Certification Problem

Input (TPDB SRS_Standard/Zantema_04/z124)

The rewrite relation of the following TRS is considered.

q0(a(x1))	→	x(q1(x1))	(1)
q1(a(x1))	→	a(q1(x1))	(2)
q1(y(x1))	→	y(q1(x1))	(3)
a(q1(b(x1)))	→	q2(a(y(x1)))	(4)
a(q2(a(x1)))	→	q2(a(a(x1)))	(5)
a(q2(y(x1)))	→	q2(a(y(x1)))	(6)
y(q1(b(x1)))	→	q2(y(y(x1)))	(7)
y(q2(a(x1)))	→	q2(y(a(x1)))	(8)
y(q2(y(x1)))	→	q2(y(y(x1)))	(9)
q2(x(x1))	→	x(q0(x1))	(10)
q0(y(x1))	→	y(q3(x1))	(11)
q3(y(x1))	→	y(q3(x1))	(12)
q3(bl(x1))	→	bl(q4(x1))	(13)

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.

q1^#(a(x1))	→	q1^#(x1)	(14)
y^#(q2(y(x1)))	→	y^#(y(x1))	(15)
y^#(q1(b(x1)))	→	y^#(x1)	(16)
q3^#(y(x1))	→	y^#(q3(x1))	(17)
y^#(q1(b(x1)))	→	q2^#(y(y(x1)))	(18)
a^#(q1(b(x1)))	→	y^#(x1)	(19)
a^#(q1(b(x1)))	→	a^#(y(x1))	(20)
y^#(q1(b(x1)))	→	y^#(y(x1))	(21)
q2^#(x(x1))	→	q0^#(x1)	(22)
y^#(q2(a(x1)))	→	q2^#(y(a(x1)))	(23)
q0^#(y(x1))	→	y^#(q3(x1))	(24)
q3^#(y(x1))	→	q3^#(x1)	(25)
y^#(q2(y(x1)))	→	q2^#(y(y(x1)))	(26)
a^#(q1(b(x1)))	→	q2^#(a(y(x1)))	(27)
q1^#(y(x1))	→	y^#(q1(x1))	(28)
q0^#(a(x1))	→	q1^#(x1)	(29)
y^#(q2(a(x1)))	→	y^#(a(x1))	(30)
a^#(q2(a(x1)))	→	a^#(a(x1))	(31)
a^#(q2(y(x1)))	→	q2^#(a(y(x1)))	(32)
q1^#(a(x1))	→	a^#(q1(x1))	(33)
q0^#(y(x1))	→	q3^#(x1)	(34)
a^#(q2(a(x1)))	→	q2^#(a(a(x1)))	(35)
a^#(q2(y(x1)))	→	a^#(y(x1))	(36)
q1^#(y(x1))	→	q1^#(x1)	(37)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

q1^#(y(x1))	→	q1^#(x1)	(37)
a^#(q2(y(x1)))	→	a^#(y(x1))	(36)
q2^#(x(x1))	→	q0^#(x1)	(22)
a^#(q2(a(x1)))	→	q2^#(a(a(x1)))	(35)
y^#(q1(b(x1)))	→	y^#(y(x1))	(21)
q0^#(y(x1))	→	q3^#(x1)	(34)
q1^#(a(x1))	→	a^#(q1(x1))	(33)
a^#(q1(b(x1)))	→	a^#(y(x1))	(20)
a^#(q2(y(x1)))	→	q2^#(a(y(x1)))	(32)
a^#(q1(b(x1)))	→	y^#(x1)	(19)
a^#(q2(a(x1)))	→	a^#(a(x1))	(31)
y^#(q2(a(x1)))	→	y^#(a(x1))	(30)
y^#(q1(b(x1)))	→	q2^#(y(y(x1)))	(18)
q3^#(y(x1))	→	y^#(q3(x1))	(17)
y^#(q1(b(x1)))	→	y^#(x1)	(16)
q0^#(a(x1))	→	q1^#(x1)	(29)
q1^#(y(x1))	→	y^#(q1(x1))	(28)
y^#(q2(y(x1)))	→	y^#(y(x1))	(15)
a^#(q1(b(x1)))	→	q2^#(a(y(x1)))	(27)
y^#(q2(y(x1)))	→	q2^#(y(y(x1)))	(26)
q3^#(y(x1))	→	q3^#(x1)	(25)
q0^#(y(x1))	→	y^#(q3(x1))	(24)
q1^#(a(x1))	→	q1^#(x1)	(14)
y^#(q2(a(x1)))	→	q2^#(y(a(x1)))	(23)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	8947
[y^#(x₁)]	=	8948
[b(x₁)]	=	1
[q3(x₁)]	=	8946
[q4(x₁)]	=	1
[y(x₁)]	=	8947
[q1^#(x₁)]	=	8950
[q0^#(x₁)]	=	8951
[x(x₁)]	=	8952
[q1(x₁)]	=	8946
[q2(x₁)]	=	x₁ + 0
[bl(x₁)]	=	8947
[a^#(x₁)]	=	8949
[q3^#(x₁)]	=	8949
[q0(x₁)]	=	x₁ + 1
[q2^#(x₁)]	=	x₁ + 0

together with the usable rules

a(q1(b(x1)))	→	q2(a(y(x1)))	(4)
y(q2(a(x1)))	→	q2(y(a(x1)))	(8)
a(q2(a(x1)))	→	q2(a(a(x1)))	(5)
q2(x(x1))	→	x(q0(x1))	(10)
y(q1(b(x1)))	→	q2(y(y(x1)))	(7)
y(q2(y(x1)))	→	q2(y(y(x1)))	(9)
a(q2(y(x1)))	→	q2(a(y(x1)))	(6)

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

q2^#(x(x1))	→	q0^#(x1)	(22)
a^#(q2(a(x1)))	→	q2^#(a(a(x1)))	(35)
q0^#(y(x1))	→	q3^#(x1)	(34)
q1^#(a(x1))	→	a^#(q1(x1))	(33)
a^#(q2(y(x1)))	→	q2^#(a(y(x1)))	(32)
a^#(q1(b(x1)))	→	y^#(x1)	(19)
y^#(q1(b(x1)))	→	q2^#(y(y(x1)))	(18)
q3^#(y(x1))	→	y^#(q3(x1))	(17)
q0^#(a(x1))	→	q1^#(x1)	(29)
q1^#(y(x1))	→	y^#(q1(x1))	(28)
a^#(q1(b(x1)))	→	q2^#(a(y(x1)))	(27)
y^#(q2(y(x1)))	→	q2^#(y(y(x1)))	(26)
q0^#(y(x1))	→	y^#(q3(x1))	(24)
y^#(q2(a(x1)))	→	q2^#(y(a(x1)))	(23)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

q3^#(y(x1))

→

q3^#(x1)

(25)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	8010
[y^#(x₁)]	=	8948
[b(x₁)]	=	1
[q3(x₁)]	=	0
[q4(x₁)]	=	1
[y(x₁)]	=	x₁ + 1
[q1^#(x₁)]	=	8950
[q0^#(x₁)]	=	8951
[x(x₁)]	=	8952
[q1(x₁)]	=	8009
[q2(x₁)]	=	x₁ + 0
[bl(x₁)]	=	1
[a^#(x₁)]	=	8949
[q3^#(x₁)]	=	x₁ + 8949
[q0(x₁)]	=	x₁ + 1
[q2^#(x₁)]	=	0

together with the usable rule

q2(x(x1))

→

x(q0(x1))

(10)

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

q3^#(y(x1))

→

q3^#(x1)

(25)

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

y^#(q2(a(x1)))	→	y^#(a(x1))	(30)
y^#(q1(b(x1)))	→	y^#(x1)	(16)
y^#(q1(b(x1)))	→	y^#(y(x1))	(21)
y^#(q2(y(x1)))	→	y^#(y(x1))	(15)

1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[y^#(x₁)]	=	x₁ + 8948
[b(x₁)]	=	x₁ + 0
[q3(x₁)]	=	x₁ + 0
[q4(x₁)]	=	1
[y(x₁)]	=	x₁ + 1
[q1^#(x₁)]	=	8950
[q0^#(x₁)]	=	8951
[x(x₁)]	=	8952
[q1(x₁)]	=	x₁ + 8009
[q2(x₁)]	=	x₁ + 1
[bl(x₁)]	=	19428
[a^#(x₁)]	=	8949
[q3^#(x₁)]	=	8949
[q0(x₁)]	=	x₁ + 1
[q2^#(x₁)]	=	0

together with the usable rules

a(q1(b(x1)))	→	q2(a(y(x1)))	(4)
y(q2(a(x1)))	→	q2(y(a(x1)))	(8)
q1(y(x1))	→	y(q1(x1))	(3)
a(q2(a(x1)))	→	q2(a(a(x1)))	(5)
q2(x(x1))	→	x(q0(x1))	(10)
y(q1(b(x1)))	→	q2(y(y(x1)))	(7)
q3(y(x1))	→	y(q3(x1))	(12)
y(q2(y(x1)))	→	q2(y(y(x1)))	(9)
q3(bl(x1))	→	bl(q4(x1))	(13)
a(q2(y(x1)))	→	q2(a(y(x1)))	(6)
q1(a(x1))	→	a(q1(x1))	(2)

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

y^#(q2(a(x1)))	→	y^#(a(x1))	(30)
y^#(q1(b(x1)))	→	y^#(x1)	(16)
y^#(q1(b(x1)))	→	y^#(y(x1))	(21)
y^#(q2(y(x1)))	→	y^#(y(x1))	(15)

could be deleted.

1.1.1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

a^#(q1(b(x1)))	→	a^#(y(x1))	(20)
a^#(q2(a(x1)))	→	a^#(a(x1))	(31)
a^#(q2(y(x1)))	→	a^#(y(x1))	(36)

1.1.1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[y^#(x₁)]	=	x₁ + 8948
[b(x₁)]	=	x₁ + 0
[q3(x₁)]	=	x₁ + 0
[q4(x₁)]	=	1
[y(x₁)]	=	x₁ + 1
[q1^#(x₁)]	=	8950
[q0^#(x₁)]	=	8951
[x(x₁)]	=	3
[q1(x₁)]	=	x₁ + 8009
[q2(x₁)]	=	x₁ + 1
[bl(x₁)]	=	19428
[a^#(x₁)]	=	x₁ + 8949
[q3^#(x₁)]	=	8949
[q0(x₁)]	=	x₁ + 1
[q2^#(x₁)]	=	0

together with the usable rules

a(q1(b(x1)))	→	q2(a(y(x1)))	(4)
y(q2(a(x1)))	→	q2(y(a(x1)))	(8)
q1(y(x1))	→	y(q1(x1))	(3)
a(q2(a(x1)))	→	q2(a(a(x1)))	(5)
q2(x(x1))	→	x(q0(x1))	(10)
y(q1(b(x1)))	→	q2(y(y(x1)))	(7)
q3(y(x1))	→	y(q3(x1))	(12)
y(q2(y(x1)))	→	q2(y(y(x1)))	(9)
q3(bl(x1))	→	bl(q4(x1))	(13)
a(q2(y(x1)))	→	q2(a(y(x1)))	(6)
q1(a(x1))	→	a(q1(x1))	(2)

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

a^#(q1(b(x1)))	→	a^#(y(x1))	(20)
a^#(q2(a(x1)))	→	a^#(a(x1))	(31)
a^#(q2(y(x1)))	→	a^#(y(x1))	(36)

could be deleted.

1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

q1^#(y(x1))	→	q1^#(x1)	(37)
q1^#(a(x1))	→	q1^#(x1)	(14)

1.1.1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a(x₁)]	=	x₁ + 1
[y^#(x₁)]	=	x₁ + 8948
[b(x₁)]	=	x₁ + 0
[q3(x₁)]	=	x₁ + 0
[q4(x₁)]	=	1
[y(x₁)]	=	x₁ + 1
[q1^#(x₁)]	=	x₁ + 8950
[q0^#(x₁)]	=	8951
[x(x₁)]	=	3
[q1(x₁)]	=	x₁ + 8009
[q2(x₁)]	=	x₁ + 7141
[bl(x₁)]	=	19428
[a^#(x₁)]	=	x₁ + 8949
[q3^#(x₁)]	=	8949
[q0(x₁)]	=	x₁ + 1
[q2^#(x₁)]	=	0

together with the usable rules

a(q1(b(x1)))	→	q2(a(y(x1)))	(4)
y(q2(a(x1)))	→	q2(y(a(x1)))	(8)
q1(y(x1))	→	y(q1(x1))	(3)
a(q2(a(x1)))	→	q2(a(a(x1)))	(5)
q2(x(x1))	→	x(q0(x1))	(10)
y(q1(b(x1)))	→	q2(y(y(x1)))	(7)
q3(y(x1))	→	y(q3(x1))	(12)
y(q2(y(x1)))	→	q2(y(y(x1)))	(9)
q3(bl(x1))	→	bl(q4(x1))	(13)
a(q2(y(x1)))	→	q2(a(y(x1)))	(6)
q1(a(x1))	→	a(q1(x1))	(2)

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

q1^#(y(x1))	→	q1^#(x1)	(37)
q1^#(a(x1))	→	q1^#(x1)	(14)

could be deleted.

1.1.1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.