The rewrite relation of the following TRS is considered.
a(x1) | → | b(c(x1)) | (1) |
b(a(b(x1))) | → | x1 | (2) |
c(c(x1)) | → | a(a(a(b(x1)))) | (3) |
{c(☐), b(☐), a(☐)}
We obtain the transformed TRSc(a(x1)) | → | c(b(c(x1))) | (4) |
c(b(a(b(x1)))) | → | c(x1) | (5) |
c(c(c(x1))) | → | c(a(a(a(b(x1))))) | (6) |
b(a(x1)) | → | b(b(c(x1))) | (7) |
b(b(a(b(x1)))) | → | b(x1) | (8) |
b(c(c(x1))) | → | b(a(a(a(b(x1))))) | (9) |
a(a(x1)) | → | a(b(c(x1))) | (10) |
a(b(a(b(x1)))) | → | a(x1) | (11) |
a(c(c(x1))) | → | a(a(a(a(b(x1))))) | (12) |
As carrier we take the set {0,1,2}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 3):
[c(x1)] | = | 3x1 + 0 |
[b(x1)] | = | 3x1 + 1 |
[a(x1)] | = | 3x1 + 2 |
a2(a2(x1)) | → | a1(b0(c2(x1))) | (13) |
a2(a1(x1)) | → | a1(b0(c1(x1))) | (14) |
a2(a0(x1)) | → | a1(b0(c0(x1))) | (15) |
b2(a2(x1)) | → | b1(b0(c2(x1))) | (16) |
b2(a1(x1)) | → | b1(b0(c1(x1))) | (17) |
b2(a0(x1)) | → | b1(b0(c0(x1))) | (18) |
c2(a2(x1)) | → | c1(b0(c2(x1))) | (19) |
c2(a1(x1)) | → | c1(b0(c1(x1))) | (20) |
c2(a0(x1)) | → | c1(b0(c0(x1))) | (21) |
a1(b2(a1(b2(x1)))) | → | a2(x1) | (22) |
a1(b2(a1(b1(x1)))) | → | a1(x1) | (23) |
a1(b2(a1(b0(x1)))) | → | a0(x1) | (24) |
b1(b2(a1(b2(x1)))) | → | b2(x1) | (25) |
b1(b2(a1(b1(x1)))) | → | b1(x1) | (26) |
b1(b2(a1(b0(x1)))) | → | b0(x1) | (27) |
c1(b2(a1(b2(x1)))) | → | c2(x1) | (28) |
c1(b2(a1(b1(x1)))) | → | c1(x1) | (29) |
c1(b2(a1(b0(x1)))) | → | c0(x1) | (30) |
a0(c0(c2(x1))) | → | a2(a2(a2(a1(b2(x1))))) | (31) |
a0(c0(c1(x1))) | → | a2(a2(a2(a1(b1(x1))))) | (32) |
a0(c0(c0(x1))) | → | a2(a2(a2(a1(b0(x1))))) | (33) |
b0(c0(c2(x1))) | → | b2(a2(a2(a1(b2(x1))))) | (34) |
b0(c0(c1(x1))) | → | b2(a2(a2(a1(b1(x1))))) | (35) |
b0(c0(c0(x1))) | → | b2(a2(a2(a1(b0(x1))))) | (36) |
c0(c0(c2(x1))) | → | c2(a2(a2(a1(b2(x1))))) | (37) |
c0(c0(c1(x1))) | → | c2(a2(a2(a1(b1(x1))))) | (38) |
c0(c0(c0(x1))) | → | c2(a2(a2(a1(b0(x1))))) | (39) |
[c0(x1)] | = |
x1 +
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[c1(x1)] | = |
x1 +
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[c2(x1)] | = |
x1 +
|
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[b0(x1)] | = |
x1 +
|
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[b1(x1)] | = |
x1 +
|
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[b2(x1)] | = |
x1 +
|
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[a0(x1)] | = |
x1 +
|
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[a1(x1)] | = |
x1 +
|
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[a2(x1)] | = |
x1 +
|
a1(b2(a1(b2(x1)))) | → | a2(x1) | (22) |
a1(b2(a1(b1(x1)))) | → | a1(x1) | (23) |
b1(b2(a1(b2(x1)))) | → | b2(x1) | (25) |
b1(b2(a1(b1(x1)))) | → | b1(x1) | (26) |
b1(b2(a1(b0(x1)))) | → | b0(x1) | (27) |
c1(b2(a1(b2(x1)))) | → | c2(x1) | (28) |
c1(b2(a1(b1(x1)))) | → | c1(x1) | (29) |
a0(c0(c2(x1))) | → | a2(a2(a2(a1(b2(x1))))) | (31) |
a0(c0(c1(x1))) | → | a2(a2(a2(a1(b1(x1))))) | (32) |
a0(c0(c0(x1))) | → | a2(a2(a2(a1(b0(x1))))) | (33) |
b0(c0(c0(x1))) | → | b2(a2(a2(a1(b0(x1))))) | (36) |
c0(c0(c2(x1))) | → | c2(a2(a2(a1(b2(x1))))) | (37) |
c0(c0(c1(x1))) | → | c2(a2(a2(a1(b1(x1))))) | (38) |
c0(c0(c0(x1))) | → | c2(a2(a2(a1(b0(x1))))) | (39) |
c2#(a0(x1)) | → | c1#(b0(c0(x1))) | (40) |
c2#(a0(x1)) | → | b0#(c0(x1)) | (41) |
c2#(a1(x1)) | → | c1#(x1) | (42) |
c2#(a1(x1)) | → | c1#(b0(c1(x1))) | (43) |
c2#(a1(x1)) | → | b0#(c1(x1)) | (44) |
c2#(a2(x1)) | → | c1#(b0(c2(x1))) | (45) |
c2#(a2(x1)) | → | c2#(x1) | (46) |
c2#(a2(x1)) | → | b0#(c2(x1)) | (47) |
b0#(c0(c1(x1))) | → | b2#(a2(a2(a1(b1(x1))))) | (48) |
b0#(c0(c1(x1))) | → | a1#(b1(x1)) | (49) |
b0#(c0(c1(x1))) | → | a2#(a1(b1(x1))) | (50) |
b0#(c0(c1(x1))) | → | a2#(a2(a1(b1(x1)))) | (51) |
b0#(c0(c2(x1))) | → | b2#(x1) | (52) |
b0#(c0(c2(x1))) | → | b2#(a2(a2(a1(b2(x1))))) | (53) |
b0#(c0(c2(x1))) | → | a1#(b2(x1)) | (54) |
b0#(c0(c2(x1))) | → | a2#(a1(b2(x1))) | (55) |
b0#(c0(c2(x1))) | → | a2#(a2(a1(b2(x1)))) | (56) |
b2#(a0(x1)) | → | b0#(c0(x1)) | (57) |
b2#(a1(x1)) | → | c1#(x1) | (58) |
b2#(a1(x1)) | → | b0#(c1(x1)) | (59) |
b2#(a2(x1)) | → | c2#(x1) | (60) |
b2#(a2(x1)) | → | b0#(c2(x1)) | (61) |
a2#(a0(x1)) | → | b0#(c0(x1)) | (62) |
a2#(a0(x1)) | → | a1#(b0(c0(x1))) | (63) |
a2#(a1(x1)) | → | c1#(x1) | (64) |
a2#(a1(x1)) | → | b0#(c1(x1)) | (65) |
a2#(a1(x1)) | → | a1#(b0(c1(x1))) | (66) |
a2#(a2(x1)) | → | c2#(x1) | (67) |
a2#(a2(x1)) | → | b0#(c2(x1)) | (68) |
a2#(a2(x1)) | → | a1#(b0(c2(x1))) | (69) |
[c0(x1)] | = |
x1 +
|
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[c1(x1)] | = |
x1 +
|
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[c2(x1)] | = |
x1 +
|
||||
[b0(x1)] | = |
x1 +
|
||||
[b1(x1)] | = |
x1 +
|
||||
[b2(x1)] | = |
x1 +
|
||||
[a0(x1)] | = |
x1 +
|
||||
[a1(x1)] | = |
x1 +
|
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[a2(x1)] | = |
x1 +
|
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[c1#(x1)] | = |
x1 +
|
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[c2#(x1)] | = |
x1 +
|
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[b0#(x1)] | = |
x1 +
|
||||
[b2#(x1)] | = |
x1 +
|
||||
[a1#(x1)] | = |
x1 +
|
||||
[a2#(x1)] | = |
x1 +
|
a2(a2(x1)) | → | a1(b0(c2(x1))) | (13) |
a2(a1(x1)) | → | a1(b0(c1(x1))) | (14) |
a2(a0(x1)) | → | a1(b0(c0(x1))) | (15) |
b2(a2(x1)) | → | b1(b0(c2(x1))) | (16) |
b2(a1(x1)) | → | b1(b0(c1(x1))) | (17) |
b2(a0(x1)) | → | b1(b0(c0(x1))) | (18) |
c2(a2(x1)) | → | c1(b0(c2(x1))) | (19) |
c2(a1(x1)) | → | c1(b0(c1(x1))) | (20) |
c2(a0(x1)) | → | c1(b0(c0(x1))) | (21) |
a1(b2(a1(b0(x1)))) | → | a0(x1) | (24) |
c1(b2(a1(b0(x1)))) | → | c0(x1) | (30) |
b0(c0(c2(x1))) | → | b2(a2(a2(a1(b2(x1))))) | (34) |
b0(c0(c1(x1))) | → | b2(a2(a2(a1(b1(x1))))) | (35) |
c2#(a0(x1)) | → | c1#(b0(c0(x1))) | (40) |
c2#(a0(x1)) | → | b0#(c0(x1)) | (41) |
c2#(a1(x1)) | → | c1#(x1) | (42) |
c2#(a1(x1)) | → | c1#(b0(c1(x1))) | (43) |
c2#(a1(x1)) | → | b0#(c1(x1)) | (44) |
c2#(a2(x1)) | → | c1#(b0(c2(x1))) | (45) |
c2#(a2(x1)) | → | c2#(x1) | (46) |
c2#(a2(x1)) | → | b0#(c2(x1)) | (47) |
b0#(c0(c1(x1))) | → | a1#(b1(x1)) | (49) |
b0#(c0(c1(x1))) | → | a2#(a1(b1(x1))) | (50) |
b0#(c0(c1(x1))) | → | a2#(a2(a1(b1(x1)))) | (51) |
b0#(c0(c2(x1))) | → | b2#(x1) | (52) |
b0#(c0(c2(x1))) | → | a1#(b2(x1)) | (54) |
b0#(c0(c2(x1))) | → | a2#(a1(b2(x1))) | (55) |
b0#(c0(c2(x1))) | → | a2#(a2(a1(b2(x1)))) | (56) |
b2#(a1(x1)) | → | c1#(x1) | (58) |
b2#(a2(x1)) | → | c2#(x1) | (60) |
a2#(a0(x1)) | → | b0#(c0(x1)) | (62) |
a2#(a0(x1)) | → | a1#(b0(c0(x1))) | (63) |
a2#(a1(x1)) | → | c1#(x1) | (64) |
a2#(a1(x1)) | → | b0#(c1(x1)) | (65) |
a2#(a1(x1)) | → | a1#(b0(c1(x1))) | (66) |
a2#(a2(x1)) | → | c2#(x1) | (67) |
a2#(a2(x1)) | → | b0#(c2(x1)) | (68) |
a2#(a2(x1)) | → | a1#(b0(c2(x1))) | (69) |
The dependency pairs are split into 1 component.
b0#(c0(c1(x1))) | → | b2#(a2(a2(a1(b1(x1))))) | (48) |
b2#(a0(x1)) | → | b0#(c0(x1)) | (57) |
b0#(c0(c2(x1))) | → | b2#(a2(a2(a1(b2(x1))))) | (53) |
b2#(a1(x1)) | → | b0#(c1(x1)) | (59) |
b2#(a2(x1)) | → | b0#(c2(x1)) | (61) |
[c0(x1)] | = |
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[c1(x1)] | = |
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[c2(x1)] | = |
|
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[b0(x1)] | = |
|
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[b1(x1)] | = |
|
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[b2(x1)] | = |
|
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[a0(x1)] | = |
|
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[a1(x1)] | = |
|
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[a2(x1)] | = |
|
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[b0#(x1)] | = |
|
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[b2#(x1)] | = |
|
a2(a2(x1)) | → | a1(b0(c2(x1))) | (13) |
a2(a1(x1)) | → | a1(b0(c1(x1))) | (14) |
a2(a0(x1)) | → | a1(b0(c0(x1))) | (15) |
b2(a2(x1)) | → | b1(b0(c2(x1))) | (16) |
b2(a1(x1)) | → | b1(b0(c1(x1))) | (17) |
b2(a0(x1)) | → | b1(b0(c0(x1))) | (18) |
c2(a2(x1)) | → | c1(b0(c2(x1))) | (19) |
c2(a1(x1)) | → | c1(b0(c1(x1))) | (20) |
c2(a0(x1)) | → | c1(b0(c0(x1))) | (21) |
a1(b2(a1(b0(x1)))) | → | a0(x1) | (24) |
c1(b2(a1(b0(x1)))) | → | c0(x1) | (30) |
b0(c0(c2(x1))) | → | b2(a2(a2(a1(b2(x1))))) | (34) |
b0(c0(c1(x1))) | → | b2(a2(a2(a1(b1(x1))))) | (35) |
b0#(c0(c1(x1))) | → | b2#(a2(a2(a1(b1(x1))))) | (48) |
The dependency pairs are split into 1 component.
b2#(a0(x1)) | → | b0#(c0(x1)) | (57) |
b0#(c0(c2(x1))) | → | b2#(a2(a2(a1(b2(x1))))) | (53) |
b2#(a1(x1)) | → | b0#(c1(x1)) | (59) |
b2#(a2(x1)) | → | b0#(c2(x1)) | (61) |
[c0(x1)] | = |
|
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[c1(x1)] | = |
|
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[c2(x1)] | = |
|
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[b0(x1)] | = |
|
||||||||||||
[b1(x1)] | = |
|
||||||||||||
[b2(x1)] | = |
|
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[a0(x1)] | = |
|
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[a1(x1)] | = |
|
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[a2(x1)] | = |
|
||||||||||||
[b0#(x1)] | = |
|
||||||||||||
[b2#(x1)] | = |
|
a2(a2(x1)) | → | a1(b0(c2(x1))) | (13) |
a2(a1(x1)) | → | a1(b0(c1(x1))) | (14) |
a2(a0(x1)) | → | a1(b0(c0(x1))) | (15) |
b2(a2(x1)) | → | b1(b0(c2(x1))) | (16) |
b2(a1(x1)) | → | b1(b0(c1(x1))) | (17) |
b2(a0(x1)) | → | b1(b0(c0(x1))) | (18) |
c2(a2(x1)) | → | c1(b0(c2(x1))) | (19) |
c2(a1(x1)) | → | c1(b0(c1(x1))) | (20) |
c2(a0(x1)) | → | c1(b0(c0(x1))) | (21) |
a1(b2(a1(b0(x1)))) | → | a0(x1) | (24) |
c1(b2(a1(b0(x1)))) | → | c0(x1) | (30) |
b0(c0(c2(x1))) | → | b2(a2(a2(a1(b2(x1))))) | (34) |
b0(c0(c1(x1))) | → | b2(a2(a2(a1(b1(x1))))) | (35) |
b2#(a2(x1)) | → | b0#(c2(x1)) | (61) |
The dependency pairs are split into 1 component.
b2#(a0(x1)) | → | b0#(c0(x1)) | (57) |
b0#(c0(c2(x1))) | → | b2#(a2(a2(a1(b2(x1))))) | (53) |
b2#(a1(x1)) | → | b0#(c1(x1)) | (59) |
[c0(x1)] | = |
|
||||||||||||||||||
[c1(x1)] | = |
|
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[c2(x1)] | = |
|
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[b0(x1)] | = |
|
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[b1(x1)] | = |
|
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[b2(x1)] | = |
|
||||||||||||||||||
[a0(x1)] | = |
|
||||||||||||||||||
[a1(x1)] | = |
|
||||||||||||||||||
[a2(x1)] | = |
|
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[b0#(x1)] | = |
|
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[b2#(x1)] | = |
|
a2(a2(x1)) | → | a1(b0(c2(x1))) | (13) |
a2(a1(x1)) | → | a1(b0(c1(x1))) | (14) |
a2(a0(x1)) | → | a1(b0(c0(x1))) | (15) |
b2(a2(x1)) | → | b1(b0(c2(x1))) | (16) |
b2(a1(x1)) | → | b1(b0(c1(x1))) | (17) |
b2(a0(x1)) | → | b1(b0(c0(x1))) | (18) |
c2(a2(x1)) | → | c1(b0(c2(x1))) | (19) |
c2(a1(x1)) | → | c1(b0(c1(x1))) | (20) |
c2(a0(x1)) | → | c1(b0(c0(x1))) | (21) |
a1(b2(a1(b0(x1)))) | → | a0(x1) | (24) |
c1(b2(a1(b0(x1)))) | → | c0(x1) | (30) |
b0(c0(c2(x1))) | → | b2(a2(a2(a1(b2(x1))))) | (34) |
b0(c0(c1(x1))) | → | b2(a2(a2(a1(b1(x1))))) | (35) |
b0#(c0(c2(x1))) | → | b2#(a2(a2(a1(b2(x1))))) | (53) |
[c0(x1)] | = |
x1 +
|
||||
[c1(x1)] | = |
x1 +
|
||||
[c2(x1)] | = |
x1 +
|
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[b0(x1)] | = |
x1 +
|
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[b1(x1)] | = |
x1 +
|
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[b2(x1)] | = |
x1 +
|
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[a0(x1)] | = |
x1 +
|
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[a1(x1)] | = |
x1 +
|
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[a2(x1)] | = |
x1 +
|
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[b0#(x1)] | = |
x1 +
|
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[b2#(x1)] | = |
x1 +
|
a2(a2(x1)) | → | a1(b0(c2(x1))) | (13) |
a2(a1(x1)) | → | a1(b0(c1(x1))) | (14) |
a2(a0(x1)) | → | a1(b0(c0(x1))) | (15) |
b2(a2(x1)) | → | b1(b0(c2(x1))) | (16) |
b2(a1(x1)) | → | b1(b0(c1(x1))) | (17) |
b2(a0(x1)) | → | b1(b0(c0(x1))) | (18) |
c2(a2(x1)) | → | c1(b0(c2(x1))) | (19) |
c2(a1(x1)) | → | c1(b0(c1(x1))) | (20) |
c2(a0(x1)) | → | c1(b0(c0(x1))) | (21) |
a1(b2(a1(b0(x1)))) | → | a0(x1) | (24) |
c1(b2(a1(b0(x1)))) | → | c0(x1) | (30) |
b0(c0(c2(x1))) | → | b2(a2(a2(a1(b2(x1))))) | (34) |
b0(c0(c1(x1))) | → | b2(a2(a2(a1(b1(x1))))) | (35) |
b2#(a0(x1)) | → | b0#(c0(x1)) | (57) |
b2#(a1(x1)) | → | b0#(c1(x1)) | (59) |
The dependency pairs are split into 0 components.