Certification Problem

Input (TPDB SRS_Standard/Waldmann_19/random-22)

The rewrite relation of the following TRS is considered.

a(b(a(b(x1))))	→	a(a(b(b(x1))))	(1)
b(b(b(b(x1))))	→	b(b(b(a(x1))))	(2)
a(a(a(a(x1))))	→	b(a(b(a(x1))))	(3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Closure Under Flat Contexts

Using the flat contexts

{a(☐), b(☐)}

We obtain the transformed TRS

a(b(a(b(x1))))	→	a(a(b(b(x1))))	(1)
b(b(b(b(x1))))	→	b(b(b(a(x1))))	(2)
a(a(a(a(a(x1)))))	→	a(b(a(b(a(x1)))))	(4)
b(a(a(a(a(x1)))))	→	b(b(a(b(a(x1)))))	(5)

1.1 Semantic Labeling

Root-labeling is applied.

We obtain the labeled TRS

a_b(b_a(a_b(b_a(x1))))	→	a_a(a_b(b_b(b_a(x1))))	(6)
a_b(b_a(a_b(b_b(x1))))	→	a_a(a_b(b_b(b_b(x1))))	(7)
b_b(b_b(b_b(b_a(x1))))	→	b_b(b_b(b_a(a_a(x1))))	(8)
b_b(b_b(b_b(b_b(x1))))	→	b_b(b_b(b_a(a_b(x1))))	(9)
a_a(a_a(a_a(a_a(a_a(x1)))))	→	a_b(b_a(a_b(b_a(a_a(x1)))))	(10)
a_a(a_a(a_a(a_a(a_b(x1)))))	→	a_b(b_a(a_b(b_a(a_b(x1)))))	(11)
b_a(a_a(a_a(a_a(a_a(x1)))))	→	b_b(b_a(a_b(b_a(a_a(x1)))))	(12)
b_a(a_a(a_a(a_a(a_b(x1)))))	→	b_b(b_a(a_b(b_a(a_b(x1)))))	(13)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a_b^#(b_a(a_b(b_a(x1))))	→	b_b^#(b_a(x1))	(14)
a_b^#(b_a(a_b(b_a(x1))))	→	a_b^#(b_b(b_a(x1)))	(15)
a_b^#(b_a(a_b(b_a(x1))))	→	a_a^#(a_b(b_b(b_a(x1))))	(16)
a_b^#(b_a(a_b(b_b(x1))))	→	b_b^#(b_b(x1))	(17)
a_b^#(b_a(a_b(b_b(x1))))	→	a_b^#(b_b(b_b(x1)))	(18)
a_b^#(b_a(a_b(b_b(x1))))	→	a_a^#(a_b(b_b(b_b(x1))))	(19)
b_b^#(b_b(b_b(b_a(x1))))	→	a_a^#(x1)	(20)
b_b^#(b_b(b_b(b_a(x1))))	→	b_a^#(a_a(x1))	(21)
b_b^#(b_b(b_b(b_a(x1))))	→	b_b^#(b_a(a_a(x1)))	(22)
b_b^#(b_b(b_b(b_a(x1))))	→	b_b^#(b_b(b_a(a_a(x1))))	(23)
b_b^#(b_b(b_b(b_b(x1))))	→	a_b^#(x1)	(24)
b_b^#(b_b(b_b(b_b(x1))))	→	b_a^#(a_b(x1))	(25)
b_b^#(b_b(b_b(b_b(x1))))	→	b_b^#(b_a(a_b(x1)))	(26)
b_b^#(b_b(b_b(b_b(x1))))	→	b_b^#(b_b(b_a(a_b(x1))))	(27)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_a(x1))	(28)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_a(x1)))	(29)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_b(b_a(a_a(x1))))	(30)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_b(b_a(a_a(x1)))))	(31)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(x1))	(32)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(x1)))	(33)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(b_a(a_b(x1))))	(34)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(b_a(a_b(x1)))))	(35)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_a(x1))	(36)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_a(x1)))	(37)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_b(b_a(a_a(x1))))	(38)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_b^#(b_a(a_b(b_a(a_a(x1)))))	(39)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(x1))	(40)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(x1)))	(41)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(b_a(a_b(x1))))	(42)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_b^#(b_a(a_b(b_a(a_b(x1)))))	(43)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_a(x1))	(36)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_b^#(b_a(a_b(b_a(a_b(x1)))))	(43)
b_b^#(b_b(b_b(b_b(x1))))	→	b_b^#(b_b(b_a(a_b(x1))))	(27)
b_b^#(b_b(b_b(b_b(x1))))	→	b_b^#(b_a(a_b(x1)))	(26)
b_b^#(b_b(b_b(b_b(x1))))	→	b_a^#(a_b(x1))	(25)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(b_a(a_b(x1))))	(42)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(x1)))	(41)
a_b^#(b_a(a_b(b_b(x1))))	→	a_a^#(a_b(b_b(b_b(x1))))	(19)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(b_a(a_b(x1)))))	(35)
a_b^#(b_a(a_b(b_b(x1))))	→	b_b^#(b_b(x1))	(17)
b_b^#(b_b(b_b(b_b(x1))))	→	a_b^#(x1)	(24)
a_b^#(b_a(a_b(b_a(x1))))	→	a_a^#(a_b(b_b(b_a(x1))))	(16)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(b_a(a_b(x1))))	(34)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(x1))	(40)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_b^#(b_a(a_b(b_a(a_a(x1)))))	(39)
b_b^#(b_b(b_b(b_a(x1))))	→	b_b^#(b_b(b_a(a_a(x1))))	(23)
b_b^#(b_b(b_b(b_a(x1))))	→	b_b^#(b_a(a_a(x1)))	(22)
b_b^#(b_b(b_b(b_a(x1))))	→	b_a^#(a_a(x1))	(21)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_b(b_a(a_a(x1))))	(38)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_a(x1)))	(37)
a_b^#(b_a(a_b(b_a(x1))))	→	b_b^#(b_a(x1))	(14)
b_b^#(b_b(b_b(b_a(x1))))	→	a_a^#(x1)	(20)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(x1)))	(33)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(x1))	(32)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_b(b_a(a_a(x1)))))	(31)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_b(b_a(a_a(x1))))	(30)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_a(x1)))	(29)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_a(x1))	(28)

1.1.1.1.1 Subterm Criterion Processor

We use the projection to multisets

π(b_a^#)	=	{ 1, 1 }
π(a_a^#)	=	{ 1 }
π(b_b^#)	=	{ 1 }
π(a_b^#)	=	{ 1 }
π(b_b)	=	{ 1, 1 }
π(a_a)	=	{ 1, 1 }
π(b_a)	=	{ 1, 1 }
π(a_b)	=	{ 1, 1 }

to remove the pairs:

b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_a(x1))	(36)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_b^#(b_a(a_b(b_a(a_b(x1)))))	(43)
b_b^#(b_b(b_b(b_b(x1))))	→	b_b^#(b_a(a_b(x1)))	(26)
b_b^#(b_b(b_b(b_b(x1))))	→	b_a^#(a_b(x1))	(25)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(b_a(a_b(x1))))	(42)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(x1)))	(41)
a_b^#(b_a(a_b(b_b(x1))))	→	b_b^#(b_b(x1))	(17)
b_b^#(b_b(b_b(b_b(x1))))	→	a_b^#(x1)	(24)
b_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(x1))	(40)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_b^#(b_a(a_b(b_a(a_a(x1)))))	(39)
b_b^#(b_b(b_b(b_a(x1))))	→	b_b^#(b_a(a_a(x1)))	(22)
b_b^#(b_b(b_b(b_a(x1))))	→	b_a^#(a_a(x1))	(21)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_b(b_a(a_a(x1))))	(38)
b_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_a(x1)))	(37)
a_b^#(b_a(a_b(b_a(x1))))	→	b_b^#(b_a(x1))	(14)
b_b^#(b_b(b_b(b_a(x1))))	→	a_a^#(x1)	(20)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(x1)))	(33)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	b_a^#(a_b(x1))	(32)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_a(x1)))	(29)
a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	b_a^#(a_a(x1))	(28)

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

b_b^#(b_b(b_b(b_b(x1))))	→	b_b^#(b_b(b_a(a_b(x1))))	(27)
b_b^#(b_b(b_b(b_a(x1))))	→	b_b^#(b_b(b_a(a_a(x1))))	(23)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the naturals

[a_a(x₁)]

0	0
0	1

· x₁ +

0	0
1	0

[b_a(x₁)]

0	0
0	1

· x₁ +

1	0
0	0

[b_b(x₁)]

0	0
1	1

· x₁ +

0	0
1	0

[b_b^#(x₁)]

0	2
0	0

· x₁ +

0	0
0	0

[a_b(x₁)]

0	0
1	0

· x₁ +

0	0
0	0

together with the usable rules

a_b(b_a(a_b(b_a(x1))))	→	a_a(a_b(b_b(b_a(x1))))	(6)
a_b(b_a(a_b(b_b(x1))))	→	a_a(a_b(b_b(b_b(x1))))	(7)
b_b(b_b(b_b(b_a(x1))))	→	b_b(b_b(b_a(a_a(x1))))	(8)
b_b(b_b(b_b(b_b(x1))))	→	b_b(b_b(b_a(a_b(x1))))	(9)
a_a(a_a(a_a(a_a(a_a(x1)))))	→	a_b(b_a(a_b(b_a(a_a(x1)))))	(10)
a_a(a_a(a_a(a_a(a_b(x1)))))	→	a_b(b_a(a_b(b_a(a_b(x1)))))	(11)
b_a(a_a(a_a(a_a(a_a(x1)))))	→	b_b(b_a(a_b(b_a(a_a(x1)))))	(12)
b_a(a_a(a_a(a_a(a_b(x1)))))	→	b_b(b_a(a_b(b_a(a_b(x1)))))	(13)

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

b_b^#(b_b(b_b(b_b(x1))))

→

b_b^#(b_b(b_a(a_b(x1))))

(27)

could be deleted.

1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[a_a(x₁)]

1	0	1
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

[b_a(x₁)]

0	0	1
0	1	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b_b(x₁)]

1	0	1
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

[b_b^#(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a_b(x₁)]

0	0	0
0	0	0
0	1	0

· x₁ +

0	0	0
1	0	0
0	0	0

together with the usable rules

a_b(b_a(a_b(b_a(x1))))	→	a_a(a_b(b_b(b_a(x1))))	(6)
a_b(b_a(a_b(b_b(x1))))	→	a_a(a_b(b_b(b_b(x1))))	(7)
b_b(b_b(b_b(b_a(x1))))	→	b_b(b_b(b_a(a_a(x1))))	(8)
b_b(b_b(b_b(b_b(x1))))	→	b_b(b_b(b_a(a_b(x1))))	(9)
a_a(a_a(a_a(a_a(a_a(x1)))))	→	a_b(b_a(a_b(b_a(a_a(x1)))))	(10)
a_a(a_a(a_a(a_a(a_b(x1)))))	→	a_b(b_a(a_b(b_a(a_b(x1)))))	(11)
b_a(a_a(a_a(a_a(a_a(x1)))))	→	b_b(b_a(a_b(b_a(a_a(x1)))))	(12)
b_a(a_a(a_a(a_a(a_b(x1)))))	→	b_b(b_a(a_b(b_a(a_b(x1)))))	(13)

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

b_b^#(b_b(b_b(b_a(x1))))

→

b_b^#(b_b(b_a(a_a(x1))))

(23)

could be deleted.

1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_b(b_a(a_a(x1)))))	(31)
a_b^#(b_a(a_b(b_b(x1))))	→	a_a^#(a_b(b_b(b_b(x1))))	(19)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(b_a(a_b(x1)))))	(35)
a_b^#(b_a(a_b(b_a(x1))))	→	a_a^#(a_b(b_b(b_a(x1))))	(16)

1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[a_a(x₁)]	=	1 · x₁ + 0
[a_a^#(x₁)]	=	1 · x₁ + 0
[b_a(x₁)]	=	-∞ · x₁ + 1
[b_b(x₁)]	=	-∞ · x₁ + 0
[a_b^#(x₁)]	=	2 · x₁ + 0
[a_b(x₁)]	=	0 · x₁ + 0

together with the usable rules

a_b(b_a(a_b(b_a(x1))))	→	a_a(a_b(b_b(b_a(x1))))	(6)
a_b(b_a(a_b(b_b(x1))))	→	a_a(a_b(b_b(b_b(x1))))	(7)
b_b(b_b(b_b(b_a(x1))))	→	b_b(b_b(b_a(a_a(x1))))	(8)
b_b(b_b(b_b(b_b(x1))))	→	b_b(b_b(b_a(a_b(x1))))	(9)
a_a(a_a(a_a(a_a(a_a(x1)))))	→	a_b(b_a(a_b(b_a(a_a(x1)))))	(10)
a_a(a_a(a_a(a_a(a_b(x1)))))	→	a_b(b_a(a_b(b_a(a_b(x1)))))	(11)
b_a(a_a(a_a(a_a(a_a(x1)))))	→	b_b(b_a(a_b(b_a(a_a(x1)))))	(12)
b_a(a_a(a_a(a_a(a_b(x1)))))	→	b_b(b_a(a_b(b_a(a_b(x1)))))	(13)

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

a_a^#(a_a(a_a(a_a(a_a(x1)))))	→	a_b^#(b_a(a_b(b_a(a_a(x1)))))	(31)
a_b^#(b_a(a_b(b_b(x1))))	→	a_a^#(a_b(b_b(b_b(x1))))	(19)
a_a^#(a_a(a_a(a_a(a_b(x1)))))	→	a_b^#(b_a(a_b(b_a(a_b(x1)))))	(35)
a_b^#(b_a(a_b(b_a(x1))))	→	a_a^#(a_b(b_b(b_a(x1))))	(16)

could be deleted.

1.1.1.1.1.1.2.1 P is empty

There are no pairs anymore.