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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
duplicate(Cons(x, xs)) → Cons(x, Cons(x, duplicate(xs)))
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duplicate(Nil) → Nil
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goal(x) → duplicate(x)
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
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(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
duplicate(Cons(z0, z1)) → Cons(z0, Cons(z0, duplicate(z1)))
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duplicate(Nil) → Nil
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goal(z0) → duplicate(z0)
Tuples:
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
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GOAL(z0) → c2(DUPLICATE(z0))
S tuples:
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
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GOAL(z0) → c2(DUPLICATE(z0))
K tuples:none
Defined Rule Symbols:
duplicate, goal
Defined Pair Symbols:
DUPLICATE, GOAL
Compound Symbols:
c, c2
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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
GOAL(z0) → c2(DUPLICATE(z0))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
duplicate(Cons(z0, z1)) → Cons(z0, Cons(z0, duplicate(z1)))
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duplicate(Nil) → Nil
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goal(z0) → duplicate(z0)
Tuples:
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
S tuples:
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
K tuples:none
Defined Rule Symbols:
duplicate, goal
Defined Pair Symbols:
DUPLICATE
Compound Symbols:
c
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(5) 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.
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
We considered the (Usable) Rules:none
And the Tuples:
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(Cons(x1, x2)) = [1] + x2
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POL(DUPLICATE(x1)) = [3]x1
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POL(c(x1)) = x1
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
duplicate(Cons(z0, z1)) → Cons(z0, Cons(z0, duplicate(z1)))
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duplicate(Nil) → Nil
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goal(z0) → duplicate(z0)
Tuples:
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
S tuples:none
K tuples:
DUPLICATE(Cons(z0, z1)) → c(DUPLICATE(z1))
Defined Rule Symbols:
duplicate, goal
Defined Pair Symbols:
DUPLICATE
Compound Symbols:
c
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(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(8) BOUNDS(O(1), O(1))
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