Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/005)

The rewrite relation of the following TRS is considered.

app(app(plus,0),y)	→	y	(1)
app(app(plus,app(s,x)),y)	→	app(s,app(app(plus,x),y))	(2)
app(app(app(curry,f),x),y)	→	app(app(f,x),y)	(3)
add	→	app(curry,plus)	(4)

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.

app^#(app(plus,app(s,x)),y)	→	app^#(app(plus,x),y)	(5)
add^#	→	app^#(curry,plus)	(6)
app^#(app(app(curry,f),x),y)	→	app^#(f,x)	(7)
app^#(app(plus,app(s,x)),y)	→	app^#(plus,x)	(8)
app^#(app(plus,app(s,x)),y)	→	app^#(s,app(app(plus,x),y))	(9)
app^#(app(app(curry,f),x),y)	→	app^#(app(f,x),y)	(10)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

app^#(app(app(curry,f),x),y)	→	app^#(app(f,x),y)	(10)
app^#(app(app(curry,f),x),y)	→	app^#(f,x)	(7)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s]	=	1
[curry]	=	51851
[0]	=	10803
[app^#(x₁, x₂)]	=	x₁ + 0
[plus]	=	41141
[add^#]	=	0
[add]	=	0
[app(x₁, x₂)]	=	x₁ + x₂ + 22248

together with the usable rules

app(app(plus,0),y)	→	y	(1)
app(app(app(curry,f),x),y)	→	app(app(f,x),y)	(3)
app(app(plus,app(s,x)),y)	→	app(s,app(app(plus,x),y))	(2)

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

app^#(app(app(curry,f),x),y)	→	app^#(app(f,x),y)	(10)
app^#(app(app(curry,f),x),y)	→	app^#(f,x)	(7)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair
app^#(app(plus,app(s,x)),y) → app^#(app(plus,x),y) (5)

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

[s] = 1

[curry] = 1

[0] = 16203

[app^#(x₁, x₂)] = x₁ + 0

[plus] = 1

[add^#] = 0

[add] = 0

[app(x₁, x₂)] = x₁ + x₂ + 29483

together with the usable rules

app(app(plus,0),y) → y (1)

app(app(app(curry,f),x),y) → app(app(f,x),y) (3)

app(app(plus,app(s,x)),y) → app(s,app(app(plus,x),y)) (2)

(w.r.t. the implicit argument filter of the reduction pair), the pair
app^#(app(plus,app(s,x)),y) → app^#(app(plus,x),y) (5)
could be deleted.
1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.