Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/AG01/#3.53)

The rewrite relation of the following TRS is considered.

minus(x,0)	→	x	(1)
minus(s(x),s(y))	→	minus(x,y)	(2)
quot(0,s(y))	→	0	(3)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(4)
app(nil,y)	→	y	(5)
app(add(n,x),y)	→	add(n,app(x,y))	(6)
reverse(nil)	→	nil	(7)
reverse(add(n,x))	→	app(reverse(x),add(n,nil))	(8)
shuffle(nil)	→	nil	(9)
shuffle(add(n,x))	→	add(n,shuffle(reverse(x)))	(10)
concat(leaf,y)	→	y	(11)
concat(cons(u,v),y)	→	cons(u,concat(v,y))	(12)
less_leaves(x,leaf)	→	false	(13)
less_leaves(leaf,cons(w,z))	→	true	(14)
less_leaves(cons(u,v),cons(w,z))	→	less_leaves(concat(u,v),concat(w,z))	(15)

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:

minus^#(z0,0)

→

(17)

originates from

minus(z0,0)

→

(16)

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

→

c1(minus^#(z0,z1))

(19)

originates from

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

→

minus(z0,z1)

(18)

quot^#(0,s(z0))

→

(21)

originates from

quot(0,s(z0))

→

(20)

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

→

c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))

(23)

originates from

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

→

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

(22)

app^#(nil,z0)

→

(25)

originates from

app(nil,z0)

→

(24)

app^#(add(z0,z1),z2)

→

c5(app^#(z1,z2))

(27)

originates from

app(add(z0,z1),z2)

→

add(z0,app(z1,z2))

(26)

reverse^#(nil)

→

(28)

originates from

reverse(nil)

→

nil

(7)

reverse^#(add(z0,z1))

→

c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))

(30)

originates from

reverse(add(z0,z1))

→

app(reverse(z1),add(z0,nil))

(29)

shuffle^#(nil)

→

(31)

originates from

shuffle(nil)

→

nil

(9)

shuffle^#(add(z0,z1))

→

c9(shuffle^#(reverse(z1)),reverse^#(z1))

(33)

originates from

shuffle(add(z0,z1))

→

add(z0,shuffle(reverse(z1)))

(32)

concat^#(leaf,z0)

→

c10

(35)

originates from

concat(leaf,z0)

→

(34)

concat^#(cons(z0,z1),z2)

→

c11(concat^#(z1,z2))

(37)

originates from

concat(cons(z0,z1),z2)

→

cons(z0,concat(z1,z2))

(36)

less_leaves^#(z0,leaf)

→

c12

(39)

originates from

less_leaves(z0,leaf)

→

false

(38)

less_leaves^#(leaf,cons(z0,z1))

→

c13

(41)

originates from

less_leaves(leaf,cons(z0,z1))

→

true

(40)

less_leaves^#(cons(z0,z1),cons(z2,z3))

→

c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))

(43)

originates from

less_leaves(cons(z0,z1),cons(z2,z3))

→

less_leaves(concat(z0,z1),concat(z2,z3))

(42)

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

minus^#(z0,0)

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

quot^#(0,s(z0))

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

app^#(nil,z0)

app^#(add(z0,z1),z2)

reverse^#(nil)

reverse^#(add(z0,z1))

shuffle^#(nil)

shuffle^#(add(z0,z1))

concat^#(leaf,z0)

concat^#(cons(z0,z1),z2)

less_leaves^#(z0,leaf)

less_leaves^#(leaf,cons(z0,z1))

less_leaves^#(cons(z0,z1),cons(z2,z3))

1.1 Usable Rules

We remove the following rules since they are not usable.

quot(0,s(z0))	→	0	(20)
quot(s(z0),s(z1))	→	s(quot(minus(z0,z1),s(z1)))	(22)
shuffle(nil)	→	nil	(9)
shuffle(add(z0,z1))	→	add(z0,shuffle(reverse(z1)))	(32)
less_leaves(z0,leaf)	→	false	(38)
less_leaves(leaf,cons(z0,z1))	→	true	(40)
less_leaves(cons(z0,z1),cons(z2,z3))	→	less_leaves(concat(z0,z1),concat(z2,z3))	(42)

1.1.1 Rule Shifting

The rules

shuffle^#(nil)	→	c8	(31)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[reverse(x₁)]	=	0
[app(x₁, x₂)]	=	3
[concat(x₁, x₂)]	=	1 + 3 · x₁ + 3 · x₂
[minus^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	3 · x₂ + 0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	2
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	2
[leaf]	=	3
[cons(x₁, x₂)]	=	3 + 1 · x₂
[nil]	=	0
[add(x₁, x₂)]	=	1 · x₁ + 0
[0]	=	3
[s(x₁)]	=	1 · x₁ + 0

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)

1.1.1.1 Rule Shifting

The rules

minus^#(z0,0)	→	c	(17)
quot^#(0,s(z0))	→	c2	(21)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 · x₁ + 0
[reverse(x₁)]	=	1
[app(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[concat(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus^#(x₁, x₂)]	=	1
[quot^#(x₁, x₂)]	=	1 · x₁ + 0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	0
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	1
[cons(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[nil]	=	1
[add(x₁, x₂)]	=	1 + 1 · x₁
[0]	=	1
[s(x₁)]	=	1 + 1 · x₁

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
concat(leaf,z0)	→	z0	(34)
concat(cons(z0,z1),z2)	→	cons(z0,concat(z1,z2))	(36)
minus(s(z0),s(z1))	→	minus(z0,z1)	(18)
minus(z0,0)	→	z0	(16)

1.1.1.1.1 Rule Shifting

The rules

quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
concat^#(leaf,z0)	→	c10	(35)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 · x₁ + 0
[reverse(x₁)]	=	1 + 1 · x₁
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[concat(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[minus^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	1 · x₁ + 0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	1 · x₁ + 0
[concat^#(x₁, x₂)]	=	1
[less_leaves^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	1
[cons(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
reverse(nil)	→	nil	(7)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(26)
concat(leaf,z0)	→	z0	(34)
app(nil,z0)	→	z0	(24)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(29)
concat(cons(z0,z1),z2)	→	cons(z0,concat(z1,z2))	(36)
minus(s(z0),s(z1))	→	minus(z0,z1)	(18)
minus(z0,0)	→	z0	(16)

1.1.1.1.1.1 Rule Shifting

The rules

reverse^#(nil)

→

(28)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 + 1 · x₁
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[concat(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[minus^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	1 · x₁ + 0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	1
[shuffle^#(x₁)]	=	1 · x₁ + 0
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	0
[leaf]	=	0
[cons(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
reverse(nil)	→	nil	(7)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(26)
app(nil,z0)	→	z0	(24)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(29)
minus(s(z0),s(z1))	→	minus(z0,z1)	(18)
minus(z0,0)	→	z0	(16)

1.1.1.1.1.1.1 Rule Shifting

The rules

shuffle^#(add(z0,z1))

→

c9(shuffle^#(reverse(z1)),reverse^#(z1))

(33)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 + 1 · x₁
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[concat(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[minus^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	1 · x₁ + 0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	1 · x₁ + 0
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	0
[cons(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[0]	=	0
[s(x₁)]	=	1 + 1 · x₁

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
reverse(nil)	→	nil	(7)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(26)
concat(leaf,z0)	→	z0	(34)
app(nil,z0)	→	z0	(24)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(29)
concat(cons(z0,z1),z2)	→	cons(z0,concat(z1,z2))	(36)
minus(s(z0),s(z1))	→	minus(z0,z1)	(18)
minus(z0,0)	→	z0	(16)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c1(minus^#(z0,z1))

(19)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 · x₁ + 0
[reverse(x₁)]	=	0
[app(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[concat(x₁, x₂)]	=	2 · x₂ + 0 + 1 · x₁ · x₁
[minus^#(x₁, x₂)]	=	1 · x₁ + 0
[quot^#(x₁, x₂)]	=	2 · x₁ · x₁ + 0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	0
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	0
[leaf]	=	1
[cons(x₁, x₂)]	=	2
[nil]	=	0
[add(x₁, x₂)]	=	0
[0]	=	2
[s(x₁)]	=	1 + 1 · x₁

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
minus(s(z0),s(z1))	→	minus(z0,z1)	(18)
minus(z0,0)	→	z0	(16)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

concat^#(cons(z0,z1),z2)

→

c11(concat^#(z1,z2))

(37)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	0
[reverse(x₁)]	=	0
[app(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[concat(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[minus^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[shuffle^#(x₁)]	=	0
[concat^#(x₁, x₂)]	=	2 · x₁ + 0 + 2 · x₂
[less_leaves^#(x₁, x₂)]	=	2 · x₂ + 0 + 2 · x₂ · x₂ + 1 · x₁ · x₁
[leaf]	=	1
[cons(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[nil]	=	0
[add(x₁, x₂)]	=	0
[0]	=	2
[s(x₁)]	=	0

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
concat(leaf,z0)	→	z0	(34)
concat(cons(z0,z1),z2)	→	cons(z0,concat(z1,z2))	(36)

1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

reverse^#(add(z0,z1))

→

c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))

(30)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 · x₁ + 0
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[concat(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	2 · x₁ · x₁ + 0
[app^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	1 · x₁ + 0
[shuffle^#(x₁)]	=	2 · x₁ · x₁ + 0
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	0
[leaf]	=	2
[cons(x₁, x₂)]	=	0
[nil]	=	0
[add(x₁, x₂)]	=	2 + 1 · x₂
[0]	=	2
[s(x₁)]	=	1 + 1 · x₁

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
reverse(nil)	→	nil	(7)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(26)
app(nil,z0)	→	z0	(24)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(29)
minus(s(z0),s(z1))	→	minus(z0,z1)	(18)
minus(z0,0)	→	z0	(16)

1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

app^#(nil,z0)

→

(25)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 · x₁ + 0
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[concat(x₁, x₂)]	=	2 + 1 · x₁ + 2 · x₁ · x₂
[minus^#(x₁, x₂)]	=	2
[quot^#(x₁, x₂)]	=	1 · x₁ · x₁ + 0
[app^#(x₁, x₂)]	=	1 + 1 · x₂
[reverse^#(x₁)]	=	2 · x₁ + 0
[shuffle^#(x₁)]	=	1 · x₁ · x₁ + 0
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	0
[leaf]	=	1
[cons(x₁, x₂)]	=	2
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₂
[0]	=	2
[s(x₁)]	=	2 + 1 · x₁

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
reverse(nil)	→	nil	(7)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(26)
app(nil,z0)	→	z0	(24)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(29)
minus(s(z0),s(z1))	→	minus(z0,z1)	(18)
minus(z0,0)	→	z0	(16)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

app^#(add(z0,z1),z2)

→

c5(app^#(z1,z2))

(27)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2]	=	0
[c3(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c4]	=	0
[c5(x₁)]	=	1 · x₁ + 0
[c6]	=	0
[c7(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c8]	=	0
[c9(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13]	=	0
[c14(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[minus(x₁, x₂)]	=	1 · x₁ · x₁ · x₁ + 0 + 1 · x₂ · x₂ · x₂
[reverse(x₁)]	=	1 · x₁ + 0
[app(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[concat(x₁, x₂)]	=	1 · x₁ · x₁ · x₁ + 0 + 1 · x₂ · x₁ · x₁ + 1 · x₂ · x₂ · x₁ + 1 · x₂ · x₂ · x₂
[minus^#(x₁, x₂)]	=	0
[quot^#(x₁, x₂)]	=	0
[app^#(x₁, x₂)]	=	1 · x₁ + 0
[reverse^#(x₁)]	=	1 + 1 · x₁ · x₁
[shuffle^#(x₁)]	=	1 · x₁ · x₁ · x₁ + 0
[concat^#(x₁, x₂)]	=	0
[less_leaves^#(x₁, x₂)]	=	0
[leaf]	=	1
[cons(x₁, x₂)]	=	0
[nil]	=	0
[add(x₁, x₂)]	=	1 + 1 · x₂
[0]	=	1
[s(x₁)]	=	0

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

minus^#(z0,0)	→	c	(17)
minus^#(s(z0),s(z1))	→	c1(minus^#(z0,z1))	(19)
quot^#(0,s(z0))	→	c2	(21)
quot^#(s(z0),s(z1))	→	c3(quot^#(minus(z0,z1),s(z1)),minus^#(z0,z1))	(23)
app^#(nil,z0)	→	c4	(25)
app^#(add(z0,z1),z2)	→	c5(app^#(z1,z2))	(27)
reverse^#(nil)	→	c6	(28)
reverse^#(add(z0,z1))	→	c7(app^#(reverse(z1),add(z0,nil)),reverse^#(z1))	(30)
shuffle^#(nil)	→	c8	(31)
shuffle^#(add(z0,z1))	→	c9(shuffle^#(reverse(z1)),reverse^#(z1))	(33)
concat^#(leaf,z0)	→	c10	(35)
concat^#(cons(z0,z1),z2)	→	c11(concat^#(z1,z2))	(37)
less_leaves^#(z0,leaf)	→	c12	(39)
less_leaves^#(leaf,cons(z0,z1))	→	c13	(41)
less_leaves^#(cons(z0,z1),cons(z2,z3))	→	c14(less_leaves^#(concat(z0,z1),concat(z2,z3)),concat^#(z0,z1),concat^#(z2,z3))	(43)
reverse(nil)	→	nil	(7)
app(add(z0,z1),z2)	→	add(z0,app(z1,z2))	(26)
app(nil,z0)	→	z0	(24)
reverse(add(z0,z1))	→	app(reverse(z1),add(z0,nil))	(29)

1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty

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