Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/ExSec4_2_DLMMU04_FR)

The rewrite relation of the following TRS is considered.

natsFrom(N)	→	cons(N,n__natsFrom(n__s(N)))	(1)
fst(pair(XS,YS))	→	XS	(2)
snd(pair(XS,YS))	→	YS	(3)
splitAt(0,XS)	→	pair(nil,XS)	(4)
splitAt(s(N),cons(X,XS))	→	u(splitAt(N,activate(XS)),N,X,activate(XS))	(5)
u(pair(YS,ZS),N,X,XS)	→	pair(cons(activate(X),YS),ZS)	(6)
head(cons(N,XS))	→	N	(7)
tail(cons(N,XS))	→	activate(XS)	(8)
sel(N,XS)	→	head(afterNth(N,XS))	(9)
take(N,XS)	→	fst(splitAt(N,XS))	(10)
afterNth(N,XS)	→	snd(splitAt(N,XS))	(11)
natsFrom(X)	→	n__natsFrom(X)	(12)
s(X)	→	n__s(X)	(13)
activate(n__natsFrom(X))	→	natsFrom(activate(X))	(14)
activate(n__s(X))	→	s(activate(X))	(15)
activate(X)	→	X	(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.

sel^#(N,XS)	→	head^#(afterNth(N,XS))	(17)
activate^#(n__natsFrom(X))	→	natsFrom^#(activate(X))	(18)
splitAt^#(s(N),cons(X,XS))	→	activate^#(XS)	(19)
take^#(N,XS)	→	splitAt^#(N,XS)	(20)
splitAt^#(s(N),cons(X,XS))	→	splitAt^#(N,activate(XS))	(21)
splitAt^#(s(N),cons(X,XS))	→	activate^#(XS)	(19)
tail^#(cons(N,XS))	→	activate^#(XS)	(22)
activate^#(n__natsFrom(X))	→	activate^#(X)	(23)
splitAt^#(s(N),cons(X,XS))	→	u^#(splitAt(N,activate(XS)),N,X,activate(XS))	(24)
afterNth^#(N,XS)	→	splitAt^#(N,XS)	(25)
activate^#(n__s(X))	→	activate^#(X)	(26)
sel^#(N,XS)	→	afterNth^#(N,XS)	(27)
u^#(pair(YS,ZS),N,X,XS)	→	activate^#(X)	(28)
take^#(N,XS)	→	fst^#(splitAt(N,XS))	(29)
activate^#(n__s(X))	→	s^#(activate(X))	(30)
afterNth^#(N,XS)	→	snd^#(splitAt(N,XS))	(31)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

splitAt^#(s(N),cons(X,XS))

→

splitAt^#(N,activate(XS))

(21)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[take^#(x₁, x₂)]	=	0
[u(x₁,...,x₄)]	=	0
[activate(x₁)]	=	x₁ + 10803
[take(x₁, x₂)]	=	0
[u^#(x₁,...,x₄)]	=	0
[pair(x₁, x₂)]	=	0
[fst(x₁)]	=	0
[activate^#(x₁)]	=	0
[natsFrom(x₁)]	=	31456
[head^#(x₁)]	=	0
[splitAt(x₁, x₂)]	=	0
[fst^#(x₁)]	=	0
[n__natsFrom(x₁)]	=	20653
[n__s(x₁)]	=	x₁ + 1
[tail(x₁)]	=	0
[0]	=	0
[sel^#(x₁, x₂)]	=	0
[sel(x₁, x₂)]	=	0
[s^#(x₁)]	=	0
[afterNth(x₁, x₂)]	=	0
[nil]	=	0
[tail^#(x₁)]	=	0
[splitAt^#(x₁, x₂)]	=	x₁ + x₂ + 0
[afterNth^#(x₁, x₂)]	=	0
[head(x₁)]	=	0
[snd^#(x₁)]	=	0
[cons(x₁, x₂)]	=	x₂ + 10803
[natsFrom^#(x₁)]	=	0
[snd(x₁)]	=	0

together with the usable rules

activate(n__s(X))	→	s(activate(X))	(15)
natsFrom(N)	→	cons(N,n__natsFrom(n__s(N)))	(1)
activate(X)	→	X	(16)
activate(n__natsFrom(X))	→	natsFrom(activate(X))	(14)
natsFrom(X)	→	n__natsFrom(X)	(12)
s(X)	→	n__s(X)	(13)

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

splitAt^#(s(N),cons(X,XS))

→

splitAt^#(N,activate(XS))

(21)

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

activate^#(n__s(X))	→	activate^#(X)	(26)
activate^#(n__natsFrom(X))	→	activate^#(X)	(23)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 0
[take^#(x₁, x₂)]	=	0
[u(x₁,...,x₄)]	=	0
[activate(x₁)]	=	x₁ + 10803
[take(x₁, x₂)]	=	0
[u^#(x₁,...,x₄)]	=	0
[pair(x₁, x₂)]	=	0
[fst(x₁)]	=	0
[activate^#(x₁)]	=	x₁ + 0
[natsFrom(x₁)]	=	10804
[head^#(x₁)]	=	0
[splitAt(x₁, x₂)]	=	0
[fst^#(x₁)]	=	0
[n__natsFrom(x₁)]	=	x₁ + 1
[n__s(x₁)]	=	x₁ + 0
[tail(x₁)]	=	0
[0]	=	0
[sel^#(x₁, x₂)]	=	0
[sel(x₁, x₂)]	=	0
[s^#(x₁)]	=	0
[afterNth(x₁, x₂)]	=	0
[nil]	=	0
[tail^#(x₁)]	=	0
[splitAt^#(x₁, x₂)]	=	0
[afterNth^#(x₁, x₂)]	=	0
[head(x₁)]	=	0
[snd^#(x₁)]	=	0
[cons(x₁, x₂)]	=	10803
[natsFrom^#(x₁)]	=	0
[snd(x₁)]	=	0

together with the usable rules

natsFrom(N)	→	cons(N,n__natsFrom(n__s(N)))	(1)
activate(X)	→	X	(16)
s(X)	→	n__s(X)	(13)

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

activate^#(n__natsFrom(X))

→

activate^#(X)

(23)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

activate^#(n__s(X))

→

activate^#(X)

(26)

1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[take^#(x₁, x₂)]	=	0
[u(x₁,...,x₄)]	=	0
[activate(x₁)]	=	x₁ + 10803
[take(x₁, x₂)]	=	0
[u^#(x₁,...,x₄)]	=	0
[pair(x₁, x₂)]	=	0
[fst(x₁)]	=	0
[activate^#(x₁)]	=	x₁ + 0
[natsFrom(x₁)]	=	10804
[head^#(x₁)]	=	0
[splitAt(x₁, x₂)]	=	0
[fst^#(x₁)]	=	0
[n__natsFrom(x₁)]	=	x₁ + 1
[n__s(x₁)]	=	x₁ + 1
[tail(x₁)]	=	0
[0]	=	0
[sel^#(x₁, x₂)]	=	0
[sel(x₁, x₂)]	=	0
[s^#(x₁)]	=	0
[afterNth(x₁, x₂)]	=	0
[nil]	=	0
[tail^#(x₁)]	=	0
[splitAt^#(x₁, x₂)]	=	0
[afterNth^#(x₁, x₂)]	=	0
[head(x₁)]	=	0
[snd^#(x₁)]	=	0
[cons(x₁, x₂)]	=	10804
[natsFrom^#(x₁)]	=	0
[snd(x₁)]	=	0

together with the usable rules

natsFrom(N)	→	cons(N,n__natsFrom(n__s(N)))	(1)
activate(X)	→	X	(16)
s(X)	→	n__s(X)	(13)

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

activate^#(n__s(X))

→

activate^#(X)

(26)

could be deleted.

1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.