YES(O(1), O(n^2)) 7.05/2.29 YES(O(1), O(n^2)) 7.43/2.37 7.43/2.37 7.43/2.37 7.43/2.37 7.43/2.37 7.43/2.37 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 7.43/2.37 7.43/2.37 7.43/2.37
7.43/2.37
7.43/2.37
7.43/2.37
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
7.43/2.37 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
7.43/2.37 
↳2 CdtProblem
7.43/2.37 
↳3 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
7.43/2.37 
↳4 CdtProblem
7.43/2.37 
↳5 CdtLeafRemovalProof (ComplexityIfPolyImplication)
7.43/2.37 
↳6 CdtProblem
7.43/2.37 
↳7 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
7.43/2.37 
↳8 CdtProblem
7.43/2.37 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
7.43/2.37 
↳10 CdtProblem
7.43/2.37 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
7.43/2.37 
↳12 CdtProblem
7.43/2.37 
↳13 CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
7.43/2.37 
↳14 CdtProblem
7.43/2.37 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
7.43/2.37 
↳16 CdtProblem
7.43/2.37 
↳17 SIsEmptyProof (BOTH BOUNDS(ID, ID))
7.43/2.37 
↳18 BOUNDS(O(1), O(1))
7.43/2.37
7.43/2.37 7.43/2.37
7.43/2.37
7.43/2.37

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

from(X) → cons(X, n__from(s(X))) 7.43/2.37
2ndspos(0, Z) → rnil 7.43/2.37
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, activate(Z))) 7.43/2.37
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, activate(Z))) 7.43/2.37
2ndsneg(0, Z) → rnil 7.43/2.37
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, activate(Z))) 7.43/2.37
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, activate(Z))) 7.43/2.37
pi(X) → 2ndspos(X, from(0)) 7.43/2.37
plus(0, Y) → Y 7.43/2.37
plus(s(X), Y) → s(plus(X, Y)) 7.43/2.37
times(0, Y) → 0 7.43/2.37
times(s(X), Y) → plus(Y, times(X, Y)) 7.43/2.37
square(X) → times(X, X) 7.43/2.37
from(X) → n__from(X) 7.43/2.37
activate(n__from(X)) → from(X) 7.43/2.37
activate(X) → X

Rewrite Strategy: INNERMOST
7.43/2.37
7.43/2.37

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

Converted CpxTRS to CDT
7.43/2.37
7.43/2.37

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
SQUARE(z0) → c13(TIMES(z0, z0)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14(FROM(z0))
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
SQUARE(z0) → c13(TIMES(z0, z0)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14(FROM(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PI, PLUS, TIMES, SQUARE, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c8, c10, c12, c13, c14

7.43/2.38
7.43/2.38

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

Removed 2 trailing tuple parts
7.43/2.38
7.43/2.38

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
SQUARE(z0) → c13(TIMES(z0, z0)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
SQUARE(z0) → c13(TIMES(z0, z0)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PLUS, TIMES, SQUARE, PI, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c10, c12, c13, c8, c14

7.43/2.38
7.43/2.38

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

SQUARE(z0) → c13(TIMES(z0, z0))
Removed 1 trailing nodes:

ACTIVATE(n__from(z0)) → c14
7.43/2.38
7.43/2.38

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PLUS, TIMES, PI, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c10, c12, c8, c14

7.43/2.38
7.43/2.38

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

The following tuples could be moved from S to K by knowledge propagation:

PI(z0) → c8(2NDSPOS(z0, from(0)))
7.43/2.38
7.43/2.38

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
K tuples:

PI(z0) → c8(2NDSPOS(z0, from(0)))
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PLUS, TIMES, PI, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c10, c12, c8, c14

7.43/2.38
7.43/2.38

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
We considered the (Usable) Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
And the Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
The order we found is given by the following interpretation:
Polynomial interpretation : 7.43/2.38

POL(0) = 0    7.43/2.38
POL(2NDSNEG(x1, x2)) = 0    7.43/2.38
POL(2NDSPOS(x1, x2)) = 0    7.43/2.38
POL(ACTIVATE(x1)) = 0    7.43/2.38
POL(PI(x1)) = [5] + [2]x1    7.43/2.38
POL(PLUS(x1, x2)) = 0    7.43/2.38
POL(TIMES(x1, x2)) = [2]x1    7.43/2.38
POL(activate(x1)) = 0    7.43/2.38
POL(c10(x1)) = x1    7.43/2.38
POL(c12(x1, x2)) = x1 + x2    7.43/2.38
POL(c14) = 0    7.43/2.38
POL(c3(x1, x2)) = x1 + x2    7.43/2.38
POL(c4(x1, x2)) = x1 + x2    7.43/2.38
POL(c6(x1, x2)) = x1 + x2    7.43/2.38
POL(c7(x1, x2)) = x1 + x2    7.43/2.38
POL(c8(x1)) = x1    7.43/2.38
POL(cons(x1, x2)) = 0    7.43/2.38
POL(cons2(x1, x2)) = 0    7.43/2.38
POL(from(x1)) = 0    7.43/2.38
POL(n__from(x1)) = 0    7.43/2.38
POL(plus(x1, x2)) = [3] + [3]x2    7.43/2.38
POL(s(x1)) = [1] + x1    7.43/2.38
POL(times(x1, x2)) = 0   
7.43/2.38
7.43/2.38

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
K tuples:

PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PLUS, TIMES, PI, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c10, c12, c8, c14

7.43/2.38
7.43/2.38

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3))
We considered the (Usable) Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
And the Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
The order we found is given by the following interpretation:
Polynomial interpretation : 7.43/2.38

POL(0) = 0    7.43/2.38
POL(2NDSNEG(x1, x2)) = [4]x1    7.43/2.38
POL(2NDSPOS(x1, x2)) = [4]x1    7.43/2.38
POL(ACTIVATE(x1)) = 0    7.43/2.38
POL(PI(x1)) = [4] + [4]x1    7.43/2.38
POL(PLUS(x1, x2)) = [2]    7.43/2.38
POL(TIMES(x1, x2)) = x1    7.43/2.38
POL(activate(x1)) = 0    7.43/2.38
POL(c10(x1)) = x1    7.43/2.38
POL(c12(x1, x2)) = x1 + x2    7.43/2.38
POL(c14) = 0    7.43/2.38
POL(c3(x1, x2)) = x1 + x2    7.43/2.38
POL(c4(x1, x2)) = x1 + x2    7.43/2.38
POL(c6(x1, x2)) = x1 + x2    7.43/2.38
POL(c7(x1, x2)) = x1 + x2    7.43/2.38
POL(c8(x1)) = x1    7.43/2.38
POL(cons(x1, x2)) = 0    7.43/2.38
POL(cons2(x1, x2)) = 0    7.43/2.38
POL(from(x1)) = 0    7.43/2.38
POL(n__from(x1)) = 0    7.43/2.38
POL(plus(x1, x2)) = [3] + [5]x2    7.43/2.38
POL(s(x1)) = [4] + x1    7.43/2.38
POL(times(x1, x2)) = 0   
7.43/2.38
7.43/2.38

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
K tuples:

PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3))
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PLUS, TIMES, PI, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c10, c12, c8, c14

7.43/2.38
7.43/2.38

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

The following tuples could be moved from S to K by knowledge propagation:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
7.43/2.38
7.43/2.38

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
S tuples:

PLUS(s(z0), z1) → c10(PLUS(z0, z1))
K tuples:

PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PLUS, TIMES, PI, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c10, c12, c8, c14

7.43/2.38
7.43/2.38

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

PLUS(s(z0), z1) → c10(PLUS(z0, z1))
We considered the (Usable) Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
And the Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
The order we found is given by the following interpretation:
Polynomial interpretation : 7.43/2.38

POL(0) = 0    7.43/2.38
POL(2NDSNEG(x1, x2)) = 0    7.43/2.38
POL(2NDSPOS(x1, x2)) = 0    7.43/2.38
POL(ACTIVATE(x1)) = 0    7.43/2.38
POL(PI(x1)) = x1 + x12    7.43/2.38
POL(PLUS(x1, x2)) = x1    7.43/2.38
POL(TIMES(x1, x2)) = x1 + x1·x2    7.43/2.38
POL(activate(x1)) = 0    7.43/2.38
POL(c10(x1)) = x1    7.43/2.38
POL(c12(x1, x2)) = x1 + x2    7.43/2.38
POL(c14) = 0    7.43/2.38
POL(c3(x1, x2)) = x1 + x2    7.43/2.38
POL(c4(x1, x2)) = x1 + x2    7.43/2.38
POL(c6(x1, x2)) = x1 + x2    7.43/2.38
POL(c7(x1, x2)) = x1 + x2    7.43/2.38
POL(c8(x1)) = x1    7.43/2.38
POL(cons(x1, x2)) = 0    7.43/2.38
POL(cons2(x1, x2)) = 0    7.43/2.38
POL(from(x1)) = 0    7.43/2.38
POL(n__from(x1)) = 0    7.43/2.38
POL(plus(x1, x2)) = [3] + [3]x1    7.43/2.38
POL(s(x1)) = [2] + x1    7.43/2.38
POL(times(x1, x2)) = [3]x22   
7.43/2.38
7.43/2.38

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0))) 7.43/2.38
from(z0) → n__from(z0) 7.43/2.38
2ndspos(0, z0) → rnil 7.43/2.38
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3))) 7.43/2.38
2ndsneg(0, z0) → rnil 7.43/2.38
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2))) 7.43/2.38
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3))) 7.43/2.38
pi(z0) → 2ndspos(z0, from(0)) 7.43/2.38
plus(0, z0) → z0 7.43/2.38
plus(s(z0), z1) → s(plus(z0, z1)) 7.43/2.38
times(0, z0) → 0 7.43/2.38
times(s(z0), z1) → plus(z1, times(z0, z1)) 7.43/2.38
square(z0) → times(z0, z0) 7.43/2.38
activate(n__from(z0)) → from(z0) 7.43/2.38
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1)) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
ACTIVATE(n__from(z0)) → c14
S tuples:none
K tuples:

PI(z0) → c8(2NDSPOS(z0, from(0))) 7.43/2.38
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 7.43/2.38
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3)) 7.43/2.38
2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), ACTIVATE(z2)) 7.43/2.38
ACTIVATE(n__from(z0)) → c14 7.43/2.38
PLUS(s(z0), z1) → c10(PLUS(z0, z1))
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PLUS, TIMES, PI, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c10, c12, c8, c14

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

The set S is empty
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(18) BOUNDS(O(1), O(1))

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7.93/2.58 EOF