Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Strategy_removed_CSR_05/Ex49_GM04)

The rewrite relation of the following TRS is considered.

minus(0,Y)	→	0	(1)
minus(s(X),s(Y))	→	minus(X,Y)	(2)
geq(X,0)	→	true	(3)
geq(0,s(Y))	→	false	(4)
geq(s(X),s(Y))	→	geq(X,Y)	(5)
div(0,s(Y))	→	0	(6)
div(s(X),s(Y))	→	if(geq(X,Y),s(div(minus(X,Y),s(Y))),0)	(7)
if(true,X,Y)	→	X	(8)
if(false,X,Y)	→	Y	(9)

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:

minus^#(0,z0)

→

(11)

originates from

minus(0,z0)

→

(10)

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

→

c1(minus^#(z0,z1))

(13)

originates from

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

→

minus(z0,z1)

(12)

geq^#(z0,0)

→

(15)

originates from

geq(z0,0)

→

true

(14)

geq^#(0,s(z0))

→

(17)

originates from

geq(0,s(z0))

→

false

(16)

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

→

c4(geq^#(z0,z1))

(19)

originates from

geq(s(z0),s(z1))

→

geq(z0,z1)

(18)

div^#(0,s(z0))

→

(21)

originates from

div(0,s(z0))

→

(20)

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

→

c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))

(23)

originates from

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

→

if(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0)

(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)

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

minus^#(0,z0)

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

geq^#(z0,0)

geq^#(0,s(z0))

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

div^#(0,s(z0))

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

if^#(true,z0,z1)

if^#(false,z0,z1)

1.1 Rule Shifting

The rules

div^#(0,s(z0))

→

(21)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[c7]	=	0
[c8]	=	0
[minus(x₁, x₂)]	=	0
[geq(x₁, x₂)]	=	0
[div(x₁, x₂)]	=	1
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[minus^#(x₁, x₂)]	=	0
[geq^#(x₁, x₂)]	=	0
[div^#(x₁, x₂)]	=	1 + 1 · x₂
[if^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[0]	=	0
[true]	=	1
[s(x₁)]	=	0
[false]	=	1

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

minus^#(0,z0)	→	c	(11)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(13)
geq^#(z0,0)	→	c2	(15)
geq^#(0,s(z0))	→	c3	(17)
geq^#(s(z0),s(z1))	→	c4(geq^#(z0,z1))	(19)
div^#(0,s(z0))	→	c5	(21)
div^#(s(z0),s(z1))	→	c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c7	(25)
if^#(false,z0,z1)	→	c8	(27)

1.1.1 Rule Shifting

The rules

geq^#(z0,0)	→	c2	(15)
geq^#(0,s(z0))	→	c3	(17)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[c7]	=	0
[c8]	=	0
[minus(x₁, x₂)]	=	0
[geq(x₁, x₂)]	=	0
[div(x₁, x₂)]	=	1
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[minus^#(x₁, x₂)]	=	0
[geq^#(x₁, x₂)]	=	1
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[0]	=	0
[true]	=	1
[s(x₁)]	=	1
[false]	=	1

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

minus^#(0,z0)	→	c	(11)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(13)
geq^#(z0,0)	→	c2	(15)
geq^#(0,s(z0))	→	c3	(17)
geq^#(s(z0),s(z1))	→	c4(geq^#(z0,z1))	(19)
div^#(0,s(z0))	→	c5	(21)
div^#(s(z0),s(z1))	→	c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c7	(25)
if^#(false,z0,z1)	→	c8	(27)
minus(s(z0),s(z1))	→	minus(z0,z1)	(12)
minus(0,z0)	→	0	(10)

1.1.1.1 Rule Shifting

The rules

geq^#(s(z0),s(z1))	→	c4(geq^#(z0,z1))	(19)
div^#(s(z0),s(z1))	→	c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[c7]	=	0
[c8]	=	0
[minus(x₁, x₂)]	=	0
[geq(x₁, x₂)]	=	1 · x₁ + 0
[div(x₁, x₂)]	=	1
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[minus^#(x₁, x₂)]	=	0
[geq^#(x₁, x₂)]	=	1 · x₁ + 0
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[0]	=	0
[true]	=	1
[s(x₁)]	=	1 + 1 · x₁
[false]	=	1

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

minus^#(0,z0)	→	c	(11)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(13)
geq^#(z0,0)	→	c2	(15)
geq^#(0,s(z0))	→	c3	(17)
geq^#(s(z0),s(z1))	→	c4(geq^#(z0,z1))	(19)
div^#(0,s(z0))	→	c5	(21)
div^#(s(z0),s(z1))	→	c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c7	(25)
if^#(false,z0,z1)	→	c8	(27)
minus(s(z0),s(z1))	→	minus(z0,z1)	(12)
minus(0,z0)	→	0	(10)

1.1.1.1.1 Rule Shifting

The rules

minus^#(0,z0)

→

(11)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[c7]	=	0
[c8]	=	0
[minus(x₁, x₂)]	=	0
[geq(x₁, x₂)]	=	1 · x₁ + 0
[div(x₁, x₂)]	=	1 + 1 · x₁
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[minus^#(x₁, x₂)]	=	1
[geq^#(x₁, x₂)]	=	0
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[0]	=	0
[true]	=	1
[s(x₁)]	=	1
[false]	=	1

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

minus^#(0,z0)	→	c	(11)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(13)
geq^#(z0,0)	→	c2	(15)
geq^#(0,s(z0))	→	c3	(17)
geq^#(s(z0),s(z1))	→	c4(geq^#(z0,z1))	(19)
div^#(0,s(z0))	→	c5	(21)
div^#(s(z0),s(z1))	→	c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c7	(25)
if^#(false,z0,z1)	→	c8	(27)
minus(s(z0),s(z1))	→	minus(z0,z1)	(12)
minus(0,z0)	→	0	(10)

1.1.1.1.1.1 Rule Shifting

The rules

if^#(true,z0,z1)	→	c7	(25)
if^#(false,z0,z1)	→	c8	(27)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[c7]	=	0
[c8]	=	0
[minus(x₁, x₂)]	=	0
[geq(x₁, x₂)]	=	0
[div(x₁, x₂)]	=	1 + 1 · x₁
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[minus^#(x₁, x₂)]	=	0
[geq^#(x₁, x₂)]	=	0
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	1 + 1 · x₃
[0]	=	0
[true]	=	0
[s(x₁)]	=	1
[false]	=	0

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

minus^#(0,z0)	→	c	(11)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(13)
geq^#(z0,0)	→	c2	(15)
geq^#(0,s(z0))	→	c3	(17)
geq^#(s(z0),s(z1))	→	c4(geq^#(z0,z1))	(19)
div^#(0,s(z0))	→	c5	(21)
div^#(s(z0),s(z1))	→	c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c7	(25)
if^#(false,z0,z1)	→	c8	(27)
minus(s(z0),s(z1))	→	minus(z0,z1)	(12)
minus(0,z0)	→	0	(10)

1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c1(minus^#(z0,z1))

(13)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3]	=	0
[c4(x₁)]	=	1 · x₁ + 0
[c5]	=	0
[c6(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[c7]	=	0
[c8]	=	0
[minus(x₁, x₂)]	=	1
[geq(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[div(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[if(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[minus^#(x₁, x₂)]	=	1 · x₁ + 0
[geq^#(x₁, x₂)]	=	0
[div^#(x₁, x₂)]	=	1 · x₁ + 0
[if^#(x₁, x₂, x₃)]	=	0
[0]	=	1
[true]	=	0
[s(x₁)]	=	1 + 1 · x₁
[false]	=	1

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

minus^#(0,z0)	→	c	(11)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(13)
geq^#(z0,0)	→	c2	(15)
geq^#(0,s(z0))	→	c3	(17)
geq^#(s(z0),s(z1))	→	c4(geq^#(z0,z1))	(19)
div^#(0,s(z0))	→	c5	(21)
div^#(s(z0),s(z1))	→	c6(if^#(geq(z0,z1),s(div(minus(z0,z1),s(z1))),0),geq^#(z0,z1),div^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
if^#(true,z0,z1)	→	c7	(25)
if^#(false,z0,z1)	→	c8	(27)
minus(s(z0),s(z1))	→	minus(z0,z1)	(12)
minus(0,z0)	→	0	(10)

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).