Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Transformed_CSR_04/Ex7_BLR02_Z)

The rewrite relation of the following TRS is considered.

from(X)	→	cons(X,n__from(s(X)))	(1)
head(cons(X,XS))	→	X	(2)
2nd(cons(X,XS))	→	head(activate(XS))	(3)
take(0,XS)	→	nil	(4)
take(s(N),cons(X,XS))	→	cons(X,n__take(N,activate(XS)))	(5)
sel(0,cons(X,XS))	→	X	(6)
sel(s(N),cons(X,XS))	→	sel(N,activate(XS))	(7)
from(X)	→	n__from(X)	(8)
take(X1,X2)	→	n__take(X1,X2)	(9)
activate(n__from(X))	→	from(X)	(10)
activate(n__take(X1,X2))	→	take(X1,X2)	(11)
activate(X)	→	X	(12)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n²).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

from^#(z0)

→

(14)

originates from

from(z0)

→

cons(z0,n__from(s(z0)))

(13)

from^#(z0)

→

(16)

originates from

from(z0)

→

n__from(z0)

(15)

head^#(cons(z0,z1))

→

(18)

originates from

head(cons(z0,z1))

→

(17)

2nd^#(cons(z0,z1))

→

c3(head^#(activate(z1)),activate^#(z1))

(20)

originates from

2nd(cons(z0,z1))

→

head(activate(z1))

(19)

take^#(0,z0)

→

(22)

originates from

take(0,z0)

→

nil

(21)

take^#(s(z0),cons(z1,z2))

→

c5(activate^#(z2))

(24)

originates from

take(s(z0),cons(z1,z2))

→

cons(z1,n__take(z0,activate(z2)))

(23)

take^#(z0,z1)

→

(26)

originates from

take(z0,z1)

→

n__take(z0,z1)

(25)

sel^#(0,cons(z0,z1))

→

(28)

originates from

sel(0,cons(z0,z1))

→

(27)

sel^#(s(z0),cons(z1,z2))

→

c8(sel^#(z0,activate(z2)),activate^#(z2))

(30)

originates from

sel(s(z0),cons(z1,z2))

→

sel(z0,activate(z2))

(29)

activate^#(n__from(z0))

→

c9(from^#(z0))

(32)

originates from

activate(n__from(z0))

→

from(z0)

(31)

activate^#(n__take(z0,z1))

→

c10(take^#(z0,z1))

(34)

originates from

activate(n__take(z0,z1))

→

take(z0,z1)

(33)

activate^#(z0)

→

c11

(36)

originates from

activate(z0)

→

(35)

Moreover, we add the following terms to the innermost strategy.

from^#(z0)

head^#(cons(z0,z1))

2nd^#(cons(z0,z1))

take^#(0,z0)

take^#(s(z0),cons(z1,z2))

take^#(z0,z1)

sel^#(0,cons(z0,z1))

sel^#(s(z0),cons(z1,z2))

activate^#(n__from(z0))

activate^#(n__take(z0,z1))

activate^#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.

head(cons(z0,z1))	→	z0	(17)
2nd(cons(z0,z1))	→	head(activate(z1))	(19)
sel(0,cons(z0,z1))	→	z0	(27)
sel(s(z0),cons(z1,z2))	→	sel(z0,activate(z2))	(29)

1.1.1 Rule Shifting

The rules

2nd^#(cons(z0,z1))	→	c3(head^#(activate(z1)),activate^#(z1))	(20)
sel^#(0,cons(z0,z1))	→	c7	(28)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁)]	=	1 · x₁ + 0
[c11]	=	0
[activate(x₁)]	=	1 + 1 · x₁
[from(x₁)]	=	1 + 1 · x₁
[take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[from^#(x₁)]	=	0
[head^#(x₁)]	=	0
[2nd^#(x₁)]	=	1 + 1 · x₁
[take^#(x₁, x₂)]	=	0
[sel^#(x₁, x₂)]	=	1
[activate^#(x₁)]	=	0
[n__from(x₁)]	=	1 + 1 · x₁
[n__take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	1
[nil]	=	1
[s(x₁)]	=	1 + 1 · x₁
[cons(x₁, x₂)]	=	1 · x₁ + 0

which has the intended complexity. Here, only the following usable rules have been considered:

from^#(z0)	→	c	(14)
from^#(z0)	→	c1	(16)
head^#(cons(z0,z1))	→	c2	(18)
2nd^#(cons(z0,z1))	→	c3(head^#(activate(z1)),activate^#(z1))	(20)
take^#(0,z0)	→	c4	(22)
take^#(s(z0),cons(z1,z2))	→	c5(activate^#(z2))	(24)
take^#(z0,z1)	→	c6	(26)
sel^#(0,cons(z0,z1))	→	c7	(28)
sel^#(s(z0),cons(z1,z2))	→	c8(sel^#(z0,activate(z2)),activate^#(z2))	(30)
activate^#(n__from(z0))	→	c9(from^#(z0))	(32)
activate^#(n__take(z0,z1))	→	c10(take^#(z0,z1))	(34)
activate^#(z0)	→	c11	(36)

1.1.1.1 Rule Shifting

The rules

head^#(cons(z0,z1))	→	c2	(18)
sel^#(s(z0),cons(z1,z2))	→	c8(sel^#(z0,activate(z2)),activate^#(z2))	(30)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁)]	=	1 · x₁ + 0
[c11]	=	0
[activate(x₁)]	=	1 + 1 · x₁
[from(x₁)]	=	1 + 1 · x₁
[take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[from^#(x₁)]	=	0
[head^#(x₁)]	=	1
[2nd^#(x₁)]	=	1 + 1 · x₁
[take^#(x₁, x₂)]	=	0
[sel^#(x₁, x₂)]	=	1 · x₁ + 0
[activate^#(x₁)]	=	0
[n__from(x₁)]	=	1 + 1 · x₁
[n__take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	1
[nil]	=	1
[s(x₁)]	=	1 + 1 · x₁
[cons(x₁, x₂)]	=	1 · x₁ + 0

which has the intended complexity. Here, only the following usable rules have been considered:

from^#(z0)	→	c	(14)
from^#(z0)	→	c1	(16)
head^#(cons(z0,z1))	→	c2	(18)
2nd^#(cons(z0,z1))	→	c3(head^#(activate(z1)),activate^#(z1))	(20)
take^#(0,z0)	→	c4	(22)
take^#(s(z0),cons(z1,z2))	→	c5(activate^#(z2))	(24)
take^#(z0,z1)	→	c6	(26)
sel^#(0,cons(z0,z1))	→	c7	(28)
sel^#(s(z0),cons(z1,z2))	→	c8(sel^#(z0,activate(z2)),activate^#(z2))	(30)
activate^#(n__from(z0))	→	c9(from^#(z0))	(32)
activate^#(n__take(z0,z1))	→	c10(take^#(z0,z1))	(34)
activate^#(z0)	→	c11	(36)

1.1.1.1.1 Rule Shifting

The rules

take^#(0,z0)	→	c4	(22)
take^#(z0,z1)	→	c6	(26)
activate^#(n__from(z0))	→	c9(from^#(z0))	(32)
activate^#(z0)	→	c11	(36)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁)]	=	1 · x₁ + 0
[c11]	=	0
[activate(x₁)]	=	1 + 1 · x₁
[from(x₁)]	=	1
[take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[from^#(x₁)]	=	0
[head^#(x₁)]	=	1 · x₁ + 0
[2nd^#(x₁)]	=	1 + 1 · x₁
[take^#(x₁, x₂)]	=	1
[sel^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[activate^#(x₁)]	=	1
[n__from(x₁)]	=	0
[n__take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	1
[nil]	=	0
[s(x₁)]	=	1 + 1 · x₁
[cons(x₁, x₂)]	=	1 + 1 · x₂

which has the intended complexity. Here, only the following usable rules have been considered:

from^#(z0)	→	c	(14)
from^#(z0)	→	c1	(16)
head^#(cons(z0,z1))	→	c2	(18)
2nd^#(cons(z0,z1))	→	c3(head^#(activate(z1)),activate^#(z1))	(20)
take^#(0,z0)	→	c4	(22)
take^#(s(z0),cons(z1,z2))	→	c5(activate^#(z2))	(24)
take^#(z0,z1)	→	c6	(26)
sel^#(0,cons(z0,z1))	→	c7	(28)
sel^#(s(z0),cons(z1,z2))	→	c8(sel^#(z0,activate(z2)),activate^#(z2))	(30)
activate^#(n__from(z0))	→	c9(from^#(z0))	(32)
activate^#(n__take(z0,z1))	→	c10(take^#(z0,z1))	(34)
activate^#(z0)	→	c11	(36)
take(z0,z1)	→	n__take(z0,z1)	(25)
from(z0)	→	cons(z0,n__from(s(z0)))	(13)
take(s(z0),cons(z1,z2))	→	cons(z1,n__take(z0,activate(z2)))	(23)
activate(n__take(z0,z1))	→	take(z0,z1)	(33)
take(0,z0)	→	nil	(21)
from(z0)	→	n__from(z0)	(15)
activate(n__from(z0))	→	from(z0)	(31)
activate(z0)	→	z0	(35)

1.1.1.1.1.1 Rule Shifting

The rules

from^#(z0)	→	c	(14)
from^#(z0)	→	c1	(16)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁)]	=	1 · x₁ + 0
[c11]	=	0
[activate(x₁)]	=	1 + 1 · x₁
[from(x₁)]	=	1
[take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[from^#(x₁)]	=	1
[head^#(x₁)]	=	1 · x₁ + 0
[2nd^#(x₁)]	=	1 + 1 · x₁
[take^#(x₁, x₂)]	=	1
[sel^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[activate^#(x₁)]	=	1
[n__from(x₁)]	=	0
[n__take(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	1
[nil]	=	0
[s(x₁)]	=	1 + 1 · x₁
[cons(x₁, x₂)]	=	1 + 1 · x₂

which has the intended complexity. Here, only the following usable rules have been considered:

from^#(z0)	→	c	(14)
from^#(z0)	→	c1	(16)
head^#(cons(z0,z1))	→	c2	(18)
2nd^#(cons(z0,z1))	→	c3(head^#(activate(z1)),activate^#(z1))	(20)
take^#(0,z0)	→	c4	(22)
take^#(s(z0),cons(z1,z2))	→	c5(activate^#(z2))	(24)
take^#(z0,z1)	→	c6	(26)
sel^#(0,cons(z0,z1))	→	c7	(28)
sel^#(s(z0),cons(z1,z2))	→	c8(sel^#(z0,activate(z2)),activate^#(z2))	(30)
activate^#(n__from(z0))	→	c9(from^#(z0))	(32)
activate^#(n__take(z0,z1))	→	c10(take^#(z0,z1))	(34)
activate^#(z0)	→	c11	(36)
take(z0,z1)	→	n__take(z0,z1)	(25)
from(z0)	→	cons(z0,n__from(s(z0)))	(13)
take(s(z0),cons(z1,z2))	→	cons(z1,n__take(z0,activate(z2)))	(23)
activate(n__take(z0,z1))	→	take(z0,z1)	(33)
take(0,z0)	→	nil	(21)
from(z0)	→	n__from(z0)	(15)
activate(n__from(z0))	→	from(z0)	(31)
activate(z0)	→	z0	(35)

1.1.1.1.1.1.1 Rule Shifting

The rules

take^#(s(z0),cons(z1,z2))	→	c5(activate^#(z2))	(24)
activate^#(n__take(z0,z1))	→	c10(take^#(z0,z1))	(34)

are strictly oriented by the following non-linear polynomial interpretation over the naturals

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁)]	=	1 · x₁ + 0
[c11]	=	0
[activate(x₁)]	=	1 · x₁ + 0
[from(x₁)]	=	0
[take(x₁, x₂)]	=	1 + 1 · x₂
[from^#(x₁)]	=	0
[head^#(x₁)]	=	0
[2nd^#(x₁)]	=	2 · x₁ + 0
[take^#(x₁, x₂)]	=	1 + 2 · x₂
[sel^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[activate^#(x₁)]	=	2 · x₁ + 0
[n__from(x₁)]	=	0
[n__take(x₁, x₂)]	=	1 + 1 · x₂
[0]	=	2
[nil]	=	1
[s(x₁)]	=	1 + 1 · x₁
[cons(x₁, x₂)]	=	1 · x₂ + 0

which has the intended complexity. Here, only the following usable rules have been considered:

from^#(z0)	→	c	(14)
from^#(z0)	→	c1	(16)
head^#(cons(z0,z1))	→	c2	(18)
2nd^#(cons(z0,z1))	→	c3(head^#(activate(z1)),activate^#(z1))	(20)
take^#(0,z0)	→	c4	(22)
take^#(s(z0),cons(z1,z2))	→	c5(activate^#(z2))	(24)
take^#(z0,z1)	→	c6	(26)
sel^#(0,cons(z0,z1))	→	c7	(28)
sel^#(s(z0),cons(z1,z2))	→	c8(sel^#(z0,activate(z2)),activate^#(z2))	(30)
activate^#(n__from(z0))	→	c9(from^#(z0))	(32)
activate^#(n__take(z0,z1))	→	c10(take^#(z0,z1))	(34)
activate^#(z0)	→	c11	(36)
take(z0,z1)	→	n__take(z0,z1)	(25)
from(z0)	→	cons(z0,n__from(s(z0)))	(13)
take(s(z0),cons(z1,z2))	→	cons(z1,n__take(z0,activate(z2)))	(23)
activate(n__take(z0,z1))	→	take(z0,z1)	(33)
take(0,z0)	→	nil	(21)
from(z0)	→	n__from(z0)	(15)
activate(n__from(z0))	→	from(z0)	(31)
activate(z0)	→	z0	(35)

1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS R. Hence, R/S has complexity O(1).