Term Rewriting System R:
[x, y, z]
+(0, 0) -> 0
+(0, 1) -> 1
+(0, 2) -> 2
+(0, 3) -> 3
+(0, 4) -> 4
+(0, 5) -> 5
+(0, 6) -> 6
+(0, 7) -> 7
+(0, 8) -> 8
+(0, 9) -> 9
+(1, 0) -> 1
+(1, 1) -> 2
+(1, 2) -> 3
+(1, 3) -> 4
+(1, 4) -> 5
+(1, 5) -> 6
+(1, 6) -> 7
+(1, 7) -> 8
+(1, 8) -> 9
+(1, 9) -> c(1, 0)
+(2, 0) -> 2
+(2, 1) -> 3
+(2, 2) -> 4
+(2, 3) -> 5
+(2, 4) -> 6
+(2, 5) -> 7
+(2, 6) -> 8
+(2, 7) -> 9
+(2, 8) -> c(1, 0)
+(2, 9) -> c(1, 1)
+(3, 0) -> 3
+(3, 1) -> 4
+(3, 2) -> 5
+(3, 3) -> 6
+(3, 4) -> 7
+(3, 5) -> 8
+(3, 6) -> 9
+(3, 7) -> c(1, 0)
+(3, 8) -> c(1, 1)
+(3, 9) -> c(1, 2)
+(4, 0) -> 4
+(4, 1) -> 5
+(4, 2) -> 6
+(4, 3) -> 7
+(4, 4) -> 8
+(4, 5) -> 9
+(4, 6) -> c(1, 0)
+(4, 7) -> c(1, 1)
+(4, 8) -> c(1, 2)
+(4, 9) -> c(1, 3)
+(5, 0) -> 5
+(5, 1) -> 6
+(5, 2) -> 7
+(5, 3) -> 8
+(5, 4) -> 9
+(5, 5) -> c(1, 0)
+(5, 6) -> c(1, 1)
+(5, 7) -> c(1, 2)
+(5, 8) -> c(1, 3)
+(5, 9) -> c(1, 4)
+(6, 0) -> 6
+(6, 1) -> 7
+(6, 2) -> 8
+(6, 3) -> 9
+(6, 4) -> c(1, 0)
+(6, 5) -> c(1, 1)
+(6, 6) -> c(1, 2)
+(6, 7) -> c(1, 3)
+(6, 8) -> c(1, 4)
+(6, 9) -> c(1, 5)
+(7, 0) -> 7
+(7, 1) -> 8
+(7, 2) -> 9
+(7, 3) -> c(1, 0)
+(7, 4) -> c(1, 1)
+(7, 5) -> c(1, 2)
+(7, 6) -> c(1, 3)
+(7, 7) -> c(1, 4)
+(7, 8) -> c(1, 5)
+(7, 9) -> c(1, 6)
+(8, 0) -> 8
+(8, 1) -> 9
+(8, 2) -> c(1, 0)
+(8, 3) -> c(1, 1)
+(8, 4) -> c(1, 2)
+(8, 5) -> c(1, 3)
+(8, 6) -> c(1, 4)
+(8, 7) -> c(1, 5)
+(8, 8) -> c(1, 6)
+(8, 9) -> c(1, 7)
+(9, 0) -> 9
+(9, 1) -> c(1, 0)
+(9, 2) -> c(1, 1)
+(9, 3) -> c(1, 2)
+(9, 4) -> c(1, 3)
+(9, 5) -> c(1, 4)
+(9, 6) -> c(1, 5)
+(9, 7) -> c(1, 6)
+(9, 8) -> c(1, 7)
+(9, 9) -> c(1, 8)
+(x, c(y, z)) -> c(y, +(x, z))
+(c(x, y), z) -> c(x, +(y, z))
c(0, x) -> x
c(x, c(y, z)) -> c(+(x, y), z)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(0, 0) -> 0
+(0, 1) -> 1
+(0, 2) -> 2
+(0, 3) -> 3
+(0, 4) -> 4
+(0, 5) -> 5
+(0, 6) -> 6
+(0, 7) -> 7
+(0, 8) -> 8
+(0, 9) -> 9
+(1, 0) -> 1
+(1, 1) -> 2
+(1, 2) -> 3
+(1, 3) -> 4
+(1, 4) -> 5
+(1, 5) -> 6
+(1, 6) -> 7
+(1, 7) -> 8
+(1, 8) -> 9
+(2, 0) -> 2
+(2, 1) -> 3
+(2, 2) -> 4
+(2, 3) -> 5
+(2, 4) -> 6
+(2, 5) -> 7
+(2, 6) -> 8
+(2, 7) -> 9
+(3, 0) -> 3
+(3, 1) -> 4
+(3, 2) -> 5
+(3, 3) -> 6
+(3, 4) -> 7
+(3, 5) -> 8
+(3, 6) -> 9
+(4, 0) -> 4
+(4, 1) -> 5
+(4, 2) -> 6
+(4, 3) -> 7
+(4, 4) -> 8
+(4, 5) -> 9
+(5, 0) -> 5
+(5, 1) -> 6
+(5, 2) -> 7
+(5, 3) -> 8
+(5, 4) -> 9
+(6, 0) -> 6
+(6, 1) -> 7
+(6, 2) -> 8
+(6, 3) -> 9
+(7, 0) -> 7
+(7, 1) -> 8
+(7, 2) -> 9
+(8, 0) -> 8
+(8, 1) -> 9
+(9, 0) -> 9
c(0, x) -> x
where the Polynomial interpretation:
| POL(4) | = 1 |
| POL(6) | = 1 |
| POL(c(x1, x2)) | = x1 + x2 |
| POL(0) | = 1 |
| POL(1) | = 1 |
| POL(8) | = 1 |
| POL(5) | = 1 |
| POL(3) | = 1 |
| POL(9) | = 1 |
| POL(2) | = 1 |
| POL(+(x1, x2)) | = x1 + x2 |
| POL(7) | = 1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(9, 4) -> c(1, 3)
+(9, 2) -> c(1, 1)
+(9, 6) -> c(1, 5)
+(8, 9) -> c(1, 7)
+(9, 9) -> c(1, 8)
+(4, 9) -> c(1, 3)
+(6, 9) -> c(1, 5)
+(9, 5) -> c(1, 4)
+(9, 8) -> c(1, 7)
+(2, 9) -> c(1, 1)
+(9, 7) -> c(1, 6)
+(7, 9) -> c(1, 6)
+(1, 9) -> c(1, 0)
+(9, 3) -> c(1, 2)
+(9, 1) -> c(1, 0)
+(5, 9) -> c(1, 4)
+(3, 9) -> c(1, 2)
where the Polynomial interpretation:
| POL(4) | = 0 |
| POL(6) | = 0 |
| POL(c(x1, x2)) | = x1 + x2 |
| POL(0) | = 0 |
| POL(1) | = 0 |
| POL(8) | = 0 |
| POL(5) | = 0 |
| POL(3) | = 0 |
| POL(9) | = 1 |
| POL(2) | = 0 |
| POL(+(x1, x2)) | = x1 + x2 |
| POL(7) | = 0 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(7, 3) -> c(1, 0)
+(7, 8) -> c(1, 5)
+(6, 7) -> c(1, 3)
+(7, 6) -> c(1, 3)
+(7, 7) -> c(1, 4)
+(3, 7) -> c(1, 0)
+(4, 7) -> c(1, 1)
+(5, 7) -> c(1, 2)
+(7, 5) -> c(1, 2)
+(7, 4) -> c(1, 1)
+(8, 7) -> c(1, 5)
where the Polynomial interpretation:
| POL(4) | = 0 |
| POL(6) | = 0 |
| POL(c(x1, x2)) | = x1 + x2 |
| POL(0) | = 0 |
| POL(8) | = 0 |
| POL(1) | = 0 |
| POL(5) | = 0 |
| POL(3) | = 0 |
| POL(2) | = 0 |
| POL(+(x1, x2)) | = x1 + x2 |
| POL(7) | = 1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(5, 8) -> c(1, 3)
+(8, 5) -> c(1, 3)
+(5, 6) -> c(1, 1)
+(5, 5) -> c(1, 0)
+(6, 5) -> c(1, 1)
where the Polynomial interpretation:
| POL(4) | = 0 |
| POL(6) | = 0 |
| POL(c(x1, x2)) | = x1 + x2 |
| POL(0) | = 0 |
| POL(8) | = 0 |
| POL(1) | = 0 |
| POL(5) | = 1 |
| POL(3) | = 0 |
| POL(2) | = 0 |
| POL(+(x1, x2)) | = x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(8, 8) -> c(1, 6)
+(6, 8) -> c(1, 4)
+(8, 4) -> c(1, 2)
+(2, 8) -> c(1, 0)
+(8, 3) -> c(1, 1)
+(4, 8) -> c(1, 2)
+(8, 2) -> c(1, 0)
+(3, 8) -> c(1, 1)
+(8, 6) -> c(1, 4)
where the Polynomial interpretation:
| POL(4) | = 0 |
| POL(6) | = 0 |
| POL(c(x1, x2)) | = x1 + x2 |
| POL(0) | = 0 |
| POL(8) | = 1 |
| POL(1) | = 0 |
| POL(3) | = 0 |
| POL(2) | = 0 |
| POL(+(x1, x2)) | = x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(4, 6) -> c(1, 0)
+(6, 6) -> c(1, 2)
+(6, 4) -> c(1, 0)
where the Polynomial interpretation:
| POL(4) | = 0 |
| POL(6) | = 1 |
| POL(c(x1, x2)) | = x1 + x2 |
| POL(0) | = 0 |
| POL(1) | = 0 |
| POL(2) | = 0 |
| POL(+(x1, x2)) | = x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS7
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
c(x, c(y, z)) -> c(+(x, y), z)
where the Polynomial interpretation:
| POL(c(x1, x2)) | = 1 + x1 + x2 |
| POL(+(x1, x2)) | = x1 + x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS8
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(x, c(y, z)) -> c(y, +(x, z))
where the Polynomial interpretation:
| POL(c(x1, x2)) | = 1 + x1 + x2 |
| POL(+(x1, x2)) | = x1 + 2·x2 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS9
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(c(x, y), z) -> c(x, +(y, z))
where the Polynomial interpretation:
| POL(c(x1, x2)) | = 1 + x1 + x2 |
| POL(+(x1, x2)) | = 2·x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS10
↳Overlay and local confluence Check
The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS11
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes