Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z066)

The rewrite relation of the following TRS is considered.

a(b(c(x1)))	→	c(b(a(x1)))	(1)
C(B(A(x1)))	→	A(B(C(x1)))	(2)
b(a(C(x1)))	→	C(a(b(x1)))	(3)
c(A(B(x1)))	→	B(A(c(x1)))	(4)
A(c(b(x1)))	→	b(c(A(x1)))	(5)
B(C(a(x1)))	→	a(C(B(x1)))	(6)
a(A(x1))	→	x1	(7)
A(a(x1))	→	x1	(8)
b(B(x1))	→	x1	(9)
B(b(x1))	→	x1	(10)
c(C(x1))	→	x1	(11)
C(c(x1))	→	x1	(12)

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^#(B(A(x1)))	→	B^#(C(x1))	(13)
B^#(C(a(x1)))	→	B^#(x1)	(14)
a^#(b(c(x1)))	→	c^#(b(a(x1)))	(15)
A^#(c(b(x1)))	→	A^#(x1)	(16)
b^#(a(C(x1)))	→	a^#(b(x1))	(17)
b^#(a(C(x1)))	→	b^#(x1)	(18)
a^#(b(c(x1)))	→	b^#(a(x1))	(19)
A^#(c(b(x1)))	→	b^#(c(A(x1)))	(20)
b^#(a(C(x1)))	→	C^#(a(b(x1)))	(21)
B^#(C(a(x1)))	→	C^#(B(x1))	(22)
c^#(A(B(x1)))	→	A^#(c(x1))	(23)
c^#(A(B(x1)))	→	B^#(A(c(x1)))	(24)
C^#(B(A(x1)))	→	C^#(x1)	(25)
C^#(B(A(x1)))	→	A^#(B(C(x1)))	(26)
A^#(c(b(x1)))	→	c^#(A(x1))	(27)
a^#(b(c(x1)))	→	a^#(x1)	(28)
B^#(C(a(x1)))	→	a^#(C(B(x1)))	(29)
c^#(A(B(x1)))	→	c^#(x1)	(30)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

c^#(A(B(x1)))	→	c^#(x1)	(30)
B^#(C(a(x1)))	→	a^#(C(B(x1)))	(29)
a^#(b(c(x1)))	→	b^#(a(x1))	(19)
a^#(b(c(x1)))	→	a^#(x1)	(28)
b^#(a(C(x1)))	→	b^#(x1)	(18)
A^#(c(b(x1)))	→	c^#(A(x1))	(27)
C^#(B(A(x1)))	→	A^#(B(C(x1)))	(26)
C^#(B(A(x1)))	→	C^#(x1)	(25)
c^#(A(B(x1)))	→	B^#(A(c(x1)))	(24)
b^#(a(C(x1)))	→	a^#(b(x1))	(17)
A^#(c(b(x1)))	→	A^#(x1)	(16)
a^#(b(c(x1)))	→	c^#(b(a(x1)))	(15)
c^#(A(B(x1)))	→	A^#(c(x1))	(23)
B^#(C(a(x1)))	→	B^#(x1)	(14)
B^#(C(a(x1)))	→	C^#(B(x1))	(22)
b^#(a(C(x1)))	→	C^#(a(b(x1)))	(21)
A^#(c(b(x1)))	→	b^#(c(A(x1)))	(20)
C^#(B(A(x1)))	→	B^#(C(x1))	(13)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[b(x₁)]	=	x₁ + 21239
[B^#(x₁)]	=	x₁ + 8855
[c(x₁)]	=	x₁ + 1
[C(x₁)]	=	x₁ + 10803
[B(x₁)]	=	x₁ + 8856
[C^#(x₁)]	=	x₁ + 10802
[A(x₁)]	=	x₁ + 44083
[A^#(x₁)]	=	x₁ + 44082
[c^#(x₁)]	=	x₁ + 0
[a^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 21238

together with the usable rules

c(A(B(x1)))	→	B(A(c(x1)))	(4)
A(a(x1))	→	x1	(8)
a(b(c(x1)))	→	c(b(a(x1)))	(1)
b(a(C(x1)))	→	C(a(b(x1)))	(3)
A(c(b(x1)))	→	b(c(A(x1)))	(5)
B(b(x1))	→	x1	(10)
a(A(x1))	→	x1	(7)
C(c(x1))	→	x1	(12)
c(C(x1))	→	x1	(11)
b(B(x1))	→	x1	(9)
B(C(a(x1)))	→	a(C(B(x1)))	(6)
C(B(A(x1)))	→	A(B(C(x1)))	(2)

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

c^#(A(B(x1)))	→	c^#(x1)	(30)
a^#(b(c(x1)))	→	b^#(a(x1))	(19)
a^#(b(c(x1)))	→	a^#(x1)	(28)
b^#(a(C(x1)))	→	b^#(x1)	(18)
A^#(c(b(x1)))	→	c^#(A(x1))	(27)
C^#(B(A(x1)))	→	C^#(x1)	(25)
b^#(a(C(x1)))	→	a^#(b(x1))	(17)
A^#(c(b(x1)))	→	A^#(x1)	(16)
c^#(A(B(x1)))	→	A^#(c(x1))	(23)
B^#(C(a(x1)))	→	B^#(x1)	(14)
B^#(C(a(x1)))	→	C^#(B(x1))	(22)
C^#(B(A(x1)))	→	B^#(C(x1))	(13)

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

c^#(A(B(x1)))	→	B^#(A(c(x1)))	(24)
a^#(b(c(x1)))	→	c^#(b(a(x1)))	(15)
B^#(C(a(x1)))	→	a^#(C(B(x1)))	(29)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a)	=	1
π(b)	=	1
π(C)	=	1
π(A)	=	1
π(A^#)	=	1
π(a^#)	=	1

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

prec(B^#)	=	3	status(B^#)	=	[1]	list-extension(B^#)	=	Lex
prec(c)	=	5	status(c)	=	[1]	list-extension(c)	=	Lex
prec(B)	=	2	status(B)	=	[1]	list-extension(B)	=	Lex
prec(C^#)	=	0	status(C^#)	=	[]	list-extension(C^#)	=	Lex
prec(c^#)	=	5	status(c^#)	=	[1]	list-extension(c^#)	=	Lex
prec(b^#)	=	0	status(b^#)	=	[]	list-extension(b^#)	=	Lex

and the following Max-polynomial interpretation

[B^#(x₁)]	=	x₁ + 0
[c(x₁)]	=	x₁ + 0
[B(x₁)]	=	x₁ + 0
[C^#(x₁)]	=	0
[c^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	0

together with the usable rules

c(A(B(x1)))	→	B(A(c(x1)))	(4)
A(a(x1))	→	x1	(8)
a(b(c(x1)))	→	c(b(a(x1)))	(1)
b(a(C(x1)))	→	C(a(b(x1)))	(3)
A(c(b(x1)))	→	b(c(A(x1)))	(5)
B(b(x1))	→	x1	(10)
a(A(x1))	→	x1	(7)
C(c(x1))	→	x1	(12)
c(C(x1))	→	x1	(11)
b(B(x1))	→	x1	(9)
B(C(a(x1)))	→	a(C(B(x1)))	(6)
C(B(A(x1)))	→	A(B(C(x1)))	(2)

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

c^#(A(B(x1)))	→	B^#(A(c(x1)))	(24)
B^#(C(a(x1)))	→	a^#(C(B(x1)))	(29)

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

b^#(a(C(x1)))	→	C^#(a(b(x1)))	(21)
A^#(c(b(x1)))	→	b^#(c(A(x1)))	(20)
C^#(B(A(x1)))	→	A^#(B(C(x1)))	(26)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the argument filter

π(a)	=	1
π(c)	=	1
π(C)	=	1
π(a^#)	=	1

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

prec(b)	=	4	status(b)	=	[1]	list-extension(b)	=	Lex
prec(B^#)	=	3	status(B^#)	=	[1]	list-extension(B^#)	=	Lex
prec(B)	=	6	status(B)	=	[1]	list-extension(B)	=	Lex
prec(C^#)	=	4	status(C^#)	=	[1]	list-extension(C^#)	=	Lex
prec(A)	=	6	status(A)	=	[1]	list-extension(A)	=	Lex
prec(A^#)	=	6	status(A^#)	=	[1]	list-extension(A^#)	=	Lex
prec(c^#)	=	5	status(c^#)	=	[1]	list-extension(c^#)	=	Lex
prec(b^#)	=	5	status(b^#)	=	[1]	list-extension(b^#)	=	Lex

and the following Max-polynomial interpretation

[b(x₁)]	=	x₁ + 0
[B^#(x₁)]	=	x₁ + 0
[B(x₁)]	=	x₁ + 0
[C^#(x₁)]	=	x₁ + 0
[A(x₁)]	=	x₁ + 0
[A^#(x₁)]	=	x₁ + 0
[c^#(x₁)]	=	x₁ + 0
[b^#(x₁)]	=	x₁ + 0

together with the usable rules

c(A(B(x1)))	→	B(A(c(x1)))	(4)
A(a(x1))	→	x1	(8)
a(b(c(x1)))	→	c(b(a(x1)))	(1)
b(a(C(x1)))	→	C(a(b(x1)))	(3)
A(c(b(x1)))	→	b(c(A(x1)))	(5)
B(b(x1))	→	x1	(10)
a(A(x1))	→	x1	(7)
C(c(x1))	→	x1	(12)
c(C(x1))	→	x1	(11)
b(B(x1))	→	x1	(9)
B(C(a(x1)))	→	a(C(B(x1)))	(6)
C(B(A(x1)))	→	A(B(C(x1)))	(2)

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

b^#(a(C(x1)))	→	C^#(a(b(x1)))	(21)
A^#(c(b(x1)))	→	b^#(c(A(x1)))	(20)
C^#(B(A(x1)))	→	A^#(B(C(x1)))	(26)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.