The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
↳3 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳6 CdtProblem
↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtUsableRulesProof (⇔, 1 ms)
↳10 CdtProblem
↳11 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 38 ms)
↳12 CdtProblem
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 BOUNDS(1, 1)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Rewrite Strategy: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of natsFrom: s, activate, natsFrom
The following defined symbols can occur below the 0th argument of head: splitAt, snd, afterNth
The following defined symbols can occur below the 0th argument of snd: splitAt
The following defined symbols can occur below the 0th argument of fst: splitAt
The following defined symbols can occur below the 0th argument of s: s, activate, natsFrom

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
splitAt(0, XS) → pair(nil, XS)
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
activate(X) → X
snd(pair(XS, YS)) → YS
take(N, XS) → fst(splitAt(N, XS))
activate(n__s(X)) → s(activate(X))
afterNth(N, XS) → snd(splitAt(N, XS))
activate(n__natsFrom(X)) → natsFrom(activate(X))
s(X) → n__s(X)
fst(pair(XS, YS)) → XS
tail(cons(N, XS)) → activate(XS)
natsFrom(X) → n__natsFrom(X)
sel(N, XS) → head(afterNth(N, XS))

Rewrite Strategy: INNERMOST

(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
splitAt(0, z0) → pair(nil, z0)
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
head(cons(z0, z1)) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
snd(pair(z0, z1)) → z1
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
s(z0) → n__s(z0)
fst(pair(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
sel(z0, z1) → head(afterNth(z0, z1))
Tuples:

NATSFROM(z0) → c
NATSFROM(z0) → c1
SPLITAT(0, z0) → c2
U(pair(z0, z1), z2, z3, z4) → c3(ACTIVATE(z3))
HEAD(cons(z0, z1)) → c4
ACTIVATE(z0) → c5
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(NATSFROM(activate(z0)), ACTIVATE(z0))
SND(pair(z0, z1)) → c8
TAKE(z0, z1) → c9(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c10(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
S(z0) → c11
FST(pair(z0, z1)) → c12
TAIL(cons(z0, z1)) → c13(ACTIVATE(z1))
SEL(z0, z1) → c14(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
S tuples:

NATSFROM(z0) → c
NATSFROM(z0) → c1
SPLITAT(0, z0) → c2
U(pair(z0, z1), z2, z3, z4) → c3(ACTIVATE(z3))
HEAD(cons(z0, z1)) → c4
ACTIVATE(z0) → c5
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(NATSFROM(activate(z0)), ACTIVATE(z0))
SND(pair(z0, z1)) → c8
TAKE(z0, z1) → c9(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c10(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
S(z0) → c11
FST(pair(z0, z1)) → c12
TAIL(cons(z0, z1)) → c13(ACTIVATE(z1))
SEL(z0, z1) → c14(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
K tuples:none
Defined Rule Symbols:

natsFrom, splitAt, u, head, activate, snd, take, afterNth, s, fst, tail, sel

Defined Pair Symbols:

NATSFROM, SPLITAT, U, HEAD, ACTIVATE, SND, TAKE, AFTERNTH, S, FST, TAIL, SEL

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

U(pair(z0, z1), z2, z3, z4) → c3(ACTIVATE(z3))
TAIL(cons(z0, z1)) → c13(ACTIVATE(z1))
Removed 11 trailing nodes:

S(z0) → c11
TAKE(z0, z1) → c9(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c10(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
SND(pair(z0, z1)) → c8
ACTIVATE(z0) → c5
SEL(z0, z1) → c14(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
NATSFROM(z0) → c1
FST(pair(z0, z1)) → c12
HEAD(cons(z0, z1)) → c4
SPLITAT(0, z0) → c2
NATSFROM(z0) → c

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
splitAt(0, z0) → pair(nil, z0)
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
head(cons(z0, z1)) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
snd(pair(z0, z1)) → z1
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
s(z0) → n__s(z0)
fst(pair(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
sel(z0, z1) → head(afterNth(z0, z1))
Tuples:

ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(NATSFROM(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(NATSFROM(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

natsFrom, splitAt, u, head, activate, snd, take, afterNth, s, fst, tail, sel

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7

(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
splitAt(0, z0) → pair(nil, z0)
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
head(cons(z0, z1)) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
snd(pair(z0, z1)) → z1
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
s(z0) → n__s(z0)
fst(pair(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
sel(z0, z1) → head(afterNth(z0, z1))
Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

natsFrom, splitAt, u, head, activate, snd, take, afterNth, s, fst, tail, sel

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
splitAt(0, z0) → pair(nil, z0)
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
head(cons(z0, z1)) → z0
activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
snd(pair(z0, z1)) → z1
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
s(z0) → n__s(z0)
fst(pair(z0, z1)) → z0
tail(cons(z0, z1)) → activate(z1)
sel(z0, z1) → head(afterNth(z0, z1))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [2]x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(n__natsFrom(x1)) = [1] + x1   
POL(n__s(x1)) = [1] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__natsFrom(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7

(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(14) BOUNDS(1, 1)