Certification Problem

Input (TPDB TRS_Standard/Various_04/10)

The rewrite relation of the following TRS is considered.

+(0,y)	→	y	(1)
+(s(x),y)	→	s(+(x,y))	(2)
++(nil,ys)	→	ys	(3)
++(:(x,xs),ys)	→	:(x,++(xs,ys))	(4)
sum(:(x,nil))	→	:(x,nil)	(5)
sum(:(x,:(y,xs)))	→	sum(:(+(x,y),xs))	(6)
sum(++(xs,:(x,:(y,ys))))	→	sum(++(xs,sum(:(x,:(y,ys)))))	(7)
-(x,0)	→	x	(8)
-(0,s(y))	→	0	(9)
-(s(x),s(y))	→	-(x,y)	(10)
quot(0,s(y))	→	0	(11)
quot(s(x),s(y))	→	s(quot(-(x,y),s(y)))	(12)
length(nil)	→	0	(13)
length(:(x,xs))	→	s(length(xs))	(14)
hd(:(x,xs))	→	x	(15)
avg(xs)	→	quot(hd(sum(xs)),length(xs))	(16)

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^#(:(x,:(y,xs)))	→	sum^#(:(+(x,y),xs))	(17)
length^#(:(x,xs))	→	length^#(xs)	(18)
avg^#(xs)	→	sum^#(xs)	(19)
sum^#(++(xs,:(x,:(y,ys))))	→	sum^#(:(x,:(y,ys)))	(20)
avg^#(xs)	→	quot^#(hd(sum(xs)),length(xs))	(21)
avg^#(xs)	→	hd^#(sum(xs))	(22)
avg^#(xs)	→	length^#(xs)	(23)
sum^#(++(xs,:(x,:(y,ys))))	→	sum^#(++(xs,sum(:(x,:(y,ys)))))	(24)
-^#(s(x),s(y))	→	-^#(x,y)	(25)
quot^#(s(x),s(y))	→	-^#(x,y)	(26)
sum^#(:(x,:(y,xs)))	→	+^#(x,y)	(27)
+^#(s(x),y)	→	+^#(x,y)	(28)
sum^#(++(xs,:(x,:(y,ys))))	→	++^#(xs,sum(:(x,:(y,ys))))	(29)
++^#(:(x,xs),ys)	→	++^#(xs,ys)	(30)
quot^#(s(x),s(y))	→	quot^#(-(x,y),s(y))	(31)

1.1 Dependency Graph Processor

The dependency pairs are split into 7 components.

The 1^st component contains the pair

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

→

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

(31)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[hd(x₁)]	=	0
[s(x₁)]	=	x₁ + 23677
[++(x₁, x₂)]	=	0
[sum(x₁)]	=	0
[avg^#(x₁)]	=	0
[0]	=	1
[quot(x₁, x₂)]	=	0
[:(x₁, x₂)]	=	0
[++^#(x₁, x₂)]	=	0
[nil]	=	0
[-(x₁, x₂)]	=	x₁ + 23676
[avg(x₁)]	=	0
[hd^#(x₁)]	=	0
[-^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	x₁ + 0
[+(x₁, x₂)]	=	0
[sum^#(x₁)]	=	0
[length(x₁)]	=	0
[length^#(x₁)]	=	0
[+^#(x₁, x₂)]	=	0

together with the usable rules

-(x,0)	→	x	(8)
-(s(x),s(y))	→	-(x,y)	(10)
-(0,s(y))	→	0	(9)

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

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

→

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

(31)

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

length^#(:(x,xs))

→

length^#(xs)

(18)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[hd(x₁)]	=	0
[s(x₁)]	=	x₁ + 1
[++(x₁, x₂)]	=	0
[sum(x₁)]	=	0
[avg^#(x₁)]	=	0
[0]	=	40651
[quot(x₁, x₂)]	=	0
[:(x₁, x₂)]	=	x₂ + 1
[++^#(x₁, x₂)]	=	0
[nil]	=	0
[-(x₁, x₂)]	=	x₁ + 23676
[avg(x₁)]	=	0
[hd^#(x₁)]	=	0
[-^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	x₁ + 0
[+(x₁, x₂)]	=	0
[sum^#(x₁)]	=	0
[length(x₁)]	=	0
[length^#(x₁)]	=	x₁ + 0
[+^#(x₁, x₂)]	=	0

together with the usable rules

-(x,0)	→	x	(8)
-(s(x),s(y))	→	-(x,y)	(10)
-(0,s(y))	→	0	(9)

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

length^#(:(x,xs))

→

length^#(xs)

(18)

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

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

→

-^#(x,y)

(25)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[hd(x₁)]	=	0
[s(x₁)]	=	x₁ + 1
[++(x₁, x₂)]	=	0
[sum(x₁)]	=	0
[avg^#(x₁)]	=	0
[0]	=	36466
[quot(x₁, x₂)]	=	0
[:(x₁, x₂)]	=	1
[++^#(x₁, x₂)]	=	0
[nil]	=	0
[-(x₁, x₂)]	=	x₁ + 0
[avg(x₁)]	=	0
[hd^#(x₁)]	=	0
[-^#(x₁, x₂)]	=	x₁ + x₂ + 0
[quot^#(x₁, x₂)]	=	x₁ + 0
[+(x₁, x₂)]	=	0
[sum^#(x₁)]	=	0
[length(x₁)]	=	0
[length^#(x₁)]	=	0
[+^#(x₁, x₂)]	=	0

together with the usable rules

-(x,0)	→	x	(8)
-(s(x),s(y))	→	-(x,y)	(10)
-(0,s(y))	→	0	(9)

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

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

→

-^#(x,y)

(25)

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^#(++(xs,:(x,:(y,ys))))

→

sum^#(++(xs,sum(:(x,:(y,ys)))))

(24)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[hd(x₁)]	=	0
[s(x₁)]	=	x₁ + 1
[++(x₁, x₂)]	=	x₁ + x₂ + 40966
[sum(x₁)]	=	3
[avg^#(x₁)]	=	0
[0]	=	2998
[quot(x₁, x₂)]	=	0
[:(x₁, x₂)]	=	x₂ + 2
[++^#(x₁, x₂)]	=	0
[nil]	=	1
[-(x₁, x₂)]	=	x₁ + 0
[avg(x₁)]	=	0
[hd^#(x₁)]	=	0
[-^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	x₁ + 0
[+(x₁, x₂)]	=	x₁ + x₂ + 1
[sum^#(x₁)]	=	x₁ + 0
[length(x₁)]	=	0
[length^#(x₁)]	=	0
[+^#(x₁, x₂)]	=	0

together with the usable rules

++(:(x,xs),ys)	→	:(x,++(xs,ys))	(4)
-(x,0)	→	x	(8)
+(0,y)	→	y	(1)
++(nil,ys)	→	ys	(3)
sum(:(x,nil))	→	:(x,nil)	(5)
-(s(x),s(y))	→	-(x,y)	(10)
sum(++(xs,:(x,:(y,ys))))	→	sum(++(xs,sum(:(x,:(y,ys)))))	(7)
-(0,s(y))	→	0	(9)
sum(:(x,:(y,xs)))	→	sum(:(+(x,y),xs))	(6)
+(s(x),y)	→	s(+(x,y))	(2)

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

sum^#(++(xs,:(x,:(y,ys))))

→

sum^#(++(xs,sum(:(x,:(y,ys)))))

(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

++^#(:(x,xs),ys)

→

++^#(xs,ys)

(30)

1.1.5 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[hd(x₁)]	=	0
[s(x₁)]	=	x₁ + 1
[++(x₁, x₂)]	=	x₁ + x₂ + 42632
[sum(x₁)]	=	57799
[avg^#(x₁)]	=	0
[0]	=	2998
[quot(x₁, x₂)]	=	0
[:(x₁, x₂)]	=	x₂ + 1
[++^#(x₁, x₂)]	=	x₁ + 0
[nil]	=	57798
[-(x₁, x₂)]	=	x₁ + 0
[avg(x₁)]	=	0
[hd^#(x₁)]	=	0
[-^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	x₁ + 0
[+(x₁, x₂)]	=	x₁ + x₂ + 1
[sum^#(x₁)]	=	x₁ + 0
[length(x₁)]	=	0
[length^#(x₁)]	=	0
[+^#(x₁, x₂)]	=	0

together with the usable rules

++(:(x,xs),ys)	→	:(x,++(xs,ys))	(4)
-(x,0)	→	x	(8)
+(0,y)	→	y	(1)
++(nil,ys)	→	ys	(3)
sum(:(x,nil))	→	:(x,nil)	(5)
-(s(x),s(y))	→	-(x,y)	(10)
sum(++(xs,:(x,:(y,ys))))	→	sum(++(xs,sum(:(x,:(y,ys)))))	(7)
-(0,s(y))	→	0	(9)
sum(:(x,:(y,xs)))	→	sum(:(+(x,y),xs))	(6)
+(s(x),y)	→	s(+(x,y))	(2)

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

++^#(:(x,xs),ys)

→

++^#(xs,ys)

(30)

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

sum^#(:(x,:(y,xs)))

→

sum^#(:(+(x,y),xs))

(17)

1.1.6 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[hd(x₁)]	=	0
[s(x₁)]	=	x₁ + 1
[++(x₁, x₂)]	=	x₁ + x₂ + 2831
[sum(x₁)]	=	842
[avg^#(x₁)]	=	0
[0]	=	12172
[quot(x₁, x₂)]	=	0
[:(x₁, x₂)]	=	x₂ + 1
[++^#(x₁, x₂)]	=	0
[nil]	=	1
[-(x₁, x₂)]	=	x₁ + 0
[avg(x₁)]	=	0
[hd^#(x₁)]	=	0
[-^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	x₁ + 0
[+(x₁, x₂)]	=	x₁ + x₂ + 50866
[sum^#(x₁)]	=	x₁ + 0
[length(x₁)]	=	0
[length^#(x₁)]	=	0
[+^#(x₁, x₂)]	=	0

together with the usable rules

++(:(x,xs),ys)	→	:(x,++(xs,ys))	(4)
-(x,0)	→	x	(8)
+(0,y)	→	y	(1)
++(nil,ys)	→	ys	(3)
sum(:(x,nil))	→	:(x,nil)	(5)
-(s(x),s(y))	→	-(x,y)	(10)
sum(++(xs,:(x,:(y,ys))))	→	sum(++(xs,sum(:(x,:(y,ys)))))	(7)
-(0,s(y))	→	0	(9)
sum(:(x,:(y,xs)))	→	sum(:(+(x,y),xs))	(6)
+(s(x),y)	→	s(+(x,y))	(2)

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

sum^#(:(x,:(y,xs)))

→

sum^#(:(+(x,y),xs))

(17)

could be deleted.

1.1.6.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 7^th component contains the pair

+^#(s(x),y)

→

+^#(x,y)

(28)

1.1.7 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[hd(x₁)]	=	0
[s(x₁)]	=	x₁ + 1
[++(x₁, x₂)]	=	x₁ + x₂ + 2831
[sum(x₁)]	=	842
[avg^#(x₁)]	=	0
[0]	=	38145
[quot(x₁, x₂)]	=	0
[:(x₁, x₂)]	=	x₂ + 1
[++^#(x₁, x₂)]	=	0
[nil]	=	1
[-(x₁, x₂)]	=	x₁ + 0
[avg(x₁)]	=	0
[hd^#(x₁)]	=	0
[-^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	x₁ + 0
[+(x₁, x₂)]	=	x₁ + x₂ + 50866
[sum^#(x₁)]	=	x₁ + 0
[length(x₁)]	=	0
[length^#(x₁)]	=	0
[+^#(x₁, x₂)]	=	x₁ + 0

together with the usable rules

++(:(x,xs),ys)	→	:(x,++(xs,ys))	(4)
-(x,0)	→	x	(8)
+(0,y)	→	y	(1)
++(nil,ys)	→	ys	(3)
sum(:(x,nil))	→	:(x,nil)	(5)
-(s(x),s(y))	→	-(x,y)	(10)
sum(++(xs,:(x,:(y,ys))))	→	sum(++(xs,sum(:(x,:(y,ys)))))	(7)
-(0,s(y))	→	0	(9)
sum(:(x,:(y,xs)))	→	sum(:(+(x,y),xs))	(6)
+(s(x),y)	→	s(+(x,y))	(2)

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

+^#(s(x),y)

→

+^#(x,y)

(28)

could be deleted.

1.1.7.1 Dependency Graph Processor

The dependency pairs are split into 0 components.