Certification Problem

Input (TPDB TRS_Standard/AProVE_07/wiehe02)

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)
plus(s(x),s(y))	→	s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))	(7)
plus(s(x),x)	→	plus(if(gt(x,x),id(x),id(x)),s(x))	(8)
plus(zero,y)	→	y	(9)
plus(id(x),s(y))	→	s(plus(x,if(gt(s(y),y),y,s(y))))	(10)
id(x)	→	x	(11)
if(true,x,y)	→	x	(12)
if(false,x,y)	→	y	(13)
not(x)	→	if(x,false,true)	(14)
gt(s(x),zero)	→	true	(15)
gt(zero,y)	→	false	(16)
gt(s(x),s(y))	→	gt(x,y)	(17)

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))))	(18)
plus^#(s(x),s(y))	→	if^#(gt(x,y),x,y)	(19)
plus^#(id(x),s(y))	→	plus^#(x,if(gt(s(y),y),y,s(y)))	(20)
plus^#(s(x),s(y))	→	not^#(gt(x,y))	(21)
plus^#(s(x),s(y))	→	id^#(x)	(22)
plus^#(s(x),x)	→	id^#(x)	(23)
plus^#(s(x),x)	→	id^#(x)	(23)
plus^#(s(x),s(y))	→	id^#(y)	(24)
plus^#(id(x),s(y))	→	if^#(gt(s(y),y),y,s(y))	(25)
plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(26)
plus^#(s(x),s(y))	→	gt^#(x,y)	(27)
plus^#(s(x),s(y))	→	gt^#(x,y)	(27)
plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(28)
plus^#(s(x),x)	→	gt^#(x,x)	(29)
sum^#(cons(x,cons(y,l)))	→	plus^#(x,y)	(30)
app^#(cons(x,l),k)	→	app^#(l,k)	(31)
plus^#(s(x),x)	→	if^#(gt(x,x),id(x),id(x))	(32)
sum^#(cons(x,cons(y,l)))	→	sum^#(cons(plus(x,y),l))	(33)
sum^#(app(l,cons(x,cons(y,k))))	→	sum^#(cons(x,cons(y,k)))	(34)
sum^#(app(l,cons(x,cons(y,k))))	→	sum^#(app(l,sum(cons(x,cons(y,k)))))	(35)
plus^#(s(x),s(y))	→	if^#(not(gt(x,y)),id(x),id(y))	(36)
plus^#(id(x),s(y))	→	gt^#(s(y),y)	(37)
not^#(x)	→	if^#(x,false,true)	(38)
gt^#(s(x),s(y))	→	gt^#(x,y)	(39)

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)))))

(35)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 2
[gt(x₁, x₂)]	=	x₂ + 1
[plus^#(x₁, x₂)]	=	0
[false]	=	2
[id^#(x₁)]	=	0
[true]	=	0
[sum(x₁)]	=	3
[not^#(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	x₃ + 2
[nil]	=	1
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + 35657
[cons(x₁, x₂)]	=	x₂ + 2
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	0
[app(x₁, x₂)]	=	x₁ + x₂ + 29283

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
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(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)))))

(35)

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 + 2
[gt(x₁, x₂)]	=	x₂ + 1
[plus^#(x₁, x₂)]	=	0
[false]	=	2
[id^#(x₁)]	=	0
[true]	=	0
[sum(x₁)]	=	31123
[not^#(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	x₃ + 2
[nil]	=	1
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 31122
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	0
[app(x₁, x₂)]	=	x₁ + x₂ + 29283

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
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(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

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

→

app^#(l,k)

(31)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 2
[gt(x₁, x₂)]	=	x₂ + 1
[plus^#(x₁, x₂)]	=	0
[false]	=	2
[id^#(x₁)]	=	0
[true]	=	0
[sum(x₁)]	=	592
[not^#(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	x₃ + 16909
[nil]	=	1
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	x₁ + 0
[plus(x₁, x₂)]	=	x₁ + 1
[cons(x₁, x₂)]	=	x₂ + 591
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	0
[app(x₁, x₂)]	=	x₁ + x₂ + 19428

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
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(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

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

→

app^#(l,k)

(31)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

plus^#(id(x),s(y))	→	plus^#(x,if(gt(s(y),y),y,s(y)))	(20)
plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(28)
plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(26)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	0
[s(x₁)]	=	x₁ + 5
[gt(x₁, x₂)]	=	max(x₂ + 1, 0)
[plus^#(x₁, x₂)]	=	max(x₁ + 1427, x₂ + 1424, 0)
[false]	=	2
[id^#(x₁)]	=	0
[true]	=	2
[sum(x₁)]	=	0
[not^#(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	max(x₂ + 1, x₃ + 0, 0)
[nil]	=	0
[gt^#(x₁, x₂)]	=	max(0)
[app^#(x₁, x₂)]	=	max(0)
[plus(x₁, x₂)]	=	max(0)
[cons(x₁, x₂)]	=	max(0)
[if^#(x₁, x₂, x₃)]	=	max(0)
[id(x₁)]	=	x₁ + 3
[sum^#(x₁)]	=	0
[not(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	max(0)

together with the usable rules

if(true,x,y)	→	x	(12)
id(x)	→	x	(11)
if(false,x,y)	→	y	(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)))	(28)
plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(26)

could be deleted.

1.1.4.1 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)))

(20)

1.1.4.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	2
[gt(x₁, x₂)]	=	x₂ + 1
[plus^#(x₁, x₂)]	=	x₁ + 0
[false]	=	21395
[id^#(x₁)]	=	0
[true]	=	0
[sum(x₁)]	=	42005
[not^#(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	x₂ + x₃ + 1
[nil]	=	42004
[gt^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	1
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	1
[app(x₁, x₂)]	=	x₁ + x₂ + 25536

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
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)
if(true,x,y)	→	x	(12)
id(x)	→	x	(11)
if(false,x,y)	→	y	(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)))

(20)

could be deleted.

1.1.4.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 5^th component contains the pair

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

→

gt^#(x,y)

(39)

1.1.5 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 1
[gt(x₁, x₂)]	=	x₂ + 1
[plus^#(x₁, x₂)]	=	x₁ + 0
[false]	=	21395
[id^#(x₁)]	=	0
[true]	=	0
[sum(x₁)]	=	2
[not^#(x₁)]	=	0
[if(x₁, x₂, x₃)]	=	x₂ + x₃ + 1
[nil]	=	1
[gt^#(x₁, x₂)]	=	x₁ + 0
[app^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	54886
[cons(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 51737
[sum^#(x₁)]	=	x₁ + 0
[not(x₁)]	=	1
[app(x₁, x₂)]	=	x₁ + x₂ + 25067

together with the usable rules

sum(cons(x,nil))	→	cons(x,nil)	(4)
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)
if(true,x,y)	→	x	(12)
id(x)	→	x	(11)
if(false,x,y)	→	y	(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

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

→

gt^#(x,y)

(39)

could be deleted.

1.1.5.1 Dependency Graph Processor

The dependency pairs are split into 0 components.