YES(O(1), O(n^1)) 6.24/2.05 YES(O(1), O(n^1)) 6.62/2.13 6.62/2.13 6.62/2.13 6.62/2.13 6.62/2.13 6.62/2.13 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 6.62/2.13 6.62/2.13 6.62/2.13
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6.62/2.13
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
6.62/2.13 
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
6.62/2.13 
↳2 CdtProblem
6.62/2.13 
↳3 CdtUnreachableProof (⇔)
6.62/2.13 
↳4 CdtProblem
6.62/2.13 
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
6.62/2.13 
↳6 CdtProblem
6.62/2.13 
↳7 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
6.62/2.13 
↳8 CdtProblem
6.62/2.13 
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.62/2.13 
↳10 CdtProblem
6.62/2.13 
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.62/2.13 
↳12 CdtProblem
6.62/2.13 
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.62/2.13 
↳14 CdtProblem
6.62/2.13 
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.62/2.14 
↳16 CdtProblem
6.62/2.14 
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
6.62/2.14 
↳18 CdtProblem
6.62/2.14 
↳19 SIsEmptyProof (BOTH BOUNDS(ID, ID))
6.62/2.14 
↳20 BOUNDS(O(1), O(1))
6.62/2.14
6.62/2.14 6.62/2.14
6.62/2.14
6.62/2.14

(0) Obligation:

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

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N))) 6.62/2.14
sqr(0) → 0 6.62/2.14
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) 6.62/2.14
dbl(0) → 0 6.62/2.14
dbl(s(X)) → s(n__s(n__dbl(activate(X)))) 6.62/2.14
add(0, X) → X 6.62/2.14
add(s(X), Y) → s(n__add(activate(X), Y)) 6.62/2.14
first(0, X) → nil 6.62/2.14
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z))) 6.62/2.14
terms(X) → n__terms(X) 6.62/2.14
s(X) → n__s(X) 6.62/2.14
add(X1, X2) → n__add(X1, X2) 6.62/2.14
sqr(X) → n__sqr(X) 6.62/2.14
dbl(X) → n__dbl(X) 6.62/2.14
first(X1, X2) → n__first(X1, X2) 6.62/2.14
activate(n__terms(X)) → terms(activate(X)) 6.62/2.14
activate(n__s(X)) → s(X) 6.62/2.14
activate(n__add(X1, X2)) → add(activate(X1), activate(X2)) 6.62/2.14
activate(n__sqr(X)) → sqr(activate(X)) 6.62/2.14
activate(n__dbl(X)) → dbl(activate(X)) 6.62/2.14
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) 6.62/2.14
activate(X) → X

Rewrite Strategy: INNERMOST
6.62/2.14
6.62/2.14

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

Converted CpxTRS to CDT
6.62/2.14
6.62/2.14

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.14
terms(z0) → n__terms(z0) 6.62/2.14
sqr(0) → 0 6.62/2.14
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.14
sqr(z0) → n__sqr(z0) 6.62/2.14
dbl(0) → 0 6.62/2.14
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.14
dbl(z0) → n__dbl(z0) 6.62/2.14
add(0, z0) → z0 6.62/2.14
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.14
add(z0, z1) → n__add(z0, z1) 6.62/2.14
first(0, z0) → nil 6.62/2.14
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.14
first(z0, z1) → n__first(z0, z1) 6.62/2.14
s(z0) → n__s(z0) 6.62/2.14
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.14
activate(n__s(z0)) → s(z0) 6.62/2.14
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.14
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.14
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.14
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.14
activate(z0) → z0
Tuples:

TERMS(z0) → c(SQR(z0)) 6.62/2.14
SQR(s(z0)) → c3(S(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))), ACTIVATE(z0), ACTIVATE(z0)) 6.62/2.14
DBL(s(z0)) → c6(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0)) 6.62/2.14
ADD(s(z0), z1) → c9(S(n__add(activate(z0), z1)), ACTIVATE(z0)) 6.62/2.14
FIRST(s(z0), cons(z1, z2)) → c12(ACTIVATE(z0), ACTIVATE(z2)) 6.62/2.14
ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__s(z0)) → c16(S(z0)) 6.62/2.14
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.14
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__first(z0, z1)) → c20(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

TERMS(z0) → c(SQR(z0)) 6.62/2.14
SQR(s(z0)) → c3(S(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))), ACTIVATE(z0), ACTIVATE(z0)) 6.62/2.14
DBL(s(z0)) → c6(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0)) 6.62/2.14
ADD(s(z0), z1) → c9(S(n__add(activate(z0), z1)), ACTIVATE(z0)) 6.62/2.14
FIRST(s(z0), cons(z1, z2)) → c12(ACTIVATE(z0), ACTIVATE(z2)) 6.62/2.14
ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__s(z0)) → c16(S(z0)) 6.62/2.14
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.14
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__first(z0, z1)) → c20(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

TERMS, SQR, DBL, ADD, FIRST, ACTIVATE

Compound Symbols:

c, c3, c6, c9, c12, c15, c16, c17, c18, c19, c20

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6.62/2.14

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

SQR(s(z0)) → c3(S(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))), ACTIVATE(z0), ACTIVATE(z0)) 6.62/2.14
DBL(s(z0)) → c6(S(n__s(n__dbl(activate(z0)))), ACTIVATE(z0)) 6.62/2.14
ADD(s(z0), z1) → c9(S(n__add(activate(z0), z1)), ACTIVATE(z0)) 6.62/2.14
FIRST(s(z0), cons(z1, z2)) → c12(ACTIVATE(z0), ACTIVATE(z2))
6.62/2.14
6.62/2.14

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.14
terms(z0) → n__terms(z0) 6.62/2.14
sqr(0) → 0 6.62/2.14
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.14
sqr(z0) → n__sqr(z0) 6.62/2.14
dbl(0) → 0 6.62/2.14
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.14
dbl(z0) → n__dbl(z0) 6.62/2.14
add(0, z0) → z0 6.62/2.14
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.14
add(z0, z1) → n__add(z0, z1) 6.62/2.14
first(0, z0) → nil 6.62/2.14
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.14
first(z0, z1) → n__first(z0, z1) 6.62/2.14
s(z0) → n__s(z0) 6.62/2.14
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.14
activate(n__s(z0)) → s(z0) 6.62/2.14
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.14
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.14
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.14
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.14
activate(z0) → z0
Tuples:

TERMS(z0) → c(SQR(z0)) 6.62/2.14
ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__s(z0)) → c16(S(z0)) 6.62/2.14
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.14
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__first(z0, z1)) → c20(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

TERMS(z0) → c(SQR(z0)) 6.62/2.14
ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__s(z0)) → c16(S(z0)) 6.62/2.14
ACTIVATE(n__add(z0, z1)) → c17(ADD(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.14
ACTIVATE(n__sqr(z0)) → c18(SQR(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__dbl(z0)) → c19(DBL(activate(z0)), ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__first(z0, z1)) → c20(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

TERMS, ACTIVATE

Compound Symbols:

c, c15, c16, c17, c18, c19, c20

6.62/2.14
6.62/2.14

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

Removed 6 trailing tuple parts
6.62/2.14
6.62/2.14

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.14
terms(z0) → n__terms(z0) 6.62/2.14
sqr(0) → 0 6.62/2.14
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.14
sqr(z0) → n__sqr(z0) 6.62/2.14
dbl(0) → 0 6.62/2.14
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.14
dbl(z0) → n__dbl(z0) 6.62/2.14
add(0, z0) → z0 6.62/2.14
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.14
add(z0, z1) → n__add(z0, z1) 6.62/2.14
first(0, z0) → nil 6.62/2.14
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.14
first(z0, z1) → n__first(z0, z1) 6.62/2.14
s(z0) → n__s(z0) 6.62/2.14
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.14
activate(n__s(z0)) → s(z0) 6.62/2.14
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.14
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.14
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.14
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.14
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.14
TERMS(z0) → c 6.62/2.14
ACTIVATE(n__s(z0)) → c16 6.62/2.14
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.14
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.14
TERMS(z0) → c 6.62/2.14
ACTIVATE(n__s(z0)) → c16 6.62/2.14
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.14
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.14
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c15, c, c16, c17, c18, c19, c20

6.62/2.14
6.62/2.14

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

Removed 2 trailing nodes:

TERMS(z0) → c 6.62/2.14
ACTIVATE(n__s(z0)) → c16
6.62/2.14
6.62/2.14

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.14
terms(z0) → n__terms(z0) 6.62/2.14
sqr(0) → 0 6.62/2.14
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.14
sqr(z0) → n__sqr(z0) 6.62/2.14
dbl(0) → 0 6.62/2.14
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.14
dbl(z0) → n__dbl(z0) 6.62/2.14
add(0, z0) → z0 6.62/2.14
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.14
add(z0, z1) → n__add(z0, z1) 6.62/2.14
first(0, z0) → nil 6.62/2.14
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.14
first(z0, z1) → n__first(z0, z1) 6.62/2.15
s(z0) → n__s(z0) 6.62/2.15
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.15
activate(n__s(z0)) → s(z0) 6.62/2.15
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.15
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.15
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.15
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.15
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.15
TERMS(z0) → c 6.62/2.15
ACTIVATE(n__s(z0)) → c16 6.62/2.15
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.15
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.15
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.15
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.15
TERMS(z0) → c 6.62/2.15
ACTIVATE(n__s(z0)) → c16 6.62/2.15
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.15
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.15
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.15
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c15, c, c16, c17, c18, c19, c20

6.62/2.15
6.62/2.15

(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.

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.15
TERMS(z0) → c
We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.15
activate(n__s(z0)) → s(z0) 6.62/2.15
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.15
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.15
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.15
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.15
activate(z0) → z0 6.62/2.15
first(0, z0) → nil 6.62/2.15
first(z0, z1) → n__first(z0, z1) 6.62/2.15
dbl(0) → 0 6.62/2.15
dbl(z0) → n__dbl(z0) 6.62/2.15
sqr(0) → 0 6.62/2.15
sqr(z0) → n__sqr(z0) 6.62/2.15
add(0, z0) → z0 6.62/2.15
add(z0, z1) → n__add(z0, z1) 6.62/2.15
s(z0) → n__s(z0) 6.62/2.15
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.15
terms(z0) → n__terms(z0)
And the Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.15
TERMS(z0) → c 6.62/2.15
ACTIVATE(n__s(z0)) → c16 6.62/2.15
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.15
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.15
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.15
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.62/2.15

POL(0) = [3]    6.62/2.15
POL(ACTIVATE(x1)) = [4]x1    6.62/2.15
POL(TERMS(x1)) = [3]    6.62/2.15
POL(activate(x1)) = 0    6.62/2.15
POL(add(x1, x2)) = [3]    6.62/2.15
POL(c) = 0    6.62/2.15
POL(c15(x1, x2)) = x1 + x2    6.62/2.15
POL(c16) = 0    6.62/2.15
POL(c17(x1, x2)) = x1 + x2    6.62/2.15
POL(c18(x1)) = x1    6.62/2.15
POL(c19(x1)) = x1    6.62/2.15
POL(c20(x1, x2)) = x1 + x2    6.62/2.15
POL(cons(x1, x2)) = [3]    6.62/2.15
POL(dbl(x1)) = [3]    6.62/2.15
POL(first(x1, x2)) = [3] + [3]x1 + [3]x2    6.62/2.15
POL(n__add(x1, x2)) = x1 + x2    6.62/2.15
POL(n__dbl(x1)) = x1    6.62/2.15
POL(n__first(x1, x2)) = x1 + x2    6.62/2.15
POL(n__s(x1)) = x1    6.62/2.15
POL(n__sqr(x1)) = x1    6.62/2.15
POL(n__terms(x1)) = [1] + x1    6.62/2.15
POL(nil) = [3]    6.62/2.15
POL(recip(x1)) = [3]    6.62/2.15
POL(s(x1)) = [3] + [3]x1    6.62/2.15
POL(sqr(x1)) = [3]    6.62/2.15
POL(terms(x1)) = [3]   
6.62/2.15
6.62/2.15

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.15
terms(z0) → n__terms(z0) 6.62/2.15
sqr(0) → 0 6.62/2.15
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.15
sqr(z0) → n__sqr(z0) 6.62/2.15
dbl(0) → 0 6.62/2.15
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.15
dbl(z0) → n__dbl(z0) 6.62/2.15
add(0, z0) → z0 6.62/2.15
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.15
add(z0, z1) → n__add(z0, z1) 6.62/2.15
first(0, z0) → nil 6.62/2.15
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.15
first(z0, z1) → n__first(z0, z1) 6.62/2.15
s(z0) → n__s(z0) 6.62/2.15
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.15
activate(n__s(z0)) → s(z0) 6.62/2.15
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.15
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.15
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.15
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.15
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.15
TERMS(z0) → c 6.62/2.15
ACTIVATE(n__s(z0)) → c16 6.62/2.15
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.15
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c15, c, c16, c17, c18, c19, c20

6.62/2.16
6.62/2.16

(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.

ACTIVATE(n__s(z0)) → c16
We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0)
And the Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.62/2.16

POL(0) = [5]    6.62/2.16
POL(ACTIVATE(x1)) = [2]x1    6.62/2.16
POL(TERMS(x1)) = 0    6.62/2.16
POL(activate(x1)) = 0    6.62/2.16
POL(add(x1, x2)) = [3]    6.62/2.16
POL(c) = 0    6.62/2.16
POL(c15(x1, x2)) = x1 + x2    6.62/2.16
POL(c16) = 0    6.62/2.16
POL(c17(x1, x2)) = x1 + x2    6.62/2.16
POL(c18(x1)) = x1    6.62/2.16
POL(c19(x1)) = x1    6.62/2.16
POL(c20(x1, x2)) = x1 + x2    6.62/2.16
POL(cons(x1, x2)) = [3]    6.62/2.16
POL(dbl(x1)) = [3]    6.62/2.16
POL(first(x1, x2)) = [3] + [3]x1 + [3]x2    6.62/2.16
POL(n__add(x1, x2)) = x1 + x2    6.62/2.16
POL(n__dbl(x1)) = x1    6.62/2.16
POL(n__first(x1, x2)) = x1 + x2    6.62/2.16
POL(n__s(x1)) = [1] + x1    6.62/2.16
POL(n__sqr(x1)) = x1    6.62/2.16
POL(n__terms(x1)) = x1    6.62/2.16
POL(nil) = [3]    6.62/2.16
POL(recip(x1)) = [3]    6.62/2.16
POL(s(x1)) = [3] + [3]x1    6.62/2.16
POL(sqr(x1)) = [3]    6.62/2.16
POL(terms(x1)) = [3]   
6.62/2.16
6.62/2.16

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c15, c, c16, c17, c18, c19, c20

6.62/2.16
6.62/2.16

(13) 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.

ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0)
And the Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.62/2.16

POL(0) = 0    6.62/2.16
POL(ACTIVATE(x1)) = [2] + [4]x1    6.62/2.16
POL(TERMS(x1)) = [5]    6.62/2.16
POL(activate(x1)) = 0    6.62/2.16
POL(add(x1, x2)) = [3]    6.62/2.16
POL(c) = 0    6.62/2.16
POL(c15(x1, x2)) = x1 + x2    6.62/2.16
POL(c16) = 0    6.62/2.16
POL(c17(x1, x2)) = x1 + x2    6.62/2.16
POL(c18(x1)) = x1    6.62/2.16
POL(c19(x1)) = x1    6.62/2.16
POL(c20(x1, x2)) = x1 + x2    6.62/2.16
POL(cons(x1, x2)) = [3]    6.62/2.16
POL(dbl(x1)) = [3]    6.62/2.16
POL(first(x1, x2)) = [3] + [3]x1 + [3]x2    6.62/2.16
POL(n__add(x1, x2)) = [1] + x1 + x2    6.62/2.16
POL(n__dbl(x1)) = x1    6.62/2.16
POL(n__first(x1, x2)) = [2] + x1 + x2    6.62/2.16
POL(n__s(x1)) = x1    6.62/2.16
POL(n__sqr(x1)) = x1    6.62/2.16
POL(n__terms(x1)) = [4] + x1    6.62/2.16
POL(nil) = [3]    6.62/2.16
POL(recip(x1)) = [3]    6.62/2.16
POL(s(x1)) = [3] + [3]x1    6.62/2.16
POL(sqr(x1)) = [3]    6.62/2.16
POL(terms(x1)) = [3]   
6.62/2.16
6.62/2.16

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
K tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c15, c, c16, c17, c18, c19, c20

6.62/2.16
6.62/2.16

(15) 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.

ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0)
And the Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.62/2.16

POL(0) = [3]    6.62/2.16
POL(ACTIVATE(x1)) = [2]x1    6.62/2.16
POL(TERMS(x1)) = 0    6.62/2.16
POL(activate(x1)) = 0    6.62/2.16
POL(add(x1, x2)) = [3]    6.62/2.16
POL(c) = 0    6.62/2.16
POL(c15(x1, x2)) = x1 + x2    6.62/2.16
POL(c16) = 0    6.62/2.16
POL(c17(x1, x2)) = x1 + x2    6.62/2.16
POL(c18(x1)) = x1    6.62/2.16
POL(c19(x1)) = x1    6.62/2.16
POL(c20(x1, x2)) = x1 + x2    6.62/2.16
POL(cons(x1, x2)) = [3]    6.62/2.16
POL(dbl(x1)) = [3]    6.62/2.16
POL(first(x1, x2)) = [3] + [3]x1 + [3]x2    6.62/2.16
POL(n__add(x1, x2)) = x1 + x2    6.62/2.16
POL(n__dbl(x1)) = x1    6.62/2.16
POL(n__first(x1, x2)) = x1 + x2    6.62/2.16
POL(n__s(x1)) = x1    6.62/2.16
POL(n__sqr(x1)) = [1] + x1    6.62/2.16
POL(n__terms(x1)) = x1    6.62/2.16
POL(nil) = [3]    6.62/2.16
POL(recip(x1)) = [3]    6.62/2.16
POL(s(x1)) = [3] + [3]x1    6.62/2.16
POL(sqr(x1)) = [3]    6.62/2.16
POL(terms(x1)) = [3]   
6.62/2.16
6.62/2.16

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
K tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0))
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c15, c, c16, c17, c18, c19, c20

6.62/2.16
6.62/2.16

(17) 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.

ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0)
And the Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation : 6.62/2.16

POL(0) = [3]    6.62/2.16
POL(ACTIVATE(x1)) = [2]x1    6.62/2.16
POL(TERMS(x1)) = 0    6.62/2.16
POL(activate(x1)) = 0    6.62/2.16
POL(add(x1, x2)) = [3]    6.62/2.16
POL(c) = 0    6.62/2.16
POL(c15(x1, x2)) = x1 + x2    6.62/2.16
POL(c16) = 0    6.62/2.16
POL(c17(x1, x2)) = x1 + x2    6.62/2.16
POL(c18(x1)) = x1    6.62/2.16
POL(c19(x1)) = x1    6.62/2.16
POL(c20(x1, x2)) = x1 + x2    6.62/2.16
POL(cons(x1, x2)) = [3]    6.62/2.16
POL(dbl(x1)) = [3]    6.62/2.16
POL(first(x1, x2)) = [3] + [3]x1 + [3]x2    6.62/2.16
POL(n__add(x1, x2)) = [3] + x1 + x2    6.62/2.16
POL(n__dbl(x1)) = [1] + x1    6.62/2.16
POL(n__first(x1, x2)) = x1 + x2    6.62/2.16
POL(n__s(x1)) = x1    6.62/2.16
POL(n__sqr(x1)) = x1    6.62/2.16
POL(n__terms(x1)) = x1    6.62/2.16
POL(nil) = [3]    6.62/2.16
POL(recip(x1)) = [3]    6.62/2.16
POL(s(x1)) = [3] + [3]x1    6.62/2.16
POL(sqr(x1)) = [3]    6.62/2.16
POL(terms(x1)) = [3]   
6.62/2.16
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(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

terms(z0) → cons(recip(sqr(z0)), n__terms(n__s(z0))) 6.62/2.16
terms(z0) → n__terms(z0) 6.62/2.16
sqr(0) → 0 6.62/2.16
sqr(s(z0)) → s(n__add(n__sqr(activate(z0)), n__dbl(activate(z0)))) 6.62/2.16
sqr(z0) → n__sqr(z0) 6.62/2.16
dbl(0) → 0 6.62/2.16
dbl(s(z0)) → s(n__s(n__dbl(activate(z0)))) 6.62/2.16
dbl(z0) → n__dbl(z0) 6.62/2.16
add(0, z0) → z0 6.62/2.16
add(s(z0), z1) → s(n__add(activate(z0), z1)) 6.62/2.16
add(z0, z1) → n__add(z0, z1) 6.62/2.16
first(0, z0) → nil 6.62/2.16
first(s(z0), cons(z1, z2)) → cons(z1, n__first(activate(z0), activate(z2))) 6.62/2.16
first(z0, z1) → n__first(z0, z1) 6.62/2.16
s(z0) → n__s(z0) 6.62/2.16
activate(n__terms(z0)) → terms(activate(z0)) 6.62/2.16
activate(n__s(z0)) → s(z0) 6.62/2.16
activate(n__add(z0, z1)) → add(activate(z0), activate(z1)) 6.62/2.16
activate(n__sqr(z0)) → sqr(activate(z0)) 6.62/2.16
activate(n__dbl(z0)) → dbl(activate(z0)) 6.62/2.16
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 6.62/2.16
activate(z0) → z0
Tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__terms(z0)) → c15(TERMS(activate(z0)), ACTIVATE(z0)) 6.62/2.16
TERMS(z0) → c 6.62/2.16
ACTIVATE(n__s(z0)) → c16 6.62/2.16
ACTIVATE(n__add(z0, z1)) → c17(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__first(z0, z1)) → c20(ACTIVATE(z0), ACTIVATE(z1)) 6.62/2.16
ACTIVATE(n__sqr(z0)) → c18(ACTIVATE(z0)) 6.62/2.16
ACTIVATE(n__dbl(z0)) → c19(ACTIVATE(z0))
Defined Rule Symbols:

terms, sqr, dbl, add, first, s, activate

Defined Pair Symbols:

ACTIVATE, TERMS

Compound Symbols:

c15, c, c16, c17, c18, c19, c20

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6.62/2.16

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

The set S is empty
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6.62/2.16

(20) BOUNDS(O(1), O(1))

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6.62/2.16
6.93/2.27 EOF