Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/HirokawaMiddeldorp_04/t014)

The rewrite relation of the following TRS is considered.

-(x,0)	→	x	(1)
-(0,s(y))	→	0	(2)
-(s(x),s(y))	→	-(x,y)	(3)
lt(x,0)	→	false	(4)
lt(0,s(y))	→	true	(5)
lt(s(x),s(y))	→	lt(x,y)	(6)
if(true,x,y)	→	x	(7)
if(false,x,y)	→	y	(8)
div(x,0)	→	0	(9)
div(0,y)	→	0	(10)
div(s(x),s(y))	→	if(lt(x,y),0,s(div(-(x,y),s(y))))	(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:

-^#(z0,0)

→

(13)

originates from

-(z0,0)

→

(12)

-^#(0,s(z0))

→

(15)

originates from

-(0,s(z0))

→

(14)

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

→

c2(-^#(z0,z1))

(17)

originates from

-(s(z0),s(z1))

→

-(z0,z1)

(16)

lt^#(z0,0)

→

(19)

originates from

lt(z0,0)

→

false

(18)

lt^#(0,s(z0))

→

(21)

originates from

lt(0,s(z0))

→

true

(20)

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

→

c5(lt^#(z0,z1))

(23)

originates from

lt(s(z0),s(z1))

→

lt(z0,z1)

(22)

if^#(true,z0,z1)

→

(25)

originates from

if(true,z0,z1)

→

(24)

if^#(false,z0,z1)

→

(27)

originates from

if(false,z0,z1)

→

(26)

div^#(z0,0)

→

(29)

originates from

div(z0,0)

→

(28)

div^#(0,z0)

→

(31)

originates from

div(0,z0)

→

(30)

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

→

c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))

(33)

originates from

div(s(z0),s(z1))

→

if(lt(z0,z1),0,s(div(-(z0,z1),s(z1))))

(32)

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

-^#(z0,0)

-^#(0,s(z0))

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

lt^#(z0,0)

lt^#(0,s(z0))

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

if^#(true,z0,z1)

if^#(false,z0,z1)

div^#(z0,0)

div^#(0,z0)

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

1.1 Usable Rules

We remove the following rules since they are not usable.

div(z0,0)

→

(28)

1.1.1 Rule Shifting

The rules

div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)

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)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[lt(x₁, x₂)]	=	1 + 1 · x₁
[div(x₁, x₂)]	=	1
[-(x₁, x₂)]	=	1 + 1 · x₂
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[-^#(x₁, x₂)]	=	0
[lt^#(x₁, x₂)]	=	0
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[div^#(x₁, x₂)]	=	1 + 1 · x₂
[0]	=	0
[false]	=	1
[s(x₁)]	=	0
[true]	=	1

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

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	c1	(15)
-^#(s(z0),s(z1))	→	c2(-^#(z0,z1))	(17)
lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(21)
lt^#(s(z0),s(z1))	→	c5(lt^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(27)
div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)
div^#(s(z0),s(z1))	→	c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))	(33)

1.1.1.1 Rule Shifting

The rules

lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(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)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[lt(x₁, x₂)]	=	1 + 1 · x₁
[div(x₁, x₂)]	=	1
[-(x₁, x₂)]	=	1 · x₁ + 0
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[-^#(x₁, x₂)]	=	0
[lt^#(x₁, x₂)]	=	1
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	0
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	1

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

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	c1	(15)
-^#(s(z0),s(z1))	→	c2(-^#(z0,z1))	(17)
lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(21)
lt^#(s(z0),s(z1))	→	c5(lt^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(27)
div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)
div^#(s(z0),s(z1))	→	c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))	(33)
-(0,s(z0))	→	0	(14)
-(s(z0),s(z1))	→	-(z0,z1)	(16)
-(z0,0)	→	z0	(12)

1.1.1.1.1 Rule Shifting

The rules

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

→

c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))

(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)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[lt(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[div(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[-(x₁, x₂)]	=	1 · x₁ + 0
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[-^#(x₁, x₂)]	=	0
[lt^#(x₁, x₂)]	=	0
[if^#(x₁, x₂, x₃)]	=	0
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	1
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	1

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

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	c1	(15)
-^#(s(z0),s(z1))	→	c2(-^#(z0,z1))	(17)
lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(21)
lt^#(s(z0),s(z1))	→	c5(lt^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(27)
div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)
div^#(s(z0),s(z1))	→	c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))	(33)
-(0,s(z0))	→	0	(14)
-(s(z0),s(z1))	→	-(z0,z1)	(16)
-(z0,0)	→	z0	(12)

1.1.1.1.1.1 Rule Shifting

The rules

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	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)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[lt(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[div(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[-(x₁, x₂)]	=	1 · x₁ + 0
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[-^#(x₁, x₂)]	=	1
[lt^#(x₁, x₂)]	=	0
[if^#(x₁, x₂, x₃)]	=	0
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	1
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	1

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

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	c1	(15)
-^#(s(z0),s(z1))	→	c2(-^#(z0,z1))	(17)
lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(21)
lt^#(s(z0),s(z1))	→	c5(lt^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(27)
div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)
div^#(s(z0),s(z1))	→	c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))	(33)
-(0,s(z0))	→	0	(14)
-(s(z0),s(z1))	→	-(z0,z1)	(16)
-(z0,0)	→	z0	(12)

1.1.1.1.1.1.1 Rule Shifting

The rules

if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(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]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[lt(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[div(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[-(x₁, x₂)]	=	1 · x₁ + 0
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[-^#(x₁, x₂)]	=	0
[lt^#(x₁, x₂)]	=	0
[if^#(x₁, x₂, x₃)]	=	1
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	1
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	1

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

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	c1	(15)
-^#(s(z0),s(z1))	→	c2(-^#(z0,z1))	(17)
lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(21)
lt^#(s(z0),s(z1))	→	c5(lt^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(27)
div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)
div^#(s(z0),s(z1))	→	c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))	(33)
-(0,s(z0))	→	0	(14)
-(s(z0),s(z1))	→	-(z0,z1)	(16)
-(z0,0)	→	z0	(12)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c5(lt^#(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)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[lt(x₁, x₂)]	=	0
[div(x₁, x₂)]	=	1 · x₁ + 0
[-(x₁, x₂)]	=	1 · x₁ + 0
[if(x₁, x₂, x₃)]	=	2 · x₂ + 0 + 1 · x₃ + 1 · x₂ · x₂
[-^#(x₁, x₂)]	=	0
[lt^#(x₁, x₂)]	=	1 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₂ · x₂
[div^#(x₁, x₂)]	=	1 · x₁ · x₁ + 0
[0]	=	0
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	1

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

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	c1	(15)
-^#(s(z0),s(z1))	→	c2(-^#(z0,z1))	(17)
lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(21)
lt^#(s(z0),s(z1))	→	c5(lt^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(27)
div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)
div^#(s(z0),s(z1))	→	c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))	(33)
-(0,s(z0))	→	0	(14)
-(s(z0),s(z1))	→	-(z0,z1)	(16)
div(s(z0),s(z1))	→	if(lt(z0,z1),0,s(div(-(z0,z1),s(z1))))	(32)
if(true,z0,z1)	→	z0	(24)
-(z0,0)	→	z0	(12)
div(0,z0)	→	0	(30)
if(false,z0,z1)	→	z1	(26)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c2(-^#(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)]	=	1 · x₁ + 0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[lt(x₁, x₂)]	=	0
[div(x₁, x₂)]	=	0
[-(x₁, x₂)]	=	1 · x₁ + 0
[if(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[-^#(x₁, x₂)]	=	1 · x₁ + 0
[lt^#(x₁, x₂)]	=	0
[if^#(x₁, x₂, x₃)]	=	2 · x₂ + 0 + 2 · x₂ · x₂
[div^#(x₁, x₂)]	=	1 · x₁ · x₁ + 0
[0]	=	0
[false]	=	1
[s(x₁)]	=	1 + 1 · x₁
[true]	=	1

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

-^#(z0,0)	→	c	(13)
-^#(0,s(z0))	→	c1	(15)
-^#(s(z0),s(z1))	→	c2(-^#(z0,z1))	(17)
lt^#(z0,0)	→	c3	(19)
lt^#(0,s(z0))	→	c4	(21)
lt^#(s(z0),s(z1))	→	c5(lt^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c6	(25)
if^#(false,z0,z1)	→	c7	(27)
div^#(z0,0)	→	c8	(29)
div^#(0,z0)	→	c9	(31)
div^#(s(z0),s(z1))	→	c10(if^#(lt(z0,z1),0,s(div(-(z0,z1),s(z1)))),lt^#(z0,z1),div^#(-(z0,z1),s(z1)),-^#(z0,z1))	(33)
-(0,s(z0))	→	0	(14)
-(s(z0),s(z1))	→	-(z0,z1)	(16)
-(z0,0)	→	z0	(12)

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).