(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:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:
U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
AND(tt, z0) → c3(ACTIVATE(z0))
FST(pair(z0, z1)) → c4
HEAD(cons(z0, z1)) → c5
NATSFROM(z0) → c6
NATSFROM(z0) → c7
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SND(pair(z0, z1)) → c9
SPLITAT(0, z0) → c10
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))
ACTIVATE(z0) → c15
S tuples:
U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
AND(tt, z0) → c3(ACTIVATE(z0))
FST(pair(z0, z1)) → c4
HEAD(cons(z0, z1)) → c5
NATSFROM(z0) → c6
NATSFROM(z0) → c7
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SND(pair(z0, z1)) → c9
SPLITAT(0, z0) → c10
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))
ACTIVATE(z0) → c15
K tuples:none
Defined Rule Symbols:
U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate
Defined Pair Symbols:
U11', U12', AFTERNTH, AND, FST, HEAD, NATSFROM, SEL, SND, SPLITAT, TAIL, TAKE, ACTIVATE
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 11 trailing nodes:
ACTIVATE(z0) → c15
NATSFROM(z0) → c6
NATSFROM(z0) → c7
FST(pair(z0, z1)) → c4
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
HEAD(cons(z0, z1)) → c5
SND(pair(z0, z1)) → c9
AND(tt, z0) → c3(ACTIVATE(z0))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
SPLITAT(0, z0) → c10
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:
U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
S tuples:
U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
K tuples:none
Defined Rule Symbols:
U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate
Defined Pair Symbols:
U11', AFTERNTH, SEL, SPLITAT, TAKE
Compound Symbols:
c, c2, c8, c11, c13
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 8 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))
S tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))
K tuples:none
Defined Rule Symbols:
U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate
Defined Pair Symbols:
U11', AFTERNTH, SEL, SPLITAT, TAKE
Compound Symbols:
c, c2, c8, c11, c13
(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 3 leading nodes:
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
S tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
K tuples:none
Defined Rule Symbols:
U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate
Defined Pair Symbols:
U11', SPLITAT
Compound Symbols:
c, c11
(9) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
Tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
S tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
K tuples:none
Defined Rule Symbols:
activate, natsFrom
Defined Pair Symbols:
U11', SPLITAT
Compound Symbols:
c, c11
(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.
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
We considered the (Usable) Rules:
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
natsFrom(z0) → n__natsFrom(z0)
And the Tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(SPLITAT(x1, x2)) = x1
POL(U11'(x1, x2, x3, x4)) = x1 + x2
POL(activate(x1)) = x1
POL(c(x1)) = x1
POL(c11(x1)) = x1
POL(cons(x1, x2)) = 0
POL(n__natsFrom(x1)) = [1]
POL(natsFrom(x1)) = [1]
POL(s(x1)) = [1] + x1
POL(tt) = [1]
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
Tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
S tuples:
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
K tuples:
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
Defined Rule Symbols:
activate, natsFrom
Defined Pair Symbols:
U11', SPLITAT
Compound Symbols:
c, c11
(13) CdtKnowledgeProof (EQUIVALENT transformation)
The following tuples could be moved from S to K by knowledge propagation:
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
Now S is empty
(14) BOUNDS(1, 1)