Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Others/strmatch)

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

prefix(Cons(x',xs'),Cons(x,xs))	→	and(!EQ(x',x),prefix(xs',xs))	(1)
domatch(Cons(x,xs),Nil,n)	→	Nil	(2)
domatch(Nil,Nil,n)	→	Cons(n,Nil)	(3)
prefix(Cons(x,xs),Nil)	→	False	(4)
prefix(Nil,cs)	→	True	(5)
domatch(patcs,Cons(x,xs),n)	→	domatch[Ite](prefix(patcs,Cons(x,xs)),patcs,Cons(x,xs),n)	(6)
eqNatList(Cons(x,xs),Cons(y,ys))	→	eqNatList[Ite](!EQ(x,y),y,ys,x,xs)	(7)
eqNatList(Cons(x,xs),Nil)	→	False	(8)
eqNatList(Nil,Cons(y,ys))	→	False	(9)
eqNatList(Nil,Nil)	→	True	(10)
notEmpty(Cons(x,xs))	→	True	(11)
notEmpty(Nil)	→	False	(12)
strmatch(patstr,str)	→	domatch(patstr,str,Nil)	(13)

and S is the following TRS.

and(False,False)	→	False	(14)
and(True,False)	→	False	(15)
and(False,True)	→	False	(16)
and(True,True)	→	True	(17)
!EQ(S(x),S(y))	→	!EQ(x,y)	(18)
!EQ(0,S(y))	→	False	(19)
!EQ(S(x),0)	→	False	(20)
!EQ(0,0)	→	True	(21)
domatch[Ite](False,patcs,Cons(x,xs),n)	→	domatch(patcs,xs,Cons(n,Cons(Nil,Nil)))	(22)
domatch[Ite](True,patcs,Cons(x,xs),n)	→	Cons(n,domatch(patcs,xs,Cons(n,Cons(Nil,Nil))))	(23)
eqNatList[Ite](False,y,ys,x,xs)	→	False	(24)
eqNatList[Ite](True,y,ys,x,xs)	→	eqNatList(xs,ys)	(25)

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:

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

→

c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))

(27)

originates from

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

→

and(!EQ(z0,z2),prefix(z1,z3))

(26)

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

→

c13

(29)

originates from

prefix(Cons(z0,z1),Nil)

→

False

(28)

prefix^#(Nil,z0)

→

c14

(31)

originates from

prefix(Nil,z0)

→

True

(30)

domatch^#(Cons(z0,z1),Nil,z2)

→

c15

(33)

originates from

domatch(Cons(z0,z1),Nil,z2)

→

Nil

(32)

domatch^#(Nil,Nil,z0)

→

c16

(35)

originates from

domatch(Nil,Nil,z0)

→

Cons(z0,Nil)

(34)

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

→

c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))

(37)

originates from

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

→

domatch[Ite](prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3)

(36)

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

→

c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))

(39)

originates from

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

→

eqNatList[Ite](!EQ(z0,z2),z2,z3,z0,z1)

(38)

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

→

c19

(41)

originates from

eqNatList(Cons(z0,z1),Nil)

→

False

(40)

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

→

c20

(43)

originates from

eqNatList(Nil,Cons(z0,z1))

→

False

(42)

eqNatList^#(Nil,Nil)

→

c21

(44)

originates from

eqNatList(Nil,Nil)

→

True

(10)

notEmpty^#(Cons(z0,z1))

→

c22

(46)

originates from

notEmpty(Cons(z0,z1))

→

True

(45)

notEmpty^#(Nil)

→

c23

(47)

originates from

notEmpty(Nil)

→

False

(12)

strmatch^#(z0,z1)

→

c24(domatch^#(z0,z1,Nil))

(49)

originates from

strmatch(z0,z1)

→

domatch(z0,z1,Nil)

(48)

and^#(False,False)

→

(50)

originates from

and(False,False)

→

False

(14)

and^#(True,False)

→

(51)

originates from

and(True,False)

→

False

(15)

and^#(False,True)

→

(52)

originates from

and(False,True)

→

False

(16)

and^#(True,True)

→

(53)

originates from

and(True,True)

→

True

(17)

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

→

c4(!EQ^#(z0,z1))

(55)

originates from

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

→

!EQ(z0,z1)

(54)

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

→

(57)

originates from

!EQ(0,S(z0))

→

False

(56)

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

→

(59)

originates from

!EQ(S(z0),0)

→

False

(58)

!EQ^#(0,0)

→

(60)

originates from

!EQ(0,0)

→

True

(21)

domatch[Ite]^#(False,z0,Cons(z1,z2),z3)

→

c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))

(62)

originates from

domatch[Ite](False,z0,Cons(z1,z2),z3)

→

domatch(z0,z2,Cons(z3,Cons(Nil,Nil)))

(61)

domatch[Ite]^#(True,z0,Cons(z1,z2),z3)

→

c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))

(64)

originates from

domatch[Ite](True,z0,Cons(z1,z2),z3)

→

Cons(z3,domatch(z0,z2,Cons(z3,Cons(Nil,Nil))))

(63)

eqNatList[Ite]^#(False,z0,z1,z2,z3)

→

c10

(66)

originates from

eqNatList[Ite](False,z0,z1,z2,z3)

→

False

(65)

eqNatList[Ite]^#(True,z0,z1,z2,z3)

→

c11(eqNatList^#(z3,z1))

(68)

originates from

eqNatList[Ite](True,z0,z1,z2,z3)

→

eqNatList(z3,z1)

(67)

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

and^#(False,False)

and^#(True,False)

and^#(False,True)

and^#(True,True)

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

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

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

!EQ^#(0,0)

domatch[Ite]^#(False,z0,Cons(z1,z2),z3)

domatch[Ite]^#(True,z0,Cons(z1,z2),z3)

eqNatList[Ite]^#(False,z0,z1,z2,z3)

eqNatList[Ite]^#(True,z0,z1,z2,z3)

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

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

prefix^#(Nil,z0)

domatch^#(Cons(z0,z1),Nil,z2)

domatch^#(Nil,Nil,z0)

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

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

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

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

eqNatList^#(Nil,Nil)

notEmpty^#(Cons(z0,z1))

notEmpty^#(Nil)

strmatch^#(z0,z1)

1.1 Usable Rules

We remove the following rules since they are not usable.

domatch[Ite](False,z0,Cons(z1,z2),z3)	→	domatch(z0,z2,Cons(z3,Cons(Nil,Nil)))	(61)
domatch[Ite](True,z0,Cons(z1,z2),z3)	→	Cons(z3,domatch(z0,z2,Cons(z3,Cons(Nil,Nil))))	(63)
eqNatList[Ite](False,z0,z1,z2,z3)	→	False	(65)
eqNatList[Ite](True,z0,z1,z2,z3)	→	eqNatList(z3,z1)	(67)
domatch(Cons(z0,z1),Nil,z2)	→	Nil	(32)
domatch(Nil,Nil,z0)	→	Cons(z0,Nil)	(34)
domatch(z0,Cons(z1,z2),z3)	→	domatch[Ite](prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3)	(36)
eqNatList(Cons(z0,z1),Cons(z2,z3))	→	eqNatList[Ite](!EQ(z0,z2),z2,z3,z0,z1)	(38)
eqNatList(Cons(z0,z1),Nil)	→	False	(40)
eqNatList(Nil,Cons(z0,z1))	→	False	(42)
eqNatList(Nil,Nil)	→	True	(10)
notEmpty(Cons(z0,z1))	→	True	(45)
notEmpty(Nil)	→	False	(12)
strmatch(z0,z1)	→	domatch(z0,z1,Nil)	(48)

1.1.1 Rule Shifting

The rules

domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c18(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24(x₁)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	1
[prefix(x₁, x₂)]	=	0
[and(x₁, x₂)]	=	1
[and^#(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[domatch[Ite]^#(x₁,...,x₄)]	=	1 · x₂ + 0
[eqNatList[Ite]^#(x₁,...,x₅)]	=	0
[prefix^#(x₁, x₂)]	=	0
[domatch^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[eqNatList^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	1 + 1 · x₁
[strmatch^#(x₁, x₂)]	=	1 + 1 · x₁
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[False]	=	0
[True]	=	0
[Cons(x₁, x₂)]	=	1
[Nil]	=	0

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

and^#(False,False)	→	c	(50)
and^#(True,False)	→	c1	(51)
and^#(False,True)	→	c2	(52)
and^#(True,True)	→	c3	(53)
!EQ^#(S(z0),S(z1))	→	c4(!EQ^#(z0,z1))	(55)
!EQ^#(0,S(z0))	→	c5	(57)
!EQ^#(S(z0),0)	→	c6	(59)
!EQ^#(0,0)	→	c7	(60)
domatch[Ite]^#(False,z0,Cons(z1,z2),z3)	→	c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(62)
domatch[Ite]^#(True,z0,Cons(z1,z2),z3)	→	c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(64)
eqNatList[Ite]^#(False,z0,z1,z2,z3)	→	c10	(66)
eqNatList[Ite]^#(True,z0,z1,z2,z3)	→	c11(eqNatList^#(z3,z1))	(68)
prefix^#(Cons(z0,z1),Cons(z2,z3))	→	c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))	(27)
prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)
domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
domatch^#(Nil,Nil,z0)	→	c16	(35)
domatch^#(z0,Cons(z1,z2),z3)	→	c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))	(37)
eqNatList^#(Cons(z0,z1),Cons(z2,z3))	→	c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))	(39)
eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

1.1.1.1 Rule Shifting

The rules

domatch^#(Nil,Nil,z0)

→

c16

(35)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c18(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24(x₁)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	1
[prefix(x₁, x₂)]	=	1
[and(x₁, x₂)]	=	1
[and^#(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[domatch[Ite]^#(x₁,...,x₄)]	=	1 · x₂ + 0 + 1 · x₃
[eqNatList[Ite]^#(x₁,...,x₅)]	=	0
[prefix^#(x₁, x₂)]	=	0
[domatch^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[eqNatList^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[strmatch^#(x₁, x₂)]	=	1 · x₁ + 0
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[False]	=	1
[True]	=	0
[Cons(x₁, x₂)]	=	0
[Nil]	=	1

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

and^#(False,False)	→	c	(50)
and^#(True,False)	→	c1	(51)
and^#(False,True)	→	c2	(52)
and^#(True,True)	→	c3	(53)
!EQ^#(S(z0),S(z1))	→	c4(!EQ^#(z0,z1))	(55)
!EQ^#(0,S(z0))	→	c5	(57)
!EQ^#(S(z0),0)	→	c6	(59)
!EQ^#(0,0)	→	c7	(60)
domatch[Ite]^#(False,z0,Cons(z1,z2),z3)	→	c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(62)
domatch[Ite]^#(True,z0,Cons(z1,z2),z3)	→	c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(64)
eqNatList[Ite]^#(False,z0,z1,z2,z3)	→	c10	(66)
eqNatList[Ite]^#(True,z0,z1,z2,z3)	→	c11(eqNatList^#(z3,z1))	(68)
prefix^#(Cons(z0,z1),Cons(z2,z3))	→	c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))	(27)
prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)
domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
domatch^#(Nil,Nil,z0)	→	c16	(35)
domatch^#(z0,Cons(z1,z2),z3)	→	c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))	(37)
eqNatList^#(Cons(z0,z1),Cons(z2,z3))	→	c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))	(39)
eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

1.1.1.1.1 Rule Shifting

The rules

eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c18(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24(x₁)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	1
[prefix(x₁, x₂)]	=	1
[and(x₁, x₂)]	=	1
[and^#(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[domatch[Ite]^#(x₁,...,x₄)]	=	1 · x₂ + 0 + 1 · x₃
[eqNatList[Ite]^#(x₁,...,x₅)]	=	1
[prefix^#(x₁, x₂)]	=	0
[domatch^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[eqNatList^#(x₁, x₂)]	=	1
[notEmpty^#(x₁)]	=	0
[strmatch^#(x₁, x₂)]	=	1 · x₁ + 0
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[False]	=	1
[True]	=	0
[Cons(x₁, x₂)]	=	0
[Nil]	=	0

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

and^#(False,False)	→	c	(50)
and^#(True,False)	→	c1	(51)
and^#(False,True)	→	c2	(52)
and^#(True,True)	→	c3	(53)
!EQ^#(S(z0),S(z1))	→	c4(!EQ^#(z0,z1))	(55)
!EQ^#(0,S(z0))	→	c5	(57)
!EQ^#(S(z0),0)	→	c6	(59)
!EQ^#(0,0)	→	c7	(60)
domatch[Ite]^#(False,z0,Cons(z1,z2),z3)	→	c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(62)
domatch[Ite]^#(True,z0,Cons(z1,z2),z3)	→	c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(64)
eqNatList[Ite]^#(False,z0,z1,z2,z3)	→	c10	(66)
eqNatList[Ite]^#(True,z0,z1,z2,z3)	→	c11(eqNatList^#(z3,z1))	(68)
prefix^#(Cons(z0,z1),Cons(z2,z3))	→	c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))	(27)
prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)
domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
domatch^#(Nil,Nil,z0)	→	c16	(35)
domatch^#(z0,Cons(z1,z2),z3)	→	c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))	(37)
eqNatList^#(Cons(z0,z1),Cons(z2,z3))	→	c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))	(39)
eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

1.1.1.1.1.1 Rule Shifting

The rules

prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c18(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24(x₁)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	1
[prefix(x₁, x₂)]	=	1
[and(x₁, x₂)]	=	1
[and^#(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[domatch[Ite]^#(x₁,...,x₄)]	=	1 · x₂ + 0 + 1 · x₃
[eqNatList[Ite]^#(x₁,...,x₅)]	=	0
[prefix^#(x₁, x₂)]	=	1
[domatch^#(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂
[eqNatList^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[strmatch^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[False]	=	1
[True]	=	0
[Cons(x₁, x₂)]	=	1 + 1 · x₂
[Nil]	=	0

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

and^#(False,False)	→	c	(50)
and^#(True,False)	→	c1	(51)
and^#(False,True)	→	c2	(52)
and^#(True,True)	→	c3	(53)
!EQ^#(S(z0),S(z1))	→	c4(!EQ^#(z0,z1))	(55)
!EQ^#(0,S(z0))	→	c5	(57)
!EQ^#(S(z0),0)	→	c6	(59)
!EQ^#(0,0)	→	c7	(60)
domatch[Ite]^#(False,z0,Cons(z1,z2),z3)	→	c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(62)
domatch[Ite]^#(True,z0,Cons(z1,z2),z3)	→	c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(64)
eqNatList[Ite]^#(False,z0,z1,z2,z3)	→	c10	(66)
eqNatList[Ite]^#(True,z0,z1,z2,z3)	→	c11(eqNatList^#(z3,z1))	(68)
prefix^#(Cons(z0,z1),Cons(z2,z3))	→	c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))	(27)
prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)
domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
domatch^#(Nil,Nil,z0)	→	c16	(35)
domatch^#(z0,Cons(z1,z2),z3)	→	c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))	(37)
eqNatList^#(Cons(z0,z1),Cons(z2,z3))	→	c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))	(39)
eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))

(39)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c18(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24(x₁)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	2
[prefix(x₁, x₂)]	=	1 + 3 · x₁ + 2 · x₂
[and(x₁, x₂)]	=	2 · x₁ + 0
[and^#(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[domatch[Ite]^#(x₁,...,x₄)]	=	0
[eqNatList[Ite]^#(x₁,...,x₅)]	=	3 · x₃ + 0
[prefix^#(x₁, x₂)]	=	0
[domatch^#(x₁, x₂, x₃)]	=	0
[eqNatList^#(x₁, x₂)]	=	3 · x₂ + 0
[notEmpty^#(x₁)]	=	0
[strmatch^#(x₁, x₂)]	=	1 + 1 · x₁
[S(x₁)]	=	1 · x₁ + 0
[0]	=	0
[False]	=	0
[True]	=	0
[Cons(x₁, x₂)]	=	3 + 1 · x₂
[Nil]	=	0

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

and^#(False,False)	→	c	(50)
and^#(True,False)	→	c1	(51)
and^#(False,True)	→	c2	(52)
and^#(True,True)	→	c3	(53)
!EQ^#(S(z0),S(z1))	→	c4(!EQ^#(z0,z1))	(55)
!EQ^#(0,S(z0))	→	c5	(57)
!EQ^#(S(z0),0)	→	c6	(59)
!EQ^#(0,0)	→	c7	(60)
domatch[Ite]^#(False,z0,Cons(z1,z2),z3)	→	c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(62)
domatch[Ite]^#(True,z0,Cons(z1,z2),z3)	→	c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(64)
eqNatList[Ite]^#(False,z0,z1,z2,z3)	→	c10	(66)
eqNatList[Ite]^#(True,z0,z1,z2,z3)	→	c11(eqNatList^#(z3,z1))	(68)
prefix^#(Cons(z0,z1),Cons(z2,z3))	→	c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))	(27)
prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)
domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
domatch^#(Nil,Nil,z0)	→	c16	(35)
domatch^#(z0,Cons(z1,z2),z3)	→	c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))	(37)
eqNatList^#(Cons(z0,z1),Cons(z2,z3))	→	c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))	(39)
eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))

(37)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c18(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24(x₁)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	1
[prefix(x₁, x₂)]	=	1
[and(x₁, x₂)]	=	1
[and^#(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[domatch[Ite]^#(x₁,...,x₄)]	=	1 · x₂ + 0 + 1 · x₃
[eqNatList[Ite]^#(x₁,...,x₅)]	=	0
[prefix^#(x₁, x₂)]	=	0
[domatch^#(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂
[eqNatList^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[strmatch^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[S(x₁)]	=	1 + 1 · x₁
[0]	=	1
[False]	=	1
[True]	=	0
[Cons(x₁, x₂)]	=	1 + 1 · x₂
[Nil]	=	0

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

and^#(False,False)	→	c	(50)
and^#(True,False)	→	c1	(51)
and^#(False,True)	→	c2	(52)
and^#(True,True)	→	c3	(53)
!EQ^#(S(z0),S(z1))	→	c4(!EQ^#(z0,z1))	(55)
!EQ^#(0,S(z0))	→	c5	(57)
!EQ^#(S(z0),0)	→	c6	(59)
!EQ^#(0,0)	→	c7	(60)
domatch[Ite]^#(False,z0,Cons(z1,z2),z3)	→	c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(62)
domatch[Ite]^#(True,z0,Cons(z1,z2),z3)	→	c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(64)
eqNatList[Ite]^#(False,z0,z1,z2,z3)	→	c10	(66)
eqNatList[Ite]^#(True,z0,z1,z2,z3)	→	c11(eqNatList^#(z3,z1))	(68)
prefix^#(Cons(z0,z1),Cons(z2,z3))	→	c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))	(27)
prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)
domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
domatch^#(Nil,Nil,z0)	→	c16	(35)
domatch^#(z0,Cons(z1,z2),z3)	→	c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))	(37)
eqNatList^#(Cons(z0,z1),Cons(z2,z3))	→	c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))	(39)
eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))

(27)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8(x₁)]	=	1 · x₁ + 0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c18(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24(x₁)]	=	1 · x₁ + 0
[!EQ(x₁, x₂)]	=	0
[prefix(x₁, x₂)]	=	0
[and(x₁, x₂)]	=	1
[and^#(x₁, x₂)]	=	0
[!EQ^#(x₁, x₂)]	=	0
[domatch[Ite]^#(x₁,...,x₄)]	=	1 · x₄ + 0 + 1 · x₃ · x₃ + 2 · x₂ · x₃
[eqNatList[Ite]^#(x₁,...,x₅)]	=	0
[prefix^#(x₁, x₂)]	=	1 · x₁ + 0
[domatch^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₃ + 2 · x₂ · x₁ + 1 · x₂ · x₂
[eqNatList^#(x₁, x₂)]	=	0
[notEmpty^#(x₁)]	=	0
[strmatch^#(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂ + 2 · x₁ · x₁
[S(x₁)]	=	0
[0]	=	2
[False]	=	0
[True]	=	0
[Cons(x₁, x₂)]	=	2 + 1 · x₂
[Nil]	=	0

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

and^#(False,False)	→	c	(50)
and^#(True,False)	→	c1	(51)
and^#(False,True)	→	c2	(52)
and^#(True,True)	→	c3	(53)
!EQ^#(S(z0),S(z1))	→	c4(!EQ^#(z0,z1))	(55)
!EQ^#(0,S(z0))	→	c5	(57)
!EQ^#(S(z0),0)	→	c6	(59)
!EQ^#(0,0)	→	c7	(60)
domatch[Ite]^#(False,z0,Cons(z1,z2),z3)	→	c8(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(62)
domatch[Ite]^#(True,z0,Cons(z1,z2),z3)	→	c9(domatch^#(z0,z2,Cons(z3,Cons(Nil,Nil))))	(64)
eqNatList[Ite]^#(False,z0,z1,z2,z3)	→	c10	(66)
eqNatList[Ite]^#(True,z0,z1,z2,z3)	→	c11(eqNatList^#(z3,z1))	(68)
prefix^#(Cons(z0,z1),Cons(z2,z3))	→	c12(and^#(!EQ(z0,z2),prefix(z1,z3)),!EQ^#(z0,z2),prefix^#(z1,z3))	(27)
prefix^#(Cons(z0,z1),Nil)	→	c13	(29)
prefix^#(Nil,z0)	→	c14	(31)
domatch^#(Cons(z0,z1),Nil,z2)	→	c15	(33)
domatch^#(Nil,Nil,z0)	→	c16	(35)
domatch^#(z0,Cons(z1,z2),z3)	→	c17(domatch[Ite]^#(prefix(z0,Cons(z1,z2)),z0,Cons(z1,z2),z3),prefix^#(z0,Cons(z1,z2)))	(37)
eqNatList^#(Cons(z0,z1),Cons(z2,z3))	→	c18(eqNatList[Ite]^#(!EQ(z0,z2),z2,z3,z0,z1),!EQ^#(z0,z2))	(39)
eqNatList^#(Cons(z0,z1),Nil)	→	c19	(41)
eqNatList^#(Nil,Cons(z0,z1))	→	c20	(43)
eqNatList^#(Nil,Nil)	→	c21	(44)
notEmpty^#(Cons(z0,z1))	→	c22	(46)
notEmpty^#(Nil)	→	c23	(47)
strmatch^#(z0,z1)	→	c24(domatch^#(z0,z1,Nil))	(49)

1.1.1.1.1.1.1.1.1.1 R is empty

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