Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/secr4)

The rewrite relation of the following TRS is considered.

a(a(x1))	→	b(b(b(x1)))	(1)
a(x1)	→	c(d(x1))	(2)
b(b(x1))	→	c(c(c(x1)))	(3)
c(c(x1))	→	d(d(d(x1)))	(4)
e(d(x1))	→	a(b(c(d(e(x1)))))	(5)
b(x1)	→	d(d(x1))	(6)
e(c(x1))	→	b(a(a(e(x1))))	(7)
c(d(d(x1)))	→	a(x1)	(8)

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.

e^#(c(x1))	→	b^#(a(a(e(x1))))	(9)
e^#(d(x1))	→	a^#(b(c(d(e(x1)))))	(10)
a^#(a(x1))	→	b^#(b(b(x1)))	(11)
e^#(d(x1))	→	e^#(x1)	(12)
b^#(b(x1))	→	c^#(c(x1))	(13)
b^#(b(x1))	→	c^#(x1)	(14)
a^#(a(x1))	→	b^#(b(x1))	(15)
e^#(d(x1))	→	b^#(c(d(e(x1))))	(16)
b^#(b(x1))	→	c^#(c(c(x1)))	(17)
e^#(c(x1))	→	e^#(x1)	(18)
c^#(d(d(x1)))	→	a^#(x1)	(19)
e^#(c(x1))	→	a^#(a(e(x1)))	(20)
a^#(x1)	→	c^#(d(x1))	(21)
e^#(d(x1))	→	c^#(d(e(x1)))	(22)
a^#(a(x1))	→	b^#(x1)	(23)
e^#(c(x1))	→	a^#(e(x1))	(24)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

e^#(d(x1))	→	e^#(x1)	(12)
e^#(c(x1))	→	e^#(x1)	(18)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	0
[d(x₁)]	=	x₁ + 1
[b(x₁)]	=	0
[e^#(x₁)]	=	x₁ + 0
[c(x₁)]	=	x₁ + 1
[c^#(x₁)]	=	0
[e(x₁)]	=	0
[a^#(x₁)]	=	0
[b^#(x₁)]	=	0

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

e^#(d(x1))	→	e^#(x1)	(12)
e^#(c(x1))	→	e^#(x1)	(18)

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

a^#(a(x1))	→	b^#(b(x1))	(15)
a^#(a(x1))	→	b^#(x1)	(23)
b^#(b(x1))	→	c^#(x1)	(14)
a^#(x1)	→	c^#(d(x1))	(21)
c^#(d(d(x1)))	→	a^#(x1)	(19)
b^#(b(x1))	→	c^#(c(x1))	(13)
a^#(a(x1))	→	b^#(b(b(x1)))	(11)
b^#(b(x1))	→	c^#(c(c(x1)))	(17)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 42
[d(x₁)]	=	x₁ + 12
[b(x₁)]	=	x₁ + 27
[e^#(x₁)]	=	0
[c(x₁)]	=	x₁ + 18
[c^#(x₁)]	=	x₁ + 0
[e(x₁)]	=	0
[a^#(x₁)]	=	x₁ + 23
[b^#(x₁)]	=	x₁ + 10

together with the usable rules

c(c(x1))	→	d(d(d(x1)))	(4)
c(d(d(x1)))	→	a(x1)	(8)
a(a(x1))	→	b(b(b(x1)))	(1)
b(b(x1))	→	c(c(c(x1)))	(3)
b(x1)	→	d(d(x1))	(6)
a(x1)	→	c(d(x1))	(2)

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

a^#(a(x1))	→	b^#(b(x1))	(15)
a^#(a(x1))	→	b^#(x1)	(23)
b^#(b(x1))	→	c^#(x1)	(14)
a^#(x1)	→	c^#(d(x1))	(21)
c^#(d(d(x1)))	→	a^#(x1)	(19)
b^#(b(x1))	→	c^#(c(x1))	(13)
a^#(a(x1))	→	b^#(b(b(x1)))	(11)
b^#(b(x1))	→	c^#(c(c(x1)))	(17)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.