Certification Problem

Input (TPDB SRS_Relative/Waldmann_06_relative/r2)

The relative rewrite relation R/S is considered where R is the following TRS

c(a(c(x1))) → x1 (1)
a(c(a(x1))) → a(a(a(a(x1)))) (2)

and S is the following TRS.

x1 → c(c(c(c(x1)))) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{c(☐), a(☐)}

We obtain the transformed TRS
c(c(a(c(x1)))) → c(x1) (4)
c(a(c(a(x1)))) → c(a(a(a(a(x1))))) (5)
a(c(a(c(x1)))) → a(x1) (6)
a(a(c(a(x1)))) → a(a(a(a(a(x1))))) (7)
c(x1) → c(c(c(c(c(x1))))) (8)
a(x1) → a(c(c(c(c(x1))))) (9)

1.1 Closure Under Flat Contexts

Using the flat contexts

{c(☐), a(☐)}

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

1.1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,...,3}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 4):

[c(x1)] = 2x1 + 0
[a(x1)] = 2x1 + 1

We obtain the labeled TRS
c0(c2(c1(a0(c0(x1))))) → c0(c0(x1)) (22)
c0(c2(c1(a0(c2(x1))))) → c0(c2(x1)) (23)
c0(c2(c1(a2(c1(x1))))) → c2(c1(x1)) (24)
c0(c2(c1(a2(c3(x1))))) → c2(c3(x1)) (25)
c1(a2(c1(a0(c0(x1))))) → c1(a0(x1)) (26)
c1(a2(c1(a0(c2(x1))))) → c1(a2(x1)) (27)
c1(a2(c1(a2(c1(x1))))) → c3(a1(x1)) (28)
c1(a2(c1(a2(c3(x1))))) → c3(a3(x1)) (29)
a0(c2(c1(a0(c0(x1))))) → a0(c0(x1)) (30)
a0(c2(c1(a0(c2(x1))))) → a0(c2(x1)) (31)
a0(c2(c1(a2(c1(x1))))) → a2(c1(x1)) (32)
a0(c2(c1(a2(c3(x1))))) → a2(c3(x1)) (33)
a1(a2(c1(a0(c0(x1))))) → a1(a0(x1)) (34)
a1(a2(c1(a0(c2(x1))))) → a1(a2(x1)) (35)
a1(a2(c1(a2(c1(x1))))) → a3(a1(x1)) (36)
a1(a2(c1(a2(c3(x1))))) → a3(a3(x1)) (37)
c2(c1(a2(c1(a0(x1))))) → c2(c3(a3(a3(a1(a0(x1)))))) (38)
c2(c1(a2(c1(a2(x1))))) → c2(c3(a3(a3(a1(a2(x1)))))) (39)
c2(c1(a2(c3(a1(x1))))) → c2(c3(a3(a3(a3(a1(x1)))))) (40)
c2(c1(a2(c3(a3(x1))))) → c2(c3(a3(a3(a3(a3(x1)))))) (41)
c3(a1(a2(c1(a0(x1))))) → c3(a3(a3(a3(a1(a0(x1)))))) (42)
c3(a1(a2(c1(a2(x1))))) → c3(a3(a3(a3(a1(a2(x1)))))) (43)
c3(a1(a2(c3(a1(x1))))) → c3(a3(a3(a3(a3(a1(x1)))))) (44)
c3(a1(a2(c3(a3(x1))))) → c3(a3(a3(a3(a3(a3(x1)))))) (45)
a2(c1(a2(c1(a0(x1))))) → a2(c3(a3(a3(a1(a0(x1)))))) (46)
a2(c1(a2(c1(a2(x1))))) → a2(c3(a3(a3(a1(a2(x1)))))) (47)
a2(c1(a2(c3(a1(x1))))) → a2(c3(a3(a3(a3(a1(x1)))))) (48)
a2(c1(a2(c3(a3(x1))))) → a2(c3(a3(a3(a3(a3(x1)))))) (49)
a3(a1(a2(c1(a0(x1))))) → a3(a3(a3(a3(a1(a0(x1)))))) (50)
a3(a1(a2(c1(a2(x1))))) → a3(a3(a3(a3(a1(a2(x1)))))) (51)
a3(a1(a2(c3(a1(x1))))) → a3(a3(a3(a3(a3(a1(x1)))))) (52)
a3(a1(a2(c3(a3(x1))))) → a3(a3(a3(a3(a3(a3(x1)))))) (53)
c0(c0(x1)) → c0(c0(c0(c0(c0(c0(x1)))))) (54)
c0(c2(x1)) → c0(c0(c0(c0(c0(c2(x1)))))) (55)
c2(c1(x1)) → c0(c0(c0(c0(c2(c1(x1)))))) (56)
c2(c3(x1)) → c0(c0(c0(c0(c2(c3(x1)))))) (57)
c1(a0(x1)) → c1(a0(c0(c0(c0(c0(x1)))))) (58)
c1(a2(x1)) → c1(a0(c0(c0(c0(c2(x1)))))) (59)
c3(a1(x1)) → c1(a0(c0(c0(c2(c1(x1)))))) (60)
c3(a3(x1)) → c1(a0(c0(c0(c2(c3(x1)))))) (61)
a0(c0(x1)) → a0(c0(c0(c0(c0(c0(x1)))))) (62)
a0(c2(x1)) → a0(c0(c0(c0(c0(c2(x1)))))) (63)
a2(c1(x1)) → a0(c0(c0(c0(c2(c1(x1)))))) (64)
a2(c3(x1)) → a0(c0(c0(c0(c2(c3(x1)))))) (65)
a1(a0(x1)) → a1(a0(c0(c0(c0(c0(x1)))))) (66)
a1(a2(x1)) → a1(a0(c0(c0(c0(c2(x1)))))) (67)
a3(a1(x1)) → a1(a0(c0(c0(c2(c1(x1)))))) (68)
a3(a3(x1)) → a1(a0(c0(c0(c2(c3(x1)))))) (69)

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
[c2(x1)] = x1 +
1
[c1(x1)] = x1 +
1
[c3(x1)] = x1 +
5/2
[a0(x1)] = x1 +
1
[a2(x1)] = x1 +
28
[a1(x1)] = x1 +
5/2
[a3(x1)] = x1 +
8
all of the following rules can be deleted.
c0(c2(c1(a0(c0(x1))))) → c0(c0(x1)) (22)
c0(c2(c1(a0(c2(x1))))) → c0(c2(x1)) (23)
c0(c2(c1(a2(c1(x1))))) → c2(c1(x1)) (24)
c0(c2(c1(a2(c3(x1))))) → c2(c3(x1)) (25)
c1(a2(c1(a0(c0(x1))))) → c1(a0(x1)) (26)
c1(a2(c1(a0(c2(x1))))) → c1(a2(x1)) (27)
c1(a2(c1(a2(c1(x1))))) → c3(a1(x1)) (28)
c1(a2(c1(a2(c3(x1))))) → c3(a3(x1)) (29)
a0(c2(c1(a0(c0(x1))))) → a0(c0(x1)) (30)
a0(c2(c1(a0(c2(x1))))) → a0(c2(x1)) (31)
a0(c2(c1(a2(c1(x1))))) → a2(c1(x1)) (32)
a0(c2(c1(a2(c3(x1))))) → a2(c3(x1)) (33)
a1(a2(c1(a0(c0(x1))))) → a1(a0(x1)) (34)
a1(a2(c1(a0(c2(x1))))) → a1(a2(x1)) (35)
a1(a2(c1(a2(c1(x1))))) → a3(a1(x1)) (36)
a1(a2(c1(a2(c3(x1))))) → a3(a3(x1)) (37)
c2(c1(a2(c1(a0(x1))))) → c2(c3(a3(a3(a1(a0(x1)))))) (38)
c2(c1(a2(c1(a2(x1))))) → c2(c3(a3(a3(a1(a2(x1)))))) (39)
c2(c1(a2(c3(a1(x1))))) → c2(c3(a3(a3(a3(a1(x1)))))) (40)
c2(c1(a2(c3(a3(x1))))) → c2(c3(a3(a3(a3(a3(x1)))))) (41)
c3(a1(a2(c1(a0(x1))))) → c3(a3(a3(a3(a1(a0(x1)))))) (42)
c3(a1(a2(c1(a2(x1))))) → c3(a3(a3(a3(a1(a2(x1)))))) (43)
c3(a1(a2(c3(a1(x1))))) → c3(a3(a3(a3(a3(a1(x1)))))) (44)
c3(a1(a2(c3(a3(x1))))) → c3(a3(a3(a3(a3(a3(x1)))))) (45)
a2(c1(a2(c1(a0(x1))))) → a2(c3(a3(a3(a1(a0(x1)))))) (46)
a2(c1(a2(c1(a2(x1))))) → a2(c3(a3(a3(a1(a2(x1)))))) (47)
a2(c1(a2(c3(a1(x1))))) → a2(c3(a3(a3(a3(a1(x1)))))) (48)
a2(c1(a2(c3(a3(x1))))) → a2(c3(a3(a3(a3(a3(x1)))))) (49)
a3(a1(a2(c1(a0(x1))))) → a3(a3(a3(a3(a1(a0(x1)))))) (50)
a3(a1(a2(c1(a2(x1))))) → a3(a3(a3(a3(a1(a2(x1)))))) (51)
a3(a1(a2(c3(a1(x1))))) → a3(a3(a3(a3(a3(a1(x1)))))) (52)
a3(a1(a2(c3(a3(x1))))) → a3(a3(a3(a3(a3(a3(x1)))))) (53)
c1(a2(x1)) → c1(a0(c0(c0(c0(c2(x1)))))) (59)
c3(a1(x1)) → c1(a0(c0(c0(c2(c1(x1)))))) (60)
c3(a3(x1)) → c1(a0(c0(c0(c2(c3(x1)))))) (61)
a2(c1(x1)) → a0(c0(c0(c0(c2(c1(x1)))))) (64)
a2(c3(x1)) → a0(c0(c0(c0(c2(c3(x1)))))) (65)
a1(a2(x1)) → a1(a0(c0(c0(c0(c2(x1)))))) (67)
a3(a1(x1)) → a1(a0(c0(c0(c2(c1(x1)))))) (68)
a3(a3(x1)) → a1(a0(c0(c0(c2(c3(x1)))))) (69)

1.1.1.1.1 R is empty

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