(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)) → +(sqr(s(x)), sum(x))
sqr(x) → *(x, x)
sum(s(x)) → +(*(s(x), s(x)), sum(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)) → +(sqr(s(z0)), sum(z0))
sum(s(z0)) → +(*(s(z0), s(z0)), sum(z0))
sqr(z0) → *(z0, z0)
Tuples:
SUM(0) → c
SUM(s(z0)) → c1(SQR(s(z0)), SUM(z0))
SUM(s(z0)) → c2(SUM(z0))
SQR(z0) → c3
S tuples:
SUM(0) → c
SUM(s(z0)) → c1(SQR(s(z0)), SUM(z0))
SUM(s(z0)) → c2(SUM(z0))
SQR(z0) → c3
K tuples:none
Defined Rule Symbols:
sum, sqr
Defined Pair Symbols:
SUM, SQR
Compound Symbols:
c, c1, c2, c3
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
SUM(0) → c
SQR(z0) → c3
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
sum(0) → 0
sum(s(z0)) → +(sqr(s(z0)), sum(z0))
sum(s(z0)) → +(*(s(z0), s(z0)), sum(z0))
sqr(z0) → *(z0, z0)
Tuples:
SUM(s(z0)) → c1(SQR(s(z0)), SUM(z0))
SUM(s(z0)) → c2(SUM(z0))
S tuples:
SUM(s(z0)) → c1(SQR(s(z0)), SUM(z0))
SUM(s(z0)) → c2(SUM(z0))
K tuples:none
Defined Rule Symbols:
sum, sqr
Defined Pair Symbols:
SUM
Compound Symbols:
c1, c2
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
sum(0) → 0
sum(s(z0)) → +(sqr(s(z0)), sum(z0))
sum(s(z0)) → +(*(s(z0), s(z0)), sum(z0))
sqr(z0) → *(z0, z0)
Tuples:
SUM(s(z0)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
S tuples:
SUM(s(z0)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
K tuples:none
Defined Rule Symbols:
sum, sqr
Defined Pair Symbols:
SUM
Compound Symbols:
c2, c1
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
sum(0) → 0
sum(s(z0)) → +(sqr(s(z0)), sum(z0))
sum(s(z0)) → +(*(s(z0), s(z0)), sum(z0))
sqr(z0) → *(z0, z0)
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
SUM(s(z0)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
S tuples:
SUM(s(z0)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
SUM
Compound Symbols:
c2, c1
(9) 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)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
We considered the (Usable) Rules:none
And the Tuples:
SUM(s(z0)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(SUM(x1)) = x12
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(s(x1)) = [2] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
SUM(s(z0)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
S tuples:none
K tuples:
SUM(s(z0)) → c2(SUM(z0))
SUM(s(z0)) → c1(SUM(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
SUM
Compound Symbols:
c2, c1
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)