Certification Problem

Input (TPDB TRS_Standard/SK90/4.32)

The rewrite relation of the following TRS is considered.

a(b(x))	→	b(a(a(x)))	(1)
b(c(x))	→	c(b(b(x)))	(2)
c(a(x))	→	a(c(c(x)))	(3)
u(a(x))	→	x	(4)
v(b(x))	→	x	(5)
w(c(x))	→	x	(6)
a(u(x))	→	x	(7)
b(v(x))	→	x	(8)
c(w(x))	→	x	(9)

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.

b^#(c(x))	→	b^#(b(x))	(10)
c^#(a(x))	→	c^#(x)	(11)
a^#(b(x))	→	a^#(x)	(12)
a^#(b(x))	→	b^#(a(a(x)))	(13)
c^#(a(x))	→	c^#(c(x))	(14)
b^#(c(x))	→	b^#(x)	(15)
b^#(c(x))	→	c^#(b(b(x)))	(16)
a^#(b(x))	→	a^#(a(x))	(17)
c^#(a(x))	→	a^#(c(c(x)))	(18)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(a(x))	→	a^#(c(c(x)))	(18)
a^#(b(x))	→	a^#(a(x))	(17)
b^#(c(x))	→	c^#(b(b(x)))	(16)
b^#(c(x))	→	b^#(x)	(15)
a^#(b(x))	→	a^#(x)	(12)
c^#(a(x))	→	c^#(x)	(11)
c^#(a(x))	→	c^#(c(x))	(14)
a^#(b(x))	→	b^#(a(a(x)))	(13)
b^#(c(x))	→	b^#(b(x))	(10)

1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(b)	=	1
π(b^#)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(a)	=	0	status(a)	=	[]	list-extension(a)	=	Lex
prec(v)	=	0	status(v)	=	[]	list-extension(v)	=	Lex
prec(w^#)	=	0	status(w^#)	=	[]	list-extension(w^#)	=	Lex
prec(u)	=	0	status(u)	=	[]	list-extension(u)	=	Lex
prec(u^#)	=	0	status(u^#)	=	[]	list-extension(u^#)	=	Lex
prec(w)	=	0	status(w)	=	[]	list-extension(w)	=	Lex
prec(c)	=	3	status(c)	=	[1]	list-extension(c)	=	Lex
prec(v^#)	=	0	status(v^#)	=	[]	list-extension(v^#)	=	Lex
prec(c^#)	=	2	status(c^#)	=	[]	list-extension(c^#)	=	Lex
prec(a^#)	=	1	status(a^#)	=	[]	list-extension(a^#)	=	Lex

and the following Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[v(x₁)]	=	x₁ + 1
[w^#(x₁)]	=	1
[u(x₁)]	=	x₁ + 10451
[u^#(x₁)]	=	1
[w(x₁)]	=	x₁ + 30613
[c(x₁)]	=	x₁ + 0
[v^#(x₁)]	=	1
[c^#(x₁)]	=	x₁ + 0
[a^#(x₁)]	=	x₁ + 0

together with the usable rules

b(v(x))	→	x	(8)
a(b(x))	→	b(a(a(x)))	(1)
c(a(x))	→	a(c(c(x)))	(3)
a(u(x))	→	x	(7)
c(w(x))	→	x	(9)
b(c(x))	→	c(b(b(x)))	(2)

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

c^#(a(x))	→	a^#(c(c(x)))	(18)
b^#(c(x))	→	c^#(b(b(x)))	(16)
b^#(c(x))	→	b^#(x)	(15)
a^#(b(x))	→	b^#(a(a(x)))	(13)
b^#(c(x))	→	b^#(b(x))	(10)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

a^#(b(x))	→	a^#(x)	(12)
a^#(b(x))	→	a^#(a(x))	(17)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a)	=	1
π(b^#)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(v)	=	0	status(v)	=	[]	list-extension(v)	=	Lex
prec(w^#)	=	0	status(w^#)	=	[]	list-extension(w^#)	=	Lex
prec(b)	=	3	status(b)	=	[1]	list-extension(b)	=	Lex
prec(u)	=	0	status(u)	=	[]	list-extension(u)	=	Lex
prec(u^#)	=	0	status(u^#)	=	[]	list-extension(u^#)	=	Lex
prec(w)	=	0	status(w)	=	[1]	list-extension(w)	=	Lex
prec(c)	=	3	status(c)	=	[]	list-extension(c)	=	Lex
prec(v^#)	=	0	status(v^#)	=	[]	list-extension(v^#)	=	Lex
prec(c^#)	=	2	status(c^#)	=	[]	list-extension(c^#)	=	Lex
prec(a^#)	=	1	status(a^#)	=	[1]	list-extension(a^#)	=	Lex

and the following Max-polynomial interpretation

[v(x₁)]	=	x₁ + 1
[w^#(x₁)]	=	1
[b(x₁)]	=	x₁ + 0
[u(x₁)]	=	x₁ + 28101
[u^#(x₁)]	=	1
[w(x₁)]	=	x₁ + 1143
[c(x₁)]	=	x₁ + 0
[v^#(x₁)]	=	1
[c^#(x₁)]	=	x₁ + 0
[a^#(x₁)]	=	x₁ + 0

together with the usable rules

b(v(x))	→	x	(8)
a(b(x))	→	b(a(a(x)))	(1)
c(a(x))	→	a(c(c(x)))	(3)
a(u(x))	→	x	(7)
c(w(x))	→	x	(9)
b(c(x))	→	c(b(b(x)))	(2)

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

a^#(b(x))	→	a^#(x)	(12)
a^#(b(x))	→	a^#(a(x))	(17)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

c^#(a(x))	→	c^#(x)	(11)
c^#(a(x))	→	c^#(c(x))	(14)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the argument filter

π(c)	=	1
π(c^#)	=	1
π(b^#)	=	1

in combination with the following Weighted Path Order with the following precedence and status

prec(a)	=	4	status(a)	=	[1]	list-extension(a)	=	Lex
prec(v)	=	0	status(v)	=	[]	list-extension(v)	=	Lex
prec(w^#)	=	0	status(w^#)	=	[]	list-extension(w^#)	=	Lex
prec(b)	=	2	status(b)	=	[]	list-extension(b)	=	Lex
prec(u)	=	0	status(u)	=	[]	list-extension(u)	=	Lex
prec(u^#)	=	0	status(u^#)	=	[]	list-extension(u^#)	=	Lex
prec(w)	=	0	status(w)	=	[1]	list-extension(w)	=	Lex
prec(v^#)	=	0	status(v^#)	=	[]	list-extension(v^#)	=	Lex
prec(a^#)	=	1	status(a^#)	=	[1]	list-extension(a^#)	=	Lex

and the following Max-polynomial interpretation

[a(x₁)]	=	x₁ + 0
[v(x₁)]	=	x₁ + 1
[w^#(x₁)]	=	1
[b(x₁)]	=	x₁ + 0
[u(x₁)]	=	x₁ + 28101
[u^#(x₁)]	=	1
[w(x₁)]	=	x₁ + 20538
[v^#(x₁)]	=	1
[a^#(x₁)]	=	x₁ + 0

together with the usable rules

b(v(x))	→	x	(8)
a(b(x))	→	b(a(a(x)))	(1)
c(a(x))	→	a(c(c(x)))	(3)
a(u(x))	→	x	(7)
c(w(x))	→	x	(9)
b(c(x))	→	c(b(b(x)))	(2)

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

c^#(a(x))	→	c^#(x)	(11)
c^#(a(x))	→	c^#(c(x))	(14)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.