Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Others/inssort)

The relative rewrite relation R/S is considered where R is the following TRS

isort(Cons(x,xs),r)	→	isort(xs,insert(x,r))	(1)
isort(Nil,r)	→	Nil	(2)
insert(S(x),r)	→	insert[Ite](<(S(x),x),S(x),r)	(3)
inssort(xs)	→	isort(xs,Nil)	(4)

and S is the following TRS.

<(S(x),S(y))	→	<(x,y)	(5)
<(0,S(y))	→	True	(6)
<(x,0)	→	False	(7)
insert[Ite](False,x',Cons(x,xs))	→	Cons(x,insert(x',xs))	(8)
insert[Ite](True,x,r)	→	Cons(x,r)	(9)

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:

isort^#(Cons(z0,z1),z2)

→

c5(isort^#(z1,insert(z0,z2)),insert^#(z0,z2))

(11)

originates from

isort(Cons(z0,z1),z2)

→

isort(z1,insert(z0,z2))

(10)

isort^#(Nil,z0)

→

(13)

originates from

isort(Nil,z0)

→

Nil

(12)

insert^#(S(z0),z1)

→

c7(insert[Ite]^#(<(S(z0),z0),S(z0),z1),<^#(S(z0),z0))

(15)

originates from

insert(S(z0),z1)

→

insert[Ite](<(S(z0),z0),S(z0),z1)

(14)

inssort^#(z0)

→

c8(isort^#(z0,Nil))

(17)

originates from

inssort(z0)

→

isort(z0,Nil)

(16)

<^#(S(z0),S(z1))

→

c(<^#(z0,z1))

(19)

originates from

<(S(z0),S(z1))

→

<(z0,z1)

(18)

<^#(0,S(z0))

→

(21)

originates from

<(0,S(z0))

→

True

(20)

<^#(z0,0)

→

(23)

originates from

<(z0,0)

→

False

(22)

insert[Ite]^#(False,z0,Cons(z1,z2))

→

c3(insert^#(z0,z2))

(25)

originates from

insert[Ite](False,z0,Cons(z1,z2))

→

Cons(z1,insert(z0,z2))

(24)

insert[Ite]^#(True,z0,z1)

→

(27)

originates from

insert[Ite](True,z0,z1)

→

Cons(z0,z1)

(26)

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

<^#(S(z0),S(z1))

<^#(0,S(z0))

<^#(z0,0)

insert[Ite]^#(False,z0,Cons(z1,z2))

insert[Ite]^#(True,z0,z1)

isort^#(Cons(z0,z1),z2)

isort^#(Nil,z0)

insert^#(S(z0),z1)

inssort^#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.

isort(Cons(z0,z1),z2)	→	isort(z1,insert(z0,z2))	(10)
isort(Nil,z0)	→	Nil	(12)
inssort(z0)	→	isort(z0,Nil)	(16)

1.1.1 Rule Shifting

The rules

isort^#(Nil,z0)

→

(13)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁)]	=	1 · x₁ + 0
[insert(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[insert[Ite](x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₂
[<^#(x₁, x₂)]	=	0
[insert[Ite]^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[isort^#(x₁, x₂)]	=	1
[insert^#(x₁, x₂)]	=	0
[inssort^#(x₁)]	=	1
[S(x₁)]	=	0
[0]	=	1
[False]	=	1
[True]	=	1
[Cons(x₁, x₂)]	=	1 + 1 · x₁
[Nil]	=	1

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

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(19)
<^#(0,S(z0))	→	c1	(21)
<^#(z0,0)	→	c2	(23)
insert[Ite]^#(False,z0,Cons(z1,z2))	→	c3(insert^#(z0,z2))	(25)
insert[Ite]^#(True,z0,z1)	→	c4	(27)
isort^#(Cons(z0,z1),z2)	→	c5(isort^#(z1,insert(z0,z2)),insert^#(z0,z2))	(11)
isort^#(Nil,z0)	→	c6	(13)
insert^#(S(z0),z1)	→	c7(insert[Ite]^#(<(S(z0),z0),S(z0),z1),<^#(S(z0),z0))	(15)
inssort^#(z0)	→	c8(isort^#(z0,Nil))	(17)

1.1.1.1 Rule Shifting

The rules

inssort^#(z0)

→

c8(isort^#(z0,Nil))

(17)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁)]	=	1 · x₁ + 0
[insert(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[insert[Ite](x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₂
[<^#(x₁, x₂)]	=	0
[insert[Ite]^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[isort^#(x₁, x₂)]	=	0
[insert^#(x₁, x₂)]	=	0
[inssort^#(x₁)]	=	1
[S(x₁)]	=	0
[0]	=	1
[False]	=	1
[True]	=	1
[Cons(x₁, x₂)]	=	1 + 1 · x₁
[Nil]	=	1

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

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(19)
<^#(0,S(z0))	→	c1	(21)
<^#(z0,0)	→	c2	(23)
insert[Ite]^#(False,z0,Cons(z1,z2))	→	c3(insert^#(z0,z2))	(25)
insert[Ite]^#(True,z0,z1)	→	c4	(27)
isort^#(Cons(z0,z1),z2)	→	c5(isort^#(z1,insert(z0,z2)),insert^#(z0,z2))	(11)
isort^#(Nil,z0)	→	c6	(13)
insert^#(S(z0),z1)	→	c7(insert[Ite]^#(<(S(z0),z0),S(z0),z1),<^#(S(z0),z0))	(15)
inssort^#(z0)	→	c8(isort^#(z0,Nil))	(17)

1.1.1.1.1 Rule Shifting

The rules

isort^#(Cons(z0,z1),z2)

→

c5(isort^#(z1,insert(z0,z2)),insert^#(z0,z2))

(11)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁)]	=	1 · x₁ + 0
[insert(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[insert[Ite](x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₂
[<^#(x₁, x₂)]	=	0
[insert[Ite]^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[isort^#(x₁, x₂)]	=	1 + 1 · x₁
[insert^#(x₁, x₂)]	=	0
[inssort^#(x₁)]	=	1 + 1 · x₁
[S(x₁)]	=	0
[0]	=	1
[False]	=	1
[True]	=	1
[Cons(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[Nil]	=	1

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

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(19)
<^#(0,S(z0))	→	c1	(21)
<^#(z0,0)	→	c2	(23)
insert[Ite]^#(False,z0,Cons(z1,z2))	→	c3(insert^#(z0,z2))	(25)
insert[Ite]^#(True,z0,z1)	→	c4	(27)
isort^#(Cons(z0,z1),z2)	→	c5(isort^#(z1,insert(z0,z2)),insert^#(z0,z2))	(11)
isort^#(Nil,z0)	→	c6	(13)
insert^#(S(z0),z1)	→	c7(insert[Ite]^#(<(S(z0),z0),S(z0),z1),<^#(S(z0),z0))	(15)
inssort^#(z0)	→	c8(isort^#(z0,Nil))	(17)

1.1.1.1.1.1 Rule Shifting

The rules

insert^#(S(z0),z1)

→

c7(insert[Ite]^#(<(S(z0),z0),S(z0),z1),<^#(S(z0),z0))

(15)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8(x₁)]	=	1 · x₁ + 0
[insert(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[insert[Ite](x₁, x₂, x₃)]	=	2 + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[<^#(x₁, x₂)]	=	0
[insert[Ite]^#(x₁, x₂, x₃)]	=	2 · x₃ + 0 + 1 · x₂ · x₃
[isort^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0 + 1 · x₁ · x₁
[insert^#(x₁, x₂)]	=	2 · x₁ + 0 + 2 · x₂ + 1 · x₁ · x₂
[inssort^#(x₁)]	=	1 + 2 · x₁ + 1 · x₁ · x₁
[S(x₁)]	=	2
[0]	=	0
[False]	=	0
[True]	=	0
[Cons(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[Nil]	=	0

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

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(19)
<^#(0,S(z0))	→	c1	(21)
<^#(z0,0)	→	c2	(23)
insert[Ite]^#(False,z0,Cons(z1,z2))	→	c3(insert^#(z0,z2))	(25)
insert[Ite]^#(True,z0,z1)	→	c4	(27)
isort^#(Cons(z0,z1),z2)	→	c5(isort^#(z1,insert(z0,z2)),insert^#(z0,z2))	(11)
isort^#(Nil,z0)	→	c6	(13)
insert^#(S(z0),z1)	→	c7(insert[Ite]^#(<(S(z0),z0),S(z0),z1),<^#(S(z0),z0))	(15)
inssort^#(z0)	→	c8(isort^#(z0,Nil))	(17)
insert[Ite](True,z0,z1)	→	Cons(z0,z1)	(26)
insert(S(z0),z1)	→	insert[Ite](<(S(z0),z0),S(z0),z1)	(14)
insert[Ite](False,z0,Cons(z1,z2))	→	Cons(z1,insert(z0,z2))	(24)

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).