Term Rewriting System R:
[Y, X]
minus(n0, Y) -> 0
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
0 -> n0
s(X) -> ns(X)
activate(n0) -> 0
activate(ns(X)) -> s(X)
activate(X) -> X
Innermost Termination of R to be shown.
R
↳Removing Redundant Rules for Innermost Termination
Removing the following rules from R which left hand sides contain non normal subterms
div(0, ns(Y)) -> 0
div(s(X), ns(Y)) -> if(geq(X, activate(Y)), ns(div(minus(X, activate(Y)), ns(activate(Y)))), n0)
R
↳RRRI
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
minus(n0, Y) -> 0
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(0) | = 0 |
| POL(geq(x1, x2)) | = x1 + x2 |
| POL(false) | = 0 |
| POL(minus(x1, x2)) | = 1 + x1 + x2 |
| POL(n__s(x1)) | = x1 |
| POL(true) | = 0 |
| POL(n__0) | = 0 |
| POL(s(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
0 -> n0
geq(X, n0) -> true
geq(n0, ns(Y)) -> false
where the Polynomial interpretation:
| POL(activate(x1)) | = 2·x1 |
| POL(if(x1, x2, x3)) | = x1 + 2·x2 + 2·x3 |
| POL(0) | = 2 |
| POL(geq(x1, x2)) | = x1 + x2 |
| POL(false) | = 0 |
| POL(minus(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(n__s(x1)) | = 2·x1 |
| POL(n__0) | = 1 |
| POL(s(x1)) | = 2·x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
if(false, X, Y) -> activate(Y)
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(0) | = 0 |
| POL(geq(x1, x2)) | = x1 + x2 |
| POL(false) | = 1 |
| POL(minus(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(n__s(x1)) | = x1 |
| POL(n__0) | = 0 |
| POL(s(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(n0) -> 0
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(geq(x1, x2)) | = x1 + x2 |
| POL(0) | = 0 |
| POL(minus(x1, x2)) | = x1 + x2 |
| POL(n__s(x1)) | = x1 |
| POL(true) | = 0 |
| POL(s(x1)) | = x1 |
| POL(n__0) | = 1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
s(X) -> ns(X)
geq(ns(X), ns(Y)) -> geq(activate(X), activate(Y))
minus(ns(X), ns(Y)) -> minus(activate(X), activate(Y))
where the Polynomial interpretation:
| POL(activate(x1)) | = 2·x1 |
| POL(if(x1, x2, x3)) | = x1 + 2·x2 + x3 |
| POL(geq(x1, x2)) | = x1 + x2 |
| POL(minus(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(n__s(x1)) | = 1 + 2·x1 |
| POL(s(x1)) | = 2 + 2·x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS7
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
if(true, X, Y) -> activate(X)
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(n__s(x1)) | = x1 |
| POL(true) | = 1 |
| POL(s(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS8
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(ns(X)) -> s(X)
activate(X) -> X
where the Polynomial interpretation:
| POL(activate(x1)) | = 1 + x1 |
| POL(n__s(x1)) | = x1 |
| POL(s(x1)) | = x1 |
was used.
All Rules of R can be deleted.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS9
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes