YES(O(1), O(n^2)) 2.80/1.11 YES(O(1), O(n^2)) 2.80/1.14 2.80/1.14 2.80/1.14
2.80/1.14 2.80/1.140 CpxTRS2.80/1.14
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))2.80/1.14
↳2 CdtProblem2.80/1.14
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))2.80/1.14
↳4 CdtProblem2.80/1.14
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication)2.80/1.14
↳6 CdtProblem2.80/1.14
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))2.80/1.14
↳8 CdtProblem2.80/1.14
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))2.80/1.14
↳10 CdtProblem2.80/1.14
↳11 CdtKnowledgeProof (⇔)2.80/1.14
↳12 BOUNDS(O(1), O(1))2.80/1.14
from(X) → cons(X, n__from(s(X))) 2.80/1.14
head(cons(X, XS)) → X 2.80/1.14
2nd(cons(X, XS)) → head(activate(XS)) 2.80/1.14
take(0, XS) → nil 2.80/1.14
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS))) 2.80/1.14
sel(0, cons(X, XS)) → X 2.80/1.14
sel(s(N), cons(X, XS)) → sel(N, activate(XS)) 2.80/1.14
from(X) → n__from(X) 2.80/1.14
take(X1, X2) → n__take(X1, X2) 2.80/1.14
activate(n__from(X)) → from(X) 2.80/1.14
activate(n__take(X1, X2)) → take(X1, X2) 2.80/1.14
activate(X) → X
Tuples:
from(z0) → cons(z0, n__from(s(z0))) 2.80/1.14
from(z0) → n__from(z0) 2.80/1.14
head(cons(z0, z1)) → z0 2.80/1.14
2nd(cons(z0, z1)) → head(activate(z1)) 2.80/1.14
take(0, z0) → nil 2.80/1.14
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 2.80/1.14
take(z0, z1) → n__take(z0, z1) 2.80/1.14
sel(0, cons(z0, z1)) → z0 2.80/1.14
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.80/1.14
activate(n__from(z0)) → from(z0) 2.80/1.14
activate(n__take(z0, z1)) → take(z0, z1) 2.80/1.14
activate(z0) → z0
S tuples:
2ND(cons(z0, z1)) → c3(HEAD(activate(z1)), ACTIVATE(z1)) 2.80/1.14
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9(FROM(z0)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1))
K tuples:none
2ND(cons(z0, z1)) → c3(HEAD(activate(z1)), ACTIVATE(z1)) 2.80/1.14
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9(FROM(z0)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1))
from, head, 2nd, take, sel, activate
2ND, TAKE, SEL, ACTIVATE
c3, c5, c8, c9, c10
Tuples:
from(z0) → cons(z0, n__from(s(z0))) 2.80/1.14
from(z0) → n__from(z0) 2.80/1.14
head(cons(z0, z1)) → z0 2.80/1.14
2nd(cons(z0, z1)) → head(activate(z1)) 2.80/1.14
take(0, z0) → nil 2.80/1.14
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 2.80/1.14
take(z0, z1) → n__take(z0, z1) 2.80/1.14
sel(0, cons(z0, z1)) → z0 2.80/1.14
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.80/1.14
activate(n__from(z0)) → from(z0) 2.80/1.14
activate(n__take(z0, z1)) → take(z0, z1) 2.80/1.14
activate(z0) → z0
S tuples:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
2ND(cons(z0, z1)) → c3(ACTIVATE(z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
K tuples:none
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
2ND(cons(z0, z1)) → c3(ACTIVATE(z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
from, head, 2nd, take, sel, activate
TAKE, SEL, ACTIVATE, 2ND
c5, c8, c10, c3, c9
Removed 1 trailing nodes:
2ND(cons(z0, z1)) → c3(ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c9
Tuples:
from(z0) → cons(z0, n__from(s(z0))) 2.80/1.14
from(z0) → n__from(z0) 2.80/1.14
head(cons(z0, z1)) → z0 2.80/1.14
2nd(cons(z0, z1)) → head(activate(z1)) 2.80/1.14
take(0, z0) → nil 2.80/1.14
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 2.80/1.14
take(z0, z1) → n__take(z0, z1) 2.80/1.14
sel(0, cons(z0, z1)) → z0 2.80/1.14
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.80/1.14
activate(n__from(z0)) → from(z0) 2.80/1.14
activate(n__take(z0, z1)) → take(z0, z1) 2.80/1.14
activate(z0) → z0
S tuples:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
K tuples:none
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
from, head, 2nd, take, sel, activate
TAKE, SEL, ACTIVATE
c5, c8, c10, c9
We considered the (Usable) Rules:
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
And the Tuples:
activate(n__from(z0)) → from(z0) 2.80/1.14
activate(n__take(z0, z1)) → take(z0, z1) 2.80/1.14
activate(z0) → z0 2.80/1.14
take(0, z0) → nil 2.80/1.14
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 2.80/1.14
take(z0, z1) → n__take(z0, z1) 2.80/1.14
from(z0) → cons(z0, n__from(s(z0))) 2.80/1.14
from(z0) → n__from(z0)
The order we found is given by the following interpretation:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
POL(0) = [3] 2.80/1.14
POL(ACTIVATE(x1)) = [4] 2.80/1.14
POL(SEL(x1, x2)) = [4]x1 2.80/1.14
POL(TAKE(x1, x2)) = [4] 2.80/1.14
POL(activate(x1)) = [4]x1 2.80/1.14
POL(c10(x1)) = x1 2.80/1.14
POL(c5(x1)) = x1 2.80/1.14
POL(c8(x1, x2)) = x1 + x2 2.80/1.14
POL(c9) = 0 2.80/1.14
POL(cons(x1, x2)) = [4] 2.80/1.14
POL(from(x1)) = [4] + [2]x1 2.80/1.14
POL(n__from(x1)) = [2] + x1 2.80/1.14
POL(n__take(x1, x2)) = [2] + x2 2.80/1.14
POL(nil) = [1] 2.80/1.14
POL(s(x1)) = [2] + x1 2.80/1.14
POL(take(x1, x2)) = [2] + [2]x2
Tuples:
from(z0) → cons(z0, n__from(s(z0))) 2.80/1.14
from(z0) → n__from(z0) 2.80/1.14
head(cons(z0, z1)) → z0 2.80/1.14
2nd(cons(z0, z1)) → head(activate(z1)) 2.80/1.14
take(0, z0) → nil 2.80/1.14
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 2.80/1.14
take(z0, z1) → n__take(z0, z1) 2.80/1.14
sel(0, cons(z0, z1)) → z0 2.80/1.14
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.80/1.14
activate(n__from(z0)) → from(z0) 2.80/1.14
activate(n__take(z0, z1)) → take(z0, z1) 2.80/1.14
activate(z0) → z0
S tuples:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
K tuples:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1))
Defined Rule Symbols:
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
from, head, 2nd, take, sel, activate
TAKE, SEL, ACTIVATE
c5, c8, c10, c9
We considered the (Usable) Rules:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2))
And the Tuples:
activate(n__from(z0)) → from(z0) 2.80/1.14
activate(n__take(z0, z1)) → take(z0, z1) 2.80/1.14
activate(z0) → z0 2.80/1.14
take(0, z0) → nil 2.80/1.14
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 2.80/1.14
take(z0, z1) → n__take(z0, z1) 2.80/1.14
from(z0) → cons(z0, n__from(s(z0))) 2.80/1.14
from(z0) → n__from(z0)
The order we found is given by the following interpretation:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
POL(0) = [3] 2.80/1.14
POL(ACTIVATE(x1)) = [2]x1 2.80/1.14
POL(SEL(x1, x2)) = x1·x2 2.80/1.14
POL(TAKE(x1, x2)) = [2]x2 2.80/1.14
POL(activate(x1)) = [1] + x1 2.80/1.14
POL(c10(x1)) = x1 2.80/1.14
POL(c5(x1)) = x1 2.80/1.14
POL(c8(x1, x2)) = x1 + x2 2.80/1.14
POL(c9) = 0 2.80/1.14
POL(cons(x1, x2)) = [1] + x2 2.80/1.14
POL(from(x1)) = [2] 2.80/1.14
POL(n__from(x1)) = [1] 2.80/1.14
POL(n__take(x1, x2)) = x2 2.80/1.14
POL(nil) = 0 2.80/1.14
POL(s(x1)) = [2] + x1 2.80/1.14
POL(take(x1, x2)) = [1] + x2
Tuples:
from(z0) → cons(z0, n__from(s(z0))) 2.80/1.14
from(z0) → n__from(z0) 2.80/1.14
head(cons(z0, z1)) → z0 2.80/1.14
2nd(cons(z0, z1)) → head(activate(z1)) 2.80/1.14
take(0, z0) → nil 2.80/1.14
take(s(z0), cons(z1, z2)) → cons(z1, n__take(z0, activate(z2))) 2.80/1.14
take(z0, z1) → n__take(z0, z1) 2.80/1.14
sel(0, cons(z0, z1)) → z0 2.80/1.14
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2)) 2.80/1.14
activate(n__from(z0)) → from(z0) 2.80/1.14
activate(n__take(z0, z1)) → take(z0, z1) 2.80/1.14
activate(z0) → z0
S tuples:
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2)) 2.80/1.14
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9
K tuples:
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1))
Defined Rule Symbols:
SEL(s(z0), cons(z1, z2)) → c8(SEL(z0, activate(z2)), ACTIVATE(z2)) 2.80/1.14
ACTIVATE(n__from(z0)) → c9 2.80/1.14
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2))
from, head, 2nd, take, sel, activate
TAKE, SEL, ACTIVATE
c5, c8, c10, c9
Now S is empty
ACTIVATE(n__take(z0, z1)) → c10(TAKE(z0, z1)) 2.80/1.14
TAKE(s(z0), cons(z1, z2)) → c5(ACTIVATE(z2))