Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Glenstrup/mergelists)

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

merge(Cons(x,xs),Nil)	→	Cons(x,xs)	(1)
merge(Cons(x',xs'),Cons(x,xs))	→	merge[Ite](<=(x',x),Cons(x',xs'),Cons(x,xs))	(2)
merge(Nil,ys)	→	ys	(3)
goal(xs,ys)	→	merge(xs,ys)	(4)

and S is the following TRS.

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

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:

merge^#(Cons(z0,z1),Nil)

→

(11)

originates from

merge(Cons(z0,z1),Nil)

→

Cons(z0,z1)

(10)

merge^#(Cons(z0,z1),Cons(z2,z3))

→

c6(merge[Ite]^#(<=(z0,z2),Cons(z0,z1),Cons(z2,z3)),<=^#(z0,z2))

(13)

originates from

merge(Cons(z0,z1),Cons(z2,z3))

→

merge[Ite](<=(z0,z2),Cons(z0,z1),Cons(z2,z3))

(12)

merge^#(Nil,z0)

→

(15)

originates from

merge(Nil,z0)

→

(14)

goal^#(z0,z1)

→

c8(merge^#(z0,z1))

(17)

originates from

goal(z0,z1)

→

merge(z0,z1)

(16)

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

→

c(<=^#(z0,z1))

(19)

originates from

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

→

<=(z0,z1)

(18)

<=^#(0,z0)

→

(21)

originates from

<=(0,z0)

→

True

(20)

<=^#(S(z0),0)

→

(23)

originates from

<=(S(z0),0)

→

False

(22)

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

→

c3(merge^#(z0,z2))

(25)

originates from

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

→

Cons(z1,merge(z0,z2))

(24)

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

→

c4(merge^#(z1,z2))

(27)

originates from

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

→

Cons(z0,merge(z1,z2))

(26)

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

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

<=^#(0,z0)

<=^#(S(z0),0)

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

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

merge^#(Cons(z0,z1),Nil)

merge^#(Cons(z0,z1),Cons(z2,z3))

merge^#(Nil,z0)

goal^#(z0,z1)

1.1 Usable Rules

We remove the following rules since they are not usable.

merge[Ite](False,z0,Cons(z1,z2))	→	Cons(z1,merge(z0,z2))	(24)
merge[Ite](True,Cons(z0,z1),z2)	→	Cons(z0,merge(z1,z2))	(26)
merge(Cons(z0,z1),Nil)	→	Cons(z0,z1)	(10)
merge(Cons(z0,z1),Cons(z2,z3))	→	merge[Ite](<=(z0,z2),Cons(z0,z1),Cons(z2,z3))	(12)
merge(Nil,z0)	→	z0	(14)
goal(z0,z1)	→	merge(z0,z1)	(16)

1.1.1 Rule Shifting

The rules

goal^#(z0,z1)

→

c8(merge^#(z0,z1))

(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(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[<=(x₁, x₂)]	=	0
[<=^#(x₁, x₂)]	=	0
[merge[Ite]^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[merge^#(x₁, x₂)]	=	0
[goal^#(x₁, x₂)]	=	1
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[True]	=	0
[False]	=	0
[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,z0)	→	c1	(21)
<=^#(S(z0),0)	→	c2	(23)
merge[Ite]^#(False,z0,Cons(z1,z2))	→	c3(merge^#(z0,z2))	(25)
merge[Ite]^#(True,Cons(z0,z1),z2)	→	c4(merge^#(z1,z2))	(27)
merge^#(Cons(z0,z1),Nil)	→	c5	(11)
merge^#(Cons(z0,z1),Cons(z2,z3))	→	c6(merge[Ite]^#(<=(z0,z2),Cons(z0,z1),Cons(z2,z3)),<=^#(z0,z2))	(13)
merge^#(Nil,z0)	→	c7	(15)
goal^#(z0,z1)	→	c8(merge^#(z0,z1))	(17)
<=(0,z0)	→	True	(20)
<=(S(z0),S(z1))	→	<=(z0,z1)	(18)
<=(S(z0),0)	→	False	(22)

1.1.1.1 Rule Shifting

The rules

merge^#(Cons(z0,z1),Nil)

→

(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(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[<=(x₁, x₂)]	=	3
[<=^#(x₁, x₂)]	=	0
[merge[Ite]^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[merge^#(x₁, x₂)]	=	1 · x₂ + 0
[goal^#(x₁, x₂)]	=	1 + 1 · x₂
[S(x₁)]	=	3 + 1 · x₁
[0]	=	3
[True]	=	3
[False]	=	3
[Cons(x₁, x₂)]	=	1 · x₂ + 0
[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,z0)	→	c1	(21)
<=^#(S(z0),0)	→	c2	(23)
merge[Ite]^#(False,z0,Cons(z1,z2))	→	c3(merge^#(z0,z2))	(25)
merge[Ite]^#(True,Cons(z0,z1),z2)	→	c4(merge^#(z1,z2))	(27)
merge^#(Cons(z0,z1),Nil)	→	c5	(11)
merge^#(Cons(z0,z1),Cons(z2,z3))	→	c6(merge[Ite]^#(<=(z0,z2),Cons(z0,z1),Cons(z2,z3)),<=^#(z0,z2))	(13)
merge^#(Nil,z0)	→	c7	(15)
goal^#(z0,z1)	→	c8(merge^#(z0,z1))	(17)

1.1.1.1.1 Rule Shifting

The rules

merge^#(Cons(z0,z1),Cons(z2,z3))	→	c6(merge[Ite]^#(<=(z0,z2),Cons(z0,z1),Cons(z2,z3)),<=^#(z0,z2))	(13)
merge^#(Nil,z0)	→	c7	(15)

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(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[<=(x₁, x₂)]	=	3
[<=^#(x₁, x₂)]	=	0
[merge[Ite]^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 3 · x₃
[merge^#(x₁, x₂)]	=	2 + 1 · x₁ + 3 · x₂
[goal^#(x₁, x₂)]	=	2 + 1 · x₁ + 3 · x₂
[S(x₁)]	=	3 + 1 · x₁
[0]	=	3
[True]	=	3
[False]	=	3
[Cons(x₁, x₂)]	=	3 + 1 · x₂
[Nil]	=	3

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

<=^#(S(z0),S(z1))	→	c(<=^#(z0,z1))	(19)
<=^#(0,z0)	→	c1	(21)
<=^#(S(z0),0)	→	c2	(23)
merge[Ite]^#(False,z0,Cons(z1,z2))	→	c3(merge^#(z0,z2))	(25)
merge[Ite]^#(True,Cons(z0,z1),z2)	→	c4(merge^#(z1,z2))	(27)
merge^#(Cons(z0,z1),Nil)	→	c5	(11)
merge^#(Cons(z0,z1),Cons(z2,z3))	→	c6(merge[Ite]^#(<=(z0,z2),Cons(z0,z1),Cons(z2,z3)),<=^#(z0,z2))	(13)
merge^#(Nil,z0)	→	c7	(15)
goal^#(z0,z1)	→	c8(merge^#(z0,z1))	(17)

1.1.1.1.1.1 R is empty

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