Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Rubio_04/division)

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)	→	ifMinus(le(s(X),Y),s(X),Y)	(5)
ifMinus(true,s(X),Y)	→	0	(6)
ifMinus(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)

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)

→

(11)

originates from

le(0,z0)

→

true

(10)

le^#(s(z0),0)

→

(13)

originates from

le(s(z0),0)

→

false

(12)

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

→

c2(le^#(z0,z1))

(15)

originates from

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

→

le(z0,z1)

(14)

minus^#(0,z0)

→

(17)

originates from

minus(0,z0)

→

(16)

minus^#(s(z0),z1)

→

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

(19)

originates from

minus(s(z0),z1)

→

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

(18)

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

→

(21)

originates from

ifMinus(true,s(z0),z1)

→

(20)

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

→

c6(minus^#(z0,z1))

(23)

originates from

ifMinus(false,s(z0),z1)

→

s(minus(z0,z1))

(22)

quot^#(0,s(z0))

→

(25)

originates from

quot(0,s(z0))

→

(24)

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

→

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

(27)

originates from

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

→

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

(26)

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)

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

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

quot^#(0,s(z0))

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

1.1 Usable Rules

We remove the following rules since they are not usable.

quot(0,s(z0))	→	0	(24)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(26)

1.1.1 Rule Shifting

The rules

quot^#(0,s(z0))

→

(25)

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
[le(x₁, x₂)]	=	1 + 1 · x₂
[minus(x₁, x₂)]	=	1 + 1 · x₂
[ifMinus(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[ifMinus^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 + 1 · x₂
[0]	=	1
[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	(11)
le^#(s(z0),0)	→	c1	(13)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(15)
minus^#(0,z0)	→	c3	(17)
minus^#(s(z0),z1)	→	c4(ifMinus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(19)
ifMinus^#(true,s(z0),z1)	→	c5	(21)
ifMinus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(23)
quot^#(0,s(z0))	→	c7	(25)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(27)

1.1.1.1 Rule Shifting

The rules

minus^#(0,z0)	→	c3	(17)
ifMinus^#(true,s(z0),z1)	→	c5	(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(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[le(x₁, x₂)]	=	1 + 1 · x₂
[minus(x₁, x₂)]	=	1 · x₁ + 0
[ifMinus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	1
[ifMinus^#(x₁, x₂, x₃)]	=	1
[quot^#(x₁, x₂)]	=	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	(11)
le^#(s(z0),0)	→	c1	(13)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(15)
minus^#(0,z0)	→	c3	(17)
minus^#(s(z0),z1)	→	c4(ifMinus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(19)
ifMinus^#(true,s(z0),z1)	→	c5	(21)
ifMinus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(23)
quot^#(0,s(z0))	→	c7	(25)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(27)
ifMinus(true,s(z0),z1)	→	0	(20)
ifMinus(false,s(z0),z1)	→	s(minus(z0,z1))	(22)
minus(s(z0),z1)	→	ifMinus(le(s(z0),z1),s(z0),z1)	(18)
minus(0,z0)	→	0	(16)

1.1.1.1.1 Rule Shifting

The rules

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

→

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

(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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 + 1 · x₁
[ifMinus(x₁, x₂, x₃)]	=	1 + 1 · x₂
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	3
[ifMinus^#(x₁, x₂, x₃)]	=	3
[quot^#(x₁, x₂)]	=	2 · x₁ + 0
[0]	=	0
[s(x₁)]	=	3 + 1 · x₁
[false]	=	0
[true]	=	0

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

le^#(0,z0)	→	c	(11)
le^#(s(z0),0)	→	c1	(13)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(15)
minus^#(0,z0)	→	c3	(17)
minus^#(s(z0),z1)	→	c4(ifMinus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(19)
ifMinus^#(true,s(z0),z1)	→	c5	(21)
ifMinus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(23)
quot^#(0,s(z0))	→	c7	(25)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(27)
ifMinus(true,s(z0),z1)	→	0	(20)
ifMinus(false,s(z0),z1)	→	s(minus(z0,z1))	(22)
minus(s(z0),z1)	→	ifMinus(le(s(z0),z1),s(z0),z1)	(18)
minus(0,z0)	→	0	(16)

1.1.1.1.1.1 Rule Shifting

The rules

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

→

c6(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(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[ifMinus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	2 · x₁ + 0
[ifMinus^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · 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	(11)
le^#(s(z0),0)	→	c1	(13)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(15)
minus^#(0,z0)	→	c3	(17)
minus^#(s(z0),z1)	→	c4(ifMinus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(19)
ifMinus^#(true,s(z0),z1)	→	c5	(21)
ifMinus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(23)
quot^#(0,s(z0))	→	c7	(25)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(27)
ifMinus(true,s(z0),z1)	→	0	(20)
ifMinus(false,s(z0),z1)	→	s(minus(z0,z1))	(22)
minus(s(z0),z1)	→	ifMinus(le(s(z0),z1),s(z0),z1)	(18)
minus(0,z0)	→	0	(16)

1.1.1.1.1.1.1 Rule Shifting

The rules

minus^#(s(z0),z1)

→

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

(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(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[ifMinus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[le^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	1 + 1 · x₁
[ifMinus^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · x₁ · x₁ + 0
[0]	=	0
[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	(11)
le^#(s(z0),0)	→	c1	(13)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(15)
minus^#(0,z0)	→	c3	(17)
minus^#(s(z0),z1)	→	c4(ifMinus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(19)
ifMinus^#(true,s(z0),z1)	→	c5	(21)
ifMinus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(23)
quot^#(0,s(z0))	→	c7	(25)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(27)
ifMinus(true,s(z0),z1)	→	0	(20)
ifMinus(false,s(z0),z1)	→	s(minus(z0,z1))	(22)
minus(s(z0),z1)	→	ifMinus(le(s(z0),z1),s(z0),z1)	(18)
minus(0,z0)	→	0	(16)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

le^#(0,z0)	→	c	(11)
le^#(s(z0),0)	→	c1	(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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[ifMinus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[le^#(x₁, x₂)]	=	1
[minus^#(x₁, x₂)]	=	1 + 1 · x₁
[ifMinus^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · x₁ · x₁ + 0
[0]	=	0
[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	(11)
le^#(s(z0),0)	→	c1	(13)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(15)
minus^#(0,z0)	→	c3	(17)
minus^#(s(z0),z1)	→	c4(ifMinus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(19)
ifMinus^#(true,s(z0),z1)	→	c5	(21)
ifMinus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(23)
quot^#(0,s(z0))	→	c7	(25)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(27)
ifMinus(true,s(z0),z1)	→	0	(20)
ifMinus(false,s(z0),z1)	→	s(minus(z0,z1))	(22)
minus(s(z0),z1)	→	ifMinus(le(s(z0),z1),s(z0),z1)	(18)
minus(0,z0)	→	0	(16)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c2(le^#(z0,z1))

(15)

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
[le(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	1 · x₁ + 0
[ifMinus(x₁, x₂, x₃)]	=	1 · x₂ + 0
[le^#(x₁, x₂)]	=	1 + 1 · x₁
[minus^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[ifMinus^#(x₁, x₂, x₃)]	=	1 · x₂ · x₂ + 0
[quot^#(x₁, x₂)]	=	1 · x₁ · x₁ · x₁ + 0
[0]	=	0
[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	(11)
le^#(s(z0),0)	→	c1	(13)
le^#(s(z0),s(z1))	→	c2(le^#(z0,z1))	(15)
minus^#(0,z0)	→	c3	(17)
minus^#(s(z0),z1)	→	c4(ifMinus^#(le(s(z0),z1),s(z0),z1),le^#(s(z0),z1))	(19)
ifMinus^#(true,s(z0),z1)	→	c5	(21)
ifMinus^#(false,s(z0),z1)	→	c6(minus^#(z0,z1))	(23)
quot^#(0,s(z0))	→	c7	(25)
quot^#(s(z0),s(z1))	→	c8(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(27)
ifMinus(true,s(z0),z1)	→	0	(20)
ifMinus(false,s(z0),z1)	→	s(minus(z0,z1))	(22)
minus(s(z0),z1)	→	ifMinus(le(s(z0),z1),s(z0),z1)	(18)
minus(0,z0)	→	0	(16)

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).