Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/raML/appendAll.raml)

The rewrite relation of the following TRS is considered.

append(@l1,@l2)	→	append#1(@l1,@l2)	(1)
append#1(::(@x,@xs),@l2)	→	::(@x,append(@xs,@l2))	(2)
append#1(nil,@l2)	→	@l2	(3)
appendAll(@l)	→	appendAll#1(@l)	(4)
appendAll#1(::(@l1,@ls))	→	append(@l1,appendAll(@ls))	(5)
appendAll#1(nil)	→	nil	(6)
appendAll2(@l)	→	appendAll2#1(@l)	(7)
appendAll2#1(::(@l1,@ls))	→	append(appendAll(@l1),appendAll2(@ls))	(8)
appendAll2#1(nil)	→	nil	(9)
appendAll3(@l)	→	appendAll3#1(@l)	(10)
appendAll3#1(::(@l1,@ls))	→	append(appendAll2(@l1),appendAll3(@ls))	(11)
appendAll3#1(nil)	→	nil	(12)

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:

append^#(z0,z1)

→

c(append#1^#(z0,z1))

(14)

originates from

append(z0,z1)

→

append#1(z0,z1)

(13)

append#1^#(::(z0,z1),z2)

→

c1(append^#(z1,z2))

(16)

originates from

append#1(::(z0,z1),z2)

→

::(z0,append(z1,z2))

(15)

append#1^#(nil,z0)

→

(18)

originates from

append#1(nil,z0)

→

(17)

appendAll^#(z0)

→

c3(appendAll#1^#(z0))

(20)

originates from

appendAll(z0)

→

appendAll#1(z0)

(19)

appendAll#1^#(::(z0,z1))

→

c4(append^#(z0,appendAll(z1)),appendAll^#(z1))

(22)

originates from

appendAll#1(::(z0,z1))

→

append(z0,appendAll(z1))

(21)

appendAll#1^#(nil)

→

(23)

originates from

appendAll#1(nil)

→

nil

(6)

appendAll2^#(z0)

→

c6(appendAll2#1^#(z0))

(25)

originates from

appendAll2(z0)

→

appendAll2#1(z0)

(24)

appendAll2#1^#(::(z0,z1))

→

c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))

(27)

originates from

appendAll2#1(::(z0,z1))

→

append(appendAll(z0),appendAll2(z1))

(26)

appendAll2#1^#(nil)

→

(28)

originates from

appendAll2#1(nil)

→

nil

(9)

appendAll3^#(z0)

→

c9(appendAll3#1^#(z0))

(30)

originates from

appendAll3(z0)

→

appendAll3#1(z0)

(29)

appendAll3#1^#(::(z0,z1))

→

c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))

(32)

originates from

appendAll3#1(::(z0,z1))

→

append(appendAll2(z0),appendAll3(z1))

(31)

appendAll3#1^#(nil)

→

c11

(33)

originates from

appendAll3#1(nil)

→

nil

(12)

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

append^#(z0,z1)

append#1^#(::(z0,z1),z2)

append#1^#(nil,z0)

appendAll^#(z0)

appendAll#1^#(::(z0,z1))

appendAll#1^#(nil)

appendAll2^#(z0)

appendAll2#1^#(::(z0,z1))

appendAll2#1^#(nil)

appendAll3^#(z0)

appendAll3#1^#(::(z0,z1))

appendAll3#1^#(nil)

1.1 Rule Shifting

The rules

appendAll#1^#(nil)	→	c5	(23)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3#1^#(nil)	→	c11	(33)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c11]	=	0
[append(x₁, x₂)]	=	0
[append#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendAll(x₁)]	=	1 · x₁ + 0
[appendAll#1(x₁)]	=	1 + 1 · x₁
[appendAll2(x₁)]	=	1 · x₁ + 0
[appendAll2#1(x₁)]	=	1 + 1 · x₁
[appendAll3(x₁)]	=	1 + 1 · x₁
[appendAll3#1(x₁)]	=	1 + 1 · x₁
[append^#(x₁, x₂)]	=	0
[append#1^#(x₁, x₂)]	=	0
[appendAll^#(x₁)]	=	1 · x₁ + 0
[appendAll#1^#(x₁)]	=	1 · x₁ + 0
[appendAll2^#(x₁)]	=	1 · x₁ + 0
[appendAll2#1^#(x₁)]	=	1 · x₁ + 0
[appendAll3^#(x₁)]	=	1 · x₁ + 0
[appendAll3#1^#(x₁)]	=	1 · x₁ + 0
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[nil]	=	1

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

append^#(z0,z1)	→	c(append#1^#(z0,z1))	(14)
append#1^#(::(z0,z1),z2)	→	c1(append^#(z1,z2))	(16)
append#1^#(nil,z0)	→	c2	(18)
appendAll^#(z0)	→	c3(appendAll#1^#(z0))	(20)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll#1^#(nil)	→	c5	(23)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3^#(z0)	→	c9(appendAll3#1^#(z0))	(30)
appendAll3#1^#(::(z0,z1))	→	c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))	(32)
appendAll3#1^#(nil)	→	c11	(33)

1.1.1 Rule Shifting

The rules

appendAll3^#(z0)

→

c9(appendAll3#1^#(z0))

(30)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c11]	=	0
[append(x₁, x₂)]	=	0
[append#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendAll(x₁)]	=	1 · x₁ + 0
[appendAll#1(x₁)]	=	1 + 1 · x₁
[appendAll2(x₁)]	=	1 · x₁ + 0
[appendAll2#1(x₁)]	=	1 + 1 · x₁
[appendAll3(x₁)]	=	1 + 1 · x₁
[appendAll3#1(x₁)]	=	1 + 1 · x₁
[append^#(x₁, x₂)]	=	0
[append#1^#(x₁, x₂)]	=	0
[appendAll^#(x₁)]	=	0
[appendAll#1^#(x₁)]	=	0
[appendAll2^#(x₁)]	=	0
[appendAll2#1^#(x₁)]	=	0
[appendAll3^#(x₁)]	=	1 + 1 · x₁
[appendAll3#1^#(x₁)]	=	1 · x₁ + 0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	1

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

append^#(z0,z1)	→	c(append#1^#(z0,z1))	(14)
append#1^#(::(z0,z1),z2)	→	c1(append^#(z1,z2))	(16)
append#1^#(nil,z0)	→	c2	(18)
appendAll^#(z0)	→	c3(appendAll#1^#(z0))	(20)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll#1^#(nil)	→	c5	(23)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3^#(z0)	→	c9(appendAll3#1^#(z0))	(30)
appendAll3#1^#(::(z0,z1))	→	c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))	(32)
appendAll3#1^#(nil)	→	c11	(33)

1.1.1.1 Rule Shifting

The rules

appendAll3#1^#(::(z0,z1))

→

c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))

(32)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c11]	=	0
[append(x₁, x₂)]	=	0
[append#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendAll(x₁)]	=	1 · x₁ + 0
[appendAll#1(x₁)]	=	1 + 1 · x₁
[appendAll2(x₁)]	=	1 · x₁ + 0
[appendAll2#1(x₁)]	=	1 + 1 · x₁
[appendAll3(x₁)]	=	1 + 1 · x₁
[appendAll3#1(x₁)]	=	1 + 1 · x₁
[append^#(x₁, x₂)]	=	0
[append#1^#(x₁, x₂)]	=	0
[appendAll^#(x₁)]	=	0
[appendAll#1^#(x₁)]	=	0
[appendAll2^#(x₁)]	=	0
[appendAll2#1^#(x₁)]	=	0
[appendAll3^#(x₁)]	=	1 · x₁ + 0
[appendAll3#1^#(x₁)]	=	1 · x₁ + 0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	1

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

append^#(z0,z1)	→	c(append#1^#(z0,z1))	(14)
append#1^#(::(z0,z1),z2)	→	c1(append^#(z1,z2))	(16)
append#1^#(nil,z0)	→	c2	(18)
appendAll^#(z0)	→	c3(appendAll#1^#(z0))	(20)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll#1^#(nil)	→	c5	(23)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3^#(z0)	→	c9(appendAll3#1^#(z0))	(30)
appendAll3#1^#(::(z0,z1))	→	c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))	(32)
appendAll3#1^#(nil)	→	c11	(33)

1.1.1.1.1 Rule Shifting

The rules

append#1^#(nil,z0)	→	c2	(18)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c11]	=	0
[append(x₁, x₂)]	=	2 + 1 · x₂
[append#1(x₁, x₂)]	=	3 + 3 · x₂
[appendAll(x₁)]	=	0
[appendAll#1(x₁)]	=	3 + 3 · x₁
[appendAll2(x₁)]	=	0
[appendAll2#1(x₁)]	=	3 + 3 · x₁
[appendAll3(x₁)]	=	2 + 3 · x₁
[appendAll3#1(x₁)]	=	1 + 3 · x₁
[append^#(x₁, x₂)]	=	3
[append#1^#(x₁, x₂)]	=	3
[appendAll^#(x₁)]	=	1 + 3 · x₁
[appendAll#1^#(x₁)]	=	1 + 3 · x₁
[appendAll2^#(x₁)]	=	3 + 3 · x₁
[appendAll2#1^#(x₁)]	=	1 + 3 · x₁
[appendAll3^#(x₁)]	=	2 + 3 · x₁
[appendAll3#1^#(x₁)]	=	1 + 3 · x₁
[::(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂
[nil]	=	0

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

append^#(z0,z1)	→	c(append#1^#(z0,z1))	(14)
append#1^#(::(z0,z1),z2)	→	c1(append^#(z1,z2))	(16)
append#1^#(nil,z0)	→	c2	(18)
appendAll^#(z0)	→	c3(appendAll#1^#(z0))	(20)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll#1^#(nil)	→	c5	(23)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3^#(z0)	→	c9(appendAll3#1^#(z0))	(30)
appendAll3#1^#(::(z0,z1))	→	c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))	(32)
appendAll3#1^#(nil)	→	c11	(33)

1.1.1.1.1.1 Rule Shifting

The rules

appendAll^#(z0)

→

c3(appendAll#1^#(z0))

(20)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c11]	=	0
[append(x₁, x₂)]	=	0
[append#1(x₁, x₂)]	=	3 + 3 · x₂
[appendAll(x₁)]	=	0
[appendAll#1(x₁)]	=	3 + 3 · x₁
[appendAll2(x₁)]	=	2
[appendAll2#1(x₁)]	=	2
[appendAll3(x₁)]	=	3 + 3 · x₁
[appendAll3#1(x₁)]	=	3 + 3 · x₁
[append^#(x₁, x₂)]	=	2
[append#1^#(x₁, x₂)]	=	2
[appendAll^#(x₁)]	=	2 + 3 · x₁
[appendAll#1^#(x₁)]	=	1 + 3 · x₁
[appendAll2^#(x₁)]	=	3 · x₁ + 0
[appendAll2#1^#(x₁)]	=	3 · x₁ + 0
[appendAll3^#(x₁)]	=	2 + 3 · x₁
[appendAll3#1^#(x₁)]	=	1 + 3 · x₁
[::(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂
[nil]	=	0

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

append^#(z0,z1)	→	c(append#1^#(z0,z1))	(14)
append#1^#(::(z0,z1),z2)	→	c1(append^#(z1,z2))	(16)
append#1^#(nil,z0)	→	c2	(18)
appendAll^#(z0)	→	c3(appendAll#1^#(z0))	(20)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll#1^#(nil)	→	c5	(23)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3^#(z0)	→	c9(appendAll3#1^#(z0))	(30)
appendAll3#1^#(::(z0,z1))	→	c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))	(32)
appendAll3#1^#(nil)	→	c11	(33)

1.1.1.1.1.1.1 Rule Shifting

The rules

append#1^#(::(z0,z1),z2)

→

c1(append^#(z1,z2))

(16)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c11]	=	0
[append(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[append#1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[appendAll(x₁)]	=	1 · x₁ + 0
[appendAll#1(x₁)]	=	1 · x₁ + 0
[appendAll2(x₁)]	=	1 · x₁ + 0
[appendAll2#1(x₁)]	=	1 · x₁ + 0
[appendAll3(x₁)]	=	3 + 3 · x₁
[appendAll3#1(x₁)]	=	3 + 3 · x₁
[append^#(x₁, x₂)]	=	1 · x₁ + 0
[append#1^#(x₁, x₂)]	=	1 · x₁ + 0
[appendAll^#(x₁)]	=	1 · x₁ + 0
[appendAll#1^#(x₁)]	=	1 · x₁ + 0
[appendAll2^#(x₁)]	=	2 · x₁ + 0
[appendAll2#1^#(x₁)]	=	2 · x₁ + 0
[appendAll3^#(x₁)]	=	2 + 3 · x₁
[appendAll3#1^#(x₁)]	=	2 + 3 · x₁
[::(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[nil]	=	0

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

append^#(z0,z1)	→	c(append#1^#(z0,z1))	(14)
append#1^#(::(z0,z1),z2)	→	c1(append^#(z1,z2))	(16)
append#1^#(nil,z0)	→	c2	(18)
appendAll^#(z0)	→	c3(appendAll#1^#(z0))	(20)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll#1^#(nil)	→	c5	(23)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3^#(z0)	→	c9(appendAll3#1^#(z0))	(30)
appendAll3#1^#(::(z0,z1))	→	c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))	(32)
appendAll3#1^#(nil)	→	c11	(33)
appendAll#1(::(z0,z1))	→	append(z0,appendAll(z1))	(21)
append(z0,z1)	→	append#1(z0,z1)	(13)
append#1(nil,z0)	→	z0	(17)
appendAll(z0)	→	appendAll#1(z0)	(19)
appendAll2(z0)	→	appendAll2#1(z0)	(24)
appendAll2#1(::(z0,z1))	→	append(appendAll(z0),appendAll2(z1))	(26)
appendAll2#1(nil)	→	nil	(9)
appendAll#1(nil)	→	nil	(6)
append#1(::(z0,z1),z2)	→	::(z0,append(z1,z2))	(15)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

append^#(z0,z1)

→

c(append#1^#(z0,z1))

(14)

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

[c(x₁)]	=	1 · x₁ + 0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c11]	=	0
[append(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[append#1(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[appendAll(x₁)]	=	1 · x₁ + 0
[appendAll#1(x₁)]	=	1 · x₁ + 0
[appendAll2(x₁)]	=	1 · x₁ + 0
[appendAll2#1(x₁)]	=	1 · x₁ + 0
[appendAll3(x₁)]	=	2 + 3 · x₁
[appendAll3#1(x₁)]	=	1 + 3 · x₁
[append^#(x₁, x₂)]	=	1 + 1 · x₁
[append#1^#(x₁, x₂)]	=	1 · x₁ + 0
[appendAll^#(x₁)]	=	3 + 1 · x₁
[appendAll#1^#(x₁)]	=	3 + 1 · x₁
[appendAll2^#(x₁)]	=	1 + 2 · x₁
[appendAll2#1^#(x₁)]	=	2 · x₁ + 0
[appendAll3^#(x₁)]	=	2 + 3 · x₁
[appendAll3#1^#(x₁)]	=	1 + 3 · x₁
[::(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂
[nil]	=	0

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

append^#(z0,z1)	→	c(append#1^#(z0,z1))	(14)
append#1^#(::(z0,z1),z2)	→	c1(append^#(z1,z2))	(16)
append#1^#(nil,z0)	→	c2	(18)
appendAll^#(z0)	→	c3(appendAll#1^#(z0))	(20)
appendAll#1^#(::(z0,z1))	→	c4(append^#(z0,appendAll(z1)),appendAll^#(z1))	(22)
appendAll#1^#(nil)	→	c5	(23)
appendAll2^#(z0)	→	c6(appendAll2#1^#(z0))	(25)
appendAll2#1^#(::(z0,z1))	→	c7(append^#(appendAll(z0),appendAll2(z1)),appendAll^#(z0),appendAll2^#(z1))	(27)
appendAll2#1^#(nil)	→	c8	(28)
appendAll3^#(z0)	→	c9(appendAll3#1^#(z0))	(30)
appendAll3#1^#(::(z0,z1))	→	c10(append^#(appendAll2(z0),appendAll3(z1)),appendAll2^#(z0),appendAll3^#(z1))	(32)
appendAll3#1^#(nil)	→	c11	(33)
appendAll#1(::(z0,z1))	→	append(z0,appendAll(z1))	(21)
append(z0,z1)	→	append#1(z0,z1)	(13)
append#1(nil,z0)	→	z0	(17)
appendAll(z0)	→	appendAll#1(z0)	(19)
appendAll2(z0)	→	appendAll2#1(z0)	(24)
appendAll2#1(::(z0,z1))	→	append(appendAll(z0),appendAll2(z1))	(26)
appendAll2#1(nil)	→	nil	(9)
appendAll#1(nil)	→	nil	(6)
append#1(::(z0,z1),z2)	→	::(z0,append(z1,z2))	(15)

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).