Certification Problem
Input (TPDB SRS_Standard/Zantema_04/z123)
The rewrite relation of the following TRS is considered.
|
a(a(x1)) |
→ |
b(b(b(x1))) |
(1) |
|
a(x1) |
→ |
d(c(d(x1))) |
(2) |
|
b(b(b(x1))) |
→ |
a(f(x1)) |
(3) |
|
b(b(x1)) |
→ |
c(c(c(x1))) |
(4) |
|
c(c(x1)) |
→ |
d(d(d(x1))) |
(5) |
|
c(d(d(x1))) |
→ |
f(x1) |
(6) |
|
f(f(x1)) |
→ |
f(a(x1)) |
(7) |
Property / Task
Prove or disprove termination.Answer / Result
Yes.Proof (by matchbox @ termCOMP 2023)
1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [f(x1)] |
= |
x1 +
|
| [d(x1)] |
= |
x1 +
|
| [c(x1)] |
= |
x1 +
|
| [b(x1)] |
= |
x1 +
|
| [a(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
b(b(x1)) |
→ |
c(c(c(x1))) |
(4) |
|
c(c(x1)) |
→ |
d(d(d(x1))) |
(5) |
1.1 Closure Under Flat Contexts
Using the flat contexts
{f(☐), d(☐), c(☐), b(☐), a(☐)}
We obtain the transformed TRS
|
f(a(a(x1))) |
→ |
f(b(b(b(x1)))) |
(8) |
|
f(a(x1)) |
→ |
f(d(c(d(x1)))) |
(9) |
|
f(b(b(b(x1)))) |
→ |
f(a(f(x1))) |
(10) |
|
f(c(d(d(x1)))) |
→ |
f(f(x1)) |
(11) |
|
f(f(f(x1))) |
→ |
f(f(a(x1))) |
(12) |
|
d(a(a(x1))) |
→ |
d(b(b(b(x1)))) |
(13) |
|
d(a(x1)) |
→ |
d(d(c(d(x1)))) |
(14) |
|
d(b(b(b(x1)))) |
→ |
d(a(f(x1))) |
(15) |
|
d(c(d(d(x1)))) |
→ |
d(f(x1)) |
(16) |
|
d(f(f(x1))) |
→ |
d(f(a(x1))) |
(17) |
|
c(a(a(x1))) |
→ |
c(b(b(b(x1)))) |
(18) |
|
c(a(x1)) |
→ |
c(d(c(d(x1)))) |
(19) |
|
c(b(b(b(x1)))) |
→ |
c(a(f(x1))) |
(20) |
|
c(c(d(d(x1)))) |
→ |
c(f(x1)) |
(21) |
|
c(f(f(x1))) |
→ |
c(f(a(x1))) |
(22) |
|
b(a(a(x1))) |
→ |
b(b(b(b(x1)))) |
(23) |
|
b(a(x1)) |
→ |
b(d(c(d(x1)))) |
(24) |
|
b(b(b(b(x1)))) |
→ |
b(a(f(x1))) |
(25) |
|
b(c(d(d(x1)))) |
→ |
b(f(x1)) |
(26) |
|
b(f(f(x1))) |
→ |
b(f(a(x1))) |
(27) |
|
a(a(a(x1))) |
→ |
a(b(b(b(x1)))) |
(28) |
|
a(a(x1)) |
→ |
a(d(c(d(x1)))) |
(29) |
|
a(b(b(b(x1)))) |
→ |
a(a(f(x1))) |
(30) |
|
a(c(d(d(x1)))) |
→ |
a(f(x1)) |
(31) |
|
a(f(f(x1))) |
→ |
a(f(a(x1))) |
(32) |
1.1.1 Semantic Labeling
The following interpretations form a
model
of the rules.
As carrier we take the set
{0,...,4}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 5):
| [f(x1)] |
= |
5x1 + 0 |
| [d(x1)] |
= |
5x1 + 1 |
| [c(x1)] |
= |
5x1 + 2 |
| [b(x1)] |
= |
5x1 + 3 |
| [a(x1)] |
= |
5x1 + 4 |
We obtain the labeled TRS
There are 125 ruless (increase limit for explicit display).
1.1.1.1 Rule Removal
Using the
matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [f0(x1)] |
= |
x1 +
|
| [f1(x1)] |
= |
x1 +
|
| [f2(x1)] |
= |
x1 +
|
| [f3(x1)] |
= |
x1 +
|
| [f4(x1)] |
= |
x1 +
|
| [d0(x1)] |
= |
x1 +
|
| [d1(x1)] |
= |
x1 +
|
| [d2(x1)] |
= |
x1 +
|
| [d3(x1)] |
= |
x1 +
|
| [d4(x1)] |
= |
x1 +
|
| [c0(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [c2(x1)] |
= |
x1 +
|
| [c3(x1)] |
= |
x1 +
|
| [c4(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b1(x1)] |
= |
x1 +
|
| [b2(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [b4(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a2(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
| [a4(x1)] |
= |
x1 +
|
all of the following rules can be deleted.
|
a4(a4(a4(x1))) |
→ |
a3(b3(b3(b4(x1)))) |
(33) |
|
a4(a4(a3(x1))) |
→ |
a3(b3(b3(b3(x1)))) |
(34) |
|
a4(a4(a1(x1))) |
→ |
a3(b3(b3(b1(x1)))) |
(35) |
|
a4(a4(a2(x1))) |
→ |
a3(b3(b3(b2(x1)))) |
(36) |
|
b4(a4(a4(x1))) |
→ |
b3(b3(b3(b4(x1)))) |
(38) |
|
b4(a4(a3(x1))) |
→ |
b3(b3(b3(b3(x1)))) |
(39) |
|
b4(a4(a1(x1))) |
→ |
b3(b3(b3(b1(x1)))) |
(40) |
|
b4(a4(a2(x1))) |
→ |
b3(b3(b3(b2(x1)))) |
(41) |
|
d4(a4(a4(x1))) |
→ |
d3(b3(b3(b4(x1)))) |
(43) |
|
d4(a4(a3(x1))) |
→ |
d3(b3(b3(b3(x1)))) |
(44) |
|
d4(a4(a1(x1))) |
→ |
d3(b3(b3(b1(x1)))) |
(45) |
|
d4(a4(a2(x1))) |
→ |
d3(b3(b3(b2(x1)))) |
(46) |
|
c4(a4(a4(x1))) |
→ |
c3(b3(b3(b4(x1)))) |
(48) |
|
c4(a4(a3(x1))) |
→ |
c3(b3(b3(b3(x1)))) |
(49) |
|
c4(a4(a1(x1))) |
→ |
c3(b3(b3(b1(x1)))) |
(50) |
|
c4(a4(a2(x1))) |
→ |
c3(b3(b3(b2(x1)))) |
(51) |
|
f4(a4(a4(x1))) |
→ |
f3(b3(b3(b4(x1)))) |
(53) |
|
f4(a4(a3(x1))) |
→ |
f3(b3(b3(b3(x1)))) |
(54) |
|
f4(a4(a1(x1))) |
→ |
f3(b3(b3(b1(x1)))) |
(55) |
|
f4(a4(a2(x1))) |
→ |
f3(b3(b3(b2(x1)))) |
(56) |
|
a4(a2(x1)) |
→ |
a1(d2(c1(d2(x1)))) |
(61) |
|
a4(a0(x1)) |
→ |
a1(d2(c1(d0(x1)))) |
(62) |
|
b4(a4(x1)) |
→ |
b1(d2(c1(d4(x1)))) |
(63) |
|
b4(a3(x1)) |
→ |
b1(d2(c1(d3(x1)))) |
(64) |
|
b4(a1(x1)) |
→ |
b1(d2(c1(d1(x1)))) |
(65) |
|
b4(a2(x1)) |
→ |
b1(d2(c1(d2(x1)))) |
(66) |
|
b4(a0(x1)) |
→ |
b1(d2(c1(d0(x1)))) |
(67) |
|
d4(a2(x1)) |
→ |
d1(d2(c1(d2(x1)))) |
(71) |
|
d4(a0(x1)) |
→ |
d1(d2(c1(d0(x1)))) |
(72) |
|
c4(a4(x1)) |
→ |
c1(d2(c1(d4(x1)))) |
(73) |
|
c4(a3(x1)) |
→ |
c1(d2(c1(d3(x1)))) |
(74) |
|
c4(a1(x1)) |
→ |
c1(d2(c1(d1(x1)))) |
(75) |
|
c4(a2(x1)) |
→ |
c1(d2(c1(d2(x1)))) |
(76) |
|
c4(a0(x1)) |
→ |
c1(d2(c1(d0(x1)))) |
(77) |
|
f4(a2(x1)) |
→ |
f1(d2(c1(d2(x1)))) |
(81) |
|
f4(a0(x1)) |
→ |
f1(d2(c1(d0(x1)))) |
(82) |
|
a3(b3(b3(b4(x1)))) |
→ |
a4(a0(f4(x1))) |
(83) |
|
a3(b3(b3(b3(x1)))) |
→ |
a4(a0(f3(x1))) |
(84) |
|
a3(b3(b3(b1(x1)))) |
→ |
a4(a0(f1(x1))) |
(85) |
|
a3(b3(b3(b2(x1)))) |
→ |
a4(a0(f2(x1))) |
(86) |
|
b3(b3(b3(b4(x1)))) |
→ |
b4(a0(f4(x1))) |
(88) |
|
b3(b3(b3(b3(x1)))) |
→ |
b4(a0(f3(x1))) |
(89) |
|
b3(b3(b3(b1(x1)))) |
→ |
b4(a0(f1(x1))) |
(90) |
|
b3(b3(b3(b2(x1)))) |
→ |
b4(a0(f2(x1))) |
(91) |
|
d3(b3(b3(b4(x1)))) |
→ |
d4(a0(f4(x1))) |
(93) |
|
d3(b3(b3(b3(x1)))) |
→ |
d4(a0(f3(x1))) |
(94) |
|
d3(b3(b3(b1(x1)))) |
→ |
d4(a0(f1(x1))) |
(95) |
|
d3(b3(b3(b2(x1)))) |
→ |
d4(a0(f2(x1))) |
(96) |
|
c3(b3(b3(b4(x1)))) |
→ |
c4(a0(f4(x1))) |
(98) |
|
c3(b3(b3(b3(x1)))) |
→ |
c4(a0(f3(x1))) |
(99) |
|
c3(b3(b3(b1(x1)))) |
→ |
c4(a0(f1(x1))) |
(100) |
|
c3(b3(b3(b2(x1)))) |
→ |
c4(a0(f2(x1))) |
(101) |
|
f3(b3(b3(b4(x1)))) |
→ |
f4(a0(f4(x1))) |
(103) |
|
f3(b3(b3(b3(x1)))) |
→ |
f4(a0(f3(x1))) |
(104) |
|
f3(b3(b3(b1(x1)))) |
→ |
f4(a0(f1(x1))) |
(105) |
|
f3(b3(b3(b2(x1)))) |
→ |
f4(a0(f2(x1))) |
(106) |
|
a2(c1(d1(d4(x1)))) |
→ |
a0(f4(x1)) |
(108) |
|
a2(c1(d1(d3(x1)))) |
→ |
a0(f3(x1)) |
(109) |
|
a2(c1(d1(d1(x1)))) |
→ |
a0(f1(x1)) |
(110) |
|
a2(c1(d1(d2(x1)))) |
→ |
a0(f2(x1)) |
(111) |
|
a2(c1(d1(d0(x1)))) |
→ |
a0(f0(x1)) |
(112) |
|
b2(c1(d1(d4(x1)))) |
→ |
b0(f4(x1)) |
(113) |
|
b2(c1(d1(d3(x1)))) |
→ |
b0(f3(x1)) |
(114) |
|
b2(c1(d1(d1(x1)))) |
→ |
b0(f1(x1)) |
(115) |
|
b2(c1(d1(d2(x1)))) |
→ |
b0(f2(x1)) |
(116) |
|
b2(c1(d1(d0(x1)))) |
→ |
b0(f0(x1)) |
(117) |
|
d2(c1(d1(d4(x1)))) |
→ |
d0(f4(x1)) |
(118) |
|
d2(c1(d1(d3(x1)))) |
→ |
d0(f3(x1)) |
(119) |
|
d2(c1(d1(d1(x1)))) |
→ |
d0(f1(x1)) |
(120) |
|
c2(c1(d1(d4(x1)))) |
→ |
c0(f4(x1)) |
(123) |
|
c2(c1(d1(d3(x1)))) |
→ |
c0(f3(x1)) |
(124) |
|
c2(c1(d1(d1(x1)))) |
→ |
c0(f1(x1)) |
(125) |
|
c2(c1(d1(d2(x1)))) |
→ |
c0(f2(x1)) |
(126) |
|
c2(c1(d1(d0(x1)))) |
→ |
c0(f0(x1)) |
(127) |
|
f2(c1(d1(d4(x1)))) |
→ |
f0(f4(x1)) |
(128) |
|
f2(c1(d1(d3(x1)))) |
→ |
f0(f3(x1)) |
(129) |
|
f2(c1(d1(d1(x1)))) |
→ |
f0(f1(x1)) |
(130) |
|
a0(f0(f2(x1))) |
→ |
a0(f4(a2(x1))) |
(136) |
|
a0(f0(f0(x1))) |
→ |
a0(f4(a0(x1))) |
(137) |
|
b0(f0(f2(x1))) |
→ |
b0(f4(a2(x1))) |
(141) |
|
b0(f0(f0(x1))) |
→ |
b0(f4(a0(x1))) |
(142) |
|
d0(f0(f2(x1))) |
→ |
d0(f4(a2(x1))) |
(146) |
|
d0(f0(f0(x1))) |
→ |
d0(f4(a0(x1))) |
(147) |
|
c0(f0(f2(x1))) |
→ |
c0(f4(a2(x1))) |
(151) |
|
c0(f0(f0(x1))) |
→ |
c0(f4(a0(x1))) |
(152) |
|
f0(f0(f2(x1))) |
→ |
f0(f4(a2(x1))) |
(156) |
|
f0(f0(f0(x1))) |
→ |
f0(f4(a0(x1))) |
(157) |
1.1.1.1.1 String Reversal
Since only unary symbols occur, one can reverse all terms and obtains the TRS
|
a0(a4(a4(x1))) |
→ |
b0(b3(b3(a3(x1)))) |
(158) |
|
a0(a4(b4(x1))) |
→ |
b0(b3(b3(b3(x1)))) |
(159) |
|
a0(a4(d4(x1))) |
→ |
b0(b3(b3(d3(x1)))) |
(160) |
|
a0(a4(c4(x1))) |
→ |
b0(b3(b3(c3(x1)))) |
(161) |
|
a0(a4(f4(x1))) |
→ |
b0(b3(b3(f3(x1)))) |
(162) |
|
a4(a4(x1)) |
→ |
d4(c1(d2(a1(x1)))) |
(163) |
|
a3(a4(x1)) |
→ |
d3(c1(d2(a1(x1)))) |
(164) |
|
a1(a4(x1)) |
→ |
d1(c1(d2(a1(x1)))) |
(165) |
|
a4(d4(x1)) |
→ |
d4(c1(d2(d1(x1)))) |
(166) |
|
a3(d4(x1)) |
→ |
d3(c1(d2(d1(x1)))) |
(167) |
|
a1(d4(x1)) |
→ |
d1(c1(d2(d1(x1)))) |
(168) |
|
a4(f4(x1)) |
→ |
d4(c1(d2(f1(x1)))) |
(169) |
|
a3(f4(x1)) |
→ |
d3(c1(d2(f1(x1)))) |
(170) |
|
a1(f4(x1)) |
→ |
d1(c1(d2(f1(x1)))) |
(171) |
|
b0(b3(b3(a3(x1)))) |
→ |
f0(a0(a4(x1))) |
(172) |
|
b0(b3(b3(b3(x1)))) |
→ |
f0(a0(b4(x1))) |
(173) |
|
b0(b3(b3(d3(x1)))) |
→ |
f0(a0(d4(x1))) |
(174) |
|
b0(b3(b3(c3(x1)))) |
→ |
f0(a0(c4(x1))) |
(175) |
|
b0(b3(b3(f3(x1)))) |
→ |
f0(a0(f4(x1))) |
(176) |
|
d2(d1(c1(d2(x1)))) |
→ |
f2(d0(x1)) |
(177) |
|
d0(d1(c1(d2(x1)))) |
→ |
f0(d0(x1)) |
(178) |
|
d2(d1(c1(f2(x1)))) |
→ |
f2(f0(x1)) |
(179) |
|
d0(d1(c1(f2(x1)))) |
→ |
f0(f0(x1)) |
(180) |
|
f4(f0(a0(x1))) |
→ |
a4(f4(a0(x1))) |
(181) |
|
f3(f0(a0(x1))) |
→ |
a3(f4(a0(x1))) |
(182) |
|
f1(f0(a0(x1))) |
→ |
a1(f4(a0(x1))) |
(183) |
|
f4(f0(b0(x1))) |
→ |
a4(f4(b0(x1))) |
(184) |
|
f3(f0(b0(x1))) |
→ |
a3(f4(b0(x1))) |
(185) |
|
f1(f0(b0(x1))) |
→ |
a1(f4(b0(x1))) |
(186) |
|
f4(f0(d0(x1))) |
→ |
a4(f4(d0(x1))) |
(187) |
|
f3(f0(d0(x1))) |
→ |
a3(f4(d0(x1))) |
(188) |
|
f1(f0(d0(x1))) |
→ |
a1(f4(d0(x1))) |
(189) |
|
f4(f0(c0(x1))) |
→ |
a4(f4(c0(x1))) |
(190) |
|
f3(f0(c0(x1))) |
→ |
a3(f4(c0(x1))) |
(191) |
|
f1(f0(c0(x1))) |
→ |
a1(f4(c0(x1))) |
(192) |
|
f4(f0(f0(x1))) |
→ |
a4(f4(f0(x1))) |
(193) |
|
f3(f0(f0(x1))) |
→ |
a3(f4(f0(x1))) |
(194) |
|
f1(f0(f0(x1))) |
→ |
a1(f4(f0(x1))) |
(195) |
1.1.1.1.1.1 Dependency Pair Transformation
The following set of initial dependency pairs has been identified.
|
f1#(f0(f0(x1))) |
→ |
f4#(f0(x1)) |
(196) |
|
f1#(f0(f0(x1))) |
→ |
a1#(f4(f0(x1))) |
(197) |
|
f1#(f0(d0(x1))) |
→ |
f4#(d0(x1)) |
(198) |
|
f1#(f0(d0(x1))) |
→ |
a1#(f4(d0(x1))) |
(199) |
|
f1#(f0(c0(x1))) |
→ |
f4#(c0(x1)) |
(200) |
|
f1#(f0(c0(x1))) |
→ |
a1#(f4(c0(x1))) |
(201) |
|
f1#(f0(b0(x1))) |
→ |
f4#(b0(x1)) |
(202) |
|
f1#(f0(b0(x1))) |
→ |
a1#(f4(b0(x1))) |
(203) |
|
f1#(f0(a0(x1))) |
→ |
f4#(a0(x1)) |
(204) |
|
f1#(f0(a0(x1))) |
→ |
a1#(f4(a0(x1))) |
(205) |
|
f3#(f0(f0(x1))) |
→ |
f4#(f0(x1)) |
(206) |
|
f3#(f0(f0(x1))) |
→ |
a3#(f4(f0(x1))) |
(207) |
|
f3#(f0(d0(x1))) |
→ |
f4#(d0(x1)) |
(208) |
|
f3#(f0(d0(x1))) |
→ |
a3#(f4(d0(x1))) |
(209) |
|
f3#(f0(c0(x1))) |
→ |
f4#(c0(x1)) |
(210) |
|
f3#(f0(c0(x1))) |
→ |
a3#(f4(c0(x1))) |
(211) |
|
f3#(f0(b0(x1))) |
→ |
f4#(b0(x1)) |
(212) |
|
f3#(f0(b0(x1))) |
→ |
a3#(f4(b0(x1))) |
(213) |
|
f3#(f0(a0(x1))) |
→ |
f4#(a0(x1)) |
(214) |
|
f3#(f0(a0(x1))) |
→ |
a3#(f4(a0(x1))) |
(215) |
|
f4#(f0(f0(x1))) |
→ |
f4#(f0(x1)) |
(216) |
|
f4#(f0(f0(x1))) |
→ |
a4#(f4(f0(x1))) |
(217) |
|
f4#(f0(d0(x1))) |
→ |
f4#(d0(x1)) |
(218) |
|
f4#(f0(d0(x1))) |
→ |
a4#(f4(d0(x1))) |
(219) |
|
f4#(f0(c0(x1))) |
→ |
f4#(c0(x1)) |
(220) |
|
f4#(f0(c0(x1))) |
→ |
a4#(f4(c0(x1))) |
(221) |
|
f4#(f0(b0(x1))) |
→ |
f4#(b0(x1)) |
(222) |
|
f4#(f0(b0(x1))) |
→ |
a4#(f4(b0(x1))) |
(223) |
|
f4#(f0(a0(x1))) |
→ |
f4#(a0(x1)) |
(224) |
|
f4#(f0(a0(x1))) |
→ |
a4#(f4(a0(x1))) |
(225) |
|
d0#(d1(c1(d2(x1)))) |
→ |
d0#(x1) |
(226) |
|
d2#(d1(c1(d2(x1)))) |
→ |
d0#(x1) |
(227) |
|
b0#(b3(b3(f3(x1)))) |
→ |
f4#(x1) |
(228) |
|
b0#(b3(b3(f3(x1)))) |
→ |
a0#(f4(x1)) |
(229) |
|
b0#(b3(b3(d3(x1)))) |
→ |
a0#(d4(x1)) |
(230) |
|
b0#(b3(b3(c3(x1)))) |
→ |
a0#(c4(x1)) |
(231) |
|
b0#(b3(b3(b3(x1)))) |
→ |
a0#(b4(x1)) |
(232) |
|
b0#(b3(b3(a3(x1)))) |
→ |
a0#(a4(x1)) |
(233) |
|
b0#(b3(b3(a3(x1)))) |
→ |
a4#(x1) |
(234) |
|
a0#(a4(f4(x1))) |
→ |
f3#(x1) |
(235) |
|
a0#(a4(f4(x1))) |
→ |
b0#(b3(b3(f3(x1)))) |
(236) |
|
a0#(a4(d4(x1))) |
→ |
b0#(b3(b3(d3(x1)))) |
(237) |
|
a0#(a4(c4(x1))) |
→ |
b0#(b3(b3(c3(x1)))) |
(238) |
|
a0#(a4(b4(x1))) |
→ |
b0#(b3(b3(b3(x1)))) |
(239) |
|
a0#(a4(a4(x1))) |
→ |
b0#(b3(b3(a3(x1)))) |
(240) |
|
a0#(a4(a4(x1))) |
→ |
a3#(x1) |
(241) |
|
a1#(f4(x1)) |
→ |
f1#(x1) |
(242) |
|
a1#(f4(x1)) |
→ |
d2#(f1(x1)) |
(243) |
|
a1#(d4(x1)) |
→ |
d2#(d1(x1)) |
(244) |
|
a1#(a4(x1)) |
→ |
d2#(a1(x1)) |
(245) |
|
a1#(a4(x1)) |
→ |
a1#(x1) |
(246) |
|
a3#(f4(x1)) |
→ |
f1#(x1) |
(247) |
|
a3#(f4(x1)) |
→ |
d2#(f1(x1)) |
(248) |
|
a3#(d4(x1)) |
→ |
d2#(d1(x1)) |
(249) |
|
a3#(a4(x1)) |
→ |
d2#(a1(x1)) |
(250) |
|
a3#(a4(x1)) |
→ |
a1#(x1) |
(251) |
|
a4#(f4(x1)) |
→ |
f1#(x1) |
(252) |
|
a4#(f4(x1)) |
→ |
d2#(f1(x1)) |
(253) |
|
a4#(d4(x1)) |
→ |
d2#(d1(x1)) |
(254) |
|
a4#(a4(x1)) |
→ |
d2#(a1(x1)) |
(255) |
|
a4#(a4(x1)) |
→ |
a1#(x1) |
(256) |
1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
| [f0(x1)] |
= |
x1 +
|
| [f1(x1)] |
= |
x1 +
|
| [f2(x1)] |
= |
x1 +
|
| [f3(x1)] |
= |
x1 +
|
| [f4(x1)] |
= |
x1 +
|
| [d0(x1)] |
= |
x1 +
|
| [d1(x1)] |
= |
x1 +
|
| [d2(x1)] |
= |
x1 +
|
| [d3(x1)] |
= |
x1 +
|
| [d4(x1)] |
= |
x1 +
|
| [c0(x1)] |
= |
x1 +
|
| [c1(x1)] |
= |
x1 +
|
| [c3(x1)] |
= |
x1 +
|
| [c4(x1)] |
= |
x1 +
|
| [b0(x1)] |
= |
x1 +
|
| [b3(x1)] |
= |
x1 +
|
| [b4(x1)] |
= |
x1 +
|
| [a0(x1)] |
= |
x1 +
|
| [a1(x1)] |
= |
x1 +
|
| [a3(x1)] |
= |
x1 +
|
| [a4(x1)] |
= |
x1 +
|
| [f1#(x1)] |
= |
x1 +
|
| [f3#(x1)] |
= |
x1 +
|
| [f4#(x1)] |
= |
x1 +
|
| [d0#(x1)] |
= |
x1 +
|
| [d2#(x1)] |
= |
x1 +
|
| [b0#(x1)] |
= |
x1 +
|
| [a0#(x1)] |
= |
x1 +
|
| [a1#(x1)] |
= |
x1 +
|
| [a3#(x1)] |
= |
x1 +
|
| [a4#(x1)] |
= |
x1 +
|
together with the usable
rules
|
a0(a4(a4(x1))) |
→ |
b0(b3(b3(a3(x1)))) |
(158) |
|
a0(a4(b4(x1))) |
→ |
b0(b3(b3(b3(x1)))) |
(159) |
|
a0(a4(d4(x1))) |
→ |
b0(b3(b3(d3(x1)))) |
(160) |
|
a0(a4(c4(x1))) |
→ |
b0(b3(b3(c3(x1)))) |
(161) |
|
a0(a4(f4(x1))) |
→ |
b0(b3(b3(f3(x1)))) |
(162) |
|
a4(a4(x1)) |
→ |
d4(c1(d2(a1(x1)))) |
(163) |
|
a3(a4(x1)) |
→ |
d3(c1(d2(a1(x1)))) |
(164) |
|
a1(a4(x1)) |
→ |
d1(c1(d2(a1(x1)))) |
(165) |
|
a4(d4(x1)) |
→ |
d4(c1(d2(d1(x1)))) |
(166) |
|
a3(d4(x1)) |
→ |
d3(c1(d2(d1(x1)))) |
(167) |
|
a1(d4(x1)) |
→ |
d1(c1(d2(d1(x1)))) |
(168) |
|
a4(f4(x1)) |
→ |
d4(c1(d2(f1(x1)))) |
(169) |
|
a3(f4(x1)) |
→ |
d3(c1(d2(f1(x1)))) |
(170) |
|
a1(f4(x1)) |
→ |
d1(c1(d2(f1(x1)))) |
(171) |
|
b0(b3(b3(a3(x1)))) |
→ |
f0(a0(a4(x1))) |
(172) |
|
b0(b3(b3(b3(x1)))) |
→ |
f0(a0(b4(x1))) |
(173) |
|
b0(b3(b3(d3(x1)))) |
→ |
f0(a0(d4(x1))) |
(174) |
|
b0(b3(b3(c3(x1)))) |
→ |
f0(a0(c4(x1))) |
(175) |
|
b0(b3(b3(f3(x1)))) |
→ |
f0(a0(f4(x1))) |
(176) |
|
d2(d1(c1(d2(x1)))) |
→ |
f2(d0(x1)) |
(177) |
|
d0(d1(c1(d2(x1)))) |
→ |
f0(d0(x1)) |
(178) |
|
d2(d1(c1(f2(x1)))) |
→ |
f2(f0(x1)) |
(179) |
|
d0(d1(c1(f2(x1)))) |
→ |
f0(f0(x1)) |
(180) |
|
f4(f0(a0(x1))) |
→ |
a4(f4(a0(x1))) |
(181) |
|
f3(f0(a0(x1))) |
→ |
a3(f4(a0(x1))) |
(182) |
|
f1(f0(a0(x1))) |
→ |
a1(f4(a0(x1))) |
(183) |
|
f4(f0(b0(x1))) |
→ |
a4(f4(b0(x1))) |
(184) |
|
f3(f0(b0(x1))) |
→ |
a3(f4(b0(x1))) |
(185) |
|
f1(f0(b0(x1))) |
→ |
a1(f4(b0(x1))) |
(186) |
|
f4(f0(d0(x1))) |
→ |
a4(f4(d0(x1))) |
(187) |
|
f3(f0(d0(x1))) |
→ |
a3(f4(d0(x1))) |
(188) |
|
f1(f0(d0(x1))) |
→ |
a1(f4(d0(x1))) |
(189) |
|
f4(f0(c0(x1))) |
→ |
a4(f4(c0(x1))) |
(190) |
|
f3(f0(c0(x1))) |
→ |
a3(f4(c0(x1))) |
(191) |
|
f1(f0(c0(x1))) |
→ |
a1(f4(c0(x1))) |
(192) |
|
f4(f0(f0(x1))) |
→ |
a4(f4(f0(x1))) |
(193) |
|
f3(f0(f0(x1))) |
→ |
a3(f4(f0(x1))) |
(194) |
|
f1(f0(f0(x1))) |
→ |
a1(f4(f0(x1))) |
(195) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
|
f1#(f0(f0(x1))) |
→ |
f4#(f0(x1)) |
(196) |
|
f1#(f0(f0(x1))) |
→ |
a1#(f4(f0(x1))) |
(197) |
|
f1#(f0(d0(x1))) |
→ |
f4#(d0(x1)) |
(198) |
|
f1#(f0(d0(x1))) |
→ |
a1#(f4(d0(x1))) |
(199) |
|
f1#(f0(c0(x1))) |
→ |
f4#(c0(x1)) |
(200) |
|
f1#(f0(c0(x1))) |
→ |
a1#(f4(c0(x1))) |
(201) |
|
f1#(f0(b0(x1))) |
→ |
f4#(b0(x1)) |
(202) |
|
f1#(f0(b0(x1))) |
→ |
a1#(f4(b0(x1))) |
(203) |
|
f1#(f0(a0(x1))) |
→ |
f4#(a0(x1)) |
(204) |
|
f1#(f0(a0(x1))) |
→ |
a1#(f4(a0(x1))) |
(205) |
|
f3#(f0(f0(x1))) |
→ |
f4#(f0(x1)) |
(206) |
|
f3#(f0(f0(x1))) |
→ |
a3#(f4(f0(x1))) |
(207) |
|
f3#(f0(d0(x1))) |
→ |
f4#(d0(x1)) |
(208) |
|
f3#(f0(d0(x1))) |
→ |
a3#(f4(d0(x1))) |
(209) |
|
f3#(f0(c0(x1))) |
→ |
f4#(c0(x1)) |
(210) |
|
f3#(f0(c0(x1))) |
→ |
a3#(f4(c0(x1))) |
(211) |
|
f3#(f0(b0(x1))) |
→ |
f4#(b0(x1)) |
(212) |
|
f3#(f0(b0(x1))) |
→ |
a3#(f4(b0(x1))) |
(213) |
|
f3#(f0(a0(x1))) |
→ |
f4#(a0(x1)) |
(214) |
|
f3#(f0(a0(x1))) |
→ |
a3#(f4(a0(x1))) |
(215) |
|
f4#(f0(f0(x1))) |
→ |
f4#(f0(x1)) |
(216) |
|
f4#(f0(f0(x1))) |
→ |
a4#(f4(f0(x1))) |
(217) |
|
f4#(f0(d0(x1))) |
→ |
f4#(d0(x1)) |
(218) |
|
f4#(f0(d0(x1))) |
→ |
a4#(f4(d0(x1))) |
(219) |
|
f4#(f0(c0(x1))) |
→ |
f4#(c0(x1)) |
(220) |
|
f4#(f0(c0(x1))) |
→ |
a4#(f4(c0(x1))) |
(221) |
|
f4#(f0(b0(x1))) |
→ |
f4#(b0(x1)) |
(222) |
|
f4#(f0(b0(x1))) |
→ |
a4#(f4(b0(x1))) |
(223) |
|
f4#(f0(a0(x1))) |
→ |
f4#(a0(x1)) |
(224) |
|
f4#(f0(a0(x1))) |
→ |
a4#(f4(a0(x1))) |
(225) |
|
d0#(d1(c1(d2(x1)))) |
→ |
d0#(x1) |
(226) |
|
d2#(d1(c1(d2(x1)))) |
→ |
d0#(x1) |
(227) |
|
b0#(b3(b3(f3(x1)))) |
→ |
f4#(x1) |
(228) |
|
b0#(b3(b3(f3(x1)))) |
→ |
a0#(f4(x1)) |
(229) |
|
b0#(b3(b3(d3(x1)))) |
→ |
a0#(d4(x1)) |
(230) |
|
b0#(b3(b3(c3(x1)))) |
→ |
a0#(c4(x1)) |
(231) |
|
b0#(b3(b3(b3(x1)))) |
→ |
a0#(b4(x1)) |
(232) |
|
b0#(b3(b3(a3(x1)))) |
→ |
a0#(a4(x1)) |
(233) |
|
b0#(b3(b3(a3(x1)))) |
→ |
a4#(x1) |
(234) |
|
a0#(a4(f4(x1))) |
→ |
f3#(x1) |
(235) |
|
a0#(a4(f4(x1))) |
→ |
b0#(b3(b3(f3(x1)))) |
(236) |
|
a0#(a4(d4(x1))) |
→ |
b0#(b3(b3(d3(x1)))) |
(237) |
|
a0#(a4(c4(x1))) |
→ |
b0#(b3(b3(c3(x1)))) |
(238) |
|
a0#(a4(b4(x1))) |
→ |
b0#(b3(b3(b3(x1)))) |
(239) |
|
a0#(a4(a4(x1))) |
→ |
b0#(b3(b3(a3(x1)))) |
(240) |
|
a0#(a4(a4(x1))) |
→ |
a3#(x1) |
(241) |
|
a1#(f4(x1)) |
→ |
f1#(x1) |
(242) |
|
a1#(f4(x1)) |
→ |
d2#(f1(x1)) |
(243) |
|
a1#(d4(x1)) |
→ |
d2#(d1(x1)) |
(244) |
|
a1#(a4(x1)) |
→ |
d2#(a1(x1)) |
(245) |
|
a1#(a4(x1)) |
→ |
a1#(x1) |
(246) |
|
a3#(f4(x1)) |
→ |
f1#(x1) |
(247) |
|
a3#(f4(x1)) |
→ |
d2#(f1(x1)) |
(248) |
|
a3#(d4(x1)) |
→ |
d2#(d1(x1)) |
(249) |
|
a3#(a4(x1)) |
→ |
d2#(a1(x1)) |
(250) |
|
a3#(a4(x1)) |
→ |
a1#(x1) |
(251) |
|
a4#(f4(x1)) |
→ |
f1#(x1) |
(252) |
|
a4#(f4(x1)) |
→ |
d2#(f1(x1)) |
(253) |
|
a4#(d4(x1)) |
→ |
d2#(d1(x1)) |
(254) |
|
a4#(a4(x1)) |
→ |
d2#(a1(x1)) |
(255) |
|
a4#(a4(x1)) |
→ |
a1#(x1) |
(256) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0
components.