The rewrite relation of the following TRS is considered.
As carrier we take the set
{0,1,2}.
Symbols are labeled by the interpretation of their arguments using the interpretations
(modulo 3):
a2(a2(x1)) |
→ |
a2(x1) |
(17) |
a2(a1(x1)) |
→ |
a1(x1) |
(18) |
a2(a0(x1)) |
→ |
a0(x1) |
(19) |
b2(a2(x1)) |
→ |
b2(x1) |
(20) |
b2(a1(x1)) |
→ |
b1(x1) |
(21) |
b2(a0(x1)) |
→ |
b0(x1) |
(22) |
c2(a2(x1)) |
→ |
c2(x1) |
(23) |
c2(a1(x1)) |
→ |
c1(x1) |
(24) |
c2(a0(x1)) |
→ |
c0(x1) |
(25) |
a2(a1(b2(x1))) |
→ |
a1(b2(a0(c1(b2(a0(c2(x1))))))) |
(26) |
a2(a1(b1(x1))) |
→ |
a1(b2(a0(c1(b2(a0(c1(x1))))))) |
(27) |
a2(a1(b0(x1))) |
→ |
a1(b2(a0(c1(b2(a0(c0(x1))))))) |
(28) |
b2(a1(b2(x1))) |
→ |
b1(b2(a0(c1(b2(a0(c2(x1))))))) |
(29) |
b2(a1(b1(x1))) |
→ |
b1(b2(a0(c1(b2(a0(c1(x1))))))) |
(30) |
b2(a1(b0(x1))) |
→ |
b1(b2(a0(c1(b2(a0(c0(x1))))))) |
(31) |
c2(a1(b2(x1))) |
→ |
c1(b2(a0(c1(b2(a0(c2(x1))))))) |
(32) |
c2(a1(b1(x1))) |
→ |
c1(b2(a0(c1(b2(a0(c1(x1))))))) |
(33) |
c2(a1(b0(x1))) |
→ |
c1(b2(a0(c1(b2(a0(c0(x1))))))) |
(34) |
a1(b2(x1)) |
→ |
a2(x1) |
(35) |
a1(b1(x1)) |
→ |
a1(x1) |
(36) |
a1(b0(x1)) |
→ |
a0(x1) |
(37) |
b1(b2(x1)) |
→ |
b2(x1) |
(38) |
b1(b1(x1)) |
→ |
b1(x1) |
(39) |
b1(b0(x1)) |
→ |
b0(x1) |
(40) |
c1(b2(x1)) |
→ |
c2(x1) |
(41) |
c1(b1(x1)) |
→ |
c1(x1) |
(42) |
c1(b0(x1)) |
→ |
c0(x1) |
(43) |
a0(c0(c2(x1))) |
→ |
a2(x1) |
(44) |
a0(c0(c1(x1))) |
→ |
a1(x1) |
(45) |
a0(c0(c0(x1))) |
→ |
a0(x1) |
(46) |
b0(c0(c2(x1))) |
→ |
b2(x1) |
(47) |
b0(c0(c1(x1))) |
→ |
b1(x1) |
(48) |
b0(c0(c0(x1))) |
→ |
b0(x1) |
(49) |
c0(c0(c2(x1))) |
→ |
c2(x1) |
(50) |
c0(c0(c1(x1))) |
→ |
c1(x1) |
(51) |
c0(c0(c0(x1))) |
→ |
c0(x1) |
(52) |
a0(b2(x1)) |
→ |
b0(x1) |
(53) |
a0(c2(x1)) |
→ |
c0(x1) |
(54) |
b2(a1(a2(x1))) |
→ |
c2(a0(b2(c1(a0(b2(a1(x1))))))) |
(55) |
b1(a1(a2(x1))) |
→ |
c1(a0(b2(c1(a0(b2(a1(x1))))))) |
(56) |
b0(a1(a2(x1))) |
→ |
c0(a0(b2(c1(a0(b2(a1(x1))))))) |
(57) |
b2(a1(b2(x1))) |
→ |
c2(a0(b2(c1(a0(b2(b1(x1))))))) |
(58) |
b1(a1(b2(x1))) |
→ |
c1(a0(b2(c1(a0(b2(b1(x1))))))) |
(59) |
b0(a1(b2(x1))) |
→ |
c0(a0(b2(c1(a0(b2(b1(x1))))))) |
(60) |
b2(a1(c2(x1))) |
→ |
c2(a0(b2(c1(a0(b2(c1(x1))))))) |
(61) |
b1(a1(c2(x1))) |
→ |
c1(a0(b2(c1(a0(b2(c1(x1))))))) |
(62) |
b0(a1(c2(x1))) |
→ |
c0(a0(b2(c1(a0(b2(c1(x1))))))) |
(63) |
b2(a1(x1)) |
→ |
a2(x1) |
(64) |
b1(a1(x1)) |
→ |
a1(x1) |
(65) |
b2(b1(x1)) |
→ |
b2(x1) |
(66) |
b1(b1(x1)) |
→ |
b1(x1) |
(39) |
b0(b1(x1)) |
→ |
b0(x1) |
(67) |
b2(c1(x1)) |
→ |
c2(x1) |
(68) |
b1(c1(x1)) |
→ |
c1(x1) |
(69) |
b0(c1(x1)) |
→ |
c0(x1) |
(70) |
c2(c0(a0(x1))) |
→ |
a2(x1) |
(71) |
c1(c0(a0(x1))) |
→ |
a1(x1) |
(72) |
b0#(b1(x1)) |
→ |
b0#(x1) |
(73) |
b0#(a1(c2(x1))) |
→ |
c1#(x1) |
(74) |
b0#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(x1)))) |
(75) |
b0#(a1(c2(x1))) |
→ |
b2#(c1(x1)) |
(76) |
b0#(a1(c2(x1))) |
→ |
b2#(c1(a0(b2(c1(x1))))) |
(77) |
b0#(a1(c2(x1))) |
→ |
a0#(b2(c1(x1))) |
(78) |
b0#(a1(c2(x1))) |
→ |
a0#(b2(c1(a0(b2(c1(x1)))))) |
(79) |
b0#(a1(b2(x1))) |
→ |
c1#(a0(b2(b1(x1)))) |
(80) |
b0#(a1(b2(x1))) |
→ |
b1#(x1) |
(81) |
b0#(a1(b2(x1))) |
→ |
b2#(c1(a0(b2(b1(x1))))) |
(82) |
b0#(a1(b2(x1))) |
→ |
b2#(b1(x1)) |
(83) |
b0#(a1(b2(x1))) |
→ |
a0#(b2(c1(a0(b2(b1(x1)))))) |
(84) |
b0#(a1(b2(x1))) |
→ |
a0#(b2(b1(x1))) |
(85) |
b0#(a1(a2(x1))) |
→ |
c1#(a0(b2(a1(x1)))) |
(86) |
b0#(a1(a2(x1))) |
→ |
b2#(c1(a0(b2(a1(x1))))) |
(87) |
b0#(a1(a2(x1))) |
→ |
b2#(a1(x1)) |
(88) |
b0#(a1(a2(x1))) |
→ |
a0#(b2(c1(a0(b2(a1(x1)))))) |
(89) |
b0#(a1(a2(x1))) |
→ |
a0#(b2(a1(x1))) |
(90) |
b1#(a1(c2(x1))) |
→ |
c1#(x1) |
(91) |
b1#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(x1)))) |
(92) |
b1#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(a0(b2(c1(x1))))))) |
(93) |
b1#(a1(c2(x1))) |
→ |
b2#(c1(x1)) |
(94) |
b1#(a1(c2(x1))) |
→ |
b2#(c1(a0(b2(c1(x1))))) |
(95) |
b1#(a1(c2(x1))) |
→ |
a0#(b2(c1(x1))) |
(96) |
b1#(a1(c2(x1))) |
→ |
a0#(b2(c1(a0(b2(c1(x1)))))) |
(97) |
b1#(a1(b2(x1))) |
→ |
c1#(a0(b2(c1(a0(b2(b1(x1))))))) |
(98) |
b1#(a1(b2(x1))) |
→ |
c1#(a0(b2(b1(x1)))) |
(99) |
b1#(a1(b2(x1))) |
→ |
b1#(x1) |
(100) |
b1#(a1(b2(x1))) |
→ |
b2#(c1(a0(b2(b1(x1))))) |
(101) |
b1#(a1(b2(x1))) |
→ |
b2#(b1(x1)) |
(102) |
b1#(a1(b2(x1))) |
→ |
a0#(b2(c1(a0(b2(b1(x1)))))) |
(103) |
b1#(a1(b2(x1))) |
→ |
a0#(b2(b1(x1))) |
(104) |
b1#(a1(a2(x1))) |
→ |
c1#(a0(b2(c1(a0(b2(a1(x1))))))) |
(105) |
b1#(a1(a2(x1))) |
→ |
c1#(a0(b2(a1(x1)))) |
(106) |
b1#(a1(a2(x1))) |
→ |
b2#(c1(a0(b2(a1(x1))))) |
(107) |
b1#(a1(a2(x1))) |
→ |
b2#(a1(x1)) |
(108) |
b1#(a1(a2(x1))) |
→ |
a0#(b2(c1(a0(b2(a1(x1)))))) |
(109) |
b1#(a1(a2(x1))) |
→ |
a0#(b2(a1(x1))) |
(110) |
b2#(c1(x1)) |
→ |
c2#(x1) |
(111) |
b2#(b1(x1)) |
→ |
b2#(x1) |
(112) |
b2#(a1(c2(x1))) |
→ |
c1#(x1) |
(113) |
b2#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(x1)))) |
(114) |
b2#(a1(c2(x1))) |
→ |
c2#(a0(b2(c1(a0(b2(c1(x1))))))) |
(115) |
b2#(a1(c2(x1))) |
→ |
b2#(c1(x1)) |
(116) |
b2#(a1(c2(x1))) |
→ |
b2#(c1(a0(b2(c1(x1))))) |
(117) |
b2#(a1(c2(x1))) |
→ |
a0#(b2(c1(x1))) |
(118) |
b2#(a1(c2(x1))) |
→ |
a0#(b2(c1(a0(b2(c1(x1)))))) |
(119) |
b2#(a1(b2(x1))) |
→ |
c1#(a0(b2(b1(x1)))) |
(120) |
b2#(a1(b2(x1))) |
→ |
c2#(a0(b2(c1(a0(b2(b1(x1))))))) |
(121) |
b2#(a1(b2(x1))) |
→ |
b1#(x1) |
(122) |
b2#(a1(b2(x1))) |
→ |
b2#(c1(a0(b2(b1(x1))))) |
(123) |
b2#(a1(b2(x1))) |
→ |
b2#(b1(x1)) |
(124) |
b2#(a1(b2(x1))) |
→ |
a0#(b2(c1(a0(b2(b1(x1)))))) |
(125) |
b2#(a1(b2(x1))) |
→ |
a0#(b2(b1(x1))) |
(126) |
b2#(a1(a2(x1))) |
→ |
c1#(a0(b2(a1(x1)))) |
(127) |
b2#(a1(a2(x1))) |
→ |
c2#(a0(b2(c1(a0(b2(a1(x1))))))) |
(128) |
b2#(a1(a2(x1))) |
→ |
b2#(c1(a0(b2(a1(x1))))) |
(129) |
b2#(a1(a2(x1))) |
→ |
b2#(a1(x1)) |
(130) |
b2#(a1(a2(x1))) |
→ |
a0#(b2(c1(a0(b2(a1(x1)))))) |
(131) |
b2#(a1(a2(x1))) |
→ |
a0#(b2(a1(x1))) |
(132) |
a0#(b2(x1)) |
→ |
b0#(x1) |
(133) |
a0(b2(x1)) |
→ |
b0(x1) |
(53) |
a0(c2(x1)) |
→ |
c0(x1) |
(54) |
b2(a1(a2(x1))) |
→ |
c2(a0(b2(c1(a0(b2(a1(x1))))))) |
(55) |
b1(a1(a2(x1))) |
→ |
c1(a0(b2(c1(a0(b2(a1(x1))))))) |
(56) |
b0(a1(a2(x1))) |
→ |
c0(a0(b2(c1(a0(b2(a1(x1))))))) |
(57) |
b2(a1(b2(x1))) |
→ |
c2(a0(b2(c1(a0(b2(b1(x1))))))) |
(58) |
b1(a1(b2(x1))) |
→ |
c1(a0(b2(c1(a0(b2(b1(x1))))))) |
(59) |
b0(a1(b2(x1))) |
→ |
c0(a0(b2(c1(a0(b2(b1(x1))))))) |
(60) |
b2(a1(c2(x1))) |
→ |
c2(a0(b2(c1(a0(b2(c1(x1))))))) |
(61) |
b1(a1(c2(x1))) |
→ |
c1(a0(b2(c1(a0(b2(c1(x1))))))) |
(62) |
b0(a1(c2(x1))) |
→ |
c0(a0(b2(c1(a0(b2(c1(x1))))))) |
(63) |
b2(a1(x1)) |
→ |
a2(x1) |
(64) |
b1(a1(x1)) |
→ |
a1(x1) |
(65) |
b2(b1(x1)) |
→ |
b2(x1) |
(66) |
b1(b1(x1)) |
→ |
b1(x1) |
(39) |
b0(b1(x1)) |
→ |
b0(x1) |
(67) |
b2(c1(x1)) |
→ |
c2(x1) |
(68) |
b1(c1(x1)) |
→ |
c1(x1) |
(69) |
b0(c1(x1)) |
→ |
c0(x1) |
(70) |
c2(c0(a0(x1))) |
→ |
a2(x1) |
(71) |
c1(c0(a0(x1))) |
→ |
a1(x1) |
(72) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
b0#(a1(c2(x1))) |
→ |
c1#(x1) |
(74) |
b0#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(x1)))) |
(75) |
b0#(a1(c2(x1))) |
→ |
b2#(c1(x1)) |
(76) |
b0#(a1(c2(x1))) |
→ |
b2#(c1(a0(b2(c1(x1))))) |
(77) |
b0#(a1(c2(x1))) |
→ |
a0#(b2(c1(x1))) |
(78) |
b0#(a1(c2(x1))) |
→ |
a0#(b2(c1(a0(b2(c1(x1)))))) |
(79) |
b0#(a1(b2(x1))) |
→ |
c1#(a0(b2(b1(x1)))) |
(80) |
b0#(a1(b2(x1))) |
→ |
b1#(x1) |
(81) |
b0#(a1(b2(x1))) |
→ |
b2#(c1(a0(b2(b1(x1))))) |
(82) |
b0#(a1(b2(x1))) |
→ |
b2#(b1(x1)) |
(83) |
b0#(a1(b2(x1))) |
→ |
a0#(b2(c1(a0(b2(b1(x1)))))) |
(84) |
b0#(a1(b2(x1))) |
→ |
a0#(b2(b1(x1))) |
(85) |
b0#(a1(a2(x1))) |
→ |
c1#(a0(b2(a1(x1)))) |
(86) |
b0#(a1(a2(x1))) |
→ |
b2#(c1(a0(b2(a1(x1))))) |
(87) |
b0#(a1(a2(x1))) |
→ |
b2#(a1(x1)) |
(88) |
b0#(a1(a2(x1))) |
→ |
a0#(b2(c1(a0(b2(a1(x1)))))) |
(89) |
b0#(a1(a2(x1))) |
→ |
a0#(b2(a1(x1))) |
(90) |
b1#(a1(c2(x1))) |
→ |
c1#(x1) |
(91) |
b1#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(x1)))) |
(92) |
b1#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(a0(b2(c1(x1))))))) |
(93) |
b1#(a1(c2(x1))) |
→ |
b2#(c1(x1)) |
(94) |
b1#(a1(c2(x1))) |
→ |
b2#(c1(a0(b2(c1(x1))))) |
(95) |
b1#(a1(c2(x1))) |
→ |
a0#(b2(c1(x1))) |
(96) |
b1#(a1(c2(x1))) |
→ |
a0#(b2(c1(a0(b2(c1(x1)))))) |
(97) |
b1#(a1(b2(x1))) |
→ |
c1#(a0(b2(c1(a0(b2(b1(x1))))))) |
(98) |
b1#(a1(b2(x1))) |
→ |
c1#(a0(b2(b1(x1)))) |
(99) |
b1#(a1(b2(x1))) |
→ |
b1#(x1) |
(100) |
b1#(a1(b2(x1))) |
→ |
b2#(c1(a0(b2(b1(x1))))) |
(101) |
b1#(a1(b2(x1))) |
→ |
b2#(b1(x1)) |
(102) |
b1#(a1(b2(x1))) |
→ |
a0#(b2(c1(a0(b2(b1(x1)))))) |
(103) |
b1#(a1(b2(x1))) |
→ |
a0#(b2(b1(x1))) |
(104) |
b1#(a1(a2(x1))) |
→ |
c1#(a0(b2(c1(a0(b2(a1(x1))))))) |
(105) |
b1#(a1(a2(x1))) |
→ |
c1#(a0(b2(a1(x1)))) |
(106) |
b1#(a1(a2(x1))) |
→ |
b2#(c1(a0(b2(a1(x1))))) |
(107) |
b1#(a1(a2(x1))) |
→ |
b2#(a1(x1)) |
(108) |
b1#(a1(a2(x1))) |
→ |
a0#(b2(c1(a0(b2(a1(x1)))))) |
(109) |
b1#(a1(a2(x1))) |
→ |
a0#(b2(a1(x1))) |
(110) |
b2#(c1(x1)) |
→ |
c2#(x1) |
(111) |
b2#(a1(c2(x1))) |
→ |
c1#(x1) |
(113) |
b2#(a1(c2(x1))) |
→ |
c1#(a0(b2(c1(x1)))) |
(114) |
b2#(a1(c2(x1))) |
→ |
c2#(a0(b2(c1(a0(b2(c1(x1))))))) |
(115) |
b2#(a1(c2(x1))) |
→ |
b2#(c1(x1)) |
(116) |
b2#(a1(c2(x1))) |
→ |
b2#(c1(a0(b2(c1(x1))))) |
(117) |
b2#(a1(c2(x1))) |
→ |
a0#(b2(c1(x1))) |
(118) |
b2#(a1(c2(x1))) |
→ |
a0#(b2(c1(a0(b2(c1(x1)))))) |
(119) |
b2#(a1(b2(x1))) |
→ |
c1#(a0(b2(b1(x1)))) |
(120) |
b2#(a1(b2(x1))) |
→ |
c2#(a0(b2(c1(a0(b2(b1(x1))))))) |
(121) |
b2#(a1(b2(x1))) |
→ |
b1#(x1) |
(122) |
b2#(a1(b2(x1))) |
→ |
b2#(c1(a0(b2(b1(x1))))) |
(123) |
b2#(a1(b2(x1))) |
→ |
b2#(b1(x1)) |
(124) |
b2#(a1(b2(x1))) |
→ |
a0#(b2(c1(a0(b2(b1(x1)))))) |
(125) |
b2#(a1(b2(x1))) |
→ |
a0#(b2(b1(x1))) |
(126) |
b2#(a1(a2(x1))) |
→ |
c1#(a0(b2(a1(x1)))) |
(127) |
b2#(a1(a2(x1))) |
→ |
c2#(a0(b2(c1(a0(b2(a1(x1))))))) |
(128) |
b2#(a1(a2(x1))) |
→ |
b2#(c1(a0(b2(a1(x1))))) |
(129) |
b2#(a1(a2(x1))) |
→ |
b2#(a1(x1)) |
(130) |
b2#(a1(a2(x1))) |
→ |
a0#(b2(c1(a0(b2(a1(x1)))))) |
(131) |
b2#(a1(a2(x1))) |
→ |
a0#(b2(a1(x1))) |
(132) |
a0#(b2(x1)) |
→ |
b0#(x1) |
(133) |
and
no rules
could be deleted.
The dependency pairs are split into 2
components.