Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.5a)

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)
pred(s(x))	→	x	(4)
minus(x,0)	→	x	(5)
minus(x,s(y))	→	pred(minus(x,y))	(6)
mod(0,y)	→	0	(7)
mod(s(x),0)	→	0	(8)
mod(s(x),s(y))	→	if_mod(le(y,x),s(x),s(y))	(9)
if_mod(true,s(x),s(y))	→	mod(minus(x,y),s(y))	(10)
if_mod(false,s(x),s(y))	→	s(x)	(11)

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:

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)

pred^#(s(z0))

→

(19)

originates from

pred(s(z0))

→

(18)

minus^#(z0,0)

→

(21)

originates from

minus(z0,0)

→

(20)

minus^#(z0,s(z1))

→

c5(pred^#(minus(z0,z1)),minus^#(z0,z1))

(23)

originates from

minus(z0,s(z1))

→

pred(minus(z0,z1))

(22)

mod^#(0,z0)

→

(25)

originates from

mod(0,z0)

→

(24)

mod^#(s(z0),0)

→

(27)

originates from

mod(s(z0),0)

→

(26)

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

→

c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))

(29)

originates from

mod(s(z0),s(z1))

→

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

(28)

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

→

c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))

(31)

originates from

if_mod(true,s(z0),s(z1))

→

mod(minus(z0,z1),s(z1))

(30)

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

→

c10

(33)

originates from

if_mod(false,s(z0),s(z1))

→

s(z0)

(32)

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

le^#(0,z0)

le^#(s(z0),0)

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

pred^#(s(z0))

minus^#(z0,0)

minus^#(z0,s(z1))

mod^#(0,z0)

mod^#(s(z0),0)

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

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

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

1.1 Usable Rules

We remove the following rules since they are not usable.

mod(0,z0)	→	0	(24)
mod(s(z0),0)	→	0	(26)
mod(s(z0),s(z1))	→	if_mod(le(z1,z0),s(z0),s(z1))	(28)
if_mod(true,s(z0),s(z1))	→	mod(minus(z0,z1),s(z1))	(30)
if_mod(false,s(z0),s(z1))	→	s(z0)	(32)

1.1.1 Rule Shifting

The rules

mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[le^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[minus^#(x₁, x₂)]	=	0
[mod^#(x₁, x₂)]	=	1 · x₁ + 0
[if_mod^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	0
[false]	=	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)
pred^#(s(z0))	→	c3	(19)
minus^#(z0,0)	→	c4	(21)
minus^#(z0,s(z1))	→	c5(pred^#(minus(z0,z1)),minus^#(z0,z1))	(23)
mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
mod^#(s(z0),s(z1))	→	c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(22)
le(s(z0),s(z1))	→	le(z0,z1)	(16)
le(s(z0),0)	→	false	(14)
le(0,z0)	→	true	(12)
minus(z0,0)	→	z0	(20)
pred(s(z0))	→	z0	(18)

1.1.1.1 Rule Shifting

The rules

minus^#(z0,0)

→

(21)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[le^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[minus^#(x₁, x₂)]	=	1
[mod^#(x₁, x₂)]	=	1 · x₁ + 0
[if_mod^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	0
[false]	=	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)
pred^#(s(z0))	→	c3	(19)
minus^#(z0,0)	→	c4	(21)
minus^#(z0,s(z1))	→	c5(pred^#(minus(z0,z1)),minus^#(z0,z1))	(23)
mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
mod^#(s(z0),s(z1))	→	c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(22)
le(s(z0),s(z1))	→	le(z0,z1)	(16)
le(s(z0),0)	→	false	(14)
le(0,z0)	→	true	(12)
minus(z0,0)	→	z0	(20)
pred(s(z0))	→	z0	(18)

1.1.1.1.1 Rule Shifting

The rules

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

→

c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))

(29)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[le^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[minus^#(x₁, x₂)]	=	0
[mod^#(x₁, x₂)]	=	1 + 1 · x₁
[if_mod^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	0
[false]	=	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)
pred^#(s(z0))	→	c3	(19)
minus^#(z0,0)	→	c4	(21)
minus^#(z0,s(z1))	→	c5(pred^#(minus(z0,z1)),minus^#(z0,z1))	(23)
mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
mod^#(s(z0),s(z1))	→	c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(22)
le(s(z0),s(z1))	→	le(z0,z1)	(16)
le(s(z0),0)	→	false	(14)
le(0,z0)	→	true	(12)
minus(z0,0)	→	z0	(20)
pred(s(z0))	→	z0	(18)

1.1.1.1.1.1 Rule Shifting

The rules

le^#(0,z0)	→	c	(13)
le^#(s(z0),0)	→	c1	(15)

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

[c]	=	0
[c1]	=	0
[c2(x₁)]	=	1 · x₁ + 0
[c3]	=	0
[c4]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[le^#(x₁, x₂)]	=	1
[pred^#(x₁)]	=	0
[minus^#(x₁, x₂)]	=	0
[mod^#(x₁, x₂)]	=	1 + 1 · x₁
[if_mod^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	0
[false]	=	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)
pred^#(s(z0))	→	c3	(19)
minus^#(z0,0)	→	c4	(21)
minus^#(z0,s(z1))	→	c5(pred^#(minus(z0,z1)),minus^#(z0,z1))	(23)
mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
mod^#(s(z0),s(z1))	→	c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(22)
le(s(z0),s(z1))	→	le(z0,z1)	(16)
le(s(z0),0)	→	false	(14)
le(0,z0)	→	true	(12)
minus(z0,0)	→	z0	(20)
pred(s(z0))	→	z0	(18)

1.1.1.1.1.1.1 Rule Shifting

The rules

minus^#(z0,s(z1))

→

c5(pred^#(minus(z0,z1)),minus^#(z0,z1))

(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]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[le^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	0
[minus^#(x₁, x₂)]	=	1 · x₂ + 0
[mod^#(x₁, x₂)]	=	2 · x₁ + 0 + 2 · x₁ · x₂
[if_mod^#(x₁, x₂, x₃)]	=	2 · x₂ + 0 + 2 · x₂ · x₃
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁
[true]	=	0
[false]	=	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)
pred^#(s(z0))	→	c3	(19)
minus^#(z0,0)	→	c4	(21)
minus^#(z0,s(z1))	→	c5(pred^#(minus(z0,z1)),minus^#(z0,z1))	(23)
mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
mod^#(s(z0),s(z1))	→	c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(22)
minus(z0,0)	→	z0	(20)
pred(s(z0))	→	z0	(18)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

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]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[le^#(x₁, x₂)]	=	1 · x₂ + 0
[pred^#(x₁)]	=	0
[minus^#(x₁, x₂)]	=	0
[mod^#(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₁ · x₁
[if_mod^#(x₁, x₂, x₃)]	=	1 · x₂ · x₂ + 0
[0]	=	0
[s(x₁)]	=	2 + 1 · x₁
[true]	=	0
[false]	=	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)
pred^#(s(z0))	→	c3	(19)
minus^#(z0,0)	→	c4	(21)
minus^#(z0,s(z1))	→	c5(pred^#(minus(z0,z1)),minus^#(z0,z1))	(23)
mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
mod^#(s(z0),s(z1))	→	c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(22)
minus(z0,0)	→	z0	(20)
pred(s(z0))	→	z0	(18)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

pred^#(s(z0))

→

(19)

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]	=	0
[c5(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c6]	=	0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[pred(x₁)]	=	1 · x₁ + 0
[le(x₁, x₂)]	=	0
[le^#(x₁, x₂)]	=	0
[pred^#(x₁)]	=	1
[minus^#(x₁, x₂)]	=	1 · x₂ + 0
[mod^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₁ · x₂
[if_mod^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₂ · x₃
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁
[true]	=	0
[false]	=	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)
pred^#(s(z0))	→	c3	(19)
minus^#(z0,0)	→	c4	(21)
minus^#(z0,s(z1))	→	c5(pred^#(minus(z0,z1)),minus^#(z0,z1))	(23)
mod^#(0,z0)	→	c6	(25)
mod^#(s(z0),0)	→	c7	(27)
mod^#(s(z0),s(z1))	→	c8(if_mod^#(le(z1,z0),s(z0),s(z1)),le^#(z1,z0))	(29)
if_mod^#(true,s(z0),s(z1))	→	c9(mod^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(31)
if_mod^#(false,s(z0),s(z1))	→	c10	(33)
minus(z0,s(z1))	→	pred(minus(z0,z1))	(22)
minus(z0,0)	→	z0	(20)
pred(s(z0))	→	z0	(18)

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).