Certification Problem
Input (TPDB TRS_Standard/Transformed_CSR_04/PALINDROME_complete-noand_L)
The rewrite relation of the following TRS is considered.
|
__(__(X,Y),Z) |
→ |
__(X,__(Y,Z)) |
(1) |
|
__(X,nil) |
→ |
X |
(2) |
|
__(nil,X) |
→ |
X |
(3) |
|
U11(tt) |
→ |
U12(isPalListKind) |
(4) |
|
U12(tt) |
→ |
U13(isNeList) |
(5) |
|
U13(tt) |
→ |
tt |
(6) |
|
U21(tt) |
→ |
U22(isPalListKind) |
(7) |
|
U22(tt) |
→ |
U23(isPalListKind) |
(8) |
|
U23(tt) |
→ |
U24(isPalListKind) |
(9) |
|
U24(tt) |
→ |
U25(isList) |
(10) |
|
U25(tt) |
→ |
U26(isList) |
(11) |
|
U26(tt) |
→ |
tt |
(12) |
|
U31(tt) |
→ |
U32(isPalListKind) |
(13) |
|
U32(tt) |
→ |
U33(isQid) |
(14) |
|
U33(tt) |
→ |
tt |
(15) |
|
U41(tt) |
→ |
U42(isPalListKind) |
(16) |
|
U42(tt) |
→ |
U43(isPalListKind) |
(17) |
|
U43(tt) |
→ |
U44(isPalListKind) |
(18) |
|
U44(tt) |
→ |
U45(isList) |
(19) |
|
U45(tt) |
→ |
U46(isNeList) |
(20) |
|
U46(tt) |
→ |
tt |
(21) |
|
U51(tt) |
→ |
U52(isPalListKind) |
(22) |
|
U52(tt) |
→ |
U53(isPalListKind) |
(23) |
|
U53(tt) |
→ |
U54(isPalListKind) |
(24) |
|
U54(tt) |
→ |
U55(isNeList) |
(25) |
|
U55(tt) |
→ |
U56(isList) |
(26) |
|
U56(tt) |
→ |
tt |
(27) |
|
U61(tt) |
→ |
U62(isPalListKind) |
(28) |
|
U62(tt) |
→ |
U63(isQid) |
(29) |
|
U63(tt) |
→ |
tt |
(30) |
|
U71(tt) |
→ |
U72(isPalListKind) |
(31) |
|
U72(tt) |
→ |
U73(isPal) |
(32) |
|
U73(tt) |
→ |
U74(isPalListKind) |
(33) |
|
U74(tt) |
→ |
tt |
(34) |
|
U81(tt) |
→ |
U82(isPalListKind) |
(35) |
|
U82(tt) |
→ |
U83(isNePal) |
(36) |
|
U83(tt) |
→ |
tt |
(37) |
|
U91(tt) |
→ |
U92(isPalListKind) |
(38) |
|
U92(tt) |
→ |
tt |
(39) |
| isList |
→ |
U11(isPalListKind) |
(40) |
| isList |
→ |
tt |
(41) |
| isList |
→ |
U21(isPalListKind) |
(42) |
| isNeList |
→ |
U31(isPalListKind) |
(43) |
| isNeList |
→ |
U41(isPalListKind) |
(44) |
| isNeList |
→ |
U51(isPalListKind) |
(45) |
| isNePal |
→ |
U61(isPalListKind) |
(46) |
| isNePal |
→ |
U71(isQid) |
(47) |
| isPal |
→ |
U81(isPalListKind) |
(48) |
| isPal |
→ |
tt |
(49) |
| isPalListKind |
→ |
tt |
(50) |
| isPalListKind |
→ |
U91(isPalListKind) |
(51) |
| isQid |
→ |
tt |
(52) |
Property / Task
Prove or disprove termination.Answer / Result
No.Proof (by AProVE @ termCOMP 2023)
1 Rule Removal
The following rules have been removed.
|
__(X,nil) |
→ |
X |
(2) |
|
__(nil,X) |
→ |
X |
(3) |
1.1 Rule Removal
The following rules have been removed.
|
__(__(X,Y),Z) |
→ |
__(X,__(Y,Z)) |
(1) |
1.1.1 Rule Removal
The following rules have been removed.
|
U62(tt) |
→ |
U63(isQid) |
(29) |
| isPal |
→ |
tt |
(49) |
1.1.1.1 Rule Removal
The following rules have been removed.
1.1.1.1.1 Rule Removal
The following rules have been removed.
|
U61(tt) |
→ |
U62(isPalListKind) |
(28) |
1.1.1.1.1.1 Rule Removal
The following rules have been removed.
| isNePal |
→ |
U61(isPalListKind) |
(46) |
1.1.1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
U11#(tt) |
→ |
U12#(isPalListKind) |
(53) |
|
U11#(tt) |
→ |
isPalListKind# |
(54) |
|
U12#(tt) |
→ |
U13#(isNeList) |
(55) |
|
U12#(tt) |
→ |
isNeList# |
(56) |
|
U21#(tt) |
→ |
U22#(isPalListKind) |
(57) |
|
U21#(tt) |
→ |
isPalListKind# |
(58) |
|
U22#(tt) |
→ |
U23#(isPalListKind) |
(59) |
|
U22#(tt) |
→ |
isPalListKind# |
(60) |
|
U23#(tt) |
→ |
U24#(isPalListKind) |
(61) |
|
U23#(tt) |
→ |
isPalListKind# |
(62) |
|
U24#(tt) |
→ |
U25#(isList) |
(63) |
|
U24#(tt) |
→ |
isList# |
(64) |
|
U25#(tt) |
→ |
U26#(isList) |
(65) |
|
U25#(tt) |
→ |
isList# |
(66) |
|
U31#(tt) |
→ |
U32#(isPalListKind) |
(67) |
|
U31#(tt) |
→ |
isPalListKind# |
(68) |
|
U32#(tt) |
→ |
U33#(isQid) |
(69) |
|
U32#(tt) |
→ |
isQid# |
(70) |
|
U41#(tt) |
→ |
U42#(isPalListKind) |
(71) |
|
U41#(tt) |
→ |
isPalListKind# |
(72) |
|
U42#(tt) |
→ |
U43#(isPalListKind) |
(73) |
|
U42#(tt) |
→ |
isPalListKind# |
(74) |
|
U43#(tt) |
→ |
U44#(isPalListKind) |
(75) |
|
U43#(tt) |
→ |
isPalListKind# |
(76) |
|
U44#(tt) |
→ |
U45#(isList) |
(77) |
|
U44#(tt) |
→ |
isList# |
(78) |
|
U45#(tt) |
→ |
U46#(isNeList) |
(79) |
|
U45#(tt) |
→ |
isNeList# |
(80) |
|
U51#(tt) |
→ |
U52#(isPalListKind) |
(81) |
|
U51#(tt) |
→ |
isPalListKind# |
(82) |
|
U52#(tt) |
→ |
U53#(isPalListKind) |
(83) |
|
U52#(tt) |
→ |
isPalListKind# |
(84) |
|
U53#(tt) |
→ |
U54#(isPalListKind) |
(85) |
|
U53#(tt) |
→ |
isPalListKind# |
(86) |
|
U54#(tt) |
→ |
U55#(isNeList) |
(87) |
|
U54#(tt) |
→ |
isNeList# |
(88) |
|
U55#(tt) |
→ |
U56#(isList) |
(89) |
|
U55#(tt) |
→ |
isList# |
(90) |
|
U71#(tt) |
→ |
U72#(isPalListKind) |
(91) |
|
U71#(tt) |
→ |
isPalListKind# |
(92) |
|
U72#(tt) |
→ |
U73#(isPal) |
(93) |
|
U72#(tt) |
→ |
isPal# |
(94) |
|
U73#(tt) |
→ |
U74#(isPalListKind) |
(95) |
|
U73#(tt) |
→ |
isPalListKind# |
(96) |
|
U81#(tt) |
→ |
U82#(isPalListKind) |
(97) |
|
U81#(tt) |
→ |
isPalListKind# |
(98) |
|
U82#(tt) |
→ |
U83#(isNePal) |
(99) |
|
U82#(tt) |
→ |
isNePal# |
(100) |
|
U91#(tt) |
→ |
U92#(isPalListKind) |
(101) |
|
U91#(tt) |
→ |
isPalListKind# |
(102) |
| isList# |
→ |
U11#(isPalListKind) |
(103) |
| isList# |
→ |
isPalListKind# |
(104) |
| isList# |
→ |
U21#(isPalListKind) |
(105) |
| isNeList# |
→ |
U31#(isPalListKind) |
(106) |
| isNeList# |
→ |
isPalListKind# |
(107) |
| isNeList# |
→ |
U41#(isPalListKind) |
(108) |
| isNeList# |
→ |
U51#(isPalListKind) |
(109) |
| isNePal# |
→ |
U71#(isQid) |
(110) |
| isNePal# |
→ |
isQid# |
(111) |
| isPal# |
→ |
U81#(isPalListKind) |
(112) |
| isPal# |
→ |
isPalListKind# |
(113) |
| isPalListKind# |
→ |
U91#(isPalListKind) |
(114) |
| isPalListKind# |
→ |
isPalListKind# |
(115) |
It remains to prove infiniteness of the resulting DP problem.
1.1.1.1.1.1.1.1 Pair and Rule Removal
Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem.
The following pairs have been deleted.
|
U11#(tt) |
→ |
U12#(isPalListKind) |
(53) |
|
U11#(tt) |
→ |
isPalListKind# |
(54) |
|
U12#(tt) |
→ |
U13#(isNeList) |
(55) |
|
U12#(tt) |
→ |
isNeList# |
(56) |
|
U21#(tt) |
→ |
U22#(isPalListKind) |
(57) |
|
U21#(tt) |
→ |
isPalListKind# |
(58) |
|
U22#(tt) |
→ |
U23#(isPalListKind) |
(59) |
|
U22#(tt) |
→ |
isPalListKind# |
(60) |
|
U23#(tt) |
→ |
U24#(isPalListKind) |
(61) |
|
U23#(tt) |
→ |
isPalListKind# |
(62) |
|
U24#(tt) |
→ |
U25#(isList) |
(63) |
|
U24#(tt) |
→ |
isList# |
(64) |
|
U25#(tt) |
→ |
U26#(isList) |
(65) |
|
U25#(tt) |
→ |
isList# |
(66) |
|
U31#(tt) |
→ |
U32#(isPalListKind) |
(67) |
|
U31#(tt) |
→ |
isPalListKind# |
(68) |
|
U32#(tt) |
→ |
U33#(isQid) |
(69) |
|
U32#(tt) |
→ |
isQid# |
(70) |
|
U41#(tt) |
→ |
U42#(isPalListKind) |
(71) |
|
U41#(tt) |
→ |
isPalListKind# |
(72) |
|
U42#(tt) |
→ |
U43#(isPalListKind) |
(73) |
|
U42#(tt) |
→ |
isPalListKind# |
(74) |
|
U43#(tt) |
→ |
U44#(isPalListKind) |
(75) |
|
U43#(tt) |
→ |
isPalListKind# |
(76) |
|
U44#(tt) |
→ |
U45#(isList) |
(77) |
|
U44#(tt) |
→ |
isList# |
(78) |
|
U45#(tt) |
→ |
U46#(isNeList) |
(79) |
|
U45#(tt) |
→ |
isNeList# |
(80) |
|
U51#(tt) |
→ |
U52#(isPalListKind) |
(81) |
|
U51#(tt) |
→ |
isPalListKind# |
(82) |
|
U52#(tt) |
→ |
U53#(isPalListKind) |
(83) |
|
U52#(tt) |
→ |
isPalListKind# |
(84) |
|
U53#(tt) |
→ |
U54#(isPalListKind) |
(85) |
|
U53#(tt) |
→ |
isPalListKind# |
(86) |
|
U54#(tt) |
→ |
U55#(isNeList) |
(87) |
|
U54#(tt) |
→ |
isNeList# |
(88) |
|
U55#(tt) |
→ |
U56#(isList) |
(89) |
|
U55#(tt) |
→ |
isList# |
(90) |
|
U71#(tt) |
→ |
U72#(isPalListKind) |
(91) |
|
U71#(tt) |
→ |
isPalListKind# |
(92) |
|
U72#(tt) |
→ |
U73#(isPal) |
(93) |
|
U72#(tt) |
→ |
isPal# |
(94) |
|
U73#(tt) |
→ |
U74#(isPalListKind) |
(95) |
|
U73#(tt) |
→ |
isPalListKind# |
(96) |
|
U81#(tt) |
→ |
U82#(isPalListKind) |
(97) |
|
U81#(tt) |
→ |
isPalListKind# |
(98) |
|
U82#(tt) |
→ |
U83#(isNePal) |
(99) |
|
U82#(tt) |
→ |
isNePal# |
(100) |
|
U91#(tt) |
→ |
U92#(isPalListKind) |
(101) |
| isList# |
→ |
U11#(isPalListKind) |
(103) |
| isList# |
→ |
isPalListKind# |
(104) |
| isList# |
→ |
U21#(isPalListKind) |
(105) |
| isNeList# |
→ |
U31#(isPalListKind) |
(106) |
| isNeList# |
→ |
isPalListKind# |
(107) |
| isNeList# |
→ |
U41#(isPalListKind) |
(108) |
| isNeList# |
→ |
U51#(isPalListKind) |
(109) |
| isNePal# |
→ |
U71#(isQid) |
(110) |
| isNePal# |
→ |
isQid# |
(111) |
| isPal# |
→ |
U81#(isPalListKind) |
(112) |
| isPal# |
→ |
isPalListKind# |
(113) |
and the following rules have been deleted.
1.1.1.1.1.1.1.1.1 Pair and Rule Removal
Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem.
The following pairs have been deleted.
and the following rules have been deleted.
|
U11(tt) |
→ |
U12(isPalListKind) |
(4) |
|
U12(tt) |
→ |
U13(isNeList) |
(5) |
|
U13(tt) |
→ |
tt |
(6) |
|
U21(tt) |
→ |
U22(isPalListKind) |
(7) |
|
U22(tt) |
→ |
U23(isPalListKind) |
(8) |
|
U23(tt) |
→ |
U24(isPalListKind) |
(9) |
|
U24(tt) |
→ |
U25(isList) |
(10) |
|
U25(tt) |
→ |
U26(isList) |
(11) |
|
U26(tt) |
→ |
tt |
(12) |
|
U31(tt) |
→ |
U32(isPalListKind) |
(13) |
|
U32(tt) |
→ |
U33(isQid) |
(14) |
|
U33(tt) |
→ |
tt |
(15) |
|
U41(tt) |
→ |
U42(isPalListKind) |
(16) |
|
U42(tt) |
→ |
U43(isPalListKind) |
(17) |
|
U43(tt) |
→ |
U44(isPalListKind) |
(18) |
|
U44(tt) |
→ |
U45(isList) |
(19) |
|
U45(tt) |
→ |
U46(isNeList) |
(20) |
|
U46(tt) |
→ |
tt |
(21) |
|
U51(tt) |
→ |
U52(isPalListKind) |
(22) |
|
U52(tt) |
→ |
U53(isPalListKind) |
(23) |
|
U53(tt) |
→ |
U54(isPalListKind) |
(24) |
|
U54(tt) |
→ |
U55(isNeList) |
(25) |
|
U55(tt) |
→ |
U56(isList) |
(26) |
|
U56(tt) |
→ |
tt |
(27) |
|
U71(tt) |
→ |
U72(isPalListKind) |
(31) |
|
U72(tt) |
→ |
U73(isPal) |
(32) |
|
U73(tt) |
→ |
U74(isPalListKind) |
(33) |
|
U74(tt) |
→ |
tt |
(34) |
|
U81(tt) |
→ |
U82(isPalListKind) |
(35) |
|
U82(tt) |
→ |
U83(isNePal) |
(36) |
|
U83(tt) |
→ |
tt |
(37) |
| isList |
→ |
U11(isPalListKind) |
(40) |
| isList |
→ |
tt |
(41) |
| isList |
→ |
U21(isPalListKind) |
(42) |
| isNeList |
→ |
U31(isPalListKind) |
(43) |
| isNeList |
→ |
U41(isPalListKind) |
(44) |
| isNeList |
→ |
U51(isPalListKind) |
(45) |
| isNePal |
→ |
U71(isQid) |
(47) |
| isPal |
→ |
U81(isPalListKind) |
(48) |
| isQid |
→ |
tt |
(52) |
1.1.1.1.1.1.1.1.1.1 Narrowing Processor
We consider narrowings of the pair
below position
1
to get the following set of pairs
| isPalListKind# |
→ |
U91#(tt) |
(116) |
| isPalListKind# |
→ |
U91#(U91(isPalListKind)) |
(117) |
1.1.1.1.1.1.1.1.1.1.1 Loop
The following loop proves infiniteness of the DP problem.
| t0
|
= |
isPalListKind# |
|
→P
|
isPalListKind# |
|
= |
t1
|
where t1 =
t0