Certification Problem

Input (TPDB TRS_Standard/AG01/#3.57)

The rewrite relation of the following TRS is considered.

minus(x,0)	→	x	(1)
minus(s(x),s(y))	→	minus(x,y)	(2)
quot(0,s(y))	→	0	(3)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(4)
plus(0,y)	→	y	(5)
plus(s(x),y)	→	s(plus(x,y))	(6)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(7)
app(nil,k)	→	k	(8)
app(l,nil)	→	l	(9)
app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
sum(cons(x,nil))	→	cons(x,nil)	(11)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(13)

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.

plus^#(s(x),y)	→	plus^#(x,y)	(14)
sum^#(cons(x,cons(y,l)))	→	sum^#(cons(plus(x,y),l))	(15)
minus^#(minus(x,y),z)	→	plus^#(y,z)	(16)
quot^#(s(x),s(y))	→	quot^#(minus(x,y),s(y))	(17)
quot^#(s(x),s(y))	→	minus^#(x,y)	(18)
sum^#(cons(x,cons(y,l)))	→	plus^#(x,y)	(19)
app^#(cons(x,l),k)	→	app^#(l,k)	(20)
sum^#(app(l,cons(x,cons(y,k))))	→	sum^#(cons(x,cons(y,k)))	(21)
sum^#(app(l,cons(x,cons(y,k))))	→	sum^#(app(l,sum(cons(x,cons(y,k)))))	(22)
minus^#(s(x),s(y))	→	minus^#(x,y)	(23)
minus^#(minus(x,y),z)	→	minus^#(x,plus(y,z))	(24)
sum^#(app(l,cons(x,cons(y,k))))	→	app^#(l,sum(cons(x,cons(y,k))))	(25)

1.1 Dependency Graph Processor

The dependency pairs are split into 6 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)))))

(22)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[minus(x₁, x₂)]	=	0
[plus^#(x₁, x₂)]	=	0
[sum(x₁)]	=	78937
[0]	=	34723
[quot(x₁, x₂)]	=	0
[nil]	=	39468
[app^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + x₂ + 17651
[cons(x₁, x₂)]	=	x₂ + 39469
[quot^#(x₁, x₂)]	=	0
[sum^#(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	x₁ + x₂ + 18816

together with the usable rules

app(nil,k)	→	k	(8)
plus(0,y)	→	y	(5)
app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
sum(cons(x,nil))	→	cons(x,nil)	(11)
app(l,nil)	→	l	(9)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(13)
plus(s(x),y)	→	s(plus(x,y))	(6)

(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)))))

(22)

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

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

→

quot^#(minus(x,y),s(y))

(17)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 2
[minus(x₁, x₂)]	=	x₁ + 1
[plus^#(x₁, x₂)]	=	0
[sum(x₁)]	=	78937
[0]	=	34723
[quot(x₁, x₂)]	=	0
[nil]	=	39468
[app^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + x₂ + 46823
[cons(x₁, x₂)]	=	x₂ + 1
[quot^#(x₁, x₂)]	=	x₁ + 0
[sum^#(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	x₁ + x₂ + 24867

together with the usable rules

app(nil,k)	→	k	(8)
minus(x,0)	→	x	(1)
plus(0,y)	→	y	(5)
app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(7)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
sum(cons(x,nil))	→	cons(x,nil)	(11)
app(l,nil)	→	l	(9)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(13)
plus(s(x),y)	→	s(plus(x,y))	(6)
minus(s(x),s(y))	→	minus(x,y)	(2)

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

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

→

quot^#(minus(x,y),s(y))

(17)

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

minus^#(minus(x,y),z)	→	minus^#(x,plus(y,z))	(24)
minus^#(s(x),s(y))	→	minus^#(x,y)	(23)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[minus(x₁, x₂)]	=	x₁ + x₂ + 46824
[plus^#(x₁, x₂)]	=	0
[sum(x₁)]	=	78937
[0]	=	0
[quot(x₁, x₂)]	=	0
[nil]	=	39468
[app^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	x₁ + x₂ + 0
[plus(x₁, x₂)]	=	x₁ + x₂ + 46823
[cons(x₁, x₂)]	=	x₂ + 1
[quot^#(x₁, x₂)]	=	x₁ + 0
[sum^#(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	x₁ + x₂ + 24867

together with the usable rules

app(nil,k)	→	k	(8)
minus(x,0)	→	x	(1)
plus(0,y)	→	y	(5)
app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(7)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
sum(cons(x,nil))	→	cons(x,nil)	(11)
app(l,nil)	→	l	(9)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(13)
plus(s(x),y)	→	s(plus(x,y))	(6)
minus(s(x),s(y))	→	minus(x,y)	(2)

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

minus^#(minus(x,y),z)	→	minus^#(x,plus(y,z))	(24)
minus^#(s(x),s(y))	→	minus^#(x,y)	(23)

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

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

→

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

(15)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[minus(x₁, x₂)]	=	x₁ + x₂ + 46824
[plus^#(x₁, x₂)]	=	0
[sum(x₁)]	=	59250
[0]	=	0
[quot(x₁, x₂)]	=	0
[nil]	=	19781
[app^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	x₂ + 0
[plus(x₁, x₂)]	=	x₁ + x₂ + 46823
[cons(x₁, x₂)]	=	x₂ + 1
[quot^#(x₁, x₂)]	=	x₁ + 0
[sum^#(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	x₁ + x₂ + 24867

together with the usable rules

app(nil,k)	→	k	(8)
minus(x,0)	→	x	(1)
plus(0,y)	→	y	(5)
app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(7)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
sum(cons(x,nil))	→	cons(x,nil)	(11)
app(l,nil)	→	l	(9)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(13)
plus(s(x),y)	→	s(plus(x,y))	(6)
minus(s(x),s(y))	→	minus(x,y)	(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))

(15)

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)

(20)

1.1.5 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[minus(x₁, x₂)]	=	x₁ + x₂ + 46824
[plus^#(x₁, x₂)]	=	0
[sum(x₁)]	=	59250
[0]	=	0
[quot(x₁, x₂)]	=	0
[nil]	=	19781
[app^#(x₁, x₂)]	=	x₁ + 0
[minus^#(x₁, x₂)]	=	x₂ + 0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[cons(x₁, x₂)]	=	x₂ + 1
[quot^#(x₁, x₂)]	=	x₁ + 0
[sum^#(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	x₁ + x₂ + 1

together with the usable rules

app(nil,k)	→	k	(8)
minus(x,0)	→	x	(1)
plus(0,y)	→	y	(5)
app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(7)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
sum(cons(x,nil))	→	cons(x,nil)	(11)
app(l,nil)	→	l	(9)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(13)
plus(s(x),y)	→	s(plus(x,y))	(6)
minus(s(x),s(y))	→	minus(x,y)	(2)

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

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

→

app^#(l,k)

(20)

could be deleted.

1.1.5.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 6^th component contains the pair

plus^#(s(x),y)

→

plus^#(x,y)

(14)

1.1.6 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[minus(x₁, x₂)]	=	x₁ + x₂ + 46824
[plus^#(x₁, x₂)]	=	x₁ + 0
[sum(x₁)]	=	2
[0]	=	0
[quot(x₁, x₂)]	=	0
[nil]	=	1
[app^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	x₂ + 0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[cons(x₁, x₂)]	=	x₂ + 1
[quot^#(x₁, x₂)]	=	x₁ + 0
[sum^#(x₁)]	=	x₁ + 0
[app(x₁, x₂)]	=	x₁ + x₂ + 34725

together with the usable rules

app(nil,k)	→	k	(8)
minus(x,0)	→	x	(1)
plus(0,y)	→	y	(5)
app(cons(x,l),k)	→	cons(x,app(l,k))	(10)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(7)
sum(cons(x,cons(y,l)))	→	sum(cons(plus(x,y),l))	(12)
sum(cons(x,nil))	→	cons(x,nil)	(11)
app(l,nil)	→	l	(9)
sum(app(l,cons(x,cons(y,k))))	→	sum(app(l,sum(cons(x,cons(y,k)))))	(13)
plus(s(x),y)	→	s(plus(x,y))	(6)
minus(s(x),s(y))	→	minus(x,y)	(2)

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

plus^#(s(x),y)

→

plus^#(x,y)

(14)

could be deleted.

1.1.6.1 Dependency Graph Processor

The dependency pairs are split into 0 components.