(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^2).
The TRS R consists of the following rules:
sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, x)))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
sum(0) → 0
sum(s(z0)) → +(sum(z0), s(z0))
sum1(0) → 0
sum1(s(z0)) → s(+(sum1(z0), +(z0, z0)))
Tuples:
SUM(0) → c
SUM(s(z0)) → c1(SUM(z0))
SUM1(0) → c2
SUM1(s(z0)) → c3(SUM1(z0))
S tuples:
SUM(0) → c
SUM(s(z0)) → c1(SUM(z0))
SUM1(0) → c2
SUM1(s(z0)) → c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:
sum, sum1
Defined Pair Symbols:
SUM, SUM1
Compound Symbols:
c, c1, c2, c3
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
SUM(0) → c
SUM1(0) → c2
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
sum(0) → 0
sum(s(z0)) → +(sum(z0), s(z0))
sum1(0) → 0
sum1(s(z0)) → s(+(sum1(z0), +(z0, z0)))
Tuples:
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
S tuples:
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:
sum, sum1
Defined Pair Symbols:
SUM, SUM1
Compound Symbols:
c1, c3
(5) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
sum(0) → 0
sum(s(z0)) → +(sum(z0), s(z0))
sum1(0) → 0
sum1(s(z0)) → s(+(sum1(z0), +(z0, z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
S tuples:
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
SUM, SUM1
Compound Symbols:
c1, c3
(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
We considered the (Usable) Rules:none
And the Tuples:
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(SUM(x1)) = x12
POL(SUM1(x1)) = x1
POL(c1(x1)) = x1
POL(c3(x1)) = x1
POL(s(x1)) = [1] + x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
S tuples:none
K tuples:
SUM(s(z0)) → c1(SUM(z0))
SUM1(s(z0)) → c3(SUM1(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
SUM, SUM1
Compound Symbols:
c1, c3
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)