(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
average(s(x), y) → average(x, s(y))
average(x, s(s(s(y)))) → s(average(s(x), y))
average(0, 0) → 0
average(0, s(0)) → 0
average(0, s(s(0))) → s(0)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
average(s(z0), z1) → average(z0, s(z1))
average(z0, s(s(s(z1)))) → s(average(s(z0), z1))
average(0, 0) → 0
average(0, s(0)) → 0
average(0, s(s(0))) → s(0)
Tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
AVERAGE(0, 0) → c2
AVERAGE(0, s(0)) → c3
AVERAGE(0, s(s(0))) → c4
S tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
AVERAGE(0, 0) → c2
AVERAGE(0, s(0)) → c3
AVERAGE(0, s(s(0))) → c4
K tuples:none
Defined Rule Symbols:
average
Defined Pair Symbols:
AVERAGE
Compound Symbols:
c, c1, c2, c3, c4
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
AVERAGE(0, 0) → c2
AVERAGE(0, s(s(0))) → c4
AVERAGE(0, s(0)) → c3
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
average(s(z0), z1) → average(z0, s(z1))
average(z0, s(s(s(z1)))) → s(average(s(z0), z1))
average(0, 0) → 0
average(0, s(0)) → 0
average(0, s(s(0))) → s(0)
Tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
S tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
K tuples:none
Defined Rule Symbols:
average
Defined Pair Symbols:
AVERAGE
Compound Symbols:
c, c1
(5) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
average(s(z0), z1) → average(z0, s(z1))
average(z0, s(s(s(z1)))) → s(average(s(z0), z1))
average(0, 0) → 0
average(0, s(0)) → 0
average(0, s(s(0))) → s(0)
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
S tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
AVERAGE
Compound Symbols:
c, c1
(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
We considered the (Usable) Rules:none
And the Tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AVERAGE(x1, x2)) = x1 + x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(s(x1)) = [1] + x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
S tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
K tuples:
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
Defined Rule Symbols:none
Defined Pair Symbols:
AVERAGE
Compound Symbols:
c, c1
(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
We considered the (Usable) Rules:none
And the Tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AVERAGE(x1, x2)) = [3]x1 + x2
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(s(x1)) = [3] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
S tuples:none
K tuples:
AVERAGE(z0, s(s(s(z1)))) → c1(AVERAGE(s(z0), z1))
AVERAGE(s(z0), z1) → c(AVERAGE(z0, s(z1)))
Defined Rule Symbols:none
Defined Pair Symbols:
AVERAGE
Compound Symbols:
c, c1
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)