Certification Problem

Input (TPDB TRS_Standard/Applicative_05/Ex8Polymorphic)

The rewrite relation of the following TRS is considered.

app(app(twice,f),x)	→	app(f,app(f,x))	(1)
app(app(map,f),nil)	→	nil	(2)
app(app(map,f),app(app(cons,h),t))	→	app(app(cons,app(f,h)),app(app(map,f),t))	(3)
app(app(fmap,nil),x)	→	nil	(4)
app(app(fmap,app(app(cons,f),t_f)),x)	→	app(app(cons,app(f,x)),app(app(fmap,t_f),x))	(5)

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.

twice	is mapped to	twice,	twice1(x₁),	twice2(x₁, x₂)
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)
fmap	is mapped to	fmap,	fmap1(x₁),	fmap2(x₁, x₂)
t_f	is mapped to	t_f

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

twice2(f,x)	→	app(f,app(f,x))	(14)
map2(f,nil)	→	nil	(15)
map2(f,cons2(h,t))	→	cons2(app(f,h),map2(f,t))	(16)
fmap2(nil,x)	→	nil	(17)
fmap2(cons2(f,t_f),x)	→	cons2(app(f,x),fmap2(t_f,x))	(18)
app(twice,y1)	→	twice1(y1)	(6)
app(twice1(x0),y1)	→	twice2(x0,y1)	(7)
app(map,y1)	→	map1(y1)	(8)
app(map1(x0),y1)	→	map2(x0,y1)	(9)
app(cons,y1)	→	cons1(y1)	(10)
app(cons1(x0),y1)	→	cons2(x0,y1)	(11)
app(fmap,y1)	→	fmap1(y1)	(12)
app(fmap1(x0),y1)	→	fmap2(x0,y1)	(13)

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.

twice2^#(f,x)	→	app^#(f,app(f,x))	(19)
twice2^#(f,x)	→	app^#(f,x)	(20)
map2^#(f,cons2(h,t))	→	app^#(f,h)	(21)
map2^#(f,cons2(h,t))	→	map2^#(f,t)	(22)
fmap2^#(cons2(f,t_f),x)	→	app^#(f,x)	(23)
fmap2^#(cons2(f,t_f),x)	→	fmap2^#(t_f,x)	(24)
app^#(twice1(x0),y1)	→	twice2^#(x0,y1)	(25)
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(26)
app^#(fmap1(x0),y1)	→	fmap2^#(x0,y1)	(27)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

app^#(twice1(x0),y1)	→	twice2^#(x0,y1)	(25)
twice2^#(f,x)	→	app^#(f,app(f,x))	(19)
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(26)
map2^#(f,cons2(h,t))	→	app^#(f,h)	(21)
app^#(fmap1(x0),y1)	→	fmap2^#(x0,y1)	(27)
fmap2^#(cons2(f,t_f),x)	→	app^#(f,x)	(23)
map2^#(f,cons2(h,t))	→	map2^#(f,t)	(22)
twice2^#(f,x)	→	app^#(f,x)	(20)

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.

app^#(twice1(x0),y1)	→	twice2^#(x0,y1)	(25)

1	>	1
2	≥	2
map2^#(f,cons2(h,t))	→	map2^#(f,t)	(22)

1	≥	1
2	>	2
map2^#(f,cons2(h,t))	→	app^#(f,h)	(21)

1	≥	1
2	>	2
fmap2^#(cons2(f,t_f),x)	→	app^#(f,x)	(23)

1	>	1
2	≥	2
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(26)

1	>	1
2	≥	2
app^#(fmap1(x0),y1)	→	fmap2^#(x0,y1)	(27)

1	>	1
2	≥	2
twice2^#(f,x)	→	app^#(f,app(f,x))	(19)

1	≥	1
twice2^#(f,x)	→	app^#(f,x)	(20)

1	≥	1
2	≥	2

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