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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
anchored(Cons(x, xs), y) → anchored(xs, Cons(Cons(Nil, Nil), y))
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anchored(Nil, y) → y
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goal(x, y) → anchored(x, y)
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:
anchored(Cons(z0, z1), z2) → anchored(z1, Cons(Cons(Nil, Nil), z2))
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anchored(Nil, z0) → z0
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goal(z0, z1) → anchored(z0, z1)
Tuples:
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
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GOAL(z0, z1) → c2(ANCHORED(z0, z1))
S tuples:
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
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GOAL(z0, z1) → c2(ANCHORED(z0, z1))
K tuples:none
Defined Rule Symbols:
anchored, goal
Defined Pair Symbols:
ANCHORED, GOAL
Compound Symbols:
c, c2
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(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
GOAL(z0, z1) → c2(ANCHORED(z0, z1))
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
anchored(Cons(z0, z1), z2) → anchored(z1, Cons(Cons(Nil, Nil), z2))
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anchored(Nil, z0) → z0
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goal(z0, z1) → anchored(z0, z1)
Tuples:
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
S tuples:
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
K tuples:none
Defined Rule Symbols:
anchored, goal
Defined Pair Symbols:
ANCHORED
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.
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
We considered the (Usable) Rules:none
And the Tuples:
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(ANCHORED(x1, x2)) = [4]x1 + [3]x2
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POL(Cons(x1, x2)) = [4] + x2
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POL(Nil) = 0
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POL(c(x1)) = x1
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
anchored(Cons(z0, z1), z2) → anchored(z1, Cons(Cons(Nil, Nil), z2))
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anchored(Nil, z0) → z0
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goal(z0, z1) → anchored(z0, z1)
Tuples:
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
S tuples:none
K tuples:
ANCHORED(Cons(z0, z1), z2) → c(ANCHORED(z1, Cons(Cons(Nil, Nil), z2)))
Defined Rule Symbols:
anchored, goal
Defined Pair Symbols:
ANCHORED
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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