Certification Problem

Input (TPDB TRS_Standard/AotoYamada_05/026)

The rewrite relation of the following TRS is considered.

app(app(map,f),nil)	→	nil	(1)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(2)
app(app(app(comp,f),g),x)	→	app(f,app(g,x))	(3)
app(twice,f)	→	app(app(comp,f),f)	(4)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

We uncurry the binary symbol app in combination with the following symbol map which also determines the applicative arities of these symbols.

map	is mapped to	map,	map1(x₁),	map2(x₁, x₂)
nil	is mapped to	nil
cons	is mapped to	cons,	cons1(x₁),	cons2(x₁, x₂)
comp	is mapped to	comp,	comp1(x₁),	comp2(x₁, x₂),	comp3(x₁, x₂, x₃)
twice	is mapped to	twice,	twice1(x₁)

There are no uncurry rules.
No rules have to be added for the eta-expansion.

Uncurrying the rules and adding the uncurrying rules yields the new set of rules

map2(f,nil)	→	nil	(13)
map2(f,cons2(x,xs))	→	cons2(app(f,x),map2(f,xs))	(14)
comp3(f,g,x)	→	app(f,app(g,x))	(15)
twice1(f)	→	comp2(f,f)	(16)
app(map,y1)	→	map1(y1)	(5)
app(map1(x0),y1)	→	map2(x0,y1)	(6)
app(cons,y1)	→	cons1(y1)	(7)
app(cons1(x0),y1)	→	cons2(x0,y1)	(8)
app(comp,y1)	→	comp1(y1)	(9)
app(comp1(x0),y1)	→	comp2(x0,y1)	(10)
app(comp2(x0,x1),y1)	→	comp3(x0,x1,y1)	(11)
app(twice,y1)	→	twice1(y1)	(12)

1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

10

Hence, it suffices to show innermost termination in the following.

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

map2^#(f,cons2(x,xs))	→	app^#(f,x)	(17)
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(18)
comp3^#(f,g,x)	→	app^#(f,app(g,x))	(19)
comp3^#(f,g,x)	→	app^#(g,x)	(20)
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(21)
app^#(comp2(x0,x1),y1)	→	comp3^#(x0,x1,y1)	(22)
app^#(twice,y1)	→	twice1^#(y1)	(23)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(map1(x0),y1)	→	map2^#(x0,y1)	(21)
map2^#(f,cons2(x,xs))	→	app^#(f,x)	(17)
app^#(comp2(x0,x1),y1)	→	comp3^#(x0,x1,y1)	(22)
comp3^#(f,g,x)	→	app^#(f,app(g,x))	(19)
comp3^#(f,g,x)	→	app^#(g,x)	(20)
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(18)

1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

map2^#(f,cons2(x,xs))	→	app^#(f,x)	(17)

1	≥	1
2	>	2
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(18)

1	≥	1
2	>	2
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(21)

1	>	1
2	≥	2
app^#(comp2(x0,x1),y1)	→	comp3^#(x0,x1,y1)	(22)

1	>	1
1	>	2
2	≥	3
comp3^#(f,g,x)	→	app^#(f,app(g,x))	(19)

1	≥	1
comp3^#(f,g,x)	→	app^#(g,x)	(20)

2	≥	1
3	≥	2

As there is no critical graph in the transitive closure, there are no infinite chains.