Certification Problem

Input (TPDB TRS_Standard/Endrullis_06/linear2)

The rewrite relation of the following TRS is considered.

f(a,f(a,f(a,f(x,b))))	→	f(f(a,f(a,f(a,x))),b)	(1)
f(f(f(a,x),b),b)	→	f(f(a,f(f(x,b),b)),b)	(2)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,x)	(3)
f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,f(a,x))	(4)
f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,f(a,f(a,x)))	(5)
f^#(a,f(a,f(a,f(x,b))))	→	f^#(f(a,f(a,f(a,x))),b)	(6)
f^#(f(f(a,x),b),b)	→	f^#(x,b)	(7)
f^#(f(f(a,x),b),b)	→	f^#(f(x,b),b)	(8)
f^#(f(f(a,x),b),b)	→	f^#(a,f(f(x,b),b))	(9)
f^#(f(f(a,x),b),b)	→	f^#(f(a,f(f(x,b),b)),b)	(10)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,f(a,f(a,x)))	(5)
f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,f(a,x))	(4)
f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,x)	(3)

1.1.1 Reduction Pair Processor with Usable Rules

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

[b]	=	1
[f^#(x₁, x₂)]	=	-∞ · x₁ + 0 · x₂ + -∞
[a]	=	5
[f(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0

together with the usable rules

f(a,f(a,f(a,f(x,b))))	→	f(f(a,f(a,f(a,x))),b)	(1)
f(f(f(a,x),b),b)	→	f(f(a,f(f(x,b),b)),b)	(2)

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

f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,f(a,x))	(4)
f^#(a,f(a,f(a,f(x,b))))	→	f^#(a,x)	(3)

could be deleted.

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

[b]

1	0	0
1	0	0
0	0	0

[f^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a]

0	0	0
0	0	0
0	0	0

[f(x₁, x₂)]

0	0	0
0	0	0
0	1	1

· x₁ +

0	0	1
1	0	0
0	1	0

· x₂ +

0	0	0
0	0	0
0	0	0

together with the usable rules

f(a,f(a,f(a,f(x,b))))	→	f(f(a,f(a,f(a,x))),b)	(1)
f(f(f(a,x),b),b)	→	f(f(a,f(f(x,b),b)),b)	(2)

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

f^#(a,f(a,f(a,f(x,b))))

→

f^#(a,f(a,f(a,x)))

(5)

could be deleted.

1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

f^#(f(f(a,x),b),b)	→	f^#(f(a,f(f(x,b),b)),b)	(10)
f^#(f(f(a,x),b),b)	→	f^#(f(x,b),b)	(8)
f^#(f(f(a,x),b),b)	→	f^#(x,b)	(7)

1.1.2 Reduction Pair Processor with Usable Rules

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

[b]	=	1
[f^#(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0
[a]	=	3
[f(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0

together with the usable rules

f(a,f(a,f(a,f(x,b))))	→	f(f(a,f(a,f(a,x))),b)	(1)
f(f(f(a,x),b),b)	→	f(f(a,f(f(x,b),b)),b)	(2)

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

f^#(f(f(a,x),b),b)	→	f^#(f(x,b),b)	(8)
f^#(f(f(a,x),b),b)	→	f^#(x,b)	(7)

could be deleted.

1.1.2.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals

[b]

0	0	0	0
1	0	0	0
0	0	0	0
0	0	0	0

[f^#(x₁, x₂)]

0	0	1	0
0	0	0	0
0	0	0	0
0	0	0	0

· x₁ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

· x₂ +

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[a]

1	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

[f(x₁, x₂)]

0	0	0
1	0	0
0	0	0
0	1	1

· x₁ +

0	0	0
0	0	0
1	0	0
0	1	1

· x₂ +

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

together with the usable rules

f(a,f(a,f(a,f(x,b))))	→	f(f(a,f(a,f(a,x))),b)	(1)
f(f(f(a,x),b),b)	→	f(f(a,f(f(x,b),b)),b)	(2)

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

f^#(f(f(a,x),b),b)

→

f^#(f(a,f(f(x,b),b)),b)

(10)

could be deleted.

1.1.2.1.1 P is empty

There are no pairs anymore.