Certification Problem

Input (TPDB SRS_Relative/Waldmann_19/random-39)

The rewrite relation of the following TRS is considered.

a(a(c(x1))) a(c(c(x1))) (1)
b(b(c(x1))) a(b(b(x1))) (2)
c(c(b(x1))) a(c(a(x1))) (3)
b(a(c(x1))) c(a(c(x1))) (4)
c(b(c(x1))) b(b(b(x1))) (5)
b(b(a(x1))) b(a(c(x1))) (6)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Split

We split R in the relative problem D/R-D and R-D, where the rules D

c(c(b(x1))) a(c(a(x1))) (3)
are deleted.

1.1 Closure Under Flat Contexts

Using the flat contexts

{c(), b(), a()}

We obtain the transformed TRS
c(c(c(b(x1)))) c(a(c(a(x1)))) (7)
b(c(c(b(x1)))) b(a(c(a(x1)))) (8)
a(c(c(b(x1)))) a(a(c(a(x1)))) (9)
c(a(a(c(x1)))) c(a(c(c(x1)))) (10)
c(b(b(c(x1)))) c(a(b(b(x1)))) (11)
c(b(a(c(x1)))) c(c(a(c(x1)))) (12)
c(c(b(c(x1)))) c(b(b(b(x1)))) (13)
c(b(b(a(x1)))) c(b(a(c(x1)))) (14)
b(a(a(c(x1)))) b(a(c(c(x1)))) (15)
b(b(b(c(x1)))) b(a(b(b(x1)))) (16)
b(b(a(c(x1)))) b(c(a(c(x1)))) (17)
b(c(b(c(x1)))) b(b(b(b(x1)))) (18)
b(b(b(a(x1)))) b(b(a(c(x1)))) (19)
a(a(a(c(x1)))) a(a(c(c(x1)))) (20)
a(b(b(c(x1)))) a(a(b(b(x1)))) (21)
a(b(a(c(x1)))) a(c(a(c(x1)))) (22)
a(c(b(c(x1)))) a(b(b(b(x1)))) (23)
a(b(b(a(x1)))) a(b(a(c(x1)))) (24)

1.1.1 Semantic Labeling

The following interpretations form a model of the rules.

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

We obtain the labeled TRS
a0(c0(c1(b2(x1)))) a2(a0(c2(a2(x1)))) (25)
a0(c0(c1(b0(x1)))) a2(a0(c2(a0(x1)))) (26)
a0(c0(c1(b1(x1)))) a2(a0(c2(a1(x1)))) (27)
c0(c0(c1(b2(x1)))) c2(a0(c2(a2(x1)))) (28)
c0(c0(c1(b0(x1)))) c2(a0(c2(a0(x1)))) (29)
c0(c0(c1(b1(x1)))) c2(a0(c2(a1(x1)))) (30)
b0(c0(c1(b2(x1)))) b2(a0(c2(a2(x1)))) (31)
b0(c0(c1(b0(x1)))) b2(a0(c2(a0(x1)))) (32)
b0(c0(c1(b1(x1)))) b2(a0(c2(a1(x1)))) (33)
a2(a2(a0(c2(x1)))) a2(a0(c0(c2(x1)))) (34)
a2(a2(a0(c0(x1)))) a2(a0(c0(c0(x1)))) (35)
a2(a2(a0(c1(x1)))) a2(a0(c0(c1(x1)))) (36)
c2(a2(a0(c2(x1)))) c2(a0(c0(c2(x1)))) (37)
c2(a2(a0(c0(x1)))) c2(a0(c0(c0(x1)))) (38)
c2(a2(a0(c1(x1)))) c2(a0(c0(c1(x1)))) (39)
b2(a2(a0(c2(x1)))) b2(a0(c0(c2(x1)))) (40)
b2(a2(a0(c0(x1)))) b2(a0(c0(c0(x1)))) (41)
b2(a2(a0(c1(x1)))) b2(a0(c0(c1(x1)))) (42)
a1(b1(b0(c2(x1)))) a2(a1(b1(b2(x1)))) (43)
a1(b1(b0(c0(x1)))) a2(a1(b1(b0(x1)))) (44)
a1(b1(b0(c1(x1)))) a2(a1(b1(b1(x1)))) (45)
c1(b1(b0(c2(x1)))) c2(a1(b1(b2(x1)))) (46)
c1(b1(b0(c0(x1)))) c2(a1(b1(b0(x1)))) (47)
c1(b1(b0(c1(x1)))) c2(a1(b1(b1(x1)))) (48)
b1(b1(b0(c2(x1)))) b2(a1(b1(b2(x1)))) (49)
b1(b1(b0(c0(x1)))) b2(a1(b1(b0(x1)))) (50)
b1(b1(b0(c1(x1)))) b2(a1(b1(b1(x1)))) (51)
a1(b2(a0(c2(x1)))) a0(c2(a0(c2(x1)))) (52)
a1(b2(a0(c0(x1)))) a0(c2(a0(c0(x1)))) (53)
a1(b2(a0(c1(x1)))) a0(c2(a0(c1(x1)))) (54)
c1(b2(a0(c2(x1)))) c0(c2(a0(c2(x1)))) (55)
c1(b2(a0(c0(x1)))) c0(c2(a0(c0(x1)))) (56)
c1(b2(a0(c1(x1)))) c0(c2(a0(c1(x1)))) (57)
b1(b2(a0(c2(x1)))) b0(c2(a0(c2(x1)))) (58)
b1(b2(a0(c0(x1)))) b0(c2(a0(c0(x1)))) (59)
b1(b2(a0(c1(x1)))) b0(c2(a0(c1(x1)))) (60)
a0(c1(b0(c2(x1)))) a1(b1(b1(b2(x1)))) (61)
a0(c1(b0(c0(x1)))) a1(b1(b1(b0(x1)))) (62)
a0(c1(b0(c1(x1)))) a1(b1(b1(b1(x1)))) (63)
c0(c1(b0(c2(x1)))) c1(b1(b1(b2(x1)))) (64)
c0(c1(b0(c0(x1)))) c1(b1(b1(b0(x1)))) (65)
c0(c1(b0(c1(x1)))) c1(b1(b1(b1(x1)))) (66)
b0(c1(b0(c2(x1)))) b1(b1(b1(b2(x1)))) (67)
b0(c1(b0(c0(x1)))) b1(b1(b1(b0(x1)))) (68)
b0(c1(b0(c1(x1)))) b1(b1(b1(b1(x1)))) (69)
a1(b1(b2(a2(x1)))) a1(b2(a0(c2(x1)))) (70)
a1(b1(b2(a0(x1)))) a1(b2(a0(c0(x1)))) (71)
a1(b1(b2(a1(x1)))) a1(b2(a0(c1(x1)))) (72)
c1(b1(b2(a2(x1)))) c1(b2(a0(c2(x1)))) (73)
c1(b1(b2(a0(x1)))) c1(b2(a0(c0(x1)))) (74)
c1(b1(b2(a1(x1)))) c1(b2(a0(c1(x1)))) (75)
b1(b1(b2(a2(x1)))) b1(b2(a0(c2(x1)))) (76)
b1(b1(b2(a0(x1)))) b1(b2(a0(c0(x1)))) (77)
b1(b1(b2(a1(x1)))) b1(b2(a0(c1(x1)))) (78)

1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] = x1 +
3/2
[c1(x1)] = x1 +
5/2
[c2(x1)] = x1 +
0
[b0(x1)] = x1 +
5/2
[b1(x1)] = x1 +
3/2
[b2(x1)] = x1 +
1
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
1
[a2(x1)] = x1 +
3/2
all of the following rules can be deleted.
a0(c0(c1(b2(x1)))) a2(a0(c2(a2(x1)))) (25)
a0(c0(c1(b0(x1)))) a2(a0(c2(a0(x1)))) (26)
a0(c0(c1(b1(x1)))) a2(a0(c2(a1(x1)))) (27)
c0(c0(c1(b2(x1)))) c2(a0(c2(a2(x1)))) (28)
c0(c0(c1(b0(x1)))) c2(a0(c2(a0(x1)))) (29)
c0(c0(c1(b1(x1)))) c2(a0(c2(a1(x1)))) (30)
b0(c0(c1(b2(x1)))) b2(a0(c2(a2(x1)))) (31)
b0(c0(c1(b0(x1)))) b2(a0(c2(a0(x1)))) (32)
b0(c0(c1(b1(x1)))) b2(a0(c2(a1(x1)))) (33)
a1(b1(b0(c1(x1)))) a2(a1(b1(b1(x1)))) (45)
c1(b1(b0(c2(x1)))) c2(a1(b1(b2(x1)))) (46)
c1(b1(b0(c0(x1)))) c2(a1(b1(b0(x1)))) (47)
c1(b1(b0(c1(x1)))) c2(a1(b1(b1(x1)))) (48)
b1(b1(b0(c2(x1)))) b2(a1(b1(b2(x1)))) (49)
b1(b1(b0(c0(x1)))) b2(a1(b1(b0(x1)))) (50)
b1(b1(b0(c1(x1)))) b2(a1(b1(b1(x1)))) (51)
a1(b2(a0(c2(x1)))) a0(c2(a0(c2(x1)))) (52)
a1(b2(a0(c0(x1)))) a0(c2(a0(c0(x1)))) (53)
a1(b2(a0(c1(x1)))) a0(c2(a0(c1(x1)))) (54)
c1(b2(a0(c2(x1)))) c0(c2(a0(c2(x1)))) (55)
c1(b2(a0(c0(x1)))) c0(c2(a0(c0(x1)))) (56)
c1(b2(a0(c1(x1)))) c0(c2(a0(c1(x1)))) (57)
a0(c1(b0(c1(x1)))) a1(b1(b1(b1(x1)))) (63)
c0(c1(b0(c1(x1)))) c1(b1(b1(b1(x1)))) (66)
b0(c1(b0(c2(x1)))) b1(b1(b1(b2(x1)))) (67)
b0(c1(b0(c0(x1)))) b1(b1(b1(b0(x1)))) (68)
b0(c1(b0(c1(x1)))) b1(b1(b1(b1(x1)))) (69)
a1(b1(b2(a2(x1)))) a1(b2(a0(c2(x1)))) (70)
c1(b1(b2(a2(x1)))) c1(b2(a0(c2(x1)))) (73)
b1(b1(b2(a2(x1)))) b1(b2(a0(c2(x1)))) (76)

1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.

1.2 Split

We split R in the relative problem D/R-D and R-D, where the rules D

b(b(a(x1))) b(a(c(x1))) (6)
are deleted.

1.2.1 Closure Under Flat Contexts

Using the flat contexts

{c(), b(), a()}

We obtain the transformed TRS
c(b(b(a(x1)))) c(b(a(c(x1)))) (14)
b(b(b(a(x1)))) b(b(a(c(x1)))) (19)
a(b(b(a(x1)))) a(b(a(c(x1)))) (24)
c(a(a(c(x1)))) c(a(c(c(x1)))) (10)
c(b(b(c(x1)))) c(a(b(b(x1)))) (11)
c(b(a(c(x1)))) c(c(a(c(x1)))) (12)
c(c(b(c(x1)))) c(b(b(b(x1)))) (13)
b(a(a(c(x1)))) b(a(c(c(x1)))) (15)
b(b(b(c(x1)))) b(a(b(b(x1)))) (16)
b(b(a(c(x1)))) b(c(a(c(x1)))) (17)
b(c(b(c(x1)))) b(b(b(b(x1)))) (18)
a(a(a(c(x1)))) a(a(c(c(x1)))) (20)
a(b(b(c(x1)))) a(a(b(b(x1)))) (21)
a(b(a(c(x1)))) a(c(a(c(x1)))) (22)
a(c(b(c(x1)))) a(b(b(b(x1)))) (23)

1.2.1.1 Semantic Labeling

The following interpretations form a model of the rules.

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

We obtain the labeled TRS
a1(b1(b2(a2(x1)))) a1(b2(a0(c2(x1)))) (70)
a1(b1(b2(a0(x1)))) a1(b2(a0(c0(x1)))) (71)
a1(b1(b2(a1(x1)))) a1(b2(a0(c1(x1)))) (72)
c1(b1(b2(a2(x1)))) c1(b2(a0(c2(x1)))) (73)
c1(b1(b2(a0(x1)))) c1(b2(a0(c0(x1)))) (74)
c1(b1(b2(a1(x1)))) c1(b2(a0(c1(x1)))) (75)
b1(b1(b2(a2(x1)))) b1(b2(a0(c2(x1)))) (76)
b1(b1(b2(a0(x1)))) b1(b2(a0(c0(x1)))) (77)
b1(b1(b2(a1(x1)))) b1(b2(a0(c1(x1)))) (78)
a2(a2(a0(c2(x1)))) a2(a0(c0(c2(x1)))) (34)
a2(a2(a0(c0(x1)))) a2(a0(c0(c0(x1)))) (35)
a2(a2(a0(c1(x1)))) a2(a0(c0(c1(x1)))) (36)
c2(a2(a0(c2(x1)))) c2(a0(c0(c2(x1)))) (37)
c2(a2(a0(c0(x1)))) c2(a0(c0(c0(x1)))) (38)
c2(a2(a0(c1(x1)))) c2(a0(c0(c1(x1)))) (39)
b2(a2(a0(c2(x1)))) b2(a0(c0(c2(x1)))) (40)
b2(a2(a0(c0(x1)))) b2(a0(c0(c0(x1)))) (41)
b2(a2(a0(c1(x1)))) b2(a0(c0(c1(x1)))) (42)
a1(b1(b0(c2(x1)))) a2(a1(b1(b2(x1)))) (43)
a1(b1(b0(c0(x1)))) a2(a1(b1(b0(x1)))) (44)
a1(b1(b0(c1(x1)))) a2(a1(b1(b1(x1)))) (45)
c1(b1(b0(c2(x1)))) c2(a1(b1(b2(x1)))) (46)
c1(b1(b0(c0(x1)))) c2(a1(b1(b0(x1)))) (47)
c1(b1(b0(c1(x1)))) c2(a1(b1(b1(x1)))) (48)
b1(b1(b0(c2(x1)))) b2(a1(b1(b2(x1)))) (49)
b1(b1(b0(c0(x1)))) b2(a1(b1(b0(x1)))) (50)
b1(b1(b0(c1(x1)))) b2(a1(b1(b1(x1)))) (51)
a1(b2(a0(c2(x1)))) a0(c2(a0(c2(x1)))) (52)
a1(b2(a0(c0(x1)))) a0(c2(a0(c0(x1)))) (53)
a1(b2(a0(c1(x1)))) a0(c2(a0(c1(x1)))) (54)
c1(b2(a0(c2(x1)))) c0(c2(a0(c2(x1)))) (55)
c1(b2(a0(c0(x1)))) c0(c2(a0(c0(x1)))) (56)
c1(b2(a0(c1(x1)))) c0(c2(a0(c1(x1)))) (57)
b1(b2(a0(c2(x1)))) b0(c2(a0(c2(x1)))) (58)
b1(b2(a0(c0(x1)))) b0(c2(a0(c0(x1)))) (59)
b1(b2(a0(c1(x1)))) b0(c2(a0(c1(x1)))) (60)
a0(c1(b0(c2(x1)))) a1(b1(b1(b2(x1)))) (61)
a0(c1(b0(c0(x1)))) a1(b1(b1(b0(x1)))) (62)
a0(c1(b0(c1(x1)))) a1(b1(b1(b1(x1)))) (63)
c0(c1(b0(c2(x1)))) c1(b1(b1(b2(x1)))) (64)
c0(c1(b0(c0(x1)))) c1(b1(b1(b0(x1)))) (65)
c0(c1(b0(c1(x1)))) c1(b1(b1(b1(x1)))) (66)
b0(c1(b0(c2(x1)))) b1(b1(b1(b2(x1)))) (67)
b0(c1(b0(c0(x1)))) b1(b1(b1(b0(x1)))) (68)
b0(c1(b0(c1(x1)))) b1(b1(b1(b1(x1)))) (69)

1.2.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] = x1 +
5/2
[c1(x1)] = x1 +
4
[c2(x1)] = x1 +
1
[b0(x1)] = x1 +
3
[b1(x1)] = x1 +
5/2
[b2(x1)] = x1 +
3/2
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
3/2
[a2(x1)] = x1 +
5/2
all of the following rules can be deleted.
a1(b1(b2(a2(x1)))) a1(b2(a0(c2(x1)))) (70)
c1(b1(b2(a2(x1)))) c1(b2(a0(c2(x1)))) (73)
b1(b1(b2(a2(x1)))) b1(b2(a0(c2(x1)))) (76)
a1(b1(b0(c1(x1)))) a2(a1(b1(b1(x1)))) (45)
c1(b1(b0(c2(x1)))) c2(a1(b1(b2(x1)))) (46)
c1(b1(b0(c0(x1)))) c2(a1(b1(b0(x1)))) (47)
c1(b1(b0(c1(x1)))) c2(a1(b1(b1(x1)))) (48)
b1(b1(b0(c2(x1)))) b2(a1(b1(b2(x1)))) (49)
b1(b1(b0(c0(x1)))) b2(a1(b1(b0(x1)))) (50)
b1(b1(b0(c1(x1)))) b2(a1(b1(b1(x1)))) (51)
a1(b2(a0(c2(x1)))) a0(c2(a0(c2(x1)))) (52)
a1(b2(a0(c0(x1)))) a0(c2(a0(c0(x1)))) (53)
a1(b2(a0(c1(x1)))) a0(c2(a0(c1(x1)))) (54)
c1(b2(a0(c2(x1)))) c0(c2(a0(c2(x1)))) (55)
c1(b2(a0(c0(x1)))) c0(c2(a0(c0(x1)))) (56)
c1(b2(a0(c1(x1)))) c0(c2(a0(c1(x1)))) (57)
a0(c1(b0(c1(x1)))) a1(b1(b1(b1(x1)))) (63)
c0(c1(b0(c1(x1)))) c1(b1(b1(b1(x1)))) (66)
b0(c1(b0(c2(x1)))) b1(b1(b1(b2(x1)))) (67)
b0(c1(b0(c0(x1)))) b1(b1(b1(b0(x1)))) (68)
b0(c1(b0(c1(x1)))) b1(b1(b1(b1(x1)))) (69)

1.2.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a0(b2(b1(a1(x1)))) c0(a0(b2(a1(x1)))) (79)
a1(b2(b1(a1(x1)))) c1(a0(b2(a1(x1)))) (80)
a0(b2(b1(c1(x1)))) c0(a0(b2(c1(x1)))) (81)
a1(b2(b1(c1(x1)))) c1(a0(b2(c1(x1)))) (82)
a0(b2(b1(b1(x1)))) c0(a0(b2(b1(x1)))) (83)
a1(b2(b1(b1(x1)))) c1(a0(b2(b1(x1)))) (84)
c2(a0(a2(a2(x1)))) c2(c0(a0(a2(x1)))) (85)
c0(a0(a2(a2(x1)))) c0(c0(a0(a2(x1)))) (86)
c1(a0(a2(a2(x1)))) c1(c0(a0(a2(x1)))) (87)
c2(a0(a2(c2(x1)))) c2(c0(a0(c2(x1)))) (88)
c0(a0(a2(c2(x1)))) c0(c0(a0(c2(x1)))) (89)
c1(a0(a2(c2(x1)))) c1(c0(a0(c2(x1)))) (90)
c2(a0(a2(b2(x1)))) c2(c0(a0(b2(x1)))) (91)
c0(a0(a2(b2(x1)))) c0(c0(a0(b2(x1)))) (92)
c1(a0(a2(b2(x1)))) c1(c0(a0(b2(x1)))) (93)
c2(b0(b1(a1(x1)))) b2(b1(a1(a2(x1)))) (94)
c0(b0(b1(a1(x1)))) b0(b1(a1(a2(x1)))) (95)
c2(a0(b2(b1(x1)))) c2(a0(c2(b0(x1)))) (96)
c0(a0(b2(b1(x1)))) c0(a0(c2(b0(x1)))) (97)
c1(a0(b2(b1(x1)))) c1(a0(c2(b0(x1)))) (98)
c2(b0(c1(a0(x1)))) b2(b1(b1(a1(x1)))) (99)
c0(b0(c1(a0(x1)))) b0(b1(b1(a1(x1)))) (100)
c2(b0(c1(c0(x1)))) b2(b1(b1(c1(x1)))) (101)
c0(b0(c1(c0(x1)))) b0(b1(b1(c1(x1)))) (102)

1.2.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals
[c0(x1)] =
2
· x1 +
1
[c1(x1)] =
1
· x1 +
0
[c2(x1)] =
2
· x1 +
0
[b0(x1)] =
2
· x1 +
1
[b1(x1)] =
2
· x1 +
1
[b2(x1)] =
2
· x1 +
0
[a0(x1)] =
2
· x1 +
1
[a1(x1)] =
1
· x1 +
0
[a2(x1)] =
2
· x1 +
1
all of the following rules can be deleted.
a0(b2(b1(a1(x1)))) c0(a0(b2(a1(x1)))) (79)
a1(b2(b1(a1(x1)))) c1(a0(b2(a1(x1)))) (80)
a0(b2(b1(c1(x1)))) c0(a0(b2(c1(x1)))) (81)
a1(b2(b1(c1(x1)))) c1(a0(b2(c1(x1)))) (82)
a0(b2(b1(b1(x1)))) c0(a0(b2(b1(x1)))) (83)
a1(b2(b1(b1(x1)))) c1(a0(b2(b1(x1)))) (84)

1.2.1.1.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] = x1 +
0
[c1(x1)] = x1 +
1
[c2(x1)] = x1 +
0
[b0(x1)] = x1 +
0
[b1(x1)] = x1 +
0
[b2(x1)] = x1 +
0
[a0(x1)] = x1 +
1
[a1(x1)] = x1 +
0
[a2(x1)] = x1 +
0
all of the following rules can be deleted.
c2(b0(c1(a0(x1)))) b2(b1(b1(a1(x1)))) (99)
c0(b0(c1(a0(x1)))) b0(b1(b1(a1(x1)))) (100)

1.2.1.1.1.1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a2(a2(a0(c2(x1)))) a2(a0(c0(c2(x1)))) (34)
a2(a2(a0(c0(x1)))) a2(a0(c0(c0(x1)))) (35)
a2(a2(a0(c1(x1)))) a2(a0(c0(c1(x1)))) (36)
c2(a2(a0(c2(x1)))) c2(a0(c0(c2(x1)))) (37)
c2(a2(a0(c0(x1)))) c2(a0(c0(c0(x1)))) (38)
c2(a2(a0(c1(x1)))) c2(a0(c0(c1(x1)))) (39)
b2(a2(a0(c2(x1)))) b2(a0(c0(c2(x1)))) (40)
b2(a2(a0(c0(x1)))) b2(a0(c0(c0(x1)))) (41)
b2(a2(a0(c1(x1)))) b2(a0(c0(c1(x1)))) (42)
a1(b1(b0(c2(x1)))) a2(a1(b1(b2(x1)))) (43)
a1(b1(b0(c0(x1)))) a2(a1(b1(b0(x1)))) (44)
b1(b2(a0(c2(x1)))) b0(c2(a0(c2(x1)))) (58)
b1(b2(a0(c0(x1)))) b0(c2(a0(c0(x1)))) (59)
b1(b2(a0(c1(x1)))) b0(c2(a0(c1(x1)))) (60)
c0(c1(b0(c2(x1)))) c1(b1(b1(b2(x1)))) (64)
c0(c1(b0(c0(x1)))) c1(b1(b1(b0(x1)))) (65)

1.2.1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.

1.2.2 Closure Under Flat Contexts

Using the flat contexts

{c(), b(), a()}

We obtain the transformed TRS
c(a(a(c(x1)))) c(a(c(c(x1)))) (10)
c(b(b(c(x1)))) c(a(b(b(x1)))) (11)
c(b(a(c(x1)))) c(c(a(c(x1)))) (12)
c(c(b(c(x1)))) c(b(b(b(x1)))) (13)
b(a(a(c(x1)))) b(a(c(c(x1)))) (15)
b(b(b(c(x1)))) b(a(b(b(x1)))) (16)
b(b(a(c(x1)))) b(c(a(c(x1)))) (17)
b(c(b(c(x1)))) b(b(b(b(x1)))) (18)
a(a(a(c(x1)))) a(a(c(c(x1)))) (20)
a(b(b(c(x1)))) a(a(b(b(x1)))) (21)
a(b(a(c(x1)))) a(c(a(c(x1)))) (22)
a(c(b(c(x1)))) a(b(b(b(x1)))) (23)

1.2.2.1 Semantic Labeling

The following interpretations form a model of the rules.

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

We obtain the labeled TRS
a2(a2(a0(c2(x1)))) a2(a0(c0(c2(x1)))) (34)
a2(a2(a0(c0(x1)))) a2(a0(c0(c0(x1)))) (35)
a2(a2(a0(c1(x1)))) a2(a0(c0(c1(x1)))) (36)
c2(a2(a0(c2(x1)))) c2(a0(c0(c2(x1)))) (37)
c2(a2(a0(c0(x1)))) c2(a0(c0(c0(x1)))) (38)
c2(a2(a0(c1(x1)))) c2(a0(c0(c1(x1)))) (39)
b2(a2(a0(c2(x1)))) b2(a0(c0(c2(x1)))) (40)
b2(a2(a0(c0(x1)))) b2(a0(c0(c0(x1)))) (41)
b2(a2(a0(c1(x1)))) b2(a0(c0(c1(x1)))) (42)
a1(b1(b0(c2(x1)))) a2(a1(b1(b2(x1)))) (43)
a1(b1(b0(c0(x1)))) a2(a1(b1(b0(x1)))) (44)
a1(b1(b0(c1(x1)))) a2(a1(b1(b1(x1)))) (45)
c1(b1(b0(c2(x1)))) c2(a1(b1(b2(x1)))) (46)
c1(b1(b0(c0(x1)))) c2(a1(b1(b0(x1)))) (47)
c1(b1(b0(c1(x1)))) c2(a1(b1(b1(x1)))) (48)
b1(b1(b0(c2(x1)))) b2(a1(b1(b2(x1)))) (49)
b1(b1(b0(c0(x1)))) b2(a1(b1(b0(x1)))) (50)
b1(b1(b0(c1(x1)))) b2(a1(b1(b1(x1)))) (51)
a1(b2(a0(c2(x1)))) a0(c2(a0(c2(x1)))) (52)
a1(b2(a0(c0(x1)))) a0(c2(a0(c0(x1)))) (53)
a1(b2(a0(c1(x1)))) a0(c2(a0(c1(x1)))) (54)
c1(b2(a0(c2(x1)))) c0(c2(a0(c2(x1)))) (55)
c1(b2(a0(c0(x1)))) c0(c2(a0(c0(x1)))) (56)
c1(b2(a0(c1(x1)))) c0(c2(a0(c1(x1)))) (57)
b1(b2(a0(c2(x1)))) b0(c2(a0(c2(x1)))) (58)
b1(b2(a0(c0(x1)))) b0(c2(a0(c0(x1)))) (59)
b1(b2(a0(c1(x1)))) b0(c2(a0(c1(x1)))) (60)
a0(c1(b0(c2(x1)))) a1(b1(b1(b2(x1)))) (61)
a0(c1(b0(c0(x1)))) a1(b1(b1(b0(x1)))) (62)
a0(c1(b0(c1(x1)))) a1(b1(b1(b1(x1)))) (63)
c0(c1(b0(c2(x1)))) c1(b1(b1(b2(x1)))) (64)
c0(c1(b0(c0(x1)))) c1(b1(b1(b0(x1)))) (65)
c0(c1(b0(c1(x1)))) c1(b1(b1(b1(x1)))) (66)
b0(c1(b0(c2(x1)))) b1(b1(b1(b2(x1)))) (67)
b0(c1(b0(c0(x1)))) b1(b1(b1(b0(x1)))) (68)
b0(c1(b0(c1(x1)))) b1(b1(b1(b1(x1)))) (69)

1.2.2.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] = x1 +
1
[c1(x1)] = x1 +
4
[c2(x1)] = x1 +
0
[b0(x1)] = x1 +
1
[b1(x1)] = x1 +
1
[b2(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
1
[a2(x1)] = x1 +
1
all of the following rules can be deleted.
a1(b1(b0(c1(x1)))) a2(a1(b1(b1(x1)))) (45)
c1(b1(b0(c2(x1)))) c2(a1(b1(b2(x1)))) (46)
c1(b1(b0(c0(x1)))) c2(a1(b1(b0(x1)))) (47)
c1(b1(b0(c1(x1)))) c2(a1(b1(b1(x1)))) (48)
b1(b1(b0(c2(x1)))) b2(a1(b1(b2(x1)))) (49)
b1(b1(b0(c0(x1)))) b2(a1(b1(b0(x1)))) (50)
b1(b1(b0(c1(x1)))) b2(a1(b1(b1(x1)))) (51)
a1(b2(a0(c2(x1)))) a0(c2(a0(c2(x1)))) (52)
a1(b2(a0(c0(x1)))) a0(c2(a0(c0(x1)))) (53)
a1(b2(a0(c1(x1)))) a0(c2(a0(c1(x1)))) (54)
c1(b2(a0(c2(x1)))) c0(c2(a0(c2(x1)))) (55)
c1(b2(a0(c0(x1)))) c0(c2(a0(c0(x1)))) (56)
c1(b2(a0(c1(x1)))) c0(c2(a0(c1(x1)))) (57)
a0(c1(b0(c2(x1)))) a1(b1(b1(b2(x1)))) (61)
a0(c1(b0(c0(x1)))) a1(b1(b1(b0(x1)))) (62)
a0(c1(b0(c1(x1)))) a1(b1(b1(b1(x1)))) (63)
c0(c1(b0(c1(x1)))) c1(b1(b1(b1(x1)))) (66)
b0(c1(b0(c2(x1)))) b1(b1(b1(b2(x1)))) (67)
b0(c1(b0(c0(x1)))) b1(b1(b1(b0(x1)))) (68)
b0(c1(b0(c1(x1)))) b1(b1(b1(b1(x1)))) (69)

1.2.2.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the naturals
[c0(x1)] =
2
· x1 +
1
[c1(x1)] =
2
· x1 +
1
[c2(x1)] =
2
· x1 +
1
[b0(x1)] =
2
· x1 +
1
[b1(x1)] =
2
· x1 +
1
[b2(x1)] =
2
· x1 +
1
[a0(x1)] =
2
· x1 +
1
[a1(x1)] =
2
· x1 +
0
[a2(x1)] =
2
· x1 +
1
all of the following rules can be deleted.
a1(b1(b0(c2(x1)))) a2(a1(b1(b2(x1)))) (43)
a1(b1(b0(c0(x1)))) a2(a1(b1(b0(x1)))) (44)

1.2.2.1.1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[c0(x1)] = x1 +
4
[c1(x1)] = x1 +
0
[c2(x1)] = x1 +
0
[b0(x1)] = x1 +
0
[b1(x1)] = x1 +
1
[b2(x1)] = x1 +
1
[a0(x1)] = x1 +
0
[a2(x1)] = x1 +
5
all of the following rules can be deleted.
a2(a2(a0(c2(x1)))) a2(a0(c0(c2(x1)))) (34)
a2(a2(a0(c0(x1)))) a2(a0(c0(c0(x1)))) (35)
a2(a2(a0(c1(x1)))) a2(a0(c0(c1(x1)))) (36)
c2(a2(a0(c2(x1)))) c2(a0(c0(c2(x1)))) (37)
c2(a2(a0(c0(x1)))) c2(a0(c0(c0(x1)))) (38)
c2(a2(a0(c1(x1)))) c2(a0(c0(c1(x1)))) (39)
b2(a2(a0(c2(x1)))) b2(a0(c0(c2(x1)))) (40)
b2(a2(a0(c0(x1)))) b2(a0(c0(c0(x1)))) (41)
b2(a2(a0(c1(x1)))) b2(a0(c0(c1(x1)))) (42)
b1(b2(a0(c2(x1)))) b0(c2(a0(c2(x1)))) (58)
b1(b2(a0(c0(x1)))) b0(c2(a0(c0(x1)))) (59)
b1(b2(a0(c1(x1)))) b0(c2(a0(c1(x1)))) (60)
c0(c1(b0(c2(x1)))) c1(b1(b1(b2(x1)))) (64)
c0(c1(b0(c0(x1)))) c1(b1(b1(b0(x1)))) (65)

1.2.2.1.1.1.1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.