Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Glenstrup/overlap)

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

overlap(Cons(x,xs),ys)	→	overlap[Ite][True][Ite](member(x,ys),Cons(x,xs),ys)	(1)
overlap(Nil,ys)	→	False	(2)
member(x',Cons(x,xs))	→	member[Ite][True][Ite](!EQ(x,x'),x',Cons(x,xs))	(3)
member(x,Nil)	→	False	(4)
notEmpty(Cons(x,xs))	→	True	(5)
notEmpty(Nil)	→	False	(6)
goal(xs,ys)	→	overlap(xs,ys)	(7)

and S is the following TRS.

!EQ(S(x),S(y))	→	!EQ(x,y)	(8)
!EQ(0,S(y))	→	False	(9)
!EQ(S(x),0)	→	False	(10)
!EQ(0,0)	→	True	(11)
overlap[Ite][True][Ite](False,Cons(x,xs),ys)	→	overlap(xs,ys)	(12)
member[Ite][True][Ite](False,x',Cons(x,xs))	→	member(x',xs)	(13)
overlap[Ite][True][Ite](True,xs,ys)	→	True	(14)
member[Ite][True][Ite](True,x,xs)	→	True	(15)

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:

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

→

c8(overlap[Ite][True][Ite]^#(member(z0,z2),Cons(z0,z1),z2),member^#(z0,z2))

(17)

originates from

overlap(Cons(z0,z1),z2)

→

overlap[Ite][True][Ite](member(z0,z2),Cons(z0,z1),z2)

(16)

overlap^#(Nil,z0)

→

(19)

originates from

overlap(Nil,z0)

→

False

(18)

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

→

c10(member[Ite][True][Ite]^#(!EQ(z1,z0),z0,Cons(z1,z2)),!EQ^#(z1,z0))

(21)

originates from

member(z0,Cons(z1,z2))

→

member[Ite][True][Ite](!EQ(z1,z0),z0,Cons(z1,z2))

(20)

member^#(z0,Nil)

→

c11

(23)

originates from

member(z0,Nil)

→

False

(22)

notEmpty^#(Cons(z0,z1))

→

c12

(25)

originates from

notEmpty(Cons(z0,z1))

→

True

(24)

notEmpty^#(Nil)

→

c13

(26)

originates from

notEmpty(Nil)

→

False

(6)

goal^#(z0,z1)

→

c14(overlap^#(z0,z1))

(28)

originates from

goal(z0,z1)

→

overlap(z0,z1)

(27)

!EQ^#(S(z0),S(z1))

→

c(!EQ^#(z0,z1))

(30)

originates from

!EQ(S(z0),S(z1))

→

!EQ(z0,z1)

(29)

!EQ^#(0,S(z0))

→

(32)

originates from

!EQ(0,S(z0))

→

False

(31)

!EQ^#(S(z0),0)

→

(34)

originates from

!EQ(S(z0),0)

→

False

(33)

!EQ^#(0,0)

→

(35)

originates from

!EQ(0,0)

→

True

(11)

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

→

c4(overlap^#(z1,z2))

(37)

originates from

overlap[Ite][True][Ite](False,Cons(z0,z1),z2)

→

overlap(z1,z2)

(36)

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

→

(39)

originates from

overlap[Ite][True][Ite](True,z0,z1)

→

True

(38)

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

→

c6(member^#(z0,z2))

(41)

originates from

member[Ite][True][Ite](False,z0,Cons(z1,z2))

→

member(z0,z2)

(40)

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

→

(43)

originates from

member[Ite][True][Ite](True,z0,z1)

→

True

(42)

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

!EQ^#(S(z0),S(z1))

!EQ^#(0,S(z0))

!EQ^#(S(z0),0)

!EQ^#(0,0)

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

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

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

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

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

overlap^#(Nil,z0)

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

member^#(z0,Nil)

notEmpty^#(Cons(z0,z1))

notEmpty^#(Nil)

goal^#(z0,z1)

1.1 Usable Rules

We remove the following rules since they are not usable.

overlap[Ite][True][Ite](False,Cons(z0,z1),z2)	→	overlap(z1,z2)	(36)
overlap[Ite][True][Ite](True,z0,z1)	→	True	(38)
overlap(Cons(z0,z1),z2)	→	overlap[Ite][True][Ite](member(z0,z2),Cons(z0,z1),z2)	(16)
overlap(Nil,z0)	→	False	(18)
notEmpty(Cons(z0,z1))	→	True	(24)
notEmpty(Nil)	→	False	(6)
goal(z0,z1)	→	overlap(z0,z1)	(27)

1.1.1 Rule Shifting

The rules

overlap^#(Nil,z0)	→	c9	(19)
notEmpty^#(Cons(z0,z1))	→	c12	(25)
notEmpty^#(Nil)	→	c13	(26)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9]	=	0
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14(x₁)]	=	1 · x₁ + 0
[member(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[member[Ite][True][Ite](x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[!EQ(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[overlap[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[member[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	0
[overlap^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[member^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	1 + 1 · x₁
[goal^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[False]	=	1
[True]	=	1
[Cons(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[Nil]	=	1

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

!EQ^#(S(z0),S(z1))	→	c(!EQ^#(z0,z1))	(30)
!EQ^#(0,S(z0))	→	c1	(32)
!EQ^#(S(z0),0)	→	c2	(34)
!EQ^#(0,0)	→	c3	(35)
overlap[Ite][True][Ite]^#(False,Cons(z0,z1),z2)	→	c4(overlap^#(z1,z2))	(37)
overlap[Ite][True][Ite]^#(True,z0,z1)	→	c5	(39)
member[Ite][True][Ite]^#(False,z0,Cons(z1,z2))	→	c6(member^#(z0,z2))	(41)
member[Ite][True][Ite]^#(True,z0,z1)	→	c7	(43)
overlap^#(Cons(z0,z1),z2)	→	c8(overlap[Ite][True][Ite]^#(member(z0,z2),Cons(z0,z1),z2),member^#(z0,z2))	(17)
overlap^#(Nil,z0)	→	c9	(19)
member^#(z0,Cons(z1,z2))	→	c10(member[Ite][True][Ite]^#(!EQ(z1,z0),z0,Cons(z1,z2)),!EQ^#(z1,z0))	(21)
member^#(z0,Nil)	→	c11	(23)
notEmpty^#(Cons(z0,z1))	→	c12	(25)
notEmpty^#(Nil)	→	c13	(26)
goal^#(z0,z1)	→	c14(overlap^#(z0,z1))	(28)

1.1.1.1 Rule Shifting

The rules

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

→

c8(overlap[Ite][True][Ite]^#(member(z0,z2),Cons(z0,z1),z2),member^#(z0,z2))

(17)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9]	=	0
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14(x₁)]	=	1 · x₁ + 0
[member(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[member[Ite][True][Ite](x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[!EQ(x₁, x₂)]	=	1
[!EQ^#(x₁, x₂)]	=	0
[overlap[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[member[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	0
[overlap^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[member^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[goal^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[S(x₁)]	=	1 + 1 · x₁
[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:

!EQ^#(S(z0),S(z1))	→	c(!EQ^#(z0,z1))	(30)
!EQ^#(0,S(z0))	→	c1	(32)
!EQ^#(S(z0),0)	→	c2	(34)
!EQ^#(0,0)	→	c3	(35)
overlap[Ite][True][Ite]^#(False,Cons(z0,z1),z2)	→	c4(overlap^#(z1,z2))	(37)
overlap[Ite][True][Ite]^#(True,z0,z1)	→	c5	(39)
member[Ite][True][Ite]^#(False,z0,Cons(z1,z2))	→	c6(member^#(z0,z2))	(41)
member[Ite][True][Ite]^#(True,z0,z1)	→	c7	(43)
overlap^#(Cons(z0,z1),z2)	→	c8(overlap[Ite][True][Ite]^#(member(z0,z2),Cons(z0,z1),z2),member^#(z0,z2))	(17)
overlap^#(Nil,z0)	→	c9	(19)
member^#(z0,Cons(z1,z2))	→	c10(member[Ite][True][Ite]^#(!EQ(z1,z0),z0,Cons(z1,z2)),!EQ^#(z1,z0))	(21)
member^#(z0,Nil)	→	c11	(23)
notEmpty^#(Cons(z0,z1))	→	c12	(25)
notEmpty^#(Nil)	→	c13	(26)
goal^#(z0,z1)	→	c14(overlap^#(z0,z1))	(28)

1.1.1.1.1 Rule Shifting

The rules

goal^#(z0,z1)

→

c14(overlap^#(z0,z1))

(28)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9]	=	0
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14(x₁)]	=	1 · x₁ + 0
[member(x₁, x₂)]	=	0
[member[Ite][True][Ite](x₁, x₂, x₃)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[overlap[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[member[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[overlap^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[member^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[goal^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[False]	=	0
[True]	=	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:

!EQ^#(S(z0),S(z1))	→	c(!EQ^#(z0,z1))	(30)
!EQ^#(0,S(z0))	→	c1	(32)
!EQ^#(S(z0),0)	→	c2	(34)
!EQ^#(0,0)	→	c3	(35)
overlap[Ite][True][Ite]^#(False,Cons(z0,z1),z2)	→	c4(overlap^#(z1,z2))	(37)
overlap[Ite][True][Ite]^#(True,z0,z1)	→	c5	(39)
member[Ite][True][Ite]^#(False,z0,Cons(z1,z2))	→	c6(member^#(z0,z2))	(41)
member[Ite][True][Ite]^#(True,z0,z1)	→	c7	(43)
overlap^#(Cons(z0,z1),z2)	→	c8(overlap[Ite][True][Ite]^#(member(z0,z2),Cons(z0,z1),z2),member^#(z0,z2))	(17)
overlap^#(Nil,z0)	→	c9	(19)
member^#(z0,Cons(z1,z2))	→	c10(member[Ite][True][Ite]^#(!EQ(z1,z0),z0,Cons(z1,z2)),!EQ^#(z1,z0))	(21)
member^#(z0,Nil)	→	c11	(23)
notEmpty^#(Cons(z0,z1))	→	c12	(25)
notEmpty^#(Nil)	→	c13	(26)
goal^#(z0,z1)	→	c14(overlap^#(z0,z1))	(28)
member(z0,Cons(z1,z2))	→	member[Ite][True][Ite](!EQ(z1,z0),z0,Cons(z1,z2))	(20)
!EQ(0,S(z0))	→	False	(31)
!EQ(S(z0),0)	→	False	(33)
member[Ite][True][Ite](False,z0,Cons(z1,z2))	→	member(z0,z2)	(40)
member[Ite][True][Ite](True,z0,z1)	→	True	(42)
!EQ(S(z0),S(z1))	→	!EQ(z0,z1)	(29)
!EQ(0,0)	→	True	(11)
member(z0,Nil)	→	False	(22)

1.1.1.1.1.1 Rule Shifting

The rules

member^#(z0,Nil)

→

c11

(23)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9]	=	0
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14(x₁)]	=	1 · x₁ + 0
[member(x₁, x₂)]	=	1 + 1 · x₁
[member[Ite][True][Ite](x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂
[!EQ(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[overlap[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[member[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	1
[overlap^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[member^#(x₁, x₂)]	=	1
[notEmpty^#(x₁)]	=	0
[goal^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[S(x₁)]	=	1 + 1 · x₁
[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:

!EQ^#(S(z0),S(z1))	→	c(!EQ^#(z0,z1))	(30)
!EQ^#(0,S(z0))	→	c1	(32)
!EQ^#(S(z0),0)	→	c2	(34)
!EQ^#(0,0)	→	c3	(35)
overlap[Ite][True][Ite]^#(False,Cons(z0,z1),z2)	→	c4(overlap^#(z1,z2))	(37)
overlap[Ite][True][Ite]^#(True,z0,z1)	→	c5	(39)
member[Ite][True][Ite]^#(False,z0,Cons(z1,z2))	→	c6(member^#(z0,z2))	(41)
member[Ite][True][Ite]^#(True,z0,z1)	→	c7	(43)
overlap^#(Cons(z0,z1),z2)	→	c8(overlap[Ite][True][Ite]^#(member(z0,z2),Cons(z0,z1),z2),member^#(z0,z2))	(17)
overlap^#(Nil,z0)	→	c9	(19)
member^#(z0,Cons(z1,z2))	→	c10(member[Ite][True][Ite]^#(!EQ(z1,z0),z0,Cons(z1,z2)),!EQ^#(z1,z0))	(21)
member^#(z0,Nil)	→	c11	(23)
notEmpty^#(Cons(z0,z1))	→	c12	(25)
notEmpty^#(Nil)	→	c13	(26)
goal^#(z0,z1)	→	c14(overlap^#(z0,z1))	(28)

1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c10(member[Ite][True][Ite]^#(!EQ(z1,z0),z0,Cons(z1,z2)),!EQ^#(z1,z0))

(21)

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

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9]	=	0
[c10(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14(x₁)]	=	1 · x₁ + 0
[member(x₁, x₂)]	=	0
[member[Ite][True][Ite](x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[!EQ(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[overlap[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	2 · x₂ · x₃ + 0 + 1 · x₂ · x₂
[member[Ite][True][Ite]^#(x₁, x₂, x₃)]	=	2 · x₃ + 0
[overlap^#(x₁, x₂)]	=	1 + 2 · x₂ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[member^#(x₁, x₂)]	=	1 + 2 · x₂
[notEmpty^#(x₁)]	=	0
[goal^#(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂ + 2 · x₂ · x₂ + 2 · x₁ · x₂ + 2 · x₁ · x₁
[S(x₁)]	=	0
[0]	=	0
[False]	=	0
[True]	=	0
[Cons(x₁, x₂)]	=	1 + 1 · x₂
[Nil]	=	0

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

!EQ^#(S(z0),S(z1))	→	c(!EQ^#(z0,z1))	(30)
!EQ^#(0,S(z0))	→	c1	(32)
!EQ^#(S(z0),0)	→	c2	(34)
!EQ^#(0,0)	→	c3	(35)
overlap[Ite][True][Ite]^#(False,Cons(z0,z1),z2)	→	c4(overlap^#(z1,z2))	(37)
overlap[Ite][True][Ite]^#(True,z0,z1)	→	c5	(39)
member[Ite][True][Ite]^#(False,z0,Cons(z1,z2))	→	c6(member^#(z0,z2))	(41)
member[Ite][True][Ite]^#(True,z0,z1)	→	c7	(43)
overlap^#(Cons(z0,z1),z2)	→	c8(overlap[Ite][True][Ite]^#(member(z0,z2),Cons(z0,z1),z2),member^#(z0,z2))	(17)
overlap^#(Nil,z0)	→	c9	(19)
member^#(z0,Cons(z1,z2))	→	c10(member[Ite][True][Ite]^#(!EQ(z1,z0),z0,Cons(z1,z2)),!EQ^#(z1,z0))	(21)
member^#(z0,Nil)	→	c11	(23)
notEmpty^#(Cons(z0,z1))	→	c12	(25)
notEmpty^#(Nil)	→	c13	(26)
goal^#(z0,z1)	→	c14(overlap^#(z0,z1))	(28)

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).