Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/212308)

The rewrite relation of the following TRS is considered.

0(0(x1)) → 0(1(2(0(1(1(x1)))))) (1)
0(0(x1)) → 1(0(1(0(1(1(x1)))))) (2)
3(0(x1)) → 1(0(1(1(1(3(x1)))))) (3)
0(0(0(x1))) → 0(0(1(1(0(x1))))) (4)
0(0(3(x1))) → 0(3(1(0(1(1(x1)))))) (5)
2(3(0(x1))) → 2(1(0(1(1(3(x1)))))) (6)
3(0(0(x1))) → 3(0(1(0(1(1(x1)))))) (7)
3(0(4(x1))) → 3(1(0(1(1(4(x1)))))) (8)
5(0(3(x1))) → 3(0(1(1(5(x1))))) (9)
5(0(3(x1))) → 3(5(0(1(1(x1))))) (10)
0(0(0(4(x1)))) → 2(0(2(0(0(4(x1)))))) (11)
0(0(1(3(x1)))) → 0(3(1(0(1(1(x1)))))) (12)
0(0(2(4(x1)))) → 0(2(2(0(4(x1))))) (13)
0(2(1(3(x1)))) → 0(1(2(3(4(x1))))) (14)
0(2(1(3(x1)))) → 2(0(1(3(4(x1))))) (15)
0(2(1(4(x1)))) → 2(0(1(1(4(x1))))) (16)
0(2(3(0(x1)))) → 2(3(4(0(0(x1))))) (17)
0(2(3(0(x1)))) → 2(0(1(2(3(0(x1)))))) (18)
0(2(5(4(x1)))) → 2(2(4(5(0(x1))))) (19)
0(2(5(4(x1)))) → 4(5(1(2(0(x1))))) (20)
0(4(1(0(x1)))) → 0(0(1(1(4(x1))))) (21)
0(5(5(4(x1)))) → 2(4(5(5(1(0(x1)))))) (22)
2(3(0(0(x1)))) → 0(1(1(2(3(0(x1)))))) (23)
2(3(0(0(x1)))) → 0(1(2(1(3(0(x1)))))) (24)
3(0(0(3(x1)))) → 3(0(3(0(1(1(x1)))))) (25)
3(0(2(0(x1)))) → 0(3(0(1(2(1(x1)))))) (26)
3(0(4(0(x1)))) → 3(0(0(1(1(4(x1)))))) (27)
3(0(5(0(x1)))) → 5(0(3(0(1(1(x1)))))) (28)
3(1(0(0(x1)))) → 1(1(1(3(0(0(x1)))))) (29)
3(1(0(4(x1)))) → 1(1(1(3(4(0(x1)))))) (30)
3(2(1(4(x1)))) → 1(1(2(2(4(3(x1)))))) (31)
3(2(1(4(x1)))) → 1(2(1(3(1(4(x1)))))) (32)
4(1(0(0(x1)))) → 0(0(1(1(4(1(x1)))))) (33)
4(1(0(4(x1)))) → 4(0(1(1(4(x1))))) (34)
5(1(0(3(x1)))) → 5(3(0(1(1(1(x1)))))) (35)
5(5(0(4(x1)))) → 5(5(1(0(4(x1))))) (36)
5(5(0(4(x1)))) → 0(1(1(5(5(4(x1)))))) (37)
5(5(4(3(x1)))) → 3(4(5(5(1(x1))))) (38)
0(1(4(3(3(x1))))) → 0(1(1(4(3(3(x1)))))) (39)
0(2(1(5(3(x1))))) → 0(5(2(1(1(3(x1)))))) (40)
0(2(5(2(3(x1))))) → 0(3(5(2(1(2(x1)))))) (41)
0(2(5(3(0(x1))))) → 0(1(2(5(3(0(x1)))))) (42)
2(5(1(0(4(x1))))) → 5(0(1(1(4(2(x1)))))) (43)
3(5(1(0(4(x1))))) → 3(5(0(1(1(4(x1)))))) (44)
4(0(1(0(3(x1))))) → 4(0(3(0(1(1(x1)))))) (45)
4(1(4(4(0(x1))))) → 4(0(4(1(1(4(x1)))))) (46)
4(2(3(5(4(x1))))) → 2(2(4(4(5(3(x1)))))) (47)
4(2(5(0(4(x1))))) → 4(4(1(0(5(2(x1)))))) (48)
4(4(5(0(0(x1))))) → 4(4(0(1(5(0(x1)))))) (49)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by matchbox @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{5(☐), 4(☐), 3(☐), 2(☐), 1(☐), 0(☐)}

We obtain the transformed TRS

There are 294 ruless (increase limit for explicit display).

1.1 Semantic Labeling

The following interpretations form a model of the rules.

As carrier we take the set {0,...,5}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 6):

[5(x1)] = 6x1 + 0
[4(x1)] = 6x1 + 1
[3(x1)] = 6x1 + 2
[2(x1)] = 6x1 + 3
[1(x1)] = 6x1 + 4
[0(x1)] = 6x1 + 5

We obtain the labeled TRS

There are 1764 ruless (increase limit for explicit display).

1.1.1 Rule Removal

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[50(x1)] = x1 +
58
[51(x1)] = x1 +
332
[52(x1)] = x1 +
46
[53(x1)] = x1 +
0
[54(x1)] = x1 +
15
[55(x1)] = x1 +
238
[40(x1)] = x1 +
134
[41(x1)] = x1 +
301
[42(x1)] = x1 +
15
[43(x1)] = x1 +
35
[44(x1)] = x1 +
0
[45(x1)] = x1 +
36
[30(x1)] = x1 +
149
[31(x1)] = x1 +
1
[32(x1)] = x1 +
35
[33(x1)] = x1 +
35
[34(x1)] = x1 +
0
[35(x1)] = x1 +
38
[20(x1)] = x1 +
198
[21(x1)] = x1 +
0
[22(x1)] = x1 +
35
[23(x1)] = x1 +
0
[24(x1)] = x1 +
0
[25(x1)] = x1 +
0
[10(x1)] = x1 +
18
[11(x1)] = x1 +
0
[12(x1)] = x1 +
6
[13(x1)] = x1 +
0
[14(x1)] = x1 +
0
[15(x1)] = x1 +
38
[00(x1)] = x1 +
312
[01(x1)] = x1 +
300
[02(x1)] = x1 +
139
[03(x1)] = x1 +
332
[04(x1)] = x1 +
1
[05(x1)] = x1 +
332
all of the following rules can be deleted.

There are 1688 ruless (increase limit for explicit display).

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 167 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
[51(x1)] = x1 +
0
[55(x1)] = x1 +
2
[40(x1)] = x1 +
0
[41(x1)] = x1 +
2
[42(x1)] = x1 +
0
[43(x1)] = x1 +
0
[44(x1)] = x1 +
2
[45(x1)] = x1 +
2
[30(x1)] = x1 +
0
[31(x1)] = x1 +
0
[32(x1)] = x1 +
0
[33(x1)] = x1 +
0
[34(x1)] = x1 +
0
[35(x1)] = x1 +
0
[21(x1)] = x1 +
0
[23(x1)] = x1 +
0
[24(x1)] = x1 +
0
[25(x1)] = x1 +
1
[11(x1)] = x1 +
0
[12(x1)] = x1 +
0
[13(x1)] = x1 +
0
[14(x1)] = x1 +
0
[15(x1)] = x1 +
0
[00(x1)] = x1 +
2
[01(x1)] = x1 +
2
[02(x1)] = x1 +
0
[03(x1)] = x1 +
0
[04(x1)] = x1 +
2
[05(x1)] = x1 +
1
[51#(x1)] = x1 +
2
[55#(x1)] = x1 +
0
[41#(x1)] = x1 +
0
[45#(x1)] = x1 +
0
[31#(x1)] = x1 +
2
[35#(x1)] = x1 +
0
[21#(x1)] = x1 +
2
[25#(x1)] = x1 +
0
[11#(x1)] = x1 +
1
[12#(x1)] = x1 +
2
[15#(x1)] = x1 +
0
[01#(x1)] = x1 +
0
[02#(x1)] = x1 +
1
[05#(x1)] = x1 +
1
together with the usable rules
12(35(04(x1))) → 14(15(04(14(14(12(34(x1))))))) (423)
05(05(05(01(45(x1))))) → 03(25(03(25(05(01(45(x1))))))) (704)
05(05(05(01(44(x1))))) → 03(25(03(25(05(01(44(x1))))))) (705)
05(05(05(01(43(x1))))) → 03(25(03(25(05(01(43(x1))))))) (706)
05(05(05(01(42(x1))))) → 03(25(03(25(05(01(42(x1))))))) (707)
05(05(05(01(41(x1))))) → 03(25(03(25(05(01(41(x1))))))) (708)
05(05(05(01(40(x1))))) → 03(25(03(25(05(01(40(x1))))))) (709)
25(05(05(01(45(x1))))) → 23(25(03(25(05(01(45(x1))))))) (716)
25(05(05(01(44(x1))))) → 23(25(03(25(05(01(44(x1))))))) (717)
25(05(05(01(43(x1))))) → 23(25(03(25(05(01(43(x1))))))) (718)
25(05(05(01(42(x1))))) → 23(25(03(25(05(01(42(x1))))))) (719)
25(05(05(01(41(x1))))) → 23(25(03(25(05(01(41(x1))))))) (720)
25(05(05(01(40(x1))))) → 23(25(03(25(05(01(40(x1))))))) (721)
02(35(03(25(04(x1))))) → 05(02(35(04(13(24(14(x1))))))) (1245)
02(35(00(55(04(x1))))) → 00(55(02(35(04(14(14(x1))))))) (1317)
12(34(15(05(05(x1))))) → 14(14(14(12(35(05(05(x1))))))) (1358)
12(34(15(05(04(x1))))) → 14(14(14(12(35(05(04(x1))))))) (1359)
12(34(15(05(03(x1))))) → 14(14(14(12(35(05(03(x1))))))) (1360)
12(34(15(05(02(x1))))) → 14(14(14(12(35(05(02(x1))))))) (1361)
12(34(15(05(01(x1))))) → 14(14(14(12(35(05(01(x1))))))) (1362)
12(34(15(05(00(x1))))) → 14(14(14(12(35(05(00(x1))))))) (1363)
11(44(15(05(04(x1))))) → 15(05(04(14(11(44(14(x1))))))) (1503)
05(04(11(42(32(35(x1)))))) → 05(04(14(11(42(32(35(x1))))))) (1712)
05(04(11(42(32(34(x1)))))) → 05(04(14(11(42(32(34(x1))))))) (1713)
05(04(11(42(32(33(x1)))))) → 05(04(14(11(42(32(33(x1))))))) (1714)
05(04(11(42(32(32(x1)))))) → 05(04(14(11(42(32(32(x1))))))) (1715)
05(04(11(42(32(31(x1)))))) → 05(04(14(11(42(32(31(x1))))))) (1716)
05(04(11(42(32(30(x1)))))) → 05(04(14(11(42(32(30(x1))))))) (1717)
15(04(11(42(32(35(x1)))))) → 15(04(14(11(42(32(35(x1))))))) (1718)
15(04(11(42(32(34(x1)))))) → 15(04(14(11(42(32(34(x1))))))) (1719)
15(04(11(42(32(33(x1)))))) → 15(04(14(11(42(32(33(x1))))))) (1720)
15(04(11(42(32(32(x1)))))) → 15(04(14(11(42(32(32(x1))))))) (1721)
15(04(11(42(32(31(x1)))))) → 15(04(14(11(42(32(31(x1))))))) (1722)
15(04(11(42(32(30(x1)))))) → 15(04(14(11(42(32(30(x1))))))) (1723)
25(04(11(42(32(35(x1)))))) → 25(04(14(11(42(32(35(x1))))))) (1724)
25(04(11(42(32(34(x1)))))) → 25(04(14(11(42(32(34(x1))))))) (1725)
25(04(11(42(32(33(x1)))))) → 25(04(14(11(42(32(33(x1))))))) (1726)
25(04(11(42(32(32(x1)))))) → 25(04(14(11(42(32(32(x1))))))) (1727)
25(04(11(42(32(31(x1)))))) → 25(04(14(11(42(32(31(x1))))))) (1728)
25(04(11(42(32(30(x1)))))) → 25(04(14(11(42(32(30(x1))))))) (1729)
35(04(11(42(32(35(x1)))))) → 35(04(14(11(42(32(35(x1))))))) (1730)
35(04(11(42(32(34(x1)))))) → 35(04(14(11(42(32(34(x1))))))) (1731)
35(04(11(42(32(33(x1)))))) → 35(04(14(11(42(32(33(x1))))))) (1732)
35(04(11(42(32(32(x1)))))) → 35(04(14(11(42(32(32(x1))))))) (1733)
35(04(11(42(32(31(x1)))))) → 35(04(14(11(42(32(31(x1))))))) (1734)
35(04(11(42(32(30(x1)))))) → 35(04(14(11(42(32(30(x1))))))) (1735)
45(04(11(42(32(35(x1)))))) → 45(04(14(11(42(32(35(x1))))))) (1736)
45(04(11(42(32(34(x1)))))) → 45(04(14(11(42(32(34(x1))))))) (1737)
45(04(11(42(32(33(x1)))))) → 45(04(14(11(42(32(33(x1))))))) (1738)
45(04(11(42(32(32(x1)))))) → 45(04(14(11(42(32(32(x1))))))) (1739)
45(04(11(42(32(31(x1)))))) → 45(04(14(11(42(32(31(x1))))))) (1740)
45(04(11(42(32(30(x1)))))) → 45(04(14(11(42(32(30(x1))))))) (1741)
55(04(11(42(32(35(x1)))))) → 55(04(14(11(42(32(35(x1))))))) (1742)
55(04(11(42(32(34(x1)))))) → 55(04(14(11(42(32(34(x1))))))) (1743)
55(04(11(42(32(33(x1)))))) → 55(04(14(11(42(32(33(x1))))))) (1744)
55(04(11(42(32(32(x1)))))) → 55(04(14(11(42(32(32(x1))))))) (1745)
55(04(11(42(32(31(x1)))))) → 55(04(14(11(42(32(31(x1))))))) (1746)
55(04(11(42(32(30(x1)))))) → 55(04(14(11(42(32(30(x1))))))) (1747)
01(45(04(15(02(35(x1)))))) → 01(45(02(35(04(14(15(x1))))))) (1928)
01(45(04(15(02(34(x1)))))) → 01(45(02(35(04(14(14(x1))))))) (1929)
11(45(04(15(02(35(x1)))))) → 11(45(02(35(04(14(15(x1))))))) (1934)
11(45(04(15(02(34(x1)))))) → 11(45(02(35(04(14(14(x1))))))) (1935)
21(45(04(15(02(35(x1)))))) → 21(45(02(35(04(14(15(x1))))))) (1940)
21(45(04(15(02(34(x1)))))) → 21(45(02(35(04(14(14(x1))))))) (1941)
31(45(04(15(02(35(x1)))))) → 31(45(02(35(04(14(15(x1))))))) (1946)
31(45(04(15(02(34(x1)))))) → 31(45(02(35(04(14(14(x1))))))) (1947)
41(45(04(15(02(35(x1)))))) → 41(45(02(35(04(14(15(x1))))))) (1952)
41(45(04(15(02(34(x1)))))) → 41(45(02(35(04(14(14(x1))))))) (1953)
51(45(04(15(02(35(x1)))))) → 51(45(02(35(04(14(15(x1))))))) (1958)
51(45(04(15(02(34(x1)))))) → 51(45(02(35(04(14(14(x1))))))) (1959)
01(44(11(41(45(01(x1)))))) → 01(45(01(44(14(11(41(x1))))))) (1968)
11(44(11(41(45(01(x1)))))) → 11(45(01(44(14(11(41(x1))))))) (1974)
21(44(11(41(45(01(x1)))))) → 21(45(01(44(14(11(41(x1))))))) (1980)
31(44(11(41(45(01(x1)))))) → 31(45(01(44(14(11(41(x1))))))) (1986)
41(44(11(41(45(01(x1)))))) → 41(45(01(44(14(11(41(x1))))))) (1992)
51(44(11(41(45(01(x1)))))) → 51(45(01(44(14(11(41(x1))))))) (1998)
(w.r.t. the implicit argument filter of the reduction pair), the pairs

There are 101 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 1 component.