YES(O(1), O(n^1)) 0.00/0.81 YES(O(1), O(n^1)) 0.00/0.84 0.00/0.84 0.00/0.84
0.00/0.84 0.00/0.840 CpxTRS0.00/0.84
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))0.00/0.84
↳2 CdtProblem0.00/0.84
↳3 CdtUnreachableProof (⇔)0.00/0.84
↳4 CdtProblem0.00/0.84
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))0.00/0.84
↳6 CdtProblem0.00/0.84
↳7 CdtLeafRemovalProof (ComplexityIfPolyImplication)0.00/0.84
↳8 CdtProblem0.00/0.84
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.84
↳10 CdtProblem0.00/0.84
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.84
↳12 CdtProblem0.00/0.84
↳13 SIsEmptyProof (BOTH BOUNDS(ID, ID))0.00/0.84
↳14 BOUNDS(O(1), O(1))0.00/0.84
natsFrom(N) → cons(N, n__natsFrom(n__s(N))) 0.00/0.84
fst(pair(XS, YS)) → XS 0.00/0.84
snd(pair(XS, YS)) → YS 0.00/0.84
splitAt(0, XS) → pair(nil, XS) 0.00/0.84
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS)) 0.00/0.84
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS) 0.00/0.84
head(cons(N, XS)) → N 0.00/0.84
tail(cons(N, XS)) → activate(XS) 0.00/0.84
sel(N, XS) → head(afterNth(N, XS)) 0.00/0.84
take(N, XS) → fst(splitAt(N, XS)) 0.00/0.84
afterNth(N, XS) → snd(splitAt(N, XS)) 0.00/0.84
natsFrom(X) → n__natsFrom(X) 0.00/0.84
s(X) → n__s(X) 0.00/0.84
activate(n__natsFrom(X)) → natsFrom(activate(X)) 0.00/0.84
activate(n__s(X)) → s(activate(X)) 0.00/0.84
activate(X) → X
Tuples:
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
S tuples:
SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 0.00/0.84
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
K tuples:none
SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2)) 0.00/0.84
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, s, activate
SPLITAT, U, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE
c5, c6, c8, c9, c10, c11, c13, c14
SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
Tuples:
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
S tuples:
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
K tuples:none
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1)) 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(NATSFROM(activate(z0)), ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(S(activate(z0)), ACTIVATE(z0))
natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, s, activate
U, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE
c6, c8, c9, c10, c11, c13, c14
Tuples:
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
S tuples:
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10 0.00/0.84
AFTERNTH(z0, z1) → c11 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
K tuples:none
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1)) 0.00/0.84
SEL(z0, z1) → c9(AFTERNTH(z0, z1)) 0.00/0.84
TAKE(z0, z1) → c10 0.00/0.84
AFTERNTH(z0, z1) → c11 0.00/0.84
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, s, activate
U, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE
c6, c8, c9, c10, c11, c13, c14
Removed 3 trailing nodes:
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3)) 0.00/0.84
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1))
AFTERNTH(z0, z1) → c11 0.00/0.84
TAKE(z0, z1) → c10 0.00/0.84
SEL(z0, z1) → c9(AFTERNTH(z0, z1))
Tuples:
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
S tuples:
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
K tuples:none
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, s, activate
ACTIVATE
c13, c14
We considered the (Usable) Rules:none
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0))
The order we found is given by the following interpretation:
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
POL(ACTIVATE(x1)) = [2]x1 0.00/0.84
POL(c13(x1)) = x1 0.00/0.84
POL(c14(x1)) = x1 0.00/0.84
POL(n__natsFrom(x1)) = [1] + x1 0.00/0.84
POL(n__s(x1)) = x1
Tuples:
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
S tuples:
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
K tuples:
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0))
natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, s, activate
ACTIVATE
c13, c14
We considered the (Usable) Rules:none
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
The order we found is given by the following interpretation:
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
POL(ACTIVATE(x1)) = [3]x1 0.00/0.84
POL(c13(x1)) = x1 0.00/0.84
POL(c14(x1)) = x1 0.00/0.84
POL(n__natsFrom(x1)) = x1 0.00/0.84
POL(n__s(x1)) = [1] + x1
Tuples:
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0))) 0.00/0.84
natsFrom(z0) → n__natsFrom(z0) 0.00/0.84
fst(pair(z0, z1)) → z0 0.00/0.84
snd(pair(z0, z1)) → z1 0.00/0.84
splitAt(0, z0) → pair(nil, z0) 0.00/0.84
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2)) 0.00/0.84
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1) 0.00/0.84
head(cons(z0, z1)) → z0 0.00/0.84
tail(cons(z0, z1)) → activate(z1) 0.00/0.84
sel(z0, z1) → head(afterNth(z0, z1)) 0.00/0.84
take(z0, z1) → fst(splitAt(z0, z1)) 0.00/0.84
afterNth(z0, z1) → snd(splitAt(z0, z1)) 0.00/0.84
s(z0) → n__s(z0) 0.00/0.84
activate(n__natsFrom(z0)) → natsFrom(activate(z0)) 0.00/0.84
activate(n__s(z0)) → s(activate(z0)) 0.00/0.84
activate(z0) → z0
S tuples:none
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__natsFrom(z0)) → c13(ACTIVATE(z0)) 0.00/0.84
ACTIVATE(n__s(z0)) → c14(ACTIVATE(z0))
natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, s, activate
ACTIVATE
c13, c14