |
→ |
0(0(2(1(2(x1))))) |
(3) |
|
0(1(0(x1))) |
→ |
0(5(2(1(2(0(x1)))))) |
(16) |
|
0(1(0(x1))) |
→ |
3(0(0(4(1(4(x1)))))) |
(21) |
|
0(1(3(0(x1)))) |
→ |
1(0(3(0(2(x1))))) |
(36) |
|
0(1(0(3(x1)))) |
→ |
1(0(3(3(0(2(x1)))))) |
(26) |
|
0(1(0(x1))) |
→ |
1(0(4(4(4(0(x1)))))) |
(19) |
|
0(1(2(0(x1)))) |
→ |
4(1(0(0(2(x1))))) |
(32) |
|
0(1(0(x1))) |
→ |
0(5(2(5(1(0(x1)))))) |
(17) |
|
0(1(0(3(x1)))) |
→ |
1(0(5(3(2(0(x1)))))) |
(27) |
|
0(1(2(0(x1)))) |
→ |
1(1(2(0(4(0(x1)))))) |
(34) |
|
0(1(0(x1))) |
→ |
4(5(1(0(2(0(x1)))))) |
(22) |
|
0(1(1(0(x1)))) |
→ |
0(4(4(1(1(0(x1)))))) |
(28) |
|
0(1(2(4(0(x1))))) |
→ |
0(0(5(4(2(1(x1)))))) |
(44) |
|
0(1(0(x1))) |
→ |
0(3(2(1(0(x1))))) |
(5) |
|
0(1(2(0(x1)))) |
→ |
0(0(4(2(5(1(x1)))))) |
(33) |
|
0(1(0(x1))) |
→ |
4(0(0(2(1(x1))))) |
(10) |
|
0(1(5(0(x1)))) |
→ |
1(5(3(0(2(0(x1)))))) |
(39) |
|
0(1(0(x1))) |
→ |
1(0(0(2(0(x1))))) |
(7) |
|
0(1(0(x1))) |
→ |
1(5(0(0(4(2(x1)))))) |
(20) |
|
0(0(1(0(x1)))) |
→ |
0(1(5(0(0(2(x1)))))) |
(25) |
|
3(0(3(1(0(x1))))) |
→ |
0(1(3(2(3(0(x1)))))) |
(49) |
|
0(1(2(0(x1)))) |
→ |
1(1(0(2(0(x1))))) |
(30) |
|
0(1(0(x1))) |
→ |
0(4(0(4(1(3(x1)))))) |
(14) |
|
0(1(2(0(x1)))) |
→ |
3(0(2(1(0(x1))))) |
(31) |
|
0(1(0(x1))) |
→ |
5(1(0(4(0(x1))))) |
(12) |
|
0(1(2(5(0(x1))))) |
→ |
1(0(2(0(5(4(x1)))))) |
(45) |
|
0(1(0(x1))) |
→ |
5(5(1(0(0(2(x1)))))) |
(23) |
|
0(0(1(0(x1)))) |
→ |
1(0(0(2(0(x1))))) |
(24) |
|
0(1(0(x1))) |
→ |
5(0(0(4(1(x1))))) |
(11) |
|
0(1(0(x1))) |
→ |
1(4(0(4(0(x1))))) |
(9) |
|
0(1(0(x1))) |
→ |
0(2(1(0(3(2(x1)))))) |
(13) |
|
0(3(1(0(x1)))) |
→ |
1(2(3(0(5(0(x1)))))) |
(40) |
|
0(1(0(x1))) |
→ |
1(0(0(0(2(x1))))) |
(6) |
|
0(1(5(0(x1)))) |
→ |
0(5(1(4(0(x1))))) |
(38) |
|
0(3(0(1(0(x1))))) |
→ |
0(3(0(0(2(1(x1)))))) |
(48) |
|
0(1(4(5(0(x1))))) |
→ |
1(5(0(0(4(2(x1)))))) |
(47) |
|
0(1(4(0(x1)))) |
→ |
0(3(4(2(1(0(x1)))))) |
(37) |
|
5(0(1(0(x1)))) |
→ |
1(4(0(0(5(1(x1)))))) |
(41) |
|
5(0(1(0(x1)))) |
→ |
2(1(0(0(4(5(x1)))))) |
(42) |
|
0(1(4(0(0(x1))))) |
→ |
0(0(0(4(1(0(x1)))))) |
(46) |
|
0(1(2(0(x1)))) |
→ |
3(0(2(1(0(4(x1)))))) |
(35) |
|
0(1(1(3(x1)))) |
→ |
3(4(5(1(1(0(x1)))))) |
(29) |
|
0(1(0(0(0(x1))))) |
→ |
0(0(5(1(0(0(x1)))))) |
(43) |
|
0(1(0(x1))) |
→ |
0(0(2(1(0(x1))))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
|
3#(0(3(1(0(x1))))) |
→ |
3#(0(x1)) |
(129) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.