Certification Problem

Input (TPDB TRS_Standard/AProVE_07/wiehe03)

The rewrite relation of the following TRS is considered.

app(nil,k)	→	k	(1)
app(l,nil)	→	l	(2)
app(cons(x,l),k)	→	cons(x,app(l,k))	(3)
sum(cons(x,nil))	→	cons(x,nil)	(4)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(5)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(6)
sum(plus(cons(0,x),cons(y,l)))	→	pred(sum(cons(s(x),cons(y,l))))	(7)
pred(cons(s(x),nil))	→	cons(x,nil)	(8)
plus(s(x),s(y))	→	s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))	(9)
plus(s(x),x)	→	plus(if(gt(x,x),id(x),id(x)),s(x))	(10)
plus(zero,y)	→	y	(11)
plus(id(x),s(y))	→	s(plus(x,if(gt(s(y),y),y,s(y))))	(12)
id(x)	→	x	(13)
if(true,x,y)	→	x	(14)
if(false,x,y)	→	y	(15)
not(x)	→	if(x,false,true)	(16)
gt(s(x),zero)	→	true	(17)
gt(zero,y)	→	false	(18)
gt(s(x),s(y))	→	gt(x,y)	(19)

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.

sum^#(app(l,cons(x,cons(y,k))))	→	app^#(l,sum(cons(x,cons(y,k))))	(20)
plus^#(s(x),s(y))	→	gt^#(x,y)	(21)
plus^#(id(x),s(y))	→	if^#(gt(s(y),y),y,s(y))	(22)
plus^#(s(x),s(y))	→	gt^#(x,y)	(21)
plus^#(s(x),s(y))	→	id^#(y)	(23)
gt^#(s(x),s(y))	→	gt^#(x,y)	(24)
sum^#(cons(x,cons(y,l)))	→	plus^#(x,y)	(25)
plus^#(id(x),s(y))	→	plus^#(x,if(gt(s(y),y),y,s(y)))	(26)
plus^#(id(x),s(y))	→	gt^#(s(y),y)	(27)
not^#(x)	→	if^#(x,false,true)	(28)
plus^#(s(x),x)	→	id^#(x)	(29)
plus^#(s(x),s(y))	→	if^#(not(gt(x,y)),id(x),id(y))	(30)
plus^#(s(x),s(y))	→	id^#(x)	(31)
plus^#(s(x),s(y))	→	if^#(gt(x,y),x,y)	(32)
sum^#(cons(x,cons(y,l)))	→	sum^#(cons(plus(x,y),l))	(33)
plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(34)
plus^#(s(x),x)	→	gt^#(x,x)	(35)
app^#(cons(x,l),k)	→	app^#(l,k)	(36)
plus^#(s(x),x)	→	id^#(x)	(29)
sum^#(plus(cons(0,x),cons(y,l)))	→	sum^#(cons(s(x),cons(y,l)))	(37)
sum^#(app(l,cons(x,cons(y,k))))	→	sum^#(cons(x,cons(y,k)))	(38)
sum^#(app(l,cons(x,cons(y,k))))	→	sum^#(app(l,sum(cons(x,cons(y,k)))))	(39)
plus^#(s(x),s(y))	→	not^#(gt(x,y))	(40)
sum^#(plus(cons(0,x),cons(y,l)))	→	pred^#(sum(cons(s(x),cons(y,l))))	(41)
plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(42)
plus^#(s(x),x)	→	if^#(gt(x,x),id(x),id(x))	(43)

1.1 Dependency Graph Processor

The dependency pairs are split into 5 components.

The 1^st component contains the pair

sum^#(app(l,cons(x,cons(y,k))))

→

sum^#(app(l,sum(cons(x,cons(y,k)))))

(39)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 33428
[gt(x₁, x₂)]	=	x₂ + 162
[plus^#(x₁, x₂)]	=	0
[false]	=	163
[id^#(x₁)]	=	0
[true]	=	164
[pred(x₁)]	=	x₁ + 0
[sum(x₁)]	=	61227
[not^#(x₁)]	=	0
[0]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + 121
[nil]	=	1
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[pred^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₂ + 30614
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	x₁ + x₂ + 33954

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
pred(cons(s(x),nil))	→	cons(x,nil)	(8)
app(nil,k)	→	k	(1)
app(cons(x,l),k)	→	cons(x,app(l,k))	(3)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(5)
sum(plus(cons(0,x),cons(y,l)))	→	pred(sum(cons(s(x),cons(y,l))))	(7)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(6)
app(l,nil)	→	l	(2)

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

sum^#(app(l,cons(x,cons(y,k))))

→

sum^#(app(l,sum(cons(x,cons(y,k)))))

(39)

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

sum^#(cons(x,cons(y,l)))

→

sum^#(cons(plus(x,y),l))

(33)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 52868
[gt(x₁, x₂)]	=	x₂ + 1
[plus^#(x₁, x₂)]	=	0
[false]	=	2
[id^#(x₁)]	=	0
[true]	=	3
[pred(x₁)]	=	x₁ + 0
[sum(x₁)]	=	30614
[not^#(x₁)]	=	0
[0]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + 21656
[nil]	=	1
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[pred^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	x₁ + 18457
[app(x₁, x₂)]	=	x₁ + x₂ + 33954

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
pred(cons(s(x),nil))	→	cons(x,nil)	(8)
app(nil,k)	→	k	(1)
app(cons(x,l),k)	→	cons(x,app(l,k))	(3)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(5)
sum(plus(cons(0,x),cons(y,l)))	→	pred(sum(cons(s(x),cons(y,l))))	(7)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(6)
app(l,nil)	→	l	(2)

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

sum^#(cons(x,cons(y,l)))

→

sum^#(cons(plus(x,y),l))

(33)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(42)
plus^#(id(x),s(y))	→	plus^#(x,if(gt(s(y),y),y,s(y)))	(26)
plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(34)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	0
[s(x₁)]	=	x₁ + 5
[gt(x₁, x₂)]	=	max(0)
[plus^#(x₁, x₂)]	=	max(x₁ + 8859, x₂ + 8856, 0)
[false]	=	0
[id^#(x₁)]	=	0
[true]	=	0
[pred(x₁)]	=	0
[sum(x₁)]	=	0
[not^#(x₁)]	=	0
[0]	=	0
[if(x₁, x₂, x₃)]	=	max(x₁ + 1, x₂ + 1, x₃ + 0, 0)
[nil]	=	0
[gt^#(x₁, x₂)]	=	max(0)
[app^#(x₁, x₂)]	=	max(0)
[plus(x₁, x₂)]	=	max(0)
[pred^#(x₁)]	=	0
[cons(x₁, x₂)]	=	max(0)
[if^#(x₁, x₂, x₃)]	=	max(0)
[id(x₁)]	=	x₁ + 3
[sum^#(x₁)]	=	0
[not(x₁)]	=	x₁ + 1
[app(x₁, x₂)]	=	max(0)

together with the usable rules

gt(zero,y)	→	false	(18)
if(false,x,y)	→	y	(15)
not(x)	→	if(x,false,true)	(16)
gt(s(x),s(y))	→	gt(x,y)	(19)
gt(s(x),zero)	→	true	(17)
if(true,x,y)	→	x	(14)
id(x)	→	x	(13)

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

plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(42)
plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(34)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

plus^#(id(x),s(y))

→

plus^#(x,if(gt(s(y),y),y,s(y)))

(26)

1.1.3.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 1032
[gt(x₁, x₂)]	=	21653
[plus^#(x₁, x₂)]	=	x₁ + 0
[false]	=	21654
[id^#(x₁)]	=	0
[true]	=	21654
[pred(x₁)]	=	x₁ + 0
[sum(x₁)]	=	2
[not^#(x₁)]	=	0
[0]	=	1
[if(x₁, x₂, x₃)]	=	x₂ + x₃ + 72
[nil]	=	1
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[pred^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	43380
[app(x₁, x₂)]	=	x₁ + x₂ + 33954

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
if(false,x,y)	→	y	(15)
pred(cons(s(x),nil))	→	cons(x,nil)	(8)
app(nil,k)	→	k	(1)
app(cons(x,l),k)	→	cons(x,app(l,k))	(3)
not(x)	→	if(x,false,true)	(16)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(5)
sum(plus(cons(0,x),cons(y,l)))	→	pred(sum(cons(s(x),cons(y,l))))	(7)
if(true,x,y)	→	x	(14)
id(x)	→	x	(13)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(6)
app(l,nil)	→	l	(2)

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

plus^#(id(x),s(y))

→

plus^#(x,if(gt(s(y),y),y,s(y)))

(26)

could be deleted.

1.1.3.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

gt^#(s(x),s(y))

→

gt^#(x,y)

(24)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 1
[gt(x₁, x₂)]	=	1
[plus^#(x₁, x₂)]	=	x₁ + 0
[false]	=	2
[id^#(x₁)]	=	0
[true]	=	5050
[pred(x₁)]	=	x₁ + 0
[sum(x₁)]	=	4426
[not^#(x₁)]	=	0
[0]	=	5957
[if(x₁, x₂, x₃)]	=	x₂ + x₃ + 13359
[nil]	=	1
[gt^#(x₁, x₂)]	=	x₁ + 0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + x₂ + 35325
[pred^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₁ + 0
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 3024
[sum^#(x₁)]	=	0
[not(x₁)]	=	21726
[app(x₁, x₂)]	=	1

together with the usable rules

if(false,x,y)	→	y	(15)
pred(cons(s(x),nil))	→	cons(x,nil)	(8)
not(x)	→	if(x,false,true)	(16)
if(true,x,y)	→	x	(14)
id(x)	→	x	(13)

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

gt^#(s(x),s(y))

→

gt^#(x,y)

(24)

could be deleted.

1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 5^th component contains the pair

app^#(cons(x,l),k)

→

app^#(l,k)

(36)

1.1.5 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 1
[gt(x₁, x₂)]	=	1
[plus^#(x₁, x₂)]	=	x₁ + 0
[false]	=	2
[id^#(x₁)]	=	0
[true]	=	5050
[pred(x₁)]	=	x₁ + 0
[sum(x₁)]	=	x₁ + 0
[not^#(x₁)]	=	0
[0]	=	26716
[if(x₁, x₂, x₃)]	=	x₂ + x₃ + 1
[nil]	=	0
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	x₁ + 0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[pred^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₁ + x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	0
[not(x₁)]	=	21726
[app(x₁, x₂)]	=	x₁ + x₂ + 0

together with the usable rules

if(false,x,y)	→	y	(15)
pred(cons(s(x),nil))	→	cons(x,nil)	(8)
not(x)	→	if(x,false,true)	(16)
if(true,x,y)	→	x	(14)
id(x)	→	x	(13)

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

app^#(cons(x,l),k)

→

app^#(l,k)

(36)

could be deleted.

1.1.5.1 Dependency Graph Processor

The dependency pairs are split into 0 components.