Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Transformed_CSR_04/MYNAT_nosorts-noand_FR)

The rewrite relation of the following TRS is considered.

U11(tt,M,N)	→	U12(tt,activate(M),activate(N))	(1)
U12(tt,M,N)	→	s(plus(activate(N),activate(M)))	(2)
U21(tt,M,N)	→	U22(tt,activate(M),activate(N))	(3)
U22(tt,M,N)	→	plus(x(activate(N),activate(M)),activate(N))	(4)
plus(N,0)	→	N	(5)
plus(N,s(M))	→	U11(tt,M,N)	(6)
x(N,0)	→	0	(7)
x(N,s(M))	→	U21(tt,M,N)	(8)
activate(X)	→	X	(9)

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:

U11^#(tt,z0,z1)

→

c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))

(11)

originates from

U11(tt,z0,z1)

→

U12(tt,activate(z0),activate(z1))

(10)

U12^#(tt,z0,z1)

→

c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))

(13)

originates from

U12(tt,z0,z1)

→

s(plus(activate(z1),activate(z0)))

(12)

U21^#(tt,z0,z1)

→

c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))

(15)

originates from

U21(tt,z0,z1)

→

U22(tt,activate(z0),activate(z1))

(14)

U22^#(tt,z0,z1)

→

c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))

(17)

originates from

U22(tt,z0,z1)

→

plus(x(activate(z1),activate(z0)),activate(z1))

(16)

plus^#(z0,0)

→

(19)

originates from

plus(z0,0)

→

(18)

plus^#(z0,s(z1))

→

c5(U11^#(tt,z1,z0))

(21)

originates from

plus(z0,s(z1))

→

U11(tt,z1,z0)

(20)

x^#(z0,0)

→

(23)

originates from

x(z0,0)

→

(22)

x^#(z0,s(z1))

→

c7(U21^#(tt,z1,z0))

(25)

originates from

x(z0,s(z1))

→

U21(tt,z1,z0)

(24)

activate^#(z0)

→

(27)

originates from

activate(z0)

→

(26)

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

U11^#(tt,z0,z1)

U12^#(tt,z0,z1)

U21^#(tt,z0,z1)

U22^#(tt,z0,z1)

plus^#(z0,0)

plus^#(z0,s(z1))

x^#(z0,0)

x^#(z0,s(z1))

activate^#(z0)

1.1 Rule Shifting

The rules

x^#(z0,0)

→

(23)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U12(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U21(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U22(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[plus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[x(x₁, x₂)]	=	1 + 1 · x₁
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	0
[U12^#(x₁, x₂, x₃)]	=	0
[U21^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[U22^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[plus^#(x₁, x₂)]	=	0
[x^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[activate^#(x₁)]	=	0
[0]	=	1
[s(x₁)]	=	1 · x₁ + 0
[tt]	=	1

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

1.1.1 Rule Shifting

The rules

x^#(z0,s(z1))

→

c7(U21^#(tt,z1,z0))

(25)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U12(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U21(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U22(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[plus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[x(x₁, x₂)]	=	1
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	0
[U12^#(x₁, x₂, x₃)]	=	0
[U21^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[U22^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[plus^#(x₁, x₂)]	=	0
[x^#(x₁, x₂)]	=	1 · x₂ + 0
[activate^#(x₁)]	=	0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[tt]	=	1

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

1.1.1.1 Rule Shifting

The rules

U21^#(tt,z0,z1)

→

c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))

(15)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U12(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U21(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U22(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[plus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[x(x₁, x₂)]	=	1
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	0
[U12^#(x₁, x₂, x₃)]	=	0
[U21^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[U22^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[plus^#(x₁, x₂)]	=	0
[x^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[activate^#(x₁)]	=	0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[tt]	=	1

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

1.1.1.1.1 Rule Shifting

The rules

U22^#(tt,z0,z1)

→

c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))

(17)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U12(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U21(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U22(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[plus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[x(x₁, x₂)]	=	1 + 1 · x₁
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	0
[U12^#(x₁, x₂, x₃)]	=	0
[U21^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[U22^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[plus^#(x₁, x₂)]	=	0
[x^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[activate^#(x₁)]	=	0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[tt]	=	1

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

1.1.1.1.1.1 Rule Shifting

The rules

plus^#(z0,0)

→

(19)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U12(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U21(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[U22(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[plus(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[x(x₁, x₂)]	=	1
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[U12^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[U21^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[U22^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[plus^#(x₁, x₂)]	=	1
[x^#(x₁, x₂)]	=	1 · x₂ + 0
[activate^#(x₁)]	=	0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[tt]	=	1

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

1.1.1.1.1.1.1 Rule Shifting

The rules

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₃ + 1 · x₁ · x₁ + 2 · x₂ · x₁ + 1 · x₂ · x₂
[U12(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 2 · x₁ · x₁
[U21(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₃ + 2 · x₁ · x₁ + 2 · x₂ · x₁ + 1 · x₂ · x₂
[U22(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 1 · x₁ · x₁
[plus(x₁, x₂)]	=	1
[x(x₁, x₂)]	=	0
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	2 + 1 · x₁ + 1 · x₂ + 2 · x₁ · x₃ + 2 · x₁ · x₁ + 1 · x₂ · x₁
[U12^#(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 1 · x₂ + 2 · x₁ · x₁
[U21^#(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 1 · x₃ + 1 · x₂ · x₃ + 2 · x₁ · x₃ + 2 · x₁ · x₁ + 1 · x₂ · x₁
[U22^#(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 1 · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₁
[plus^#(x₁, x₂)]	=	1 · x₂ + 0
[x^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[activate^#(x₁)]	=	0
[0]	=	2
[s(x₁)]	=	2 + 1 · x₁
[tt]	=	0

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

plus^#(z0,s(z1))

→

c5(U11^#(tt,z1,z0))

(21)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 2 · x₁ · x₃ + 1 · x₁ · x₁ + 2 · x₂ · x₁ + 1 · x₂ · x₂
[U12(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 1 · x₁ · x₁
[U21(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 2 · x₁ · x₃ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₂ · x₂
[U22(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 2 · x₁ · x₁
[plus(x₁, x₂)]	=	1
[x(x₁, x₂)]	=	0
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 1 · x₂ + 2 · x₁ · x₃ + 2 · x₁ · x₁ + 1 · x₂ · x₁
[U12^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₁ · x₁
[U21^#(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 1 · x₃ + 1 · x₂ · x₃ + 2 · x₁ · x₃ + 2 · x₁ · x₁ + 1 · x₂ · x₁
[U22^#(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 1 · x₃ + 1 · x₂ · x₃ + 2 · x₁ · x₁
[plus^#(x₁, x₂)]	=	1 · x₂ + 0
[x^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[activate^#(x₁)]	=	0
[0]	=	2
[s(x₁)]	=	2 + 1 · x₁
[tt]	=	0

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

activate^#(z0)

→

(27)

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

[c(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c2(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c3(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁)]	=	1 · x₁ + 0
[c8]	=	0
[U11(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₃ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₂ · x₂
[U12(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 2 · x₁ · x₁
[U21(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₃ + 1 · x₁ · x₁ + 2 · x₂ · x₁ + 1 · x₂ · x₂
[U22(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 2 · x₁ · x₁
[plus(x₁, x₂)]	=	2
[x(x₁, x₂)]	=	0
[activate(x₁)]	=	1 · x₁ + 0
[U11^#(x₁, x₂, x₃)]	=	2 + 2 · x₁ + 2 · x₂
[U12^#(x₁, x₂, x₃)]	=	1 + 1 · x₁ · x₁ + 2 · x₂ · x₁
[U21^#(x₁, x₂, x₃)]	=	2 + 2 · x₁ + 1 · x₂ + 2 · x₃ + 2 · x₂ · x₃ + 2 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₂ · x₂
[U22^#(x₁, x₂, x₃)]	=	2 + 2 · x₁ + 1 · x₂ + 2 · x₂ · x₃ + 2 · x₁ · x₃ + 1 · x₂ · x₂
[plus^#(x₁, x₂)]	=	2 · x₂ + 0
[x^#(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂
[activate^#(x₁)]	=	1
[0]	=	2
[s(x₁)]	=	2 + 1 · x₁
[tt]	=	1

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

U11^#(tt,z0,z1)	→	c(U12^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(11)
U12^#(tt,z0,z1)	→	c1(plus^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0))	(13)
U21^#(tt,z0,z1)	→	c2(U22^#(tt,activate(z0),activate(z1)),activate^#(z0),activate^#(z1))	(15)
U22^#(tt,z0,z1)	→	c3(plus^#(x(activate(z1),activate(z0)),activate(z1)),x^#(activate(z1),activate(z0)),activate^#(z1),activate^#(z0),activate^#(z1))	(17)
plus^#(z0,0)	→	c4	(19)
plus^#(z0,s(z1))	→	c5(U11^#(tt,z1,z0))	(21)
x^#(z0,0)	→	c6	(23)
x^#(z0,s(z1))	→	c7(U21^#(tt,z1,z0))	(25)
activate^#(z0)	→	c8	(27)
activate(z0)	→	z0	(26)

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).