Certification Problem
Input (TPDB TRS_Innermost/Transformed_CSR_innermost_04/PALINDROME_nokinds_noand_iGM)
The rewrite relation of the following TRS is considered.
There are 101 ruless (increase limit for explicit display).
The evaluation strategy is innermost.Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [__(x1, x2)] |
= |
1 + 2 · x1 + 1 · x2
|
| [mark(x1)] |
= |
1 · x1
|
| [nil] |
= |
0 |
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [U21(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
2 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [U41(x1, x2)] |
= |
2 + 1 · x1 + 2 · x2
|
| [U42(x1)] |
= |
1 + 1 · x1
|
| [isNeList(x1)] |
= |
2 · x1
|
| [U51(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U52(x1)] |
= |
1 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U71(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U72(x1)] |
= |
1 · x1
|
| [isPal(x1)] |
= |
1 · x1
|
| [U81(x1)] |
= |
1 · x1
|
| [isQid(x1)] |
= |
1 · x1
|
| [isNePal(x1)] |
= |
1 · x1
|
| [a] |
= |
0 |
| [e] |
= |
0 |
| [i] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
0 |
all of the following rules can be deleted.
|
active(__(__(X,Y),Z)) |
→ |
mark(__(X,__(Y,Z))) |
(1) |
|
active(__(X,nil)) |
→ |
mark(X) |
(2) |
|
active(__(nil,X)) |
→ |
mark(X) |
(3) |
|
active(U41(tt,V2)) |
→ |
mark(U42(isNeList(V2))) |
(8) |
|
active(U42(tt)) |
→ |
mark(tt) |
(9) |
|
active(isList(__(V1,V2))) |
→ |
mark(U21(isList(V1),V2)) |
(18) |
|
active(isNeList(__(V1,V2))) |
→ |
mark(U51(isNeList(V1),V2)) |
(21) |
|
active(isNePal(__(I,__(P,I)))) |
→ |
mark(U71(isQid(I),P)) |
(23) |
1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
2 + 1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
2 + 2 · x1
|
| [isList(x1)] |
= |
1 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [U51(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U52(x1)] |
= |
1 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U71(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U72(x1)] |
= |
1 · x1
|
| [isPal(x1)] |
= |
2 · x1
|
| [U81(x1)] |
= |
2 · x1
|
| [isNeList(x1)] |
= |
1 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
1 · x1
|
| [__(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [isNePal(x1)] |
= |
1 · x1
|
| [a] |
= |
0 |
| [e] |
= |
0 |
| [i] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
0 |
| [U42(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
active(U22(tt)) |
→ |
mark(tt) |
(6) |
1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
1 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [U51(x1, x2)] |
= |
2 · x1 + 2 · x2
|
| [U52(x1)] |
= |
1 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U71(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U72(x1)] |
= |
2 · x1
|
| [isPal(x1)] |
= |
1 · x1
|
| [U81(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
1 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
1 · x1
|
| [__(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [isNePal(x1)] |
= |
1 · x1
|
| [a] |
= |
0 |
| [e] |
= |
2 |
| [i] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
0 |
| [U42(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
active(isQid(e)) |
→ |
mark(tt) |
(27) |
1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
2 · x1
|
| [U31(x1)] |
= |
2 · x1
|
| [U51(x1, x2)] |
= |
2 + 1 · x1 + 2 · x2
|
| [U52(x1)] |
= |
1 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U71(x1, x2)] |
= |
2 + 1 · x1 + 1 · x2
|
| [U72(x1)] |
= |
1 · x1
|
| [isPal(x1)] |
= |
1 · x1
|
| [U81(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
2 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
1 · x1
|
| [__(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [isNePal(x1)] |
= |
1 · x1
|
| [a] |
= |
0 |
| [i] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
0 |
| [U42(x1)] |
= |
1 · x1
|
| [e] |
= |
0 |
all of the following rules can be deleted.
|
active(U51(tt,V2)) |
→ |
mark(U52(isList(V2))) |
(10) |
|
active(U71(tt,P)) |
→ |
mark(U72(isPal(P))) |
(13) |
1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
2 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [U52(x1)] |
= |
2 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U72(x1)] |
= |
1 · x1
|
| [U81(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
2 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
2 · x1
|
| [__(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [isNePal(x1)] |
= |
2 · x1
|
| [isPal(x1)] |
= |
2 + 2 · x1
|
| [a] |
= |
0 |
| [i] |
= |
1 |
| [o] |
= |
0 |
| [u] |
= |
0 |
| [U42(x1)] |
= |
2 · x1
|
| [U51(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U71(x1, x2)] |
= |
2 · x1 + 2 · x2
|
| [e] |
= |
0 |
all of the following rules can be deleted.
|
active(isPal(V)) |
→ |
mark(U81(isNePal(V))) |
(24) |
|
active(isPal(nil)) |
→ |
mark(tt) |
(25) |
|
active(isQid(i)) |
→ |
mark(tt) |
(28) |
1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
1 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [U52(x1)] |
= |
2 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U72(x1)] |
= |
1 + 1 · x1
|
| [U81(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
1 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
1 · x1
|
| [__(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [isNePal(x1)] |
= |
2 · x1
|
| [a] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
0 |
| [U42(x1)] |
= |
1 · x1
|
| [U51(x1, x2)] |
= |
2 · x1 + 1 · x2
|
| [U71(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [isPal(x1)] |
= |
2 · x1
|
| [e] |
= |
0 |
| [i] |
= |
0 |
all of the following rules can be deleted.
|
active(U72(tt)) |
→ |
mark(tt) |
(14) |
1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
2 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
2 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [U52(x1)] |
= |
1 · x1
|
| [U61(x1)] |
= |
2 · x1
|
| [U81(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
1 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
1 · x1
|
| [__(x1, x2)] |
= |
2 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [isNePal(x1)] |
= |
2 · x1
|
| [a] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
2 |
| [U42(x1)] |
= |
1 · x1
|
| [U51(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U71(x1, x2)] |
= |
2 · x1 + 2 · x2
|
| [U72(x1)] |
= |
1 · x1
|
| [isPal(x1)] |
= |
2 · x1
|
| [e] |
= |
0 |
| [i] |
= |
0 |
all of the following rules can be deleted.
|
active(isQid(u)) |
→ |
mark(tt) |
(30) |
1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
2 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [U52(x1)] |
= |
1 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U81(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
2 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
2 · x1
|
| [__(x1, x2)] |
= |
2 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [isNePal(x1)] |
= |
2 · x1
|
| [a] |
= |
0 |
| [o] |
= |
1 |
| [U42(x1)] |
= |
1 · x1
|
| [U51(x1, x2)] |
= |
2 · x1 + 1 · x2
|
| [U71(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U72(x1)] |
= |
1 · x1
|
| [isPal(x1)] |
= |
2 · x1
|
| [e] |
= |
0 |
| [i] |
= |
0 |
| [u] |
= |
0 |
all of the following rules can be deleted.
|
active(isQid(o)) |
→ |
mark(tt) |
(29) |
1.1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 · x1
|
| [tt] |
= |
1 |
| [mark(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
2 + 2 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
2 + 2 · x1
|
| [U31(x1)] |
= |
2 · x1
|
| [U52(x1)] |
= |
2 + 1 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [U81(x1)] |
= |
1 + 1 · x1
|
| [isNeList(x1)] |
= |
2 · x1
|
| [nil] |
= |
0 |
| [isQid(x1)] |
= |
1 · x1
|
| [__(x1, x2)] |
= |
2 + 1 · x1 + 1 · x2
|
| [U41(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [isNePal(x1)] |
= |
2 · x1
|
| [a] |
= |
2 |
| [U42(x1)] |
= |
2 · x1
|
| [U51(x1, x2)] |
= |
1 · x1 + 2 · x2
|
| [U71(x1, x2)] |
= |
2 · x1 + 2 · x2
|
| [U72(x1)] |
= |
1 · x1
|
| [isPal(x1)] |
= |
2 · x1
|
| [e] |
= |
0 |
| [i] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
0 |
all of the following rules can be deleted.
|
active(U21(tt,V2)) |
→ |
mark(U22(isList(V2))) |
(5) |
|
active(U31(tt)) |
→ |
mark(tt) |
(7) |
|
active(U52(tt)) |
→ |
mark(tt) |
(11) |
|
active(U81(tt)) |
→ |
mark(tt) |
(15) |
|
active(isList(V)) |
→ |
mark(U11(isNeList(V))) |
(16) |
|
active(isList(nil)) |
→ |
mark(tt) |
(17) |
|
active(isNeList(__(V1,V2))) |
→ |
mark(U41(isList(V1),V2)) |
(20) |
|
active(isQid(a)) |
→ |
mark(tt) |
(26) |
1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [U11(x1)] |
= |
1 + 1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
2 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
1 + 2 · x1
|
| [U31(x1)] |
= |
1 · x1
|
| [isQid(x1)] |
= |
1 · x1
|
| [isNePal(x1)] |
= |
2 · x1
|
| [__(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
| [nil] |
= |
0 |
| [U21(x1, x2)] |
= |
1 + 1 · x1 + 2 · x2
|
| [U22(x1)] |
= |
1 + 2 · x1
|
| [isList(x1)] |
= |
1 + 2 · x1
|
| [U41(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
| [U42(x1)] |
= |
1 + 1 · x1
|
| [U51(x1, x2)] |
= |
1 + 2 · x1 + 1 · x2
|
| [U52(x1)] |
= |
2 · x1
|
| [U71(x1, x2)] |
= |
1 + 1 · x1 + 2 · x2
|
| [U72(x1)] |
= |
1 + 1 · x1
|
| [isPal(x1)] |
= |
1 + 2 · x1
|
| [U81(x1)] |
= |
1 + 2 · x1
|
| [a] |
= |
0 |
| [e] |
= |
0 |
| [i] |
= |
0 |
| [o] |
= |
0 |
| [u] |
= |
1 |
all of the following rules can be deleted.
|
active(U11(tt)) |
→ |
mark(tt) |
(4) |
|
active(isNeList(V)) |
→ |
mark(U31(isQid(V))) |
(19) |
|
mark(__(X1,X2)) |
→ |
active(__(mark(X1),mark(X2))) |
(31) |
|
mark(U11(X)) |
→ |
active(U11(mark(X))) |
(33) |
|
mark(U21(X1,X2)) |
→ |
active(U21(mark(X1),X2)) |
(35) |
|
mark(U22(X)) |
→ |
active(U22(mark(X))) |
(36) |
|
mark(isList(X)) |
→ |
active(isList(X)) |
(37) |
|
mark(U41(X1,X2)) |
→ |
active(U41(mark(X1),X2)) |
(39) |
|
mark(U42(X)) |
→ |
active(U42(mark(X))) |
(40) |
|
mark(isNeList(X)) |
→ |
active(isNeList(X)) |
(41) |
|
mark(U51(X1,X2)) |
→ |
active(U51(mark(X1),X2)) |
(42) |
|
mark(U71(X1,X2)) |
→ |
active(U71(mark(X1),X2)) |
(45) |
|
mark(U72(X)) |
→ |
active(U72(mark(X))) |
(46) |
|
mark(isPal(X)) |
→ |
active(isPal(X)) |
(47) |
|
mark(U81(X)) |
→ |
active(U81(mark(X))) |
(48) |
|
mark(u) |
→ |
active(u) |
(55) |
1.1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 + 1 · x1
|
| [U61(x1)] |
= |
1 + 1 · x1
|
| [tt] |
= |
0 |
| [mark(x1)] |
= |
1 + 2 · x1
|
| [isNePal(x1)] |
= |
2 + 2 · x1
|
| [isQid(x1)] |
= |
1 · x1
|
| [nil] |
= |
2 |
| [U31(x1)] |
= |
2 + 2 · x1
|
| [U52(x1)] |
= |
1 + 1 · x1
|
| [a] |
= |
0 |
| [e] |
= |
1 |
| [i] |
= |
1 |
| [o] |
= |
1 |
| [__(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U11(x1)] |
= |
1 · x1
|
| [U21(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U22(x1)] |
= |
1 · x1
|
| [isList(x1)] |
= |
1 · x1
|
| [U41(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U42(x1)] |
= |
1 · x1
|
| [isNeList(x1)] |
= |
1 · x1
|
| [U51(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U71(x1, x2)] |
= |
1 · x1 + 1 · x2
|
| [U72(x1)] |
= |
1 · x1
|
| [isPal(x1)] |
= |
1 · x1
|
| [U81(x1)] |
= |
1 · x1
|
all of the following rules can be deleted.
|
active(U61(tt)) |
→ |
mark(tt) |
(12) |
|
mark(nil) |
→ |
active(nil) |
(32) |
|
mark(isNePal(X)) |
→ |
active(isNePal(X)) |
(50) |
|
mark(e) |
→ |
active(e) |
(52) |
|
mark(i) |
→ |
active(i) |
(53) |
|
mark(o) |
→ |
active(o) |
(54) |
|
__(mark(X1),X2) |
→ |
__(X1,X2) |
(56) |
|
__(X1,mark(X2)) |
→ |
__(X1,X2) |
(57) |
|
__(active(X1),X2) |
→ |
__(X1,X2) |
(58) |
|
__(X1,active(X2)) |
→ |
__(X1,X2) |
(59) |
|
U11(mark(X)) |
→ |
U11(X) |
(60) |
|
U11(active(X)) |
→ |
U11(X) |
(61) |
|
U21(mark(X1),X2) |
→ |
U21(X1,X2) |
(62) |
|
U21(X1,mark(X2)) |
→ |
U21(X1,X2) |
(63) |
|
U21(active(X1),X2) |
→ |
U21(X1,X2) |
(64) |
|
U21(X1,active(X2)) |
→ |
U21(X1,X2) |
(65) |
|
U22(mark(X)) |
→ |
U22(X) |
(66) |
|
U22(active(X)) |
→ |
U22(X) |
(67) |
|
isList(mark(X)) |
→ |
isList(X) |
(68) |
|
isList(active(X)) |
→ |
isList(X) |
(69) |
|
U31(mark(X)) |
→ |
U31(X) |
(70) |
|
U31(active(X)) |
→ |
U31(X) |
(71) |
|
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(72) |
|
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(73) |
|
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(74) |
|
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(75) |
|
U42(mark(X)) |
→ |
U42(X) |
(76) |
|
U42(active(X)) |
→ |
U42(X) |
(77) |
|
isNeList(mark(X)) |
→ |
isNeList(X) |
(78) |
|
isNeList(active(X)) |
→ |
isNeList(X) |
(79) |
|
U51(mark(X1),X2) |
→ |
U51(X1,X2) |
(80) |
|
U51(X1,mark(X2)) |
→ |
U51(X1,X2) |
(81) |
|
U51(active(X1),X2) |
→ |
U51(X1,X2) |
(82) |
|
U51(X1,active(X2)) |
→ |
U51(X1,X2) |
(83) |
|
U52(mark(X)) |
→ |
U52(X) |
(84) |
|
U52(active(X)) |
→ |
U52(X) |
(85) |
|
U61(mark(X)) |
→ |
U61(X) |
(86) |
|
U61(active(X)) |
→ |
U61(X) |
(87) |
|
U71(mark(X1),X2) |
→ |
U71(X1,X2) |
(88) |
|
U71(X1,mark(X2)) |
→ |
U71(X1,X2) |
(89) |
|
U71(active(X1),X2) |
→ |
U71(X1,X2) |
(90) |
|
U71(X1,active(X2)) |
→ |
U71(X1,X2) |
(91) |
|
U72(mark(X)) |
→ |
U72(X) |
(92) |
|
U72(active(X)) |
→ |
U72(X) |
(93) |
|
isPal(mark(X)) |
→ |
isPal(X) |
(94) |
|
isPal(active(X)) |
→ |
isPal(X) |
(95) |
|
U81(mark(X)) |
→ |
U81(X) |
(96) |
|
U81(active(X)) |
→ |
U81(X) |
(97) |
|
isQid(mark(X)) |
→ |
isQid(X) |
(98) |
|
isQid(active(X)) |
→ |
isQid(X) |
(99) |
|
isNePal(mark(X)) |
→ |
isNePal(X) |
(100) |
|
isNePal(active(X)) |
→ |
isNePal(X) |
(101) |
1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
linear polynomial interpretation over the naturals
| [active(x1)] |
= |
1 · x1
|
| [isNePal(x1)] |
= |
2 · x1
|
| [mark(x1)] |
= |
2 · x1
|
| [U61(x1)] |
= |
1 · x1
|
| [isQid(x1)] |
= |
1 · x1
|
| [tt] |
= |
0 |
| [U31(x1)] |
= |
1 + 1 · x1
|
| [U52(x1)] |
= |
1 · x1
|
| [a] |
= |
0 |
all of the following rules can be deleted.
|
mark(U31(X)) |
→ |
active(U31(mark(X))) |
(38) |
1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
| prec(isNePal) |
= |
4 |
|
stat(isNePal) |
= |
lex
|
| prec(mark) |
= |
3 |
|
stat(mark) |
= |
lex
|
| prec(isQid) |
= |
0 |
|
stat(isQid) |
= |
lex
|
| prec(tt) |
= |
1 |
|
stat(tt) |
= |
lex
|
| prec(a) |
= |
2 |
|
stat(a) |
= |
lex
|
| π(active) |
= |
1 |
| π(isNePal) |
= |
[1] |
| π(mark) |
= |
[1] |
| π(U61) |
= |
1 |
| π(isQid) |
= |
[1] |
| π(tt) |
= |
[] |
| π(U52) |
= |
1 |
| π(a) |
= |
[] |
all of the following rules can be deleted.
|
active(isNePal(V)) |
→ |
mark(U61(isQid(V))) |
(22) |
|
mark(tt) |
→ |
active(tt) |
(34) |
|
mark(isQid(X)) |
→ |
active(isQid(X)) |
(49) |
|
mark(a) |
→ |
active(a) |
(51) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
| prec(mark) |
= |
1 |
|
stat(mark) |
= |
lex
|
| prec(U52) |
= |
0 |
|
stat(U52) |
= |
lex
|
| π(mark) |
= |
[1] |
| π(U52) |
= |
[1] |
| π(active) |
= |
1 |
| π(U61) |
= |
1 |
all of the following rules can be deleted.
|
mark(U52(X)) |
→ |
active(U52(mark(X))) |
(43) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Removal
Using the
| prec(mark) |
= |
1 |
|
stat(mark) |
= |
lex
|
| prec(U61) |
= |
0 |
|
stat(U61) |
= |
lex
|
| π(mark) |
= |
[1] |
| π(U61) |
= |
[1] |
| π(active) |
= |
1 |
all of the following rules can be deleted.
|
mark(U61(X)) |
→ |
active(U61(mark(X))) |
(44) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty
There are no rules in the TRS. Hence, it is terminating.