Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Glenstrup/ordered)

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

ordered(Cons(x',Cons(x,xs)))	→	ordered[Ite](<(x',x),Cons(x',Cons(x,xs)))	(1)
ordered(Cons(x,Nil))	→	True	(2)
ordered(Nil)	→	True	(3)
notEmpty(Cons(x,xs))	→	True	(4)
notEmpty(Nil)	→	False	(5)
goal(xs)	→	ordered(xs)	(6)

and S is the following TRS.

<(S(x),S(y))	→	<(x,y)	(7)
<(0,S(y))	→	True	(8)
<(x,0)	→	False	(9)
ordered[Ite](True,Cons(x',Cons(x,xs)))	→	ordered(xs)	(10)
ordered[Ite](False,xs)	→	False	(11)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

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

→

c5(ordered[Ite]^#(<(z0,z1),Cons(z0,Cons(z1,z2))),<^#(z0,z1))

(13)

originates from

ordered(Cons(z0,Cons(z1,z2)))

→

ordered[Ite](<(z0,z1),Cons(z0,Cons(z1,z2)))

(12)

ordered^#(Cons(z0,Nil))

→

(15)

originates from

ordered(Cons(z0,Nil))

→

True

(14)

ordered^#(Nil)

→

(16)

originates from

ordered(Nil)

→

True

(3)

notEmpty^#(Cons(z0,z1))

→

(18)

originates from

notEmpty(Cons(z0,z1))

→

True

(17)

notEmpty^#(Nil)

→

(19)

originates from

notEmpty(Nil)

→

False

(5)

goal^#(z0)

→

c10(ordered^#(z0))

(21)

originates from

goal(z0)

→

ordered(z0)

(20)

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

→

c(<^#(z0,z1))

(23)

originates from

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

→

<(z0,z1)

(22)

<^#(0,S(z0))

→

(25)

originates from

<(0,S(z0))

→

True

(24)

<^#(z0,0)

→

(27)

originates from

<(z0,0)

→

False

(26)

ordered[Ite]^#(True,Cons(z0,Cons(z1,z2)))

→

c3(ordered^#(z2))

(29)

originates from

ordered[Ite](True,Cons(z0,Cons(z1,z2)))

→

ordered(z2)

(28)

ordered[Ite]^#(False,z0)

→

(31)

originates from

ordered[Ite](False,z0)

→

False

(30)

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

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

<^#(0,S(z0))

<^#(z0,0)

ordered[Ite]^#(True,Cons(z0,Cons(z1,z2)))

ordered[Ite]^#(False,z0)

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

ordered^#(Cons(z0,Nil))

ordered^#(Nil)

notEmpty^#(Cons(z0,z1))

notEmpty^#(Nil)

goal^#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.

ordered[Ite](True,Cons(z0,Cons(z1,z2)))	→	ordered(z2)	(28)
ordered[Ite](False,z0)	→	False	(30)
ordered(Cons(z0,Cons(z1,z2)))	→	ordered[Ite](<(z0,z1),Cons(z0,Cons(z1,z2)))	(12)
ordered(Cons(z0,Nil))	→	True	(14)
ordered(Nil)	→	True	(3)
notEmpty(Cons(z0,z1))	→	True	(17)
notEmpty(Nil)	→	False	(5)
goal(z0)	→	ordered(z0)	(20)

1.1.1 Rule Shifting

The rules

ordered^#(Cons(z0,Nil))	→	c6	(15)
ordered^#(Nil)	→	c7	(16)
notEmpty^#(Cons(z0,z1))	→	c8	(18)
notEmpty^#(Nil)	→	c9	(19)
goal^#(z0)	→	c10(ordered^#(z0))	(21)

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]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁)]	=	1 · x₁ + 0
[<(x₁, x₂)]	=	0
[<^#(x₁, x₂)]	=	0
[ordered[Ite]^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[ordered^#(x₁)]	=	1 · x₁ + 0
[notEmpty^#(x₁)]	=	1 + 1 · x₁
[goal^#(x₁)]	=	1 + 1 · x₁
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[True]	=	0
[False]	=	0
[Cons(x₁, x₂)]	=	1 · x₁ + 0 + 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))	(23)
<^#(0,S(z0))	→	c1	(25)
<^#(z0,0)	→	c2	(27)
ordered[Ite]^#(True,Cons(z0,Cons(z1,z2)))	→	c3(ordered^#(z2))	(29)
ordered[Ite]^#(False,z0)	→	c4	(31)
ordered^#(Cons(z0,Cons(z1,z2)))	→	c5(ordered[Ite]^#(<(z0,z1),Cons(z0,Cons(z1,z2))),<^#(z0,z1))	(13)
ordered^#(Cons(z0,Nil))	→	c6	(15)
ordered^#(Nil)	→	c7	(16)
notEmpty^#(Cons(z0,z1))	→	c8	(18)
notEmpty^#(Nil)	→	c9	(19)
goal^#(z0)	→	c10(ordered^#(z0))	(21)
<(z0,0)	→	False	(26)
<(0,S(z0))	→	True	(24)
<(S(z0),S(z1))	→	<(z0,z1)	(22)

1.1.1.1 Rule Shifting

The rules

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

→

c5(ordered[Ite]^#(<(z0,z1),Cons(z0,Cons(z1,z2))),<^#(z0,z1))

(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]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁)]	=	1 · x₁ + 0
[<(x₁, x₂)]	=	0
[<^#(x₁, x₂)]	=	0
[ordered[Ite]^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[ordered^#(x₁)]	=	1 + 1 · x₁
[notEmpty^#(x₁)]	=	0
[goal^#(x₁)]	=	1 + 1 · x₁
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[True]	=	0
[False]	=	0
[Cons(x₁, x₂)]	=	1 + 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))	(23)
<^#(0,S(z0))	→	c1	(25)
<^#(z0,0)	→	c2	(27)
ordered[Ite]^#(True,Cons(z0,Cons(z1,z2)))	→	c3(ordered^#(z2))	(29)
ordered[Ite]^#(False,z0)	→	c4	(31)
ordered^#(Cons(z0,Cons(z1,z2)))	→	c5(ordered[Ite]^#(<(z0,z1),Cons(z0,Cons(z1,z2))),<^#(z0,z1))	(13)
ordered^#(Cons(z0,Nil))	→	c6	(15)
ordered^#(Nil)	→	c7	(16)
notEmpty^#(Cons(z0,z1))	→	c8	(18)
notEmpty^#(Nil)	→	c9	(19)
goal^#(z0)	→	c10(ordered^#(z0))	(21)
<(z0,0)	→	False	(26)
<(0,S(z0))	→	True	(24)
<(S(z0),S(z1))	→	<(z0,z1)	(22)

1.1.1.1.1 R is empty

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