Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Others/inssort_better)

The relative rewrite relation R/S is considered where R is the following TRS

isort(Cons(x,xs),r) → isort(xs,insert(x,r)) (1)
insert(x',Cons(x,xs)) → insert[Ite][False][Ite](<(x',x),x',Cons(x,xs)) (2)
isort(Nil,r) → r (3)
insert(x,Nil) → Cons(x,Nil) (4)
inssort(xs) → isort(xs,Nil) (5)

and S is the following TRS.

<(S(x),S(y)) → <(x,y) (6)
<(0,S(y)) → True (7)
<(x,0) → False (8)
insert[Ite][False][Ite](False,x',Cons(x,xs)) → Cons(x,insert(x',xs)) (9)
insert[Ite][False][Ite](True,x,r) → Cons(x,r) (10)
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:
isort#(Cons(z0,z1),z2) → c5(isort#(z1,insert(z0,z2)),insert#(z0,z2)) (12)
originates from
isort(Cons(z0,z1),z2) → isort(z1,insert(z0,z2)) (11)
isort#(Nil,z0) → c6 (14)
originates from
isort(Nil,z0) → z0 (13)
insert#(z0,Cons(z1,z2)) → c7(insert[Ite][False][Ite]#(<(z0,z1),z0,Cons(z1,z2)),<#(z0,z1)) (16)
originates from
insert(z0,Cons(z1,z2)) → insert[Ite][False][Ite](<(z0,z1),z0,Cons(z1,z2)) (15)
insert#(z0,Nil) → c8 (18)
originates from
insert(z0,Nil) → Cons(z0,Nil) (17)
inssort#(z0) → c9(isort#(z0,Nil)) (20)
originates from
inssort(z0) → isort(z0,Nil) (19)
<#(S(z0),S(z1)) → c(<#(z0,z1)) (22)
originates from
<(S(z0),S(z1)) → <(z0,z1) (21)
<#(0,S(z0)) → c1 (24)
originates from
<(0,S(z0)) → True (23)
<#(z0,0) → c2 (26)
originates from
<(z0,0) → False (25)
insert[Ite][False][Ite]#(False,z0,Cons(z1,z2)) → c3(insert#(z0,z2)) (28)
originates from
insert[Ite][False][Ite](False,z0,Cons(z1,z2)) → Cons(z1,insert(z0,z2)) (27)
insert[Ite][False][Ite]#(True,z0,z1) → c4 (30)
originates from
insert[Ite][False][Ite](True,z0,z1) → Cons(z0,z1) (29)
Moreover, we add the following terms to the innermost strategy.
<#(S(z0),S(z1))
<#(0,S(z0))
<#(z0,0)
insert[Ite][False][Ite]#(False,z0,Cons(z1,z2))
insert[Ite][False][Ite]#(True,z0,z1)
isort#(Cons(z0,z1),z2)
isort#(Nil,z0)
insert#(z0,Cons(z1,z2))
insert#(z0,Nil)
inssort#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.
isort(Cons(z0,z1),z2) → isort(z1,insert(z0,z2)) (11)
isort(Nil,z0) → z0 (13)
inssort(z0) → isort(z0,Nil) (19)

1.1.1 Rule Shifting

The rules
isort#(Nil,z0) → c6 (14)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c(x1)] = 1 · x1 + 0
[c1] = 0
[c2] = 0
[c3(x1)] = 1 · x1 + 0
[c4] = 0
[c5(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c6] = 0
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8] = 0
[c9(x1)] = 1 · x1 + 0
[insert(x1, x2)] = 2 + 2 · x2
[insert[Ite][False][Ite](x1, x2, x3)] = 0
[<(x1, x2)] = 0
[<#(x1, x2)] = 0
[insert[Ite][False][Ite]#(x1, x2, x3)] = 0
[isort#(x1, x2)] = 1
[insert#(x1, x2)] = 0
[inssort#(x1)] = 1 + 3 · x1
[S(x1)] = 3 + 1 · x1
[0] = 3
[True] = 0
[False] = 0
[Cons(x1, x2)] = 0
[Nil] = 2
which has the intended complexity. Here, only the following usable rules have been considered:
<#(S(z0),S(z1)) → c(<#(z0,z1)) (22)
<#(0,S(z0)) → c1 (24)
<#(z0,0) → c2 (26)
insert[Ite][False][Ite]#(False,z0,Cons(z1,z2)) → c3(insert#(z0,z2)) (28)
insert[Ite][False][Ite]#(True,z0,z1) → c4 (30)
isort#(Cons(z0,z1),z2) → c5(isort#(z1,insert(z0,z2)),insert#(z0,z2)) (12)
isort#(Nil,z0) → c6 (14)
insert#(z0,Cons(z1,z2)) → c7(insert[Ite][False][Ite]#(<(z0,z1),z0,Cons(z1,z2)),<#(z0,z1)) (16)
insert#(z0,Nil) → c8 (18)
inssort#(z0) → c9(isort#(z0,Nil)) (20)

1.1.1.1 Rule Shifting

The rules
inssort#(z0) → c9(isort#(z0,Nil)) (20)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c(x1)] = 1 · x1 + 0
[c1] = 0
[c2] = 0
[c3(x1)] = 1 · x1 + 0
[c4] = 0
[c5(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c6] = 0
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8] = 0
[c9(x1)] = 1 · x1 + 0
[insert(x1, x2)] = 1 + 1 · x1 + 1 · x2
[insert[Ite][False][Ite](x1, x2, x3)] = 1 + 1 · x1
[<(x1, x2)] = 1 + 1 · x1
[<#(x1, x2)] = 0
[insert[Ite][False][Ite]#(x1, x2, x3)] = 0
[isort#(x1, x2)] = 0
[insert#(x1, x2)] = 0
[inssort#(x1)] = 1
[S(x1)] = 1 + 1 · x1
[0] = 1
[True] = 1
[False] = 1
[Cons(x1, x2)] = 1
[Nil] = 1
which has the intended complexity. Here, only the following usable rules have been considered:
<#(S(z0),S(z1)) → c(<#(z0,z1)) (22)
<#(0,S(z0)) → c1 (24)
<#(z0,0) → c2 (26)
insert[Ite][False][Ite]#(False,z0,Cons(z1,z2)) → c3(insert#(z0,z2)) (28)
insert[Ite][False][Ite]#(True,z0,z1) → c4 (30)
isort#(Cons(z0,z1),z2) → c5(isort#(z1,insert(z0,z2)),insert#(z0,z2)) (12)
isort#(Nil,z0) → c6 (14)
insert#(z0,Cons(z1,z2)) → c7(insert[Ite][False][Ite]#(<(z0,z1),z0,Cons(z1,z2)),<#(z0,z1)) (16)
insert#(z0,Nil) → c8 (18)
inssort#(z0) → c9(isort#(z0,Nil)) (20)

1.1.1.1.1 Rule Shifting

The rules
isort#(Cons(z0,z1),z2) → c5(isort#(z1,insert(z0,z2)),insert#(z0,z2)) (12)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c(x1)] = 1 · x1 + 0
[c1] = 0
[c2] = 0
[c3(x1)] = 1 · x1 + 0
[c4] = 0
[c5(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c6] = 0
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8] = 0
[c9(x1)] = 1 · x1 + 0
[insert(x1, x2)] = 1 + 1 · x1 + 1 · x2
[insert[Ite][False][Ite](x1, x2, x3)] = 1 + 1 · x2 + 1 · x3
[<(x1, x2)] = 1 + 1 · x1 + 1 · x2
[<#(x1, x2)] = 0
[insert[Ite][False][Ite]#(x1, x2, x3)] = 1 · x2 + 0
[isort#(x1, x2)] = 1 + 1 · x1
[insert#(x1, x2)] = 1 · x1 + 0
[inssort#(x1)] = 1 + 1 · x1
[S(x1)] = 1 + 1 · x1
[0] = 1
[True] = 1
[False] = 1
[Cons(x1, x2)] = 1 + 1 · x1 + 1 · x2
[Nil] = 1
which has the intended complexity. Here, only the following usable rules have been considered:
<#(S(z0),S(z1)) → c(<#(z0,z1)) (22)
<#(0,S(z0)) → c1 (24)
<#(z0,0) → c2 (26)
insert[Ite][False][Ite]#(False,z0,Cons(z1,z2)) → c3(insert#(z0,z2)) (28)
insert[Ite][False][Ite]#(True,z0,z1) → c4 (30)
isort#(Cons(z0,z1),z2) → c5(isort#(z1,insert(z0,z2)),insert#(z0,z2)) (12)
isort#(Nil,z0) → c6 (14)
insert#(z0,Cons(z1,z2)) → c7(insert[Ite][False][Ite]#(<(z0,z1),z0,Cons(z1,z2)),<#(z0,z1)) (16)
insert#(z0,Nil) → c8 (18)
inssort#(z0) → c9(isort#(z0,Nil)) (20)

1.1.1.1.1.1 Rule Shifting

The rules
insert#(z0,Nil) → c8 (18)
are strictly oriented by the following linear polynomial interpretation over the naturals
[c(x1)] = 1 · x1 + 0
[c1] = 0
[c2] = 0
[c3(x1)] = 1 · x1 + 0
[c4] = 0
[c5(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c6] = 0
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8] = 0
[c9(x1)] = 1 · x1 + 0
[insert(x1, x2)] = 1 + 1 · x1 + 1 · x2
[insert[Ite][False][Ite](x1, x2, x3)] = 1 + 1 · x1 + 1 · x2 + 1 · x3
[<(x1, x2)] = 0
[<#(x1, x2)] = 0
[insert[Ite][False][Ite]#(x1, x2, x3)] = 1 + 1 · x1 + 1 · x2
[isort#(x1, x2)] = 1 + 1 · x1
[insert#(x1, x2)] = 1 + 1 · x1
[inssort#(x1)] = 1 + 1 · x1
[S(x1)] = 1 + 1 · x1
[0] = 1
[True] = 0
[False] = 0
[Cons(x1, x2)] = 1 + 1 · x1 + 1 · x2
[Nil] = 1
which has the intended complexity. Here, only the following usable rules have been considered:
<#(S(z0),S(z1)) → c(<#(z0,z1)) (22)
<#(0,S(z0)) → c1 (24)
<#(z0,0) → c2 (26)
insert[Ite][False][Ite]#(False,z0,Cons(z1,z2)) → c3(insert#(z0,z2)) (28)
insert[Ite][False][Ite]#(True,z0,z1) → c4 (30)
isort#(Cons(z0,z1),z2) → c5(isort#(z1,insert(z0,z2)),insert#(z0,z2)) (12)
isort#(Nil,z0) → c6 (14)
insert#(z0,Cons(z1,z2)) → c7(insert[Ite][False][Ite]#(<(z0,z1),z0,Cons(z1,z2)),<#(z0,z1)) (16)
insert#(z0,Nil) → c8 (18)
inssort#(z0) → c9(isort#(z0,Nil)) (20)
<(z0,0) → False (25)
<(0,S(z0)) → True (23)
<(S(z0),S(z1)) → <(z0,z1) (21)

1.1.1.1.1.1.1 Rule Shifting

The rules
insert#(z0,Cons(z1,z2)) → c7(insert[Ite][False][Ite]#(<(z0,z1),z0,Cons(z1,z2)),<#(z0,z1)) (16)
are strictly oriented by the following non-linear polynomial interpretation over the naturals
[c(x1)] = 1 · x1 + 0
[c1] = 0
[c2] = 0
[c3(x1)] = 1 · x1 + 0
[c4] = 0
[c5(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c6] = 0
[c7(x1, x2)] = 1 · x1 + 0 + 1 · x2
[c8] = 0
[c9(x1)] = 1 · x1 + 0
[insert(x1, x2)] = 1 + 1 · x2
[insert[Ite][False][Ite](x1, x2, x3)] = 1 + 1 · x3
[<(x1, x2)] = 0
[<#(x1, x2)] = 0
[insert[Ite][False][Ite]#(x1, x2, x3)] = 2 · x3 + 0
[isort#(x1, x2)] = 2 · x1 + 0 + 2 · x2 + 2 · x1 · x2 + 2 · x1 · x1
[insert#(x1, x2)] = 2 + 2 · x2
[inssort#(x1)] = 2 + 2 · x1 + 2 · x1 · x1
[S(x1)] = 0
[0] = 0
[True] = 0
[False] = 0
[Cons(x1, x2)] = 1 + 1 · x2
[Nil] = 0
which has the intended complexity. Here, only the following usable rules have been considered:
<#(S(z0),S(z1)) → c(<#(z0,z1)) (22)
<#(0,S(z0)) → c1 (24)
<#(z0,0) → c2 (26)
insert[Ite][False][Ite]#(False,z0,Cons(z1,z2)) → c3(insert#(z0,z2)) (28)
insert[Ite][False][Ite]#(True,z0,z1) → c4 (30)
isort#(Cons(z0,z1),z2) → c5(isort#(z1,insert(z0,z2)),insert#(z0,z2)) (12)
isort#(Nil,z0) → c6 (14)
insert#(z0,Cons(z1,z2)) → c7(insert[Ite][False][Ite]#(<(z0,z1),z0,Cons(z1,z2)),<#(z0,z1)) (16)
insert#(z0,Nil) → c8 (18)
inssort#(z0) → c9(isort#(z0,Nil)) (20)
insert(z0,Cons(z1,z2)) → insert[Ite][False][Ite](<(z0,z1),z0,Cons(z1,z2)) (15)
insert[Ite][False][Ite](False,z0,Cons(z1,z2)) → Cons(z1,insert(z0,z2)) (27)
insert(z0,Nil) → Cons(z0,Nil) (17)
insert[Ite][False][Ite](True,z0,z1) → Cons(z0,z1) (29)

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).