Certification Problem

Input (TPDB TRS_Standard/Rubio_04/bn129)

The rewrite relation of the following TRS is considered.

plus(s(X),plus(Y,Z))	→	plus(X,plus(s(s(Y)),Z))	(1)
plus(s(X1),plus(X2,plus(X3,X4)))	→	plus(X1,plus(X3,plus(X2,X4)))	(2)

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.

plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X3,plus(X2,X4))	(3)
plus^#(s(X),plus(Y,Z))	→	plus^#(s(s(Y)),Z)	(4)
plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X2,X4)	(5)
plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X1,plus(X3,plus(X2,X4)))	(6)
plus^#(s(X),plus(Y,Z))	→	plus^#(X,plus(s(s(Y)),Z))	(7)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

plus^#(s(X),plus(Y,Z))	→	plus^#(X,plus(s(s(Y)),Z))	(7)
plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X1,plus(X3,plus(X2,X4)))	(6)
plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X2,X4)	(5)
plus^#(s(X),plus(Y,Z))	→	plus^#(s(s(Y)),Z)	(4)
plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X3,plus(X2,X4))	(3)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	1
[plus^#(x₁, x₂)]	=	x₂ + 0
[plus(x₁, x₂)]	=	x₂ + 7722

together with the usable rules

plus(s(X),plus(Y,Z))	→	plus(X,plus(s(s(Y)),Z))	(1)
plus(s(X1),plus(X2,plus(X3,X4)))	→	plus(X1,plus(X3,plus(X2,X4)))	(2)

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

plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X2,X4)	(5)
plus^#(s(X),plus(Y,Z))	→	plus^#(s(s(Y)),Z)	(4)
plus^#(s(X1),plus(X2,plus(X3,X4)))	→	plus^#(X3,plus(X2,X4))	(3)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

plus^#(s(X),plus(Y,Z)) → plus^#(X,plus(s(s(Y)),Z)) (7)

plus^#(s(X1),plus(X2,plus(X3,X4))) → plus^#(X1,plus(X3,plus(X2,X4))) (6)

1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation

[s(x₁)] = x₁ + 1

[plus^#(x₁, x₂)] = x₁ + x₂ + 0

[plus(x₁, x₂)] = x₂ + 23676

together with the usable rules

plus(s(X),plus(Y,Z)) → plus(X,plus(s(s(Y)),Z)) (1)

plus(s(X1),plus(X2,plus(X3,X4))) → plus(X1,plus(X3,plus(X2,X4))) (2)

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

plus^#(s(X),plus(Y,Z)) → plus^#(X,plus(s(s(Y)),Z)) (7)

plus^#(s(X1),plus(X2,plus(X3,X4))) → plus^#(X1,plus(X3,plus(X2,X4))) (6)

could be deleted.
1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.