Certification Problem

Input (TPDB SRS_Standard/ICFP_2010/41843)

The rewrite relation of the following TRS is considered.

0(0(1(2(x1)))) 0(3(1(0(2(x1))))) (1)
0(1(2(2(x1)))) 1(2(0(3(2(2(x1)))))) (2)
0(1(2(4(x1)))) 0(3(2(3(1(4(x1)))))) (3)
0(5(0(5(x1)))) 0(3(0(5(5(x1))))) (4)
0(5(1(2(x1)))) 1(0(1(5(2(x1))))) (5)
0(5(1(2(x1)))) 0(1(0(1(5(2(x1)))))) (6)
0(5(1(2(x1)))) 0(3(2(3(1(5(x1)))))) (7)
0(5(4(2(x1)))) 0(4(5(3(2(x1))))) (8)
0(5(5(2(x1)))) 5(0(1(5(2(x1))))) (9)
1(0(0(5(x1)))) 1(1(0(0(1(5(4(x1))))))) (10)
1(0(1(2(x1)))) 1(1(3(0(2(x1))))) (11)
1(0(1(2(x1)))) 1(1(0(3(2(2(x1)))))) (12)
1(0(1(2(x1)))) 1(1(0(3(2(3(x1)))))) (13)
1(0(5(4(x1)))) 0(1(1(5(4(x1))))) (14)
1(2(0(5(x1)))) 0(3(2(3(1(5(x1)))))) (15)
1(2(0(5(x1)))) 5(0(3(3(2(1(x1)))))) (16)
1(5(0(2(x1)))) 1(1(0(1(1(5(2(x1))))))) (17)
1(5(1(2(x1)))) 0(1(1(5(2(x1))))) (18)
1(5(1(2(x1)))) 1(0(1(5(3(2(x1)))))) (19)
5(0(0(2(x1)))) 5(0(3(0(2(x1))))) (20)
5(0(1(2(x1)))) 5(1(0(3(2(x1))))) (21)
5(0(1(2(x1)))) 5(1(0(3(2(3(x1)))))) (22)
0(0(0(1(2(x1))))) 0(2(0(1(0(3(3(4(x1)))))))) (23)
0(0(2(5(2(x1))))) 0(3(2(0(5(2(x1)))))) (24)
0(1(2(5(0(x1))))) 3(3(2(2(0(0(1(5(x1)))))))) (25)
0(1(2(5(2(x1))))) 0(3(2(1(5(3(2(x1))))))) (26)
0(3(5(2(2(x1))))) 0(4(5(3(2(2(x1)))))) (27)
0(4(2(0(5(x1))))) 0(4(0(3(2(1(5(x1))))))) (28)
0(4(2(5(2(x1))))) 0(5(4(3(3(2(2(x1))))))) (29)
0(5(0(2(2(x1))))) 0(2(5(0(3(2(x1)))))) (30)
0(5(0(5(1(x1))))) 0(1(0(3(5(5(x1)))))) (31)
0(5(1(3(0(x1))))) 0(0(1(1(5(3(x1)))))) (32)
0(5(2(2(4(x1))))) 0(5(3(2(2(4(x1)))))) (33)
0(5(2(3(1(x1))))) 0(1(5(3(2(2(2(x1))))))) (34)
0(5(2(4(1(x1))))) 0(4(3(2(5(1(x1)))))) (35)
0(5(3(5(2(x1))))) 0(0(3(5(5(2(x1)))))) (36)
0(5(5(3(1(x1))))) 5(0(1(5(3(3(2(x1))))))) (37)
1(0(5(5(1(x1))))) 0(4(5(1(5(1(x1)))))) (38)
1(1(2(2(0(x1))))) 1(1(3(2(2(0(x1)))))) (39)
1(1(2(3(4(x1))))) 1(1(3(2(2(4(x1)))))) (40)
1(1(3(5(2(x1))))) 1(1(5(3(3(2(x1)))))) (41)
1(5(0(5(0(x1))))) 0(1(5(3(5(1(0(x1))))))) (42)
1(5(5(1(2(x1))))) 1(5(1(1(5(3(2(2(x1)))))))) (43)
5(0(2(0(5(x1))))) 5(0(3(3(2(0(5(x1))))))) (44)
5(0(2(3(4(x1))))) 5(0(3(2(3(4(x1)))))) (45)
5(5(0(1(2(x1))))) 5(5(3(0(2(1(x1)))))) (46)
0(0(0(5(1(2(x1)))))) 0(0(1(5(0(2(4(x1))))))) (47)
0(0(1(2(4(1(x1)))))) 1(3(0(2(3(0(4(1(x1)))))))) (48)
0(0(2(1(2(0(x1)))))) 0(1(0(3(2(2(2(0(x1)))))))) (49)
0(0(2(3(0(5(x1)))))) 0(0(3(0(3(2(5(x1))))))) (50)
0(0(5(2(3(4(x1)))))) 0(4(0(3(1(2(5(x1))))))) (51)
0(0(5(5(3(4(x1)))))) 1(4(1(0(0(3(5(5(x1)))))))) (52)
0(1(2(0(1(2(x1)))))) 1(0(3(2(2(1(1(0(x1)))))))) (53)
0(1(2(2(0(5(x1)))))) 0(4(1(5(0(3(2(2(x1)))))))) (54)
0(1(2(5(5(5(x1)))))) 0(2(5(1(5(3(5(x1))))))) (55)
0(1(3(1(5(2(x1)))))) 1(0(1(5(3(3(2(x1))))))) (56)
0(1(4(4(0(5(x1)))))) 4(3(0(0(1(5(4(x1))))))) (57)
0(2(5(3(5(1(x1)))))) 0(3(3(2(2(1(5(5(x1)))))))) (58)
0(5(0(0(5(4(x1)))))) 0(1(0(1(0(4(5(5(x1)))))))) (59)
0(5(1(2(1(4(x1)))))) 1(1(5(3(2(2(0(4(x1)))))))) (60)
0(5(5(1(2(5(x1)))))) 0(2(1(5(5(4(5(x1))))))) (61)
1(0(0(2(3(4(x1)))))) 1(4(0(0(3(3(2(x1))))))) (62)
1(0(1(3(5(1(x1)))))) 0(1(1(1(5(3(2(2(x1)))))))) (63)
1(0(5(4(2(1(x1)))))) 0(1(1(5(3(4(2(x1))))))) (64)
1(1(0(1(2(2(x1)))))) 1(0(1(1(3(1(2(2(x1)))))))) (65)
1(2(1(2(0(0(x1)))))) 0(3(2(2(1(0(1(x1))))))) (66)
1(4(1(0(0(5(x1)))))) 0(0(1(5(4(2(1(x1))))))) (67)
1(4(3(5(0(2(x1)))))) 0(3(3(2(1(5(4(x1))))))) (68)
0(0(1(2(3(5(5(x1))))))) 5(1(5(0(3(0(2(1(x1)))))))) (69)
0(0(2(3(4(2(1(x1))))))) 0(0(4(1(3(2(3(2(x1)))))))) (70)
0(1(0(0(5(3(4(x1))))))) 0(3(0(5(0(4(3(1(x1)))))))) (71)
0(1(2(4(4(0(5(x1))))))) 5(4(0(1(0(3(2(4(x1)))))))) (72)
0(5(2(5(1(3(4(x1))))))) 1(4(5(2(0(3(1(5(x1)))))))) (73)
0(5(5(0(2(5(1(x1))))))) 5(3(0(0(1(5(2(5(x1)))))))) (74)
0(5(5(2(5(3(4(x1))))))) 0(3(2(1(4(5(5(5(x1)))))))) (75)
1(0(1(2(3(4(5(x1))))))) 3(0(2(1(5(1(3(4(x1)))))))) (76)
1(1(0(2(0(2(2(x1))))))) 1(1(0(2(0(3(2(2(x1)))))))) (77)
1(2(4(3(5(3(5(x1))))))) 5(1(3(2(2(4(3(5(x1)))))))) (78)
1(4(3(5(2(5(2(x1))))))) 5(1(3(2(2(1(5(4(x1)))))))) (79)

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 474 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 2844 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 +
90
[51(x1)] = x1 +
91
[52(x1)] = x1 +
8
[53(x1)] = x1 +
123
[54(x1)] = x1 +
82
[55(x1)] = x1 +
85
[40(x1)] = x1 +
1
[41(x1)] = x1 +
82
[42(x1)] = x1 +
0
[43(x1)] = x1 +
49
[44(x1)] = x1 +
3
[45(x1)] = x1 +
41
[30(x1)] = x1 +
82
[31(x1)] = x1 +
131
[32(x1)] = x1 +
0
[33(x1)] = x1 +
0
[34(x1)] = x1 +
0
[35(x1)] = x1 +
8
[20(x1)] = x1 +
41
[21(x1)] = x1 +
42
[22(x1)] = x1 +
41
[23(x1)] = x1 +
0
[24(x1)] = x1 +
8
[25(x1)] = x1 +
0
[10(x1)] = x1 +
0
[11(x1)] = x1 +
82
[12(x1)] = x1 +
0
[13(x1)] = x1 +
134
[14(x1)] = x1 +
0
[15(x1)] = x1 +
1
[00(x1)] = x1 +
132
[01(x1)] = x1 +
11
[02(x1)] = x1 +
0
[03(x1)] = x1 +
49
[04(x1)] = x1 +
9
[05(x1)] = x1 +
49
all of the following rules can be deleted.

There are 2808 ruless (increase limit for explicit display).

1.1.1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
25(23(13(04(15(14(04(x1))))))) 25(23(13(34(12(14(04(15(04(x1))))))))) (3398)
24(23(13(04(15(14(04(x1))))))) 24(23(13(34(12(14(04(15(04(x1))))))))) (3399)
23(23(13(04(15(14(04(x1))))))) 23(23(13(34(12(14(04(15(04(x1))))))))) (3400)
22(23(13(04(15(14(04(x1))))))) 22(23(13(34(12(14(04(15(04(x1))))))))) (3401)
21(23(13(04(15(14(04(x1))))))) 21(23(13(34(12(14(04(15(04(x1))))))))) (3402)
20(23(13(04(15(14(04(x1))))))) 20(23(13(34(12(14(04(15(04(x1))))))))) (3403)
25(23(13(04(15(14(14(x1))))))) 25(23(13(34(12(14(04(15(14(x1))))))))) (3404)
24(23(13(04(15(14(14(x1))))))) 24(23(13(34(12(14(04(15(14(x1))))))))) (3405)
23(23(13(04(15(14(14(x1))))))) 23(23(13(34(12(14(04(15(14(x1))))))))) (3406)
22(23(13(04(15(14(14(x1))))))) 22(23(13(34(12(14(04(15(14(x1))))))))) (3407)
21(23(13(04(15(14(14(x1))))))) 21(23(13(34(12(14(04(15(14(x1))))))))) (3408)
20(23(13(04(15(14(14(x1))))))) 20(23(13(34(12(14(04(15(14(x1))))))))) (3409)
25(23(13(04(15(14(24(x1))))))) 25(23(13(34(12(14(04(15(24(x1))))))))) (3410)
24(23(13(04(15(14(24(x1))))))) 24(23(13(34(12(14(04(15(24(x1))))))))) (3411)
23(23(13(04(15(14(24(x1))))))) 23(23(13(34(12(14(04(15(24(x1))))))))) (3412)
22(23(13(04(15(14(24(x1))))))) 22(23(13(34(12(14(04(15(24(x1))))))))) (3413)
21(23(13(04(15(14(24(x1))))))) 21(23(13(34(12(14(04(15(24(x1))))))))) (3414)
20(23(13(04(15(14(24(x1))))))) 20(23(13(34(12(14(04(15(24(x1))))))))) (3415)
25(23(13(04(15(14(34(x1))))))) 25(23(13(34(12(14(04(15(34(x1))))))))) (3416)
24(23(13(04(15(14(34(x1))))))) 24(23(13(34(12(14(04(15(34(x1))))))))) (3417)
23(23(13(04(15(14(34(x1))))))) 23(23(13(34(12(14(04(15(34(x1))))))))) (3418)
22(23(13(04(15(14(34(x1))))))) 22(23(13(34(12(14(04(15(34(x1))))))))) (3419)
21(23(13(04(15(14(34(x1))))))) 21(23(13(34(12(14(04(15(34(x1))))))))) (3420)
20(23(13(04(15(14(34(x1))))))) 20(23(13(34(12(14(04(15(34(x1))))))))) (3421)
25(23(13(04(15(14(44(x1))))))) 25(23(13(34(12(14(04(15(44(x1))))))))) (3422)
24(23(13(04(15(14(44(x1))))))) 24(23(13(34(12(14(04(15(44(x1))))))))) (3423)
23(23(13(04(15(14(44(x1))))))) 23(23(13(34(12(14(04(15(44(x1))))))))) (3424)
22(23(13(04(15(14(44(x1))))))) 22(23(13(34(12(14(04(15(44(x1))))))))) (3425)
21(23(13(04(15(14(44(x1))))))) 21(23(13(34(12(14(04(15(44(x1))))))))) (3426)
20(23(13(04(15(14(44(x1))))))) 20(23(13(34(12(14(04(15(44(x1))))))))) (3427)
25(23(13(04(15(14(54(x1))))))) 25(23(13(34(12(14(04(15(54(x1))))))))) (3428)
24(23(13(04(15(14(54(x1))))))) 24(23(13(34(12(14(04(15(54(x1))))))))) (3429)
23(23(13(04(15(14(54(x1))))))) 23(23(13(34(12(14(04(15(54(x1))))))))) (3430)
22(23(13(04(15(14(54(x1))))))) 22(23(13(34(12(14(04(15(54(x1))))))))) (3431)
21(23(13(04(15(14(54(x1))))))) 21(23(13(34(12(14(04(15(54(x1))))))))) (3432)
20(23(13(04(15(14(54(x1))))))) 20(23(13(34(12(14(04(15(54(x1))))))))) (3433)

1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
20#(23(13(04(15(14(54(x1))))))) 20#(23(13(34(12(14(04(15(54(x1))))))))) (3434)
20#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3435)
20#(23(13(04(15(14(44(x1))))))) 20#(23(13(34(12(14(04(15(44(x1))))))))) (3436)
20#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3437)
20#(23(13(04(15(14(34(x1))))))) 20#(23(13(34(12(14(04(15(34(x1))))))))) (3438)
20#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3439)
20#(23(13(04(15(14(24(x1))))))) 20#(23(13(34(12(14(04(15(24(x1))))))))) (3440)
20#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3441)
20#(23(13(04(15(14(14(x1))))))) 20#(23(13(34(12(14(04(15(14(x1))))))))) (3442)
20#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3443)
20#(23(13(04(15(14(04(x1))))))) 20#(23(13(34(12(14(04(15(04(x1))))))))) (3444)
20#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3445)
21#(23(13(04(15(14(54(x1))))))) 21#(23(13(34(12(14(04(15(54(x1))))))))) (3446)
21#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3447)
21#(23(13(04(15(14(44(x1))))))) 21#(23(13(34(12(14(04(15(44(x1))))))))) (3448)
21#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3449)
21#(23(13(04(15(14(34(x1))))))) 21#(23(13(34(12(14(04(15(34(x1))))))))) (3450)
21#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3451)
21#(23(13(04(15(14(24(x1))))))) 21#(23(13(34(12(14(04(15(24(x1))))))))) (3452)
21#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3453)
21#(23(13(04(15(14(14(x1))))))) 21#(23(13(34(12(14(04(15(14(x1))))))))) (3454)
21#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3455)
21#(23(13(04(15(14(04(x1))))))) 21#(23(13(34(12(14(04(15(04(x1))))))))) (3456)
21#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3457)
22#(23(13(04(15(14(54(x1))))))) 22#(23(13(34(12(14(04(15(54(x1))))))))) (3458)
22#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3459)
22#(23(13(04(15(14(44(x1))))))) 22#(23(13(34(12(14(04(15(44(x1))))))))) (3460)
22#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3461)
22#(23(13(04(15(14(34(x1))))))) 22#(23(13(34(12(14(04(15(34(x1))))))))) (3462)
22#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3463)
22#(23(13(04(15(14(24(x1))))))) 22#(23(13(34(12(14(04(15(24(x1))))))))) (3464)
22#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3465)
22#(23(13(04(15(14(14(x1))))))) 22#(23(13(34(12(14(04(15(14(x1))))))))) (3466)
22#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3467)
22#(23(13(04(15(14(04(x1))))))) 22#(23(13(34(12(14(04(15(04(x1))))))))) (3468)
22#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3469)
23#(23(13(04(15(14(54(x1))))))) 23#(23(13(34(12(14(04(15(54(x1))))))))) (3470)
23#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3471)
23#(23(13(04(15(14(44(x1))))))) 23#(23(13(34(12(14(04(15(44(x1))))))))) (3472)
23#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3473)
23#(23(13(04(15(14(34(x1))))))) 23#(23(13(34(12(14(04(15(34(x1))))))))) (3474)
23#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3475)
23#(23(13(04(15(14(24(x1))))))) 23#(23(13(34(12(14(04(15(24(x1))))))))) (3476)
23#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3477)
23#(23(13(04(15(14(14(x1))))))) 23#(23(13(34(12(14(04(15(14(x1))))))))) (3478)
23#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3479)
23#(23(13(04(15(14(04(x1))))))) 23#(23(13(34(12(14(04(15(04(x1))))))))) (3480)
23#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3481)
24#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3482)
24#(23(13(04(15(14(54(x1))))))) 24#(23(13(34(12(14(04(15(54(x1))))))))) (3483)
24#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3484)
24#(23(13(04(15(14(44(x1))))))) 24#(23(13(34(12(14(04(15(44(x1))))))))) (3485)
24#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3486)
24#(23(13(04(15(14(34(x1))))))) 24#(23(13(34(12(14(04(15(34(x1))))))))) (3487)
24#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3488)
24#(23(13(04(15(14(24(x1))))))) 24#(23(13(34(12(14(04(15(24(x1))))))))) (3489)
24#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3490)
24#(23(13(04(15(14(14(x1))))))) 24#(23(13(34(12(14(04(15(14(x1))))))))) (3491)
24#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3492)
24#(23(13(04(15(14(04(x1))))))) 24#(23(13(34(12(14(04(15(04(x1))))))))) (3493)
25#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3494)
25#(23(13(04(15(14(54(x1))))))) 25#(23(13(34(12(14(04(15(54(x1))))))))) (3495)
25#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3496)
25#(23(13(04(15(14(44(x1))))))) 25#(23(13(34(12(14(04(15(44(x1))))))))) (3497)
25#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3498)
25#(23(13(04(15(14(34(x1))))))) 25#(23(13(34(12(14(04(15(34(x1))))))))) (3499)
25#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3500)
25#(23(13(04(15(14(24(x1))))))) 25#(23(13(34(12(14(04(15(24(x1))))))))) (3501)
25#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3502)
25#(23(13(04(15(14(14(x1))))))) 25#(23(13(34(12(14(04(15(14(x1))))))))) (3503)
25#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3504)
25#(23(13(04(15(14(04(x1))))))) 25#(23(13(34(12(14(04(15(04(x1))))))))) (3505)

1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1
[54(x1)] = x1 +
0
[44(x1)] = x1 +
0
[34(x1)] = x1 +
0
[20(x1)] = x1 +
0
[21(x1)] = x1 +
0
[22(x1)] = x1 +
0
[23(x1)] = x1 +
1
[24(x1)] = x1 +
0
[25(x1)] = x1 +
0
[12(x1)] = x1 +
0
[13(x1)] = x1 +
0
[14(x1)] = x1 +
0
[15(x1)] = x1 +
0
[04(x1)] = x1 +
0
[20#(x1)] = x1 +
1
[21#(x1)] = x1 +
1
[22#(x1)] = x1 +
1
[23#(x1)] = x1 +
0
[24#(x1)] = x1 +
1
[25#(x1)] = x1 +
1
together with the usable rules
25(23(13(04(15(14(04(x1))))))) 25(23(13(34(12(14(04(15(04(x1))))))))) (3398)
24(23(13(04(15(14(04(x1))))))) 24(23(13(34(12(14(04(15(04(x1))))))))) (3399)
23(23(13(04(15(14(04(x1))))))) 23(23(13(34(12(14(04(15(04(x1))))))))) (3400)
22(23(13(04(15(14(04(x1))))))) 22(23(13(34(12(14(04(15(04(x1))))))))) (3401)
21(23(13(04(15(14(04(x1))))))) 21(23(13(34(12(14(04(15(04(x1))))))))) (3402)
20(23(13(04(15(14(04(x1))))))) 20(23(13(34(12(14(04(15(04(x1))))))))) (3403)
25(23(13(04(15(14(14(x1))))))) 25(23(13(34(12(14(04(15(14(x1))))))))) (3404)
24(23(13(04(15(14(14(x1))))))) 24(23(13(34(12(14(04(15(14(x1))))))))) (3405)
23(23(13(04(15(14(14(x1))))))) 23(23(13(34(12(14(04(15(14(x1))))))))) (3406)
22(23(13(04(15(14(14(x1))))))) 22(23(13(34(12(14(04(15(14(x1))))))))) (3407)
21(23(13(04(15(14(14(x1))))))) 21(23(13(34(12(14(04(15(14(x1))))))))) (3408)
20(23(13(04(15(14(14(x1))))))) 20(23(13(34(12(14(04(15(14(x1))))))))) (3409)
25(23(13(04(15(14(24(x1))))))) 25(23(13(34(12(14(04(15(24(x1))))))))) (3410)
24(23(13(04(15(14(24(x1))))))) 24(23(13(34(12(14(04(15(24(x1))))))))) (3411)
23(23(13(04(15(14(24(x1))))))) 23(23(13(34(12(14(04(15(24(x1))))))))) (3412)
22(23(13(04(15(14(24(x1))))))) 22(23(13(34(12(14(04(15(24(x1))))))))) (3413)
21(23(13(04(15(14(24(x1))))))) 21(23(13(34(12(14(04(15(24(x1))))))))) (3414)
20(23(13(04(15(14(24(x1))))))) 20(23(13(34(12(14(04(15(24(x1))))))))) (3415)
25(23(13(04(15(14(34(x1))))))) 25(23(13(34(12(14(04(15(34(x1))))))))) (3416)
24(23(13(04(15(14(34(x1))))))) 24(23(13(34(12(14(04(15(34(x1))))))))) (3417)
23(23(13(04(15(14(34(x1))))))) 23(23(13(34(12(14(04(15(34(x1))))))))) (3418)
22(23(13(04(15(14(34(x1))))))) 22(23(13(34(12(14(04(15(34(x1))))))))) (3419)
21(23(13(04(15(14(34(x1))))))) 21(23(13(34(12(14(04(15(34(x1))))))))) (3420)
20(23(13(04(15(14(34(x1))))))) 20(23(13(34(12(14(04(15(34(x1))))))))) (3421)
25(23(13(04(15(14(44(x1))))))) 25(23(13(34(12(14(04(15(44(x1))))))))) (3422)
24(23(13(04(15(14(44(x1))))))) 24(23(13(34(12(14(04(15(44(x1))))))))) (3423)
23(23(13(04(15(14(44(x1))))))) 23(23(13(34(12(14(04(15(44(x1))))))))) (3424)
22(23(13(04(15(14(44(x1))))))) 22(23(13(34(12(14(04(15(44(x1))))))))) (3425)
21(23(13(04(15(14(44(x1))))))) 21(23(13(34(12(14(04(15(44(x1))))))))) (3426)
20(23(13(04(15(14(44(x1))))))) 20(23(13(34(12(14(04(15(44(x1))))))))) (3427)
25(23(13(04(15(14(54(x1))))))) 25(23(13(34(12(14(04(15(54(x1))))))))) (3428)
24(23(13(04(15(14(54(x1))))))) 24(23(13(34(12(14(04(15(54(x1))))))))) (3429)
23(23(13(04(15(14(54(x1))))))) 23(23(13(34(12(14(04(15(54(x1))))))))) (3430)
22(23(13(04(15(14(54(x1))))))) 22(23(13(34(12(14(04(15(54(x1))))))))) (3431)
21(23(13(04(15(14(54(x1))))))) 21(23(13(34(12(14(04(15(54(x1))))))))) (3432)
20(23(13(04(15(14(54(x1))))))) 20(23(13(34(12(14(04(15(54(x1))))))))) (3433)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
20#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3435)
20#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3437)
20#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3439)
20#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3441)
20#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3443)
20#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3445)
21#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3447)
21#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3449)
21#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3451)
21#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3453)
21#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3455)
21#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3457)
22#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3459)
22#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3461)
22#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3463)
22#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3465)
22#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3467)
22#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3469)
23#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3471)
23#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3473)
23#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3475)
23#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3477)
23#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3479)
23#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3481)
24#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3482)
24#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3484)
24#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3486)
24#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3488)
24#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3490)
24#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3492)
25#(23(13(04(15(14(54(x1))))))) 23#(13(34(12(14(04(15(54(x1)))))))) (3494)
25#(23(13(04(15(14(44(x1))))))) 23#(13(34(12(14(04(15(44(x1)))))))) (3496)
25#(23(13(04(15(14(34(x1))))))) 23#(13(34(12(14(04(15(34(x1)))))))) (3498)
25#(23(13(04(15(14(24(x1))))))) 23#(13(34(12(14(04(15(24(x1)))))))) (3500)
25#(23(13(04(15(14(14(x1))))))) 23#(13(34(12(14(04(15(14(x1)))))))) (3502)
25#(23(13(04(15(14(04(x1))))))) 23#(13(34(12(14(04(15(04(x1)))))))) (3504)
and no rules could be deleted.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.