YES(O(1), O(n^1)) 4.69/1.67 YES(O(1), O(n^1)) 5.09/1.73 5.09/1.73 5.09/1.73 5.09/1.73 5.09/1.73 5.09/1.73 Runtime Complexity (innermost) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml.xml 5.09/1.73 5.09/1.73 5.09/1.73
5.09/1.73 5.09/1.73 5.09/1.73
5.09/1.73
5.09/1.73

(0) Obligation:

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

p(0) → 0 5.09/1.73
p(s(X)) → X 5.09/1.73
leq(0, Y) → true 5.09/1.73
leq(s(X), 0) → false 5.09/1.73
leq(s(X), s(Y)) → leq(X, Y) 5.09/1.73
if(true, X, Y) → activate(X) 5.09/1.73
if(false, X, Y) → activate(Y) 5.09/1.73
diff(X, Y) → if(leq(X, Y), n__0, n__s(n__diff(n__p(X), Y))) 5.09/1.73
0n__0 5.09/1.73
s(X) → n__s(X) 5.09/1.73
diff(X1, X2) → n__diff(X1, X2) 5.09/1.73
p(X) → n__p(X) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(X)) → s(activate(X)) 5.09/1.73
activate(n__diff(X1, X2)) → diff(activate(X1), activate(X2)) 5.09/1.73
activate(n__p(X)) → p(activate(X)) 5.09/1.73
activate(X) → X

Rewrite Strategy: INNERMOST
5.09/1.73
5.09/1.73

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

Converted CpxTRS to CDT
5.09/1.73
5.09/1.73

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
Tuples:

P(0) → c(0') 5.09/1.73
LEQ(s(z0), s(z1)) → c5(LEQ(z0, z1)) 5.09/1.73
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
S tuples:

P(0) → c(0') 5.09/1.73
LEQ(s(z0), s(z1)) → c5(LEQ(z0, z1)) 5.09/1.73
IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

P, LEQ, IF, DIFF, ACTIVATE

Compound Symbols:

c, c5, c6, c7, c8, c12, c13, c14, c15

5.09/1.73
5.09/1.73

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

P(0) → c(0') 5.09/1.73
LEQ(s(z0), s(z1)) → c5(LEQ(z0, z1))
5.09/1.73
5.09/1.73

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
Tuples:

IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
S tuples:

IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8(IF(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))), LEQ(z0, z1)) 5.09/1.73
ACTIVATE(n__0) → c12(0') 5.09/1.73
ACTIVATE(n__s(z0)) → c13(S(activate(z0)), ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(P(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

IF, DIFF, ACTIVATE

Compound Symbols:

c6, c7, c8, c12, c13, c14, c15

5.09/1.73
5.09/1.73

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

Removed 5 trailing tuple parts
5.09/1.73
5.09/1.73

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
Tuples:

IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
S tuples:

IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

IF, ACTIVATE, DIFF

Compound Symbols:

c6, c7, c14, c8, c12, c13, c15

5.09/1.73
5.09/1.73

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

IF(true, z0, z1) → c6(ACTIVATE(z0)) 5.09/1.73
IF(false, z0, z1) → c7(ACTIVATE(z1))
Removed 2 trailing nodes:

ACTIVATE(n__0) → c12 5.09/1.73
DIFF(z0, z1) → c8
5.09/1.73
5.09/1.73

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.73
p(s(z0)) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
leq(0, z0) → true 5.09/1.73
leq(s(z0), 0) → false 5.09/1.73
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.73
if(true, z0, z1) → activate(z0) 5.09/1.73
if(false, z0, z1) → activate(z1) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
0n__0 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0
Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
S tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

ACTIVATE, DIFF

Compound Symbols:

c14, c8, c12, c13, c15

5.09/1.73
5.09/1.73

(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__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
ACTIVATE(n__0) → c12
We considered the (Usable) Rules:

activate(n__0) → 0 5.09/1.73
activate(n__s(z0)) → s(activate(z0)) 5.09/1.73
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.73
activate(n__p(z0)) → p(activate(z0)) 5.09/1.73
activate(z0) → z0 5.09/1.73
p(z0) → n__p(z0) 5.09/1.73
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.73
diff(z0, z1) → n__diff(z0, z1) 5.09/1.73
s(z0) → n__s(z0) 5.09/1.73
0n__0
And the Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.73
DIFF(z0, z1) → c8 5.09/1.73
ACTIVATE(n__0) → c12 5.09/1.73
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.73
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.09/1.73

POL(0) = [3]    5.09/1.73
POL(ACTIVATE(x1)) = [4]x1    5.09/1.73
POL(DIFF(x1, x2)) = 0    5.09/1.73
POL(activate(x1)) = 0    5.09/1.73
POL(c12) = 0    5.09/1.73
POL(c13(x1)) = x1    5.09/1.73
POL(c14(x1, x2, x3)) = x1 + x2 + x3    5.09/1.73
POL(c15(x1)) = x1    5.09/1.73
POL(c8) = 0    5.09/1.73
POL(diff(x1, x2)) = [3] + [3]x1 + [3]x2    5.09/1.73
POL(if(x1, x2, x3)) = [3] + x2    5.09/1.73
POL(leq(x1, x2)) = [3] + [3]x1 + [3]x2    5.09/1.73
POL(n__0) = [1]    5.09/1.73
POL(n__diff(x1, x2)) = [3] + x1 + x2    5.09/1.73
POL(n__p(x1)) = x1    5.09/1.73
POL(n__s(x1)) = x1    5.09/1.73
POL(p(x1)) = [3]    5.09/1.73
POL(s(x1)) = [3] + [3]x1   
5.09/1.73
5.09/1.73

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
S tuples:

DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

ACTIVATE, DIFF

Compound Symbols:

c14, c8, c12, c13, c15

5.09/1.74
5.09/1.74

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

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

DIFF(z0, z1) → c8
5.09/1.74
5.09/1.74

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
K tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
DIFF(z0, z1) → c8
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

ACTIVATE, DIFF

Compound Symbols:

c14, c8, c12, c13, c15

5.09/1.74
5.09/1.74

(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__p(z0)) → c15(ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
0n__0
And the Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.09/1.74

POL(0) = [3]    5.09/1.74
POL(ACTIVATE(x1)) = [2]x1    5.09/1.74
POL(DIFF(x1, x2)) = 0    5.09/1.74
POL(activate(x1)) = 0    5.09/1.74
POL(c12) = 0    5.09/1.74
POL(c13(x1)) = x1    5.09/1.74
POL(c14(x1, x2, x3)) = x1 + x2 + x3    5.09/1.74
POL(c15(x1)) = x1    5.09/1.74
POL(c8) = 0    5.09/1.74
POL(diff(x1, x2)) = [3] + [3]x1 + [3]x2    5.09/1.74
POL(if(x1, x2, x3)) = [3] + x2    5.09/1.74
POL(leq(x1, x2)) = [3] + [3]x1 + [3]x2    5.09/1.74
POL(n__0) = 0    5.09/1.74
POL(n__diff(x1, x2)) = x1 + x2    5.09/1.74
POL(n__p(x1)) = [1] + x1    5.09/1.74
POL(n__s(x1)) = x1    5.09/1.74
POL(p(x1)) = [3]    5.09/1.74
POL(s(x1)) = [3] + [3]x1   
5.09/1.74
5.09/1.74

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
S tuples:

ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
K tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

ACTIVATE, DIFF

Compound Symbols:

c14, c8, c12, c13, c15

5.09/1.74
5.09/1.74

(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__s(z0)) → c13(ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
0n__0
And the Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation : 5.09/1.74

POL(0) = [3]    5.09/1.74
POL(ACTIVATE(x1)) = [1] + [4]x1    5.09/1.74
POL(DIFF(x1, x2)) = [5]    5.09/1.74
POL(activate(x1)) = 0    5.09/1.74
POL(c12) = 0    5.09/1.74
POL(c13(x1)) = x1    5.09/1.74
POL(c14(x1, x2, x3)) = x1 + x2 + x3    5.09/1.74
POL(c15(x1)) = x1    5.09/1.74
POL(c8) = 0    5.09/1.74
POL(diff(x1, x2)) = [3] + [3]x1 + [3]x2    5.09/1.74
POL(if(x1, x2, x3)) = [3] + x2    5.09/1.74
POL(leq(x1, x2)) = [3] + [3]x1 + [3]x2    5.09/1.74
POL(n__0) = 0    5.09/1.74
POL(n__diff(x1, x2)) = [4] + x1 + x2    5.09/1.74
POL(n__p(x1)) = x1    5.09/1.74
POL(n__s(x1)) = [1] + x1    5.09/1.74
POL(p(x1)) = [3]    5.09/1.74
POL(s(x1)) = [3] + [3]x1   
5.09/1.74
5.09/1.74

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

p(0) → 0 5.09/1.74
p(s(z0)) → z0 5.09/1.74
p(z0) → n__p(z0) 5.09/1.74
leq(0, z0) → true 5.09/1.74
leq(s(z0), 0) → false 5.09/1.74
leq(s(z0), s(z1)) → leq(z0, z1) 5.09/1.74
if(true, z0, z1) → activate(z0) 5.09/1.74
if(false, z0, z1) → activate(z1) 5.09/1.74
diff(z0, z1) → if(leq(z0, z1), n__0, n__s(n__diff(n__p(z0), z1))) 5.09/1.74
diff(z0, z1) → n__diff(z0, z1) 5.09/1.74
0n__0 5.09/1.74
s(z0) → n__s(z0) 5.09/1.74
activate(n__0) → 0 5.09/1.74
activate(n__s(z0)) → s(activate(z0)) 5.09/1.74
activate(n__diff(z0, z1)) → diff(activate(z0), activate(z1)) 5.09/1.74
activate(n__p(z0)) → p(activate(z0)) 5.09/1.74
activate(z0) → z0
Tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__diff(z0, z1)) → c14(DIFF(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 5.09/1.74
ACTIVATE(n__0) → c12 5.09/1.74
DIFF(z0, z1) → c8 5.09/1.74
ACTIVATE(n__p(z0)) → c15(ACTIVATE(z0)) 5.09/1.74
ACTIVATE(n__s(z0)) → c13(ACTIVATE(z0))
Defined Rule Symbols:

p, leq, if, diff, 0, s, activate

Defined Pair Symbols:

ACTIVATE, DIFF

Compound Symbols:

c14, c8, c12, c13, c15

5.09/1.74
5.09/1.74

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

The set S is empty
5.09/1.74
5.09/1.74

(18) BOUNDS(O(1), O(1))

5.09/1.74
5.09/1.74
5.09/1.79 EOF