Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup01)

The rewrite relation of the following TRS is considered.

a12(a12(a12(a12(x1))))	→	x1	(1)
a13(a13(a13(a13(x1))))	→	x1	(2)
a14(a14(a14(a14(x1))))	→	x1	(3)
a15(a15(a15(a15(x1))))	→	x1	(4)
a16(a16(a16(a16(x1))))	→	x1	(5)
a23(a23(a23(a23(x1))))	→	x1	(6)
a24(a24(a24(a24(x1))))	→	x1	(7)
a25(a25(a25(a25(x1))))	→	x1	(8)
a26(a26(a26(a26(x1))))	→	x1	(9)
a34(a34(a34(a34(x1))))	→	x1	(10)
a35(a35(a35(a35(x1))))	→	x1	(11)
a36(a36(a36(a36(x1))))	→	x1	(12)
a45(a45(a45(a45(x1))))	→	x1	(13)
a46(a46(a46(a46(x1))))	→	x1	(14)
a56(a56(a56(a56(x1))))	→	x1	(15)
a13(a13(x1))	→	a12(a12(a23(a23(a12(a12(x1))))))	(16)
a14(a14(x1))	→	a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))	(17)
a15(a15(x1))	→	a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))	(18)
a16(a16(x1))	→	a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))	(19)
a24(a24(x1))	→	a23(a23(a34(a34(a23(a23(x1))))))	(20)
a25(a25(x1))	→	a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))	(21)
a26(a26(x1))	→	a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))	(22)
a35(a35(x1))	→	a34(a34(a45(a45(a34(a34(x1))))))	(23)
a36(a36(x1))	→	a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))	(24)
a46(a46(x1))	→	a45(a45(a56(a56(a45(a45(x1))))))	(25)
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1))))))))))))	→	x1	(26)
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1))))))))))))	→	x1	(27)
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1))))))))))))	→	x1	(28)
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1))))))))))))	→	x1	(29)
a12(a12(a34(a34(x1))))	→	a34(a34(a12(a12(x1))))	(30)
a12(a12(a45(a45(x1))))	→	a45(a45(a12(a12(x1))))	(31)
a12(a12(a56(a56(x1))))	→	a56(a56(a12(a12(x1))))	(32)
a23(a23(a45(a45(x1))))	→	a45(a45(a23(a23(x1))))	(33)
a23(a23(a56(a56(x1))))	→	a56(a56(a23(a23(x1))))	(34)
a34(a34(a56(a56(x1))))	→	a56(a56(a34(a34(x1))))	(35)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 124 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

a12^#(a12(a45(a45(x1))))	→	a12^#(a12(x1))	(86)
a12^#(a12(a45(a45(x1))))	→	a12^#(x1)	(84)
a12^#(a12(a56(a56(x1))))	→	a12^#(x1)	(81)
a12^#(a12(a34(a34(x1))))	→	a12^#(a12(x1))	(129)
a12^#(a12(a56(a56(x1))))	→	a12^#(a12(x1))	(59)
a12^#(a12(a34(a34(x1))))	→	a12^#(x1)	(124)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a15^#(x₁)]	=	0
[a24^#(x₁)]	=	0
[a36(x₁)]	=	0
[a25(x₁)]	=	0
[a23^#(x₁)]	=	0
[a16^#(x₁)]	=	0
[a16(x₁)]	=	0
[a24(x₁)]	=	0
[a36^#(x₁)]	=	0
[a45(x₁)]	=	x₁ + 1143
[a34(x₁)]	=	x₁ + 32567
[a26(x₁)]	=	0
[a56(x₁)]	=	x₁ + 16304
[a13(x₁)]	=	0
[a25^#(x₁)]	=	0
[a12^#(x₁)]	=	x₁ + 0
[a15(x₁)]	=	0
[a14(x₁)]	=	0
[a23(x₁)]	=	x₁ + 58713
[a46^#(x₁)]	=	0
[a14^#(x₁)]	=	0
[a26^#(x₁)]	=	0
[a12(x₁)]	=	x₁ + 1
[a45^#(x₁)]	=	0
[a35(x₁)]	=	0
[a13^#(x₁)]	=	0
[a34^#(x₁)]	=	0
[a35^#(x₁)]	=	0
[a46(x₁)]	=	0
[a56^#(x₁)]	=	0

together with the usable rules

a56(a56(a56(a56(x1))))	→	x1	(15)
a12(a12(a12(a12(x1))))	→	x1	(1)
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1))))))))))))	→	x1	(26)
a12(a12(a56(a56(x1))))	→	a56(a56(a12(a12(x1))))	(32)
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1))))))))))))	→	x1	(28)
a34(a34(a34(a34(x1))))	→	x1	(10)
a12(a12(a34(a34(x1))))	→	a34(a34(a12(a12(x1))))	(30)
a12(a12(a45(a45(x1))))	→	a45(a45(a12(a12(x1))))	(31)
a45(a45(a45(a45(x1))))	→	x1	(13)
a34(a34(a56(a56(x1))))	→	a56(a56(a34(a34(x1))))	(35)
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1))))))))))))	→	x1	(29)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

a12^#(a12(a45(a45(x1))))	→	a12^#(a12(x1))	(86)
a12^#(a12(a45(a45(x1))))	→	a12^#(x1)	(84)
a12^#(a12(a56(a56(x1))))	→	a12^#(x1)	(81)
a12^#(a12(a34(a34(x1))))	→	a12^#(a12(x1))	(129)
a12^#(a12(a56(a56(x1))))	→	a12^#(a12(x1))	(59)
a12^#(a12(a34(a34(x1))))	→	a12^#(x1)	(124)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

a23^#(a23(a56(a56(x1))))	→	a23^#(x1)	(103)
a23^#(a23(a56(a56(x1))))	→	a23^#(a23(x1))	(64)
a23^#(a23(a45(a45(x1))))	→	a23^#(x1)	(122)
a23^#(a23(a45(a45(x1))))	→	a23^#(a23(x1))	(38)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a15^#(x₁)]	=	0
[a24^#(x₁)]	=	0
[a36(x₁)]	=	0
[a25(x₁)]	=	0
[a23^#(x₁)]	=	x₁ + 0
[a16^#(x₁)]	=	0
[a16(x₁)]	=	0
[a24(x₁)]	=	0
[a36^#(x₁)]	=	0
[a45(x₁)]	=	x₁ + 6879
[a34(x₁)]	=	x₁ + 45131
[a26(x₁)]	=	0
[a56(x₁)]	=	x₁ + 33694
[a13(x₁)]	=	0
[a25^#(x₁)]	=	0
[a12^#(x₁)]	=	x₁ + 0
[a15(x₁)]	=	0
[a14(x₁)]	=	0
[a23(x₁)]	=	x₁ + 53145
[a46^#(x₁)]	=	0
[a14^#(x₁)]	=	0
[a26^#(x₁)]	=	0
[a12(x₁)]	=	x₁ + 13485
[a45^#(x₁)]	=	0
[a35(x₁)]	=	0
[a13^#(x₁)]	=	0
[a34^#(x₁)]	=	0
[a35^#(x₁)]	=	0
[a46(x₁)]	=	0
[a56^#(x₁)]	=	0

together with the usable rules

a56(a56(a56(a56(x1))))	→	x1	(15)
a12(a12(a12(a12(x1))))	→	x1	(1)
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1))))))))))))	→	x1	(26)
a12(a12(a56(a56(x1))))	→	a56(a56(a12(a12(x1))))	(32)
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1))))))))))))	→	x1	(27)
a23(a23(a56(a56(x1))))	→	a56(a56(a23(a23(x1))))	(34)
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1))))))))))))	→	x1	(28)
a23(a23(a45(a45(x1))))	→	a45(a45(a23(a23(x1))))	(33)
a34(a34(a34(a34(x1))))	→	x1	(10)
a12(a12(a34(a34(x1))))	→	a34(a34(a12(a12(x1))))	(30)
a12(a12(a45(a45(x1))))	→	a45(a45(a12(a12(x1))))	(31)
a45(a45(a45(a45(x1))))	→	x1	(13)
a23(a23(a23(a23(x1))))	→	x1	(6)
a34(a34(a56(a56(x1))))	→	a56(a56(a34(a34(x1))))	(35)
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1))))))))))))	→	x1	(29)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

a23^#(a23(a56(a56(x1))))	→	a23^#(x1)	(103)
a23^#(a23(a56(a56(x1))))	→	a23^#(a23(x1))	(64)
a23^#(a23(a45(a45(x1))))	→	a23^#(x1)	(122)
a23^#(a23(a45(a45(x1))))	→	a23^#(a23(x1))	(38)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

a34^#(a34(a56(a56(x1))))	→	a34^#(x1)	(155)
a34^#(a34(a56(a56(x1))))	→	a34^#(a34(x1))	(145)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a15^#(x₁)]	=	0
[a24^#(x₁)]	=	0
[a36(x₁)]	=	0
[a25(x₁)]	=	0
[a23^#(x₁)]	=	x₁ + 0
[a16^#(x₁)]	=	0
[a16(x₁)]	=	0
[a24(x₁)]	=	0
[a36^#(x₁)]	=	0
[a45(x₁)]	=	x₁ + 27857
[a34(x₁)]	=	x₁ + 27713
[a26(x₁)]	=	0
[a56(x₁)]	=	x₁ + 35789
[a13(x₁)]	=	0
[a25^#(x₁)]	=	0
[a12^#(x₁)]	=	x₁ + 0
[a15(x₁)]	=	0
[a14(x₁)]	=	0
[a23(x₁)]	=	x₁ + 51266
[a46^#(x₁)]	=	0
[a14^#(x₁)]	=	0
[a26^#(x₁)]	=	0
[a12(x₁)]	=	x₁ + 9507
[a45^#(x₁)]	=	0
[a35(x₁)]	=	0
[a13^#(x₁)]	=	0
[a34^#(x₁)]	=	x₁ + 0
[a35^#(x₁)]	=	0
[a46(x₁)]	=	0
[a56^#(x₁)]	=	0

together with the usable rules

a56(a56(a56(a56(x1))))	→	x1	(15)
a12(a12(a12(a12(x1))))	→	x1	(1)
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1))))))))))))	→	x1	(26)
a12(a12(a56(a56(x1))))	→	a56(a56(a12(a12(x1))))	(32)
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1))))))))))))	→	x1	(27)
a23(a23(a56(a56(x1))))	→	a56(a56(a23(a23(x1))))	(34)
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1))))))))))))	→	x1	(28)
a23(a23(a45(a45(x1))))	→	a45(a45(a23(a23(x1))))	(33)
a34(a34(a34(a34(x1))))	→	x1	(10)
a12(a12(a34(a34(x1))))	→	a34(a34(a12(a12(x1))))	(30)
a12(a12(a45(a45(x1))))	→	a45(a45(a12(a12(x1))))	(31)
a45(a45(a45(a45(x1))))	→	x1	(13)
a23(a23(a23(a23(x1))))	→	x1	(6)
a34(a34(a56(a56(x1))))	→	a56(a56(a34(a34(x1))))	(35)
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1))))))))))))	→	x1	(29)

(w.r.t. the implicit argument filter of the reduction pair), the pairs

a34^#(a34(a56(a56(x1))))	→	a34^#(x1)	(155)
a34^#(a34(a56(a56(x1))))	→	a34^#(a34(x1))	(145)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.