Certification Problem

Input (TPDB TRS_Standard/AProVE_06/mapHard)

The rewrite relation of the following TRS is considered.

ap(ap(map,f),xs)	→	ap(ap(ap(if,ap(isEmpty,xs)),f),xs)	(1)
ap(ap(ap(if,true),f),xs)	→	nil	(2)
ap(ap(ap(if,false),f),xs)	→	ap(ap(cons,ap(f,ap(last,xs))),ap(ap(map,f),ap(dropLast,xs)))	(3)
ap(isEmpty,nil)	→	true	(4)
ap(isEmpty,ap(ap(cons,x),xs))	→	false	(5)
ap(last,ap(ap(cons,x),nil))	→	x	(6)
ap(last,ap(ap(cons,x),ap(ap(cons,y),ys)))	→	ap(last,ap(ap(cons,y),ys))	(7)
ap(dropLast,nil)	→	nil	(8)
ap(dropLast,ap(ap(cons,x),nil))	→	nil	(9)
ap(dropLast,ap(ap(cons,x),ap(ap(cons,y),ys)))	→	ap(ap(cons,x),ap(dropLast,ap(ap(cons,y),ys)))	(10)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

We uncurry the binary symbol ap 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)
if	is mapped to	if,	if1(x₁),	if2(x₁, x₂),	if3(x₁, x₂, x₃)
isEmpty	is mapped to	isEmpty,	isEmpty1(x₁)
true	is mapped to	true
nil	is mapped to	nil
false	is mapped to	false
cons	is mapped to	cons,	cons1(x₁),	cons2(x₁, x₂)
last	is mapped to	last,	last1(x₁)
dropLast	is mapped to	dropLast,	dropLast1(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,xs)	→	if3(isEmpty1(xs),f,xs)	(21)
if3(true,f,xs)	→	nil	(22)
if3(false,f,xs)	→	cons2(ap(f,last1(xs)),map2(f,dropLast1(xs)))	(23)
isEmpty1(nil)	→	true	(24)
isEmpty1(cons2(x,xs))	→	false	(25)
last1(cons2(x,nil))	→	x	(26)
last1(cons2(x,cons2(y,ys)))	→	last1(cons2(y,ys))	(27)
dropLast1(nil)	→	nil	(28)
dropLast1(cons2(x,nil))	→	nil	(29)
dropLast1(cons2(x,cons2(y,ys)))	→	cons2(x,dropLast1(cons2(y,ys)))	(30)
ap(map,y1)	→	map1(y1)	(11)
ap(map1(x0),y1)	→	map2(x0,y1)	(12)
ap(if,y1)	→	if1(y1)	(13)
ap(if1(x0),y1)	→	if2(x0,y1)	(14)
ap(if2(x0,x1),y1)	→	if3(x0,x1,y1)	(15)
ap(isEmpty,y1)	→	isEmpty1(y1)	(16)
ap(cons,y1)	→	cons1(y1)	(17)
ap(cons1(x0),y1)	→	cons2(x0,y1)	(18)
ap(last,y1)	→	last1(y1)	(19)
ap(dropLast,y1)	→	dropLast1(y1)	(20)

1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

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,xs)	→	if3^#(isEmpty1(xs),f,xs)	(31)
map2^#(f,xs)	→	isEmpty1^#(xs)	(32)
if3^#(false,f,xs)	→	ap^#(f,last1(xs))	(33)
if3^#(false,f,xs)	→	last1^#(xs)	(34)
if3^#(false,f,xs)	→	map2^#(f,dropLast1(xs))	(35)
if3^#(false,f,xs)	→	dropLast1^#(xs)	(36)
last1^#(cons2(x,cons2(y,ys)))	→	last1^#(cons2(y,ys))	(37)
dropLast1^#(cons2(x,cons2(y,ys)))	→	dropLast1^#(cons2(y,ys))	(38)
ap^#(map1(x0),y1)	→	map2^#(x0,y1)	(39)
ap^#(if2(x0,x1),y1)	→	if3^#(x0,x1,y1)	(40)
ap^#(isEmpty,y1)	→	isEmpty1^#(y1)	(41)
ap^#(last,y1)	→	last1^#(y1)	(42)
ap^#(dropLast,y1)	→	dropLast1^#(y1)	(43)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

if3^#(false,f,xs)	→	ap^#(f,last1(xs))	(33)
ap^#(map1(x0),y1)	→	map2^#(x0,y1)	(39)
map2^#(f,xs)	→	if3^#(isEmpty1(xs),f,xs)	(31)
if3^#(false,f,xs)	→	map2^#(f,dropLast1(xs))	(35)
ap^#(if2(x0,x1),y1)	→	if3^#(x0,x1,y1)	(40)

1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

dropLast1(nil)	→	nil	(28)
dropLast1(cons2(x,nil))	→	nil	(29)
dropLast1(cons2(x,cons2(y,ys)))	→	cons2(x,dropLast1(cons2(y,ys)))	(30)
isEmpty1(nil)	→	true	(24)
isEmpty1(cons2(x,xs))	→	false	(25)
last1(cons2(x,nil))	→	x	(26)
last1(cons2(x,cons2(y,ys)))	→	last1(cons2(y,ys))	(27)

1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

isEmpty1(nil)

isEmpty1(cons2(x0,x1))

last1(cons2(x0,nil))

last1(cons2(x0,cons2(x1,x2)))

dropLast1(nil)

dropLast1(cons2(x0,nil))

dropLast1(cons2(x0,cons2(x1,x2)))

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the

prec(false)	=	0	stat(false)	=	lex
prec(last1)	=	3	stat(last1)	=	lex
prec(map1)	=	4	stat(map1)	=	lex
prec(isEmpty1)	=	5	stat(isEmpty1)	=	lex
prec(dropLast1)	=	1	stat(dropLast1)	=	lex
prec(if2)	=	6	stat(if2)	=	lex
prec(cons2)	=	2	stat(cons2)	=	lex
prec(nil)	=	7	stat(nil)	=	lex
prec(true)	=	7	stat(true)	=	lex

π(if3^#)	=	2
π(false)	=	[]
π(ap^#)	=	1
π(last1)	=	[]
π(map1)	=	[1]
π(map2^#)	=	1
π(isEmpty1)	=	[1]
π(dropLast1)	=	[1]
π(if2)	=	[2]
π(cons2)	=	[]
π(nil)	=	[]
π(true)	=	[]

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

ap^#(map1(x0),y1)	→	map2^#(x0,y1)	(39)
ap^#(if2(x0,x1),y1)	→	if3^#(x0,x1,y1)	(40)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

if3^#(false,f,xs)	→	map2^#(f,dropLast1(xs))	(35)
map2^#(f,xs)	→	if3^#(isEmpty1(xs),f,xs)	(31)

1.1.1.1.1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

isEmpty1(nil)	→	true	(24)
isEmpty1(cons2(x,xs))	→	false	(25)
dropLast1(nil)	→	nil	(28)
dropLast1(cons2(x,nil))	→	nil	(29)
dropLast1(cons2(x,cons2(y,ys)))	→	cons2(x,dropLast1(cons2(y,ys)))	(30)

1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

isEmpty1(nil)

isEmpty1(cons2(x0,x1))

dropLast1(nil)

dropLast1(cons2(x0,nil))

dropLast1(cons2(x0,cons2(x1,x2)))

1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider all narrowings of the pair below position 1 to get the following set of pairs

map2^#(y0,nil)	→	if3^#(true,y0,nil)	(44)
map2^#(y0,cons2(x0,x1))	→	if3^#(false,y0,cons2(x0,x1))	(45)

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

map2^#(y0,cons2(x0,x1))	→	if3^#(false,y0,cons2(x0,x1))	(45)
if3^#(false,f,xs)	→	map2^#(f,dropLast1(xs))	(35)

1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

dropLast1(nil)	→	nil	(28)
dropLast1(cons2(x,nil))	→	nil	(29)
dropLast1(cons2(x,cons2(y,ys)))	→	cons2(x,dropLast1(cons2(y,ys)))	(30)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

dropLast1(nil)

dropLast1(cons2(x0,nil))

dropLast1(cons2(x0,cons2(x1,x2)))

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[dropLast1(x₁)]	=	1 · x₁
[nil]	=	1
[cons2(x₁, x₂)]	=	1 · x₁ + 2 · x₂
[map2^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[if3^#(x₁, x₂, x₃)]	=	2 · x₁ + 1 · x₂ + 1 · x₃
[false]	=	0

the rules

map2(f,xs)	→	if3(isEmpty1(xs),f,xs)	(21)
if3(true,f,xs)	→	nil	(22)
if3(false,f,xs)	→	cons2(ap(f,last1(xs)),map2(f,dropLast1(xs)))	(23)
isEmpty1(nil)	→	true	(24)
isEmpty1(cons2(x,xs))	→	false	(25)
last1(cons2(x,nil))	→	x	(26)
last1(cons2(x,cons2(y,ys)))	→	last1(cons2(y,ys))	(27)
dropLast1(cons2(x,nil))	→	nil	(29)
ap(map,y1)	→	map1(y1)	(11)
ap(map1(x0),y1)	→	map2(x0,y1)	(12)
ap(if,y1)	→	if1(y1)	(13)
ap(if1(x0),y1)	→	if2(x0,y1)	(14)
ap(if2(x0,x1),y1)	→	if3(x0,x1,y1)	(15)
ap(isEmpty,y1)	→	isEmpty1(y1)	(16)
ap(cons,y1)	→	cons1(y1)	(17)
ap(cons1(x0),y1)	→	cons2(x0,y1)	(18)
ap(last,y1)	→	last1(y1)	(19)
ap(dropLast,y1)	→	dropLast1(y1)	(20)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers

[map2^#(x₁, x₂)]

1	1
0	0

· x₁ +

1	0
0	0

· x₂

[cons2(x₁, x₂)]

0	0
0	0

· x₁ +

1	0
1	0

· x₂

[if3^#(x₁, x₂, x₃)]

1	1
0	0

· x₁ +

1	1
0	0

· x₂ +

0	1
0	0

· x₃

[false]

[dropLast1(x₁)]

0	1
1	0

· x₁

[nil]

the pair

map2^#(y0,cons2(x0,x1))

→

if3^#(false,y0,cons2(x0,x1))

(45)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

if3^#(false,f,xs)	→	map2^#(f,dropLast1(xs))	(35)

2	≥	1

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

The 2^nd component contains the pair
last1^#(cons2(x,cons2(y,ys))) → last1^#(cons2(y,ys)) (37)

1.1.1.1.2 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1.1.2.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

There are no lhss.

1.1.1.1.2.1.1 Size-Change Termination

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

last1^#(cons2(x,cons2(y,ys))) → last1^#(cons2(y,ys)) (37)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 3^rd component contains the pair
dropLast1^#(cons2(x,cons2(y,ys))) → dropLast1^#(cons2(y,ys)) (38)

1.1.1.1.3 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1.1.3.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

There are no lhss.

1.1.1.1.3.1.1 Size-Change Termination

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

dropLast1^#(cons2(x,cons2(y,ys))) → dropLast1^#(cons2(y,ys)) (38)

1 > 1

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

Certification Problem

Input (TPDB TRS_Standard/AProVE_06/mapHard)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

1.1 Switch to Innermost Termination

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Usable Rules Processor

1.1.1.1.1.1 Innermost Lhss Removal Processor

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1 Usable Rules Processor

1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1 Usable Rules Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

1.1.1.1.2 Usable Rules Processor

1.1.1.1.2.1 Innermost Lhss Removal Processor

1.1.1.1.2.1.1 Size-Change Termination

1.1.1.1.3 Usable Rules Processor

1.1.1.1.3.1 Innermost Lhss Removal Processor

1.1.1.1.3.1.1 Size-Change Termination