Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/torpa3)

The rewrite relation of the following TRS is considered.

b(b(x1))	→	c(d(x1))	(1)
c(c(x1))	→	d(d(d(x1)))	(2)
c(x1)	→	g(x1)	(3)
d(d(x1))	→	c(f(x1))	(4)
d(d(d(x1)))	→	g(c(x1))	(5)
f(x1)	→	a(g(x1))	(6)
g(x1)	→	d(a(b(x1)))	(7)
g(g(x1))	→	b(c(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.

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

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(x1)	→	g^#(x1)	(23)
d^#(d(x1))	→	c^#(f(x1))	(15)
d^#(d(x1))	→	f^#(x1)	(22)
g^#(g(x1))	→	b^#(c(x1))	(14)
d^#(d(d(x1)))	→	c^#(x1)	(21)
c^#(c(x1))	→	d^#(d(d(x1)))	(20)
c^#(c(x1))	→	d^#(x1)	(19)
b^#(b(x1))	→	d^#(x1)	(13)
c^#(x1)	→	g^#(x1)	(12)
g^#(g(x1))	→	c^#(x1)	(11)
g^#(x1)	→	b^#(x1)	(10)
b^#(b(x1))	→	c^#(d(x1))	(17)
d^#(d(d(x1)))	→	g^#(c(x1))	(16)
c^#(c(x1))	→	d^#(d(x1))	(9)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	1539
[d(x₁)]	=	x₁ + 4882
[b(x₁)]	=	x₁ + 7323
[c(x₁)]	=	x₁ + 7323
[f(x₁)]	=	1539
[d^#(x₁)]	=	x₁ + 0
[f^#(x₁)]	=	x₁ + 4
[g^#(x₁)]	=	x₁ + 3
[c^#(x₁)]	=	x₁ + 2442
[g(x₁)]	=	x₁ + 7323
[b^#(x₁)]	=	x₁ + 2

together with the usable rules

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

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

f^#(x1)	→	g^#(x1)	(23)
d^#(d(x1))	→	c^#(f(x1))	(15)
d^#(d(x1))	→	f^#(x1)	(22)
g^#(g(x1))	→	b^#(c(x1))	(14)
d^#(d(d(x1)))	→	c^#(x1)	(21)
c^#(c(x1))	→	d^#(d(d(x1)))	(20)
c^#(c(x1))	→	d^#(x1)	(19)
b^#(b(x1))	→	d^#(x1)	(13)
c^#(x1)	→	g^#(x1)	(12)
g^#(g(x1))	→	c^#(x1)	(11)
g^#(x1)	→	b^#(x1)	(10)
b^#(b(x1))	→	c^#(d(x1))	(17)
d^#(d(d(x1)))	→	g^#(c(x1))	(16)
c^#(c(x1))	→	d^#(d(x1))	(9)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.