Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.8b)

The rewrite relation of the following TRS is considered.

le(0,y)	→	true	(1)
le(s(x),0)	→	false	(2)
le(s(x),s(y))	→	le(x,y)	(3)
minus(0,y)	→	0	(4)
minus(s(x),y)	→	if_minus(le(s(x),y),s(x),y)	(5)
if_minus(true,s(x),y)	→	0	(6)
if_minus(false,s(x),y)	→	s(minus(x,y))	(7)
quot(0,s(y))	→	0	(8)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(9)
log(s(0))	→	0	(10)
log(s(s(x)))	→	s(log(s(quot(x,s(s(0))))))	(11)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n³).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

le^#(0,z0)

→

(13)

originates from

le(0,z0)

→

true

(12)

le^#(s(z0),0)

→

(15)

originates from

le(s(z0),0)

→

false

(14)

le^#(s(z0),s(z1))

→

c2(le^#(z0,z1))

(17)

originates from

le(s(z0),s(z1))

→

le(z0,z1)

(16)

minus^#(0,z0)

→

(19)

originates from

minus(0,z0)

→

(18)

minus^#(s(z0),z1)

→

c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))

(21)

originates from

minus(s(z0),z1)

→

if_minus(le(s(z0),z1),s(z0),z1)

(20)

if_minus^#(true,s(z0),z1)

→

(23)

originates from

if_minus(true,s(z0),z1)

→

(22)

if_minus^#(false,s(z0),z1)

→

c6(minus^#(z0,z1))

(25)

originates from

if_minus(false,s(z0),z1)

→

s(minus(z0,z1))

(24)

quot^#(0,s(z0))

→

(27)

originates from

quot(0,s(z0))

→

(26)

quot^#(s(z0),s(z1))

→

c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))

(29)

originates from

quot(s(z0),s(z1))

→

s(quot(minus(z0,z1),s(z1)))

(28)

log^#(s(0))

→

(30)

originates from

log(s(0))

→

(10)

log^#(s(s(z0)))

→

c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))

(32)

originates from

log(s(s(z0)))

→

s(log(s(quot(z0,s(s(0))))))

(31)

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

le^#(0,z0)

le^#(s(z0),0)

le^#(s(z0),s(z1))

minus^#(0,z0)

minus^#(s(z0),z1)

if_minus^#(true,s(z0),z1)

if_minus^#(false,s(z0),z1)

quot^#(0,s(z0))

quot^#(s(z0),s(z1))

log^#(s(0))

log^#(s(s(z0)))

1.1 Usable Rules

We remove the following rules since they are not usable.

log(s(0))	→	0	(10)
log(s(s(z0)))	→	s(log(s(quot(z0,s(s(0))))))	(31)

1.1.1 Rule Shifting

The rules

log^#(s(0))

→

(30)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	1 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₂ + 0
[if_minus(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[quot(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[if_minus^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · x₂ + 0
[log^#(x₁)]	=	1
[0]	=	0
[s(x₁)]	=	0
[false]	=	1
[true]	=	1

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)

1.1.1.1 Rule Shifting

The rules

quot^#(0,s(z0))

→

(27)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	1 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[if_minus^#(x₁, x₂, x₃)]	=	0
[quot^#(x₁, x₂)]	=	1
[log^#(x₁)]	=	1 · x₁ + 0
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁
[false]	=	1
[true]	=	1

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1 Rule Shifting

The rules

log^#(s(s(z0)))

→

c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))

(32)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	1 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[if_minus^#(x₁, x₂, x₃)]	=	0
[quot^#(x₁, x₂)]	=	0
[log^#(x₁)]	=	1 · x₁ + 0
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁
[false]	=	1
[true]	=	1

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1.1 Rule Shifting

The rules

quot^#(s(z0),s(z1))

→

c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))

(29)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[if_minus^#(x₁, x₂, x₃)]	=	0
[quot^#(x₁, x₂)]	=	2 · x₁ + 0 + 1 · x₁ · x₂
[log^#(x₁)]	=	2 · x₁ · x₁ + 0
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁
[false]	=	0
[true]	=	0

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1.1.1 Rule Shifting

The rules

minus^#(0,z0)	→	c3	(19)
if_minus^#(true,s(z0),z1)	→	c5	(23)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	1
[if_minus^#(x₁, x₂, x₃)]	=	1
[quot^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[log^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁
[false]	=	0
[true]	=	0

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	1 · x₂ + 0
[minus^#(x₁, x₂)]	=	1 · x₂ + 0 + 1 · x₁ · x₂
[if_minus^#(x₁, x₂, x₃)]	=	1 · x₂ · x₃ + 0
[quot^#(x₁, x₂)]	=	1 · x₂ · x₁ · x₁ + 0
[log^#(x₁)]	=	1 · x₁ · x₁ · x₁ + 0
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[false]	=	0
[true]	=	0

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

if_minus^#(false,s(z0),z1)

→

c6(minus^#(z0,z1))

(25)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	1 · x₁ + 0
[if_minus^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0 + 1 · x₂ · x₁ · x₁
[log^#(x₁)]	=	1 · x₁ + 0 + 1 · x₁ · x₁ · x₁
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[false]	=	0
[true]	=	0

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

le^#(0,z0)

→

(13)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	1
[minus^#(x₁, x₂)]	=	1 + 1 · x₁
[if_minus^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · x₂ · x₁ · x₁ + 0
[log^#(x₁)]	=	1 · x₁ · x₁ + 0 + 1 · x₁ · x₁ · x₁
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[false]	=	0
[true]	=	0

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

minus^#(s(z0),z1)

→

c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))

(21)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[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
[le(x₁, x₂)]	=	1 · x₂ + 0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[if_minus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot(x₁, x₂)]	=	1 · x₁ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₁ · x₂
[if_minus^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ · x₃
[quot^#(x₁, x₂)]	=	1 · x₂ · x₁ · x₁ + 0
[log^#(x₁)]	=	1 · x₁ · x₁ + 0 + 1 · x₁ · x₁ · x₁
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[false]	=	1
[true]	=	0

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

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(17)
minus^#(0,z0)	→	c3	(19)
minus^#(s(z0),z1)	→	c4(if_minus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(21)
if_minus^#(true,s(z0),z1)	→	c5	(23)
if_minus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(25)
quot^#(0,s(z0))	→	c7	(27)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(29)
log^#(s(0))	→	c9	(30)
log^#(s(s(z0)))	→	c10(log^#(s(quot(z0,s(s(0))))),quot^#(z0,s(s(0))))	(32)
minus(s(z0),z1)	→	if_minus(le(s(z0),z1),s(z0),z1)	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(28)
le(s(z0),s(z1))	→	le(z0,z1)	(16)
if_minus(false,s(z0),z1)	→	s(minus(z0,z1))	(24)
le(s(z0),0)	→	false	(14)
le(0,z0)	→	true	(12)
if_minus(true,s(z0),z1)	→	0	(22)
minus(0,z0)	→	0	(18)
quot(0,s(z0))	→	0	(26)

1.1.1.1.1.1.1.1.1.1.1.1 R is empty

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