Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup15)

The rewrite relation of the following TRS is considered.

a(a(d(d(x1)))) → d(d(b(b(x1)))) (1)
a(a(x1)) → b(b(b(b(b(b(x1)))))) (2)
b(b(d(d(b(b(x1)))))) → a(a(c(c(x1)))) (3)
c(c(x1)) → d(d(x1)) (4)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{d(☐), c(☐), b(☐), a(☐)}

We obtain the transformed TRS
d(a(a(d(d(x1))))) → d(d(d(b(b(x1))))) (5)
d(a(a(x1))) → d(b(b(b(b(b(b(x1))))))) (6)
d(b(b(d(d(b(b(x1))))))) → d(a(a(c(c(x1))))) (7)
d(c(c(x1))) → d(d(d(x1))) (8)
c(a(a(d(d(x1))))) → c(d(d(b(b(x1))))) (9)
c(a(a(x1))) → c(b(b(b(b(b(b(x1))))))) (10)
c(b(b(d(d(b(b(x1))))))) → c(a(a(c(c(x1))))) (11)
c(c(c(x1))) → c(d(d(x1))) (12)
b(a(a(d(d(x1))))) → b(d(d(b(b(x1))))) (13)
b(a(a(x1))) → b(b(b(b(b(b(b(x1))))))) (14)
b(b(b(d(d(b(b(x1))))))) → b(a(a(c(c(x1))))) (15)
b(c(c(x1))) → b(d(d(x1))) (16)
a(a(a(d(d(x1))))) → a(d(d(b(b(x1))))) (17)
a(a(a(x1))) → a(b(b(b(b(b(b(x1))))))) (18)
a(b(b(d(d(b(b(x1))))))) → a(a(a(c(c(x1))))) (19)
a(c(c(x1))) → a(d(d(x1))) (20)

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):

[d(x1)] = 4x1 + 0
[c(x1)] = 4x1 + 1
[b(x1)] = 4x1 + 2
[a(x1)] = 4x1 + 3

We obtain the labeled TRS
a3(a3(a0(d0(d3(x1))))) → a0(d0(d2(b2(b3(x1))))) (21)
a3(a3(a0(d0(d0(x1))))) → a0(d0(d2(b2(b0(x1))))) (22)
a3(a3(a0(d0(d2(x1))))) → a0(d0(d2(b2(b2(x1))))) (23)
a3(a3(a0(d0(d1(x1))))) → a0(d0(d2(b2(b1(x1))))) (24)
d3(a3(a0(d0(d3(x1))))) → d0(d0(d2(b2(b3(x1))))) (25)
d3(a3(a0(d0(d0(x1))))) → d0(d0(d2(b2(b0(x1))))) (26)
d3(a3(a0(d0(d2(x1))))) → d0(d0(d2(b2(b2(x1))))) (27)
d3(a3(a0(d0(d1(x1))))) → d0(d0(d2(b2(b1(x1))))) (28)
b3(a3(a0(d0(d3(x1))))) → b0(d0(d2(b2(b3(x1))))) (29)
b3(a3(a0(d0(d0(x1))))) → b0(d0(d2(b2(b0(x1))))) (30)
b3(a3(a0(d0(d2(x1))))) → b0(d0(d2(b2(b2(x1))))) (31)
b3(a3(a0(d0(d1(x1))))) → b0(d0(d2(b2(b1(x1))))) (32)
c3(a3(a0(d0(d3(x1))))) → c0(d0(d2(b2(b3(x1))))) (33)
c3(a3(a0(d0(d0(x1))))) → c0(d0(d2(b2(b0(x1))))) (34)
c3(a3(a0(d0(d2(x1))))) → c0(d0(d2(b2(b2(x1))))) (35)
c3(a3(a0(d0(d1(x1))))) → c0(d0(d2(b2(b1(x1))))) (36)
a3(a3(a3(x1))) → a2(b2(b2(b2(b2(b2(b3(x1))))))) (37)
a3(a3(a0(x1))) → a2(b2(b2(b2(b2(b2(b0(x1))))))) (38)
a3(a3(a2(x1))) → a2(b2(b2(b2(b2(b2(b2(x1))))))) (39)
a3(a3(a1(x1))) → a2(b2(b2(b2(b2(b2(b1(x1))))))) (40)
d3(a3(a3(x1))) → d2(b2(b2(b2(b2(b2(b3(x1))))))) (41)
d3(a3(a0(x1))) → d2(b2(b2(b2(b2(b2(b0(x1))))))) (42)
d3(a3(a2(x1))) → d2(b2(b2(b2(b2(b2(b2(x1))))))) (43)
d3(a3(a1(x1))) → d2(b2(b2(b2(b2(b2(b1(x1))))))) (44)
b3(a3(a3(x1))) → b2(b2(b2(b2(b2(b2(b3(x1))))))) (45)
b3(a3(a0(x1))) → b2(b2(b2(b2(b2(b2(b0(x1))))))) (46)
b3(a3(a2(x1))) → b2(b2(b2(b2(b2(b2(b2(x1))))))) (47)
b3(a3(a1(x1))) → b2(b2(b2(b2(b2(b2(b1(x1))))))) (48)
c3(a3(a3(x1))) → c2(b2(b2(b2(b2(b2(b3(x1))))))) (49)
c3(a3(a0(x1))) → c2(b2(b2(b2(b2(b2(b0(x1))))))) (50)
c3(a3(a2(x1))) → c2(b2(b2(b2(b2(b2(b2(x1))))))) (51)
c3(a3(a1(x1))) → c2(b2(b2(b2(b2(b2(b1(x1))))))) (52)
a2(b2(b0(d0(d2(b2(b3(x1))))))) → a3(a3(a1(c1(c3(x1))))) (53)
a2(b2(b0(d0(d2(b2(b0(x1))))))) → a3(a3(a1(c1(c0(x1))))) (54)
a2(b2(b0(d0(d2(b2(b2(x1))))))) → a3(a3(a1(c1(c2(x1))))) (55)
a2(b2(b0(d0(d2(b2(b1(x1))))))) → a3(a3(a1(c1(c1(x1))))) (56)
d2(b2(b0(d0(d2(b2(b3(x1))))))) → d3(a3(a1(c1(c3(x1))))) (57)
d2(b2(b0(d0(d2(b2(b0(x1))))))) → d3(a3(a1(c1(c0(x1))))) (58)
d2(b2(b0(d0(d2(b2(b2(x1))))))) → d3(a3(a1(c1(c2(x1))))) (59)
d2(b2(b0(d0(d2(b2(b1(x1))))))) → d3(a3(a1(c1(c1(x1))))) (60)
b2(b2(b0(d0(d2(b2(b3(x1))))))) → b3(a3(a1(c1(c3(x1))))) (61)
b2(b2(b0(d0(d2(b2(b0(x1))))))) → b3(a3(a1(c1(c0(x1))))) (62)
b2(b2(b0(d0(d2(b2(b2(x1))))))) → b3(a3(a1(c1(c2(x1))))) (63)
b2(b2(b0(d0(d2(b2(b1(x1))))))) → b3(a3(a1(c1(c1(x1))))) (64)
c2(b2(b0(d0(d2(b2(b3(x1))))))) → c3(a3(a1(c1(c3(x1))))) (65)
c2(b2(b0(d0(d2(b2(b0(x1))))))) → c3(a3(a1(c1(c0(x1))))) (66)
c2(b2(b0(d0(d2(b2(b2(x1))))))) → c3(a3(a1(c1(c2(x1))))) (67)
c2(b2(b0(d0(d2(b2(b1(x1))))))) → c3(a3(a1(c1(c1(x1))))) (68)
a1(c1(c3(x1))) → a0(d0(d3(x1))) (69)
a1(c1(c0(x1))) → a0(d0(d0(x1))) (70)
a1(c1(c2(x1))) → a0(d0(d2(x1))) (71)
a1(c1(c1(x1))) → a0(d0(d1(x1))) (72)
d1(c1(c3(x1))) → d0(d0(d3(x1))) (73)
d1(c1(c0(x1))) → d0(d0(d0(x1))) (74)
d1(c1(c2(x1))) → d0(d0(d2(x1))) (75)
d1(c1(c1(x1))) → d0(d0(d1(x1))) (76)
b1(c1(c3(x1))) → b0(d0(d3(x1))) (77)
b1(c1(c0(x1))) → b0(d0(d0(x1))) (78)
b1(c1(c2(x1))) → b0(d0(d2(x1))) (79)
b1(c1(c1(x1))) → b0(d0(d1(x1))) (80)
c1(c1(c3(x1))) → c0(d0(d3(x1))) (81)
c1(c1(c0(x1))) → c0(d0(d0(x1))) (82)
c1(c1(c2(x1))) → c0(d0(d2(x1))) (83)
c1(c1(c1(x1))) → c0(d0(d1(x1))) (84)

1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[d0(x1)] = x1 +
1
[d1(x1)] = x1 +
1
[d2(x1)] = x1 +
0
[d3(x1)] = x1 +
0
[c0(x1)] = x1 +
1
[c1(x1)] = x1 +
1
[c2(x1)] = x1 +
0
[c3(x1)] = x1 +
0
[b0(x1)] = x1 +
1
[b1(x1)] = x1 +
1
[b2(x1)] = x1 +
0
[b3(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
0
[a2(x1)] = x1 +
1
[a3(x1)] = x1 +
1
all of the following rules can be deleted.
a3(a3(a0(d0(d3(x1))))) → a0(d0(d2(b2(b3(x1))))) (21)
a3(a3(a0(d0(d0(x1))))) → a0(d0(d2(b2(b0(x1))))) (22)
a3(a3(a0(d0(d2(x1))))) → a0(d0(d2(b2(b2(x1))))) (23)
a3(a3(a0(d0(d1(x1))))) → a0(d0(d2(b2(b1(x1))))) (24)
a3(a3(a3(x1))) → a2(b2(b2(b2(b2(b2(b3(x1))))))) (37)
a3(a3(a2(x1))) → a2(b2(b2(b2(b2(b2(b2(x1))))))) (39)
d3(a3(a3(x1))) → d2(b2(b2(b2(b2(b2(b3(x1))))))) (41)
d3(a3(a2(x1))) → d2(b2(b2(b2(b2(b2(b2(x1))))))) (43)
b3(a3(a3(x1))) → b2(b2(b2(b2(b2(b2(b3(x1))))))) (45)
b3(a3(a2(x1))) → b2(b2(b2(b2(b2(b2(b2(x1))))))) (47)
c3(a3(a3(x1))) → c2(b2(b2(b2(b2(b2(b3(x1))))))) (49)
c3(a3(a2(x1))) → c2(b2(b2(b2(b2(b2(b2(x1))))))) (51)

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 173 ruless (increase limit for explicit display).

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
[d0(x1)] = x1 +
3
[d1(x1)] = x1 +
3
[d2(x1)] = x1 +
0
[d3(x1)] = x1 +
0
[c0(x1)] = x1 +
3
[c1(x1)] = x1 +
3
[c2(x1)] = x1 +
0
[c3(x1)] = x1 +
0
[b0(x1)] = x1 +
3
[b1(x1)] = x1 +
3
[b2(x1)] = x1 +
0
[b3(x1)] = x1 +
0
[a0(x1)] = x1 +
0
[a1(x1)] = x1 +
0
[a2(x1)] = x1 +
3
[a3(x1)] = x1 +
3
[d1#(x1)] = x1 +
1
[d2#(x1)] = x1 +
2
[d3#(x1)] = x1 +
2
[c1#(x1)] = x1 +
1
[c2#(x1)] = x1 +
2
[c3#(x1)] = x1 +
2
[b1#(x1)] = x1 +
1
[b2#(x1)] = x1 +
1
[b3#(x1)] = x1 +
1
[a1#(x1)] = x1 +
1
[a2#(x1)] = x1 +
2
[a3#(x1)] = x1 +
2
together with the usable rules
d3(a3(a0(d0(d3(x1))))) → d0(d0(d2(b2(b3(x1))))) (25)
d3(a3(a0(d0(d0(x1))))) → d0(d0(d2(b2(b0(x1))))) (26)
d3(a3(a0(d0(d2(x1))))) → d0(d0(d2(b2(b2(x1))))) (27)
d3(a3(a0(d0(d1(x1))))) → d0(d0(d2(b2(b1(x1))))) (28)
b3(a3(a0(d0(d3(x1))))) → b0(d0(d2(b2(b3(x1))))) (29)
b3(a3(a0(d0(d0(x1))))) → b0(d0(d2(b2(b0(x1))))) (30)
b3(a3(a0(d0(d2(x1))))) → b0(d0(d2(b2(b2(x1))))) (31)
b3(a3(a0(d0(d1(x1))))) → b0(d0(d2(b2(b1(x1))))) (32)
c3(a3(a0(d0(d3(x1))))) → c0(d0(d2(b2(b3(x1))))) (33)
c3(a3(a0(d0(d0(x1))))) → c0(d0(d2(b2(b0(x1))))) (34)
c3(a3(a0(d0(d2(x1))))) → c0(d0(d2(b2(b2(x1))))) (35)
c3(a3(a0(d0(d1(x1))))) → c0(d0(d2(b2(b1(x1))))) (36)
a3(a3(a0(x1))) → a2(b2(b2(b2(b2(b2(b0(x1))))))) (38)
a3(a3(a1(x1))) → a2(b2(b2(b2(b2(b2(b1(x1))))))) (40)
d3(a3(a0(x1))) → d2(b2(b2(b2(b2(b2(b0(x1))))))) (42)
d3(a3(a1(x1))) → d2(b2(b2(b2(b2(b2(b1(x1))))))) (44)
b3(a3(a0(x1))) → b2(b2(b2(b2(b2(b2(b0(x1))))))) (46)
b3(a3(a1(x1))) → b2(b2(b2(b2(b2(b2(b1(x1))))))) (48)
c3(a3(a0(x1))) → c2(b2(b2(b2(b2(b2(b0(x1))))))) (50)
c3(a3(a1(x1))) → c2(b2(b2(b2(b2(b2(b1(x1))))))) (52)
a2(b2(b0(d0(d2(b2(b3(x1))))))) → a3(a3(a1(c1(c3(x1))))) (53)
a2(b2(b0(d0(d2(b2(b0(x1))))))) → a3(a3(a1(c1(c0(x1))))) (54)
a2(b2(b0(d0(d2(b2(b2(x1))))))) → a3(a3(a1(c1(c2(x1))))) (55)
a2(b2(b0(d0(d2(b2(b1(x1))))))) → a3(a3(a1(c1(c1(x1))))) (56)
d2(b2(b0(d0(d2(b2(b3(x1))))))) → d3(a3(a1(c1(c3(x1))))) (57)
d2(b2(b0(d0(d2(b2(b0(x1))))))) → d3(a3(a1(c1(c0(x1))))) (58)
d2(b2(b0(d0(d2(b2(b2(x1))))))) → d3(a3(a1(c1(c2(x1))))) (59)
d2(b2(b0(d0(d2(b2(b1(x1))))))) → d3(a3(a1(c1(c1(x1))))) (60)
b2(b2(b0(d0(d2(b2(b3(x1))))))) → b3(a3(a1(c1(c3(x1))))) (61)
b2(b2(b0(d0(d2(b2(b0(x1))))))) → b3(a3(a1(c1(c0(x1))))) (62)
b2(b2(b0(d0(d2(b2(b2(x1))))))) → b3(a3(a1(c1(c2(x1))))) (63)
b2(b2(b0(d0(d2(b2(b1(x1))))))) → b3(a3(a1(c1(c1(x1))))) (64)
c2(b2(b0(d0(d2(b2(b3(x1))))))) → c3(a3(a1(c1(c3(x1))))) (65)
c2(b2(b0(d0(d2(b2(b0(x1))))))) → c3(a3(a1(c1(c0(x1))))) (66)
c2(b2(b0(d0(d2(b2(b2(x1))))))) → c3(a3(a1(c1(c2(x1))))) (67)
c2(b2(b0(d0(d2(b2(b1(x1))))))) → c3(a3(a1(c1(c1(x1))))) (68)
a1(c1(c3(x1))) → a0(d0(d3(x1))) (69)
a1(c1(c0(x1))) → a0(d0(d0(x1))) (70)
a1(c1(c2(x1))) → a0(d0(d2(x1))) (71)
a1(c1(c1(x1))) → a0(d0(d1(x1))) (72)
d1(c1(c3(x1))) → d0(d0(d3(x1))) (73)
d1(c1(c0(x1))) → d0(d0(d0(x1))) (74)
d1(c1(c2(x1))) → d0(d0(d2(x1))) (75)
d1(c1(c1(x1))) → d0(d0(d1(x1))) (76)
b1(c1(c3(x1))) → b0(d0(d3(x1))) (77)
b1(c1(c0(x1))) → b0(d0(d0(x1))) (78)
b1(c1(c2(x1))) → b0(d0(d2(x1))) (79)
b1(c1(c1(x1))) → b0(d0(d1(x1))) (80)
c1(c1(c3(x1))) → c0(d0(d3(x1))) (81)
c1(c1(c0(x1))) → c0(d0(d0(x1))) (82)
c1(c1(c2(x1))) → c0(d0(d2(x1))) (83)
c1(c1(c1(x1))) → c0(d0(d1(x1))) (84)
(w.r.t. the implicit argument filter of the reduction pair), the pairs

There are 139 ruless (increase limit for explicit display).

and no rules could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.