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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
ackin(s(X), s(Y)) → u21(ackin(s(X), Y), X)
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u21(ackout(X), Y) → u22(ackin(Y, 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:
ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
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u21(ackout(z0), z1) → u22(ackin(z1, z0))
Tuples:
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
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U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
S tuples:
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
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U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
K tuples:none
Defined Rule Symbols:
ackin, u21
Defined Pair Symbols:
ACKIN, U21
Compound Symbols:
c, c1
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(3) 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.
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
We considered the (Usable) Rules:
ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
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u21(ackout(z0), z1) → u22(ackin(z1, z0))
And the Tuples:
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
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U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(ACKIN(x1, x2)) = 0
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POL(U21(x1, x2)) = x1
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POL(ackin(x1, x2)) = 0
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POL(ackout(x1)) = [1]
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POL(c(x1, x2)) = x1 + x2
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POL(c1(x1)) = x1
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POL(s(x1)) = 0
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POL(u21(x1, x2)) = 0
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POL(u22(x1)) = 0
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
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u21(ackout(z0), z1) → u22(ackin(z1, z0))
Tuples:
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
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U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
S tuples:
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
K tuples:
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
Defined Rule Symbols:
ackin, u21
Defined Pair Symbols:
ACKIN, U21
Compound Symbols:
c, c1
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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.
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
We considered the (Usable) Rules:
ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
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u21(ackout(z0), z1) → u22(ackin(z1, z0))
And the Tuples:
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
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U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(ACKIN(x1, x2)) = [3] + [2]x2
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POL(U21(x1, x2)) = [1] + [2]x1
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POL(ackin(x1, x2)) = 0
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POL(ackout(x1)) = [3] + x1
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POL(c(x1, x2)) = x1 + x2
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POL(c1(x1)) = x1
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POL(s(x1)) = [1] + x1
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POL(u21(x1, x2)) = [4]x1
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POL(u22(x1)) = [2]
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
ackin(s(z0), s(z1)) → u21(ackin(s(z0), z1), z0)
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u21(ackout(z0), z1) → u22(ackin(z1, z0))
Tuples:
ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
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U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
S tuples:none
K tuples:
U21(ackout(z0), z1) → c1(ACKIN(z1, z0))
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ACKIN(s(z0), s(z1)) → c(U21(ackin(s(z0), z1), z0), ACKIN(s(z0), z1))
Defined Rule Symbols:
ackin, u21
Defined Pair Symbols:
ACKIN, U21
Compound Symbols:
c, c1
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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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