Certification Problem
Input (TPDB SRS_Standard/ICFP_2010/213865)
The rewrite relation of the following TRS is considered.
|
0(1(1(2(x1)))) |
→ |
0(1(0(3(1(2(x1)))))) |
(1) |
|
0(2(3(1(x1)))) |
→ |
0(3(2(0(1(x1))))) |
(2) |
|
0(2(3(1(x1)))) |
→ |
1(0(3(4(2(x1))))) |
(3) |
|
0(2(3(1(x1)))) |
→ |
4(0(3(2(1(x1))))) |
(4) |
|
0(2(3(1(x1)))) |
→ |
0(0(3(2(1(4(x1)))))) |
(5) |
|
3(0(1(1(x1)))) |
→ |
0(3(4(1(1(0(x1)))))) |
(6) |
|
3(0(1(2(x1)))) |
→ |
0(3(4(1(2(4(x1)))))) |
(7) |
|
3(0(1(2(x1)))) |
→ |
0(3(4(4(1(2(x1)))))) |
(8) |
|
3(0(2(1(x1)))) |
→ |
0(3(2(1(0(x1))))) |
(9) |
|
3(0(2(1(x1)))) |
→ |
0(3(2(1(4(x1))))) |
(10) |
|
3(0(2(1(x1)))) |
→ |
0(3(4(2(1(0(x1)))))) |
(11) |
|
0(1(0(1(2(x1))))) |
→ |
0(0(2(1(1(3(x1)))))) |
(12) |
|
0(1(1(4(2(x1))))) |
→ |
1(0(3(1(4(2(x1)))))) |
(13) |
|
0(1(2(3(1(x1))))) |
→ |
1(0(3(1(2(2(x1)))))) |
(14) |
|
0(1(2(3(1(x1))))) |
→ |
1(4(0(3(1(2(x1)))))) |
(15) |
|
0(1(2(3(1(x1))))) |
→ |
2(1(0(3(2(1(x1)))))) |
(16) |
|
0(1(5(3(1(x1))))) |
→ |
0(3(1(5(4(1(x1)))))) |
(17) |
|
0(2(3(2(1(x1))))) |
→ |
0(3(2(4(1(2(x1)))))) |
(18) |
|
0(2(4(3(1(x1))))) |
→ |
0(3(4(1(2(0(x1)))))) |
(19) |
|
0(2(4(3(1(x1))))) |
→ |
1(0(0(3(4(2(x1)))))) |
(20) |
|
0(3(3(1(2(x1))))) |
→ |
3(0(1(0(3(2(x1)))))) |
(21) |
|
0(3(3(2(1(x1))))) |
→ |
0(3(2(3(1(4(x1)))))) |
(22) |
|
0(5(2(3(1(x1))))) |
→ |
0(3(2(1(5(4(x1)))))) |
(23) |
|
0(5(2(3(1(x1))))) |
→ |
0(5(0(3(2(1(x1)))))) |
(24) |
|
0(5(4(3(1(x1))))) |
→ |
0(3(4(1(4(5(x1)))))) |
(25) |
|
1(4(0(1(2(x1))))) |
→ |
2(1(4(0(3(1(x1)))))) |
(26) |
|
3(0(1(0(2(x1))))) |
→ |
0(3(2(0(4(1(x1)))))) |
(27) |
|
3(0(2(3(1(x1))))) |
→ |
3(0(0(3(1(2(x1)))))) |
(28) |
|
3(0(2(5(1(x1))))) |
→ |
0(3(2(5(0(1(x1)))))) |
(29) |
|
3(0(2(5(1(x1))))) |
→ |
0(3(5(2(1(4(x1)))))) |
(30) |
|
3(0(2(5(1(x1))))) |
→ |
5(4(0(3(2(1(x1)))))) |
(31) |
|
3(0(4(5(1(x1))))) |
→ |
5(0(0(3(4(1(x1)))))) |
(32) |
|
3(3(0(1(4(x1))))) |
→ |
3(0(0(3(1(4(x1)))))) |
(33) |
|
3(3(1(1(1(x1))))) |
→ |
0(3(1(3(1(1(x1)))))) |
(34) |
|
3(3(1(1(2(x1))))) |
→ |
1(3(2(1(4(3(x1)))))) |
(35) |
|
3(3(1(1(4(x1))))) |
→ |
4(4(3(1(3(1(x1)))))) |
(36) |
|
3(4(3(2(1(x1))))) |
→ |
3(0(3(2(4(1(x1)))))) |
(37) |
|
3(5(0(2(1(x1))))) |
→ |
0(3(2(0(5(1(x1)))))) |
(38) |
|
3(5(0(2(1(x1))))) |
→ |
0(3(5(2(4(1(x1)))))) |
(39) |
|
4(0(1(1(4(x1))))) |
→ |
1(0(3(4(4(1(x1)))))) |
(40) |
|
4(5(3(2(1(x1))))) |
→ |
0(3(1(5(2(4(x1)))))) |
(41) |
|
5(0(1(1(4(x1))))) |
→ |
0(3(1(5(1(4(x1)))))) |
(42) |
|
5(0(2(3(1(x1))))) |
→ |
5(0(3(2(1(0(x1)))))) |
(43) |
|
5(3(0(1(1(x1))))) |
→ |
0(3(1(1(5(0(x1)))))) |
(44) |
|
5(3(0(2(1(x1))))) |
→ |
0(3(4(2(1(5(x1)))))) |
(45) |
|
5(3(0(2(1(x1))))) |
→ |
0(3(5(1(2(4(x1)))))) |
(46) |
|
5(3(1(1(2(x1))))) |
→ |
3(1(2(1(4(5(x1)))))) |
(47) |
|
5(3(1(1(4(x1))))) |
→ |
3(4(1(5(2(1(x1)))))) |
(48) |
|
5(4(3(4(1(x1))))) |
→ |
0(3(4(4(1(5(x1)))))) |
(49) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by ttt2 @ termCOMP 2023)
1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
2(1(1(0(x1)))) |
→ |
2(1(3(0(1(0(x1)))))) |
(50) |
|
1(3(2(0(x1)))) |
→ |
1(0(2(3(0(x1))))) |
(51) |
|
1(3(2(0(x1)))) |
→ |
2(4(3(0(1(x1))))) |
(52) |
|
1(3(2(0(x1)))) |
→ |
1(2(3(0(4(x1))))) |
(53) |
|
1(3(2(0(x1)))) |
→ |
4(1(2(3(0(0(x1)))))) |
(54) |
|
1(1(0(3(x1)))) |
→ |
0(1(1(4(3(0(x1)))))) |
(55) |
|
2(1(0(3(x1)))) |
→ |
4(2(1(4(3(0(x1)))))) |
(56) |
|
2(1(0(3(x1)))) |
→ |
2(1(4(4(3(0(x1)))))) |
(57) |
|
1(2(0(3(x1)))) |
→ |
0(1(2(3(0(x1))))) |
(58) |
|
1(2(0(3(x1)))) |
→ |
4(1(2(3(0(x1))))) |
(59) |
|
1(2(0(3(x1)))) |
→ |
0(1(2(4(3(0(x1)))))) |
(60) |
|
2(1(0(1(0(x1))))) |
→ |
3(1(1(2(0(0(x1)))))) |
(61) |
|
2(4(1(1(0(x1))))) |
→ |
2(4(1(3(0(1(x1)))))) |
(62) |
|
1(3(2(1(0(x1))))) |
→ |
2(2(1(3(0(1(x1)))))) |
(63) |
|
1(3(2(1(0(x1))))) |
→ |
2(1(3(0(4(1(x1)))))) |
(64) |
|
1(3(2(1(0(x1))))) |
→ |
1(2(3(0(1(2(x1)))))) |
(65) |
|
1(3(5(1(0(x1))))) |
→ |
1(4(5(1(3(0(x1)))))) |
(66) |
|
1(2(3(2(0(x1))))) |
→ |
2(1(4(2(3(0(x1)))))) |
(67) |
|
1(3(4(2(0(x1))))) |
→ |
0(2(1(4(3(0(x1)))))) |
(68) |
|
1(3(4(2(0(x1))))) |
→ |
2(4(3(0(0(1(x1)))))) |
(69) |
|
2(1(3(3(0(x1))))) |
→ |
2(3(0(1(0(3(x1)))))) |
(70) |
|
1(2(3(3(0(x1))))) |
→ |
4(1(3(2(3(0(x1)))))) |
(71) |
|
1(3(2(5(0(x1))))) |
→ |
4(5(1(2(3(0(x1)))))) |
(72) |
|
1(3(2(5(0(x1))))) |
→ |
1(2(3(0(5(0(x1)))))) |
(73) |
|
1(3(4(5(0(x1))))) |
→ |
5(4(1(4(3(0(x1)))))) |
(74) |
|
2(1(0(4(1(x1))))) |
→ |
1(3(0(4(1(2(x1)))))) |
(75) |
|
2(0(1(0(3(x1))))) |
→ |
1(4(0(2(3(0(x1)))))) |
(76) |
|
1(3(2(0(3(x1))))) |
→ |
2(1(3(0(0(3(x1)))))) |
(77) |
|
1(5(2(0(3(x1))))) |
→ |
1(0(5(2(3(0(x1)))))) |
(78) |
|
1(5(2(0(3(x1))))) |
→ |
4(1(2(5(3(0(x1)))))) |
(79) |
|
1(5(2(0(3(x1))))) |
→ |
1(2(3(0(4(5(x1)))))) |
(80) |
|
1(5(4(0(3(x1))))) |
→ |
1(4(3(0(0(5(x1)))))) |
(81) |
|
4(1(0(3(3(x1))))) |
→ |
4(1(3(0(0(3(x1)))))) |
(82) |
|
1(1(1(3(3(x1))))) |
→ |
1(1(3(1(3(0(x1)))))) |
(83) |
|
2(1(1(3(3(x1))))) |
→ |
3(4(1(2(3(1(x1)))))) |
(84) |
|
4(1(1(3(3(x1))))) |
→ |
1(3(1(3(4(4(x1)))))) |
(85) |
|
1(2(3(4(3(x1))))) |
→ |
1(4(2(3(0(3(x1)))))) |
(86) |
|
1(2(0(5(3(x1))))) |
→ |
1(5(0(2(3(0(x1)))))) |
(87) |
|
1(2(0(5(3(x1))))) |
→ |
1(4(2(5(3(0(x1)))))) |
(88) |
|
4(1(1(0(4(x1))))) |
→ |
1(4(4(3(0(1(x1)))))) |
(89) |
|
1(2(3(5(4(x1))))) |
→ |
4(2(5(1(3(0(x1)))))) |
(90) |
|
4(1(1(0(5(x1))))) |
→ |
4(1(5(1(3(0(x1)))))) |
(91) |
|
1(3(2(0(5(x1))))) |
→ |
0(1(2(3(0(5(x1)))))) |
(92) |
|
1(1(0(3(5(x1))))) |
→ |
0(5(1(1(3(0(x1)))))) |
(93) |
|
1(2(0(3(5(x1))))) |
→ |
5(1(2(4(3(0(x1)))))) |
(94) |
|
1(2(0(3(5(x1))))) |
→ |
4(2(1(5(3(0(x1)))))) |
(95) |
|
2(1(1(3(5(x1))))) |
→ |
5(4(1(2(1(3(x1)))))) |
(96) |
|
4(1(1(3(5(x1))))) |
→ |
1(2(5(1(4(3(x1)))))) |
(97) |
|
1(4(3(4(5(x1))))) |
→ |
5(1(4(4(3(0(x1)))))) |
(98) |
1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
There are 144 ruless (increase limit for explicit display).
1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
|
4#(1(1(3(3(x1))))) |
→ |
4#(4(x1)) |
(201) |
|
4#(1(1(0(4(x1))))) |
→ |
1#(x1) |
(212) |
|
1#(3(4(2(0(x1))))) |
→ |
1#(x1) |
(156) |
|
1#(3(2(1(0(x1))))) |
→ |
1#(2(x1)) |
(143) |
|
1#(3(2(1(0(x1))))) |
→ |
2#(x1) |
(142) |
|
2#(1(1(3(5(x1))))) |
→ |
4#(1(2(1(3(x1))))) |
(235) |
|
4#(1(1(3(3(x1))))) |
→ |
1#(3(1(3(4(4(x1)))))) |
(203) |
|
1#(3(2(1(0(x1))))) |
→ |
4#(1(x1)) |
(139) |
|
4#(1(1(3(3(x1))))) |
→ |
1#(3(4(4(x1)))) |
(202) |
|
1#(3(2(1(0(x1))))) |
→ |
1#(x1) |
(135) |
|
1#(3(2(0(x1)))) |
→ |
4#(x1) |
(106) |
|
4#(1(1(3(3(x1))))) |
→ |
4#(x1) |
(200) |
|
1#(3(2(0(x1)))) |
→ |
1#(x1) |
(103) |
|
2#(1(1(3(5(x1))))) |
→ |
1#(2(1(3(x1)))) |
(234) |
|
2#(1(1(3(5(x1))))) |
→ |
2#(1(3(x1))) |
(233) |
|
2#(1(1(3(5(x1))))) |
→ |
1#(3(x1)) |
(232) |
|
2#(1(1(3(3(x1))))) |
→ |
4#(1(2(3(1(x1))))) |
(199) |
|
2#(1(1(3(3(x1))))) |
→ |
1#(x1) |
(196) |
|
2#(1(0(4(1(x1))))) |
→ |
4#(1(2(x1))) |
(174) |
|
2#(1(0(4(1(x1))))) |
→ |
1#(2(x1)) |
(173) |
|
2#(1(0(4(1(x1))))) |
→ |
2#(x1) |
(172) |
|
2#(4(1(1(0(x1))))) |
→ |
1#(x1) |
(131) |
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
| prec(4#) |
= |
0 |
|
stat(4#) |
= |
lex
|
| prec(1#) |
= |
0 |
|
stat(1#) |
= |
lex
|
| prec(2#) |
= |
1 |
|
stat(2#) |
= |
lex
|
| prec(5) |
= |
0 |
|
stat(5) |
= |
lex
|
| prec(4) |
= |
0 |
|
stat(4) |
= |
lex
|
| prec(3) |
= |
0 |
|
stat(3) |
= |
lex
|
| prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
| prec(1) |
= |
2 |
|
stat(1) |
= |
lex
|
| prec(2) |
= |
0 |
|
stat(2) |
= |
lex
|
| π(4#) |
= |
1 |
| π(1#) |
= |
[1] |
| π(2#) |
= |
[1] |
| π(5) |
= |
1 |
| π(4) |
= |
1 |
| π(3) |
= |
1 |
| π(0) |
= |
1 |
| π(1) |
= |
[1] |
| π(2) |
= |
1 |
together with the usable
rules
|
2(1(1(0(x1)))) |
→ |
2(1(3(0(1(0(x1)))))) |
(50) |
|
1(3(2(0(x1)))) |
→ |
1(0(2(3(0(x1))))) |
(51) |
|
1(3(2(0(x1)))) |
→ |
2(4(3(0(1(x1))))) |
(52) |
|
1(3(2(0(x1)))) |
→ |
1(2(3(0(4(x1))))) |
(53) |
|
1(3(2(0(x1)))) |
→ |
4(1(2(3(0(0(x1)))))) |
(54) |
|
1(1(0(3(x1)))) |
→ |
0(1(1(4(3(0(x1)))))) |
(55) |
|
2(1(0(3(x1)))) |
→ |
4(2(1(4(3(0(x1)))))) |
(56) |
|
2(1(0(3(x1)))) |
→ |
2(1(4(4(3(0(x1)))))) |
(57) |
|
1(2(0(3(x1)))) |
→ |
0(1(2(3(0(x1))))) |
(58) |
|
1(2(0(3(x1)))) |
→ |
4(1(2(3(0(x1))))) |
(59) |
|
1(2(0(3(x1)))) |
→ |
0(1(2(4(3(0(x1)))))) |
(60) |
|
2(1(0(1(0(x1))))) |
→ |
3(1(1(2(0(0(x1)))))) |
(61) |
|
2(4(1(1(0(x1))))) |
→ |
2(4(1(3(0(1(x1)))))) |
(62) |
|
1(3(2(1(0(x1))))) |
→ |
2(2(1(3(0(1(x1)))))) |
(63) |
|
1(3(2(1(0(x1))))) |
→ |
2(1(3(0(4(1(x1)))))) |
(64) |
|
1(3(2(1(0(x1))))) |
→ |
1(2(3(0(1(2(x1)))))) |
(65) |
|
1(3(5(1(0(x1))))) |
→ |
1(4(5(1(3(0(x1)))))) |
(66) |
|
1(2(3(2(0(x1))))) |
→ |
2(1(4(2(3(0(x1)))))) |
(67) |
|
1(3(4(2(0(x1))))) |
→ |
0(2(1(4(3(0(x1)))))) |
(68) |
|
1(3(4(2(0(x1))))) |
→ |
2(4(3(0(0(1(x1)))))) |
(69) |
|
2(1(3(3(0(x1))))) |
→ |
2(3(0(1(0(3(x1)))))) |
(70) |
|
1(2(3(3(0(x1))))) |
→ |
4(1(3(2(3(0(x1)))))) |
(71) |
|
1(3(2(5(0(x1))))) |
→ |
4(5(1(2(3(0(x1)))))) |
(72) |
|
1(3(2(5(0(x1))))) |
→ |
1(2(3(0(5(0(x1)))))) |
(73) |
|
1(3(4(5(0(x1))))) |
→ |
5(4(1(4(3(0(x1)))))) |
(74) |
|
2(1(0(4(1(x1))))) |
→ |
1(3(0(4(1(2(x1)))))) |
(75) |
|
2(0(1(0(3(x1))))) |
→ |
1(4(0(2(3(0(x1)))))) |
(76) |
|
1(3(2(0(3(x1))))) |
→ |
2(1(3(0(0(3(x1)))))) |
(77) |
|
1(5(2(0(3(x1))))) |
→ |
1(0(5(2(3(0(x1)))))) |
(78) |
|
1(5(2(0(3(x1))))) |
→ |
4(1(2(5(3(0(x1)))))) |
(79) |
|
1(5(2(0(3(x1))))) |
→ |
1(2(3(0(4(5(x1)))))) |
(80) |
|
1(5(4(0(3(x1))))) |
→ |
1(4(3(0(0(5(x1)))))) |
(81) |
|
4(1(0(3(3(x1))))) |
→ |
4(1(3(0(0(3(x1)))))) |
(82) |
|
1(1(1(3(3(x1))))) |
→ |
1(1(3(1(3(0(x1)))))) |
(83) |
|
2(1(1(3(3(x1))))) |
→ |
3(4(1(2(3(1(x1)))))) |
(84) |
|
4(1(1(3(3(x1))))) |
→ |
1(3(1(3(4(4(x1)))))) |
(85) |
|
1(2(3(4(3(x1))))) |
→ |
1(4(2(3(0(3(x1)))))) |
(86) |
|
1(2(0(5(3(x1))))) |
→ |
1(5(0(2(3(0(x1)))))) |
(87) |
|
1(2(0(5(3(x1))))) |
→ |
1(4(2(5(3(0(x1)))))) |
(88) |
|
4(1(1(0(4(x1))))) |
→ |
1(4(4(3(0(1(x1)))))) |
(89) |
|
1(2(3(5(4(x1))))) |
→ |
4(2(5(1(3(0(x1)))))) |
(90) |
|
4(1(1(0(5(x1))))) |
→ |
4(1(5(1(3(0(x1)))))) |
(91) |
|
1(3(2(0(5(x1))))) |
→ |
0(1(2(3(0(5(x1)))))) |
(92) |
|
1(1(0(3(5(x1))))) |
→ |
0(5(1(1(3(0(x1)))))) |
(93) |
|
1(2(0(3(5(x1))))) |
→ |
5(1(2(4(3(0(x1)))))) |
(94) |
|
1(2(0(3(5(x1))))) |
→ |
4(2(1(5(3(0(x1)))))) |
(95) |
|
2(1(1(3(5(x1))))) |
→ |
5(4(1(2(1(3(x1)))))) |
(96) |
|
4(1(1(3(5(x1))))) |
→ |
1(2(5(1(4(3(x1)))))) |
(97) |
|
1(4(3(4(5(x1))))) |
→ |
5(1(4(4(3(0(x1)))))) |
(98) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
4#(1(1(3(3(x1))))) |
→ |
4#(4(x1)) |
(201) |
|
4#(1(1(0(4(x1))))) |
→ |
1#(x1) |
(212) |
|
1#(3(2(1(0(x1))))) |
→ |
1#(2(x1)) |
(143) |
|
1#(3(2(1(0(x1))))) |
→ |
2#(x1) |
(142) |
|
2#(1(1(3(5(x1))))) |
→ |
4#(1(2(1(3(x1))))) |
(235) |
|
4#(1(1(3(3(x1))))) |
→ |
1#(3(1(3(4(4(x1)))))) |
(203) |
|
1#(3(2(1(0(x1))))) |
→ |
4#(1(x1)) |
(139) |
|
4#(1(1(3(3(x1))))) |
→ |
1#(3(4(4(x1)))) |
(202) |
|
1#(3(2(1(0(x1))))) |
→ |
1#(x1) |
(135) |
|
1#(3(2(0(x1)))) |
→ |
4#(x1) |
(106) |
|
4#(1(1(3(3(x1))))) |
→ |
4#(x1) |
(200) |
|
2#(1(1(3(5(x1))))) |
→ |
1#(2(1(3(x1)))) |
(234) |
|
2#(1(1(3(5(x1))))) |
→ |
2#(1(3(x1))) |
(233) |
|
2#(1(1(3(5(x1))))) |
→ |
1#(3(x1)) |
(232) |
|
2#(1(1(3(3(x1))))) |
→ |
4#(1(2(3(1(x1))))) |
(199) |
|
2#(1(1(3(3(x1))))) |
→ |
1#(x1) |
(196) |
|
2#(1(0(4(1(x1))))) |
→ |
4#(1(2(x1))) |
(174) |
|
2#(1(0(4(1(x1))))) |
→ |
1#(2(x1)) |
(173) |
|
2#(1(0(4(1(x1))))) |
→ |
2#(x1) |
(172) |
|
2#(4(1(1(0(x1))))) |
→ |
1#(x1) |
(131) |
could be deleted.
1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
|
1#(3(4(2(0(x1))))) |
→ |
1#(x1) |
(156) |
|
| 1 |
> |
1 |
|
1#(3(2(0(x1)))) |
→ |
1#(x1) |
(103) |
|
| 1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.