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(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
norm(nil) → 0
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norm(g(x, y)) → s(norm(x))
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f(x, nil) → g(nil, x)
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f(x, g(y, z)) → g(f(x, y), z)
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rem(nil, y) → nil
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rem(g(x, y), 0) → g(x, y)
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rem(g(x, y), s(z)) → rem(x, z)
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:
norm(nil) → 0
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norm(g(z0, z1)) → s(norm(z0))
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f(z0, nil) → g(nil, z0)
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f(z0, g(z1, z2)) → g(f(z0, z1), z2)
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rem(nil, z0) → nil
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rem(g(z0, z1), 0) → g(z0, z1)
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rem(g(z0, z1), s(z2)) → rem(z0, z2)
Tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
K tuples:none
Defined Rule Symbols:
norm, f, rem
Defined Pair Symbols:
NORM, F, REM
Compound Symbols:
c1, c3, c6
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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.
NORM(g(z0, z1)) → c1(NORM(z0))
We considered the (Usable) Rules:none
And the Tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(F(x1, x2)) = 0
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POL(NORM(x1)) = x1
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POL(REM(x1, x2)) = x2
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POL(c1(x1)) = x1
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POL(c3(x1)) = x1
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POL(c6(x1)) = x1
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POL(g(x1, x2)) = [1] + x1
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POL(s(x1)) = x1
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(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
norm(nil) → 0
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norm(g(z0, z1)) → s(norm(z0))
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f(z0, nil) → g(nil, z0)
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f(z0, g(z1, z2)) → g(f(z0, z1), z2)
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rem(nil, z0) → nil
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rem(g(z0, z1), 0) → g(z0, z1)
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rem(g(z0, z1), s(z2)) → rem(z0, z2)
Tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:
F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
K tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
Defined Rule Symbols:
norm, f, rem
Defined Pair Symbols:
NORM, F, REM
Compound Symbols:
c1, c3, c6
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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.
F(z0, g(z1, z2)) → c3(F(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(F(x1, x2)) = [2]x2
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POL(NORM(x1)) = [5]x1
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POL(REM(x1, x2)) = 0
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POL(c1(x1)) = x1
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POL(c3(x1)) = x1
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POL(c6(x1)) = x1
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POL(g(x1, x2)) = [1] + x1
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POL(s(x1)) = [1] + x1
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(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
norm(nil) → 0
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norm(g(z0, z1)) → s(norm(z0))
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f(z0, nil) → g(nil, z0)
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f(z0, g(z1, z2)) → g(f(z0, z1), z2)
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rem(nil, z0) → nil
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rem(g(z0, z1), 0) → g(z0, z1)
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rem(g(z0, z1), s(z2)) → rem(z0, z2)
Tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
K tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
Defined Rule Symbols:
norm, f, rem
Defined Pair Symbols:
NORM, F, REM
Compound Symbols:
c1, c3, c6
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(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :
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POL(F(x1, x2)) = 0
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POL(NORM(x1)) = 0
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POL(REM(x1, x2)) = x22
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POL(c1(x1)) = x1
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POL(c3(x1)) = x1
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POL(c6(x1)) = x1
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POL(g(x1, x2)) = 0
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POL(s(x1)) = [1] + x1
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(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
norm(nil) → 0
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norm(g(z0, z1)) → s(norm(z0))
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f(z0, nil) → g(nil, z0)
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f(z0, g(z1, z2)) → g(f(z0, z1), z2)
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rem(nil, z0) → nil
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rem(g(z0, z1), 0) → g(z0, z1)
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rem(g(z0, z1), s(z2)) → rem(z0, z2)
Tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
S tuples:none
K tuples:
NORM(g(z0, z1)) → c1(NORM(z0))
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F(z0, g(z1, z2)) → c3(F(z0, z1))
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REM(g(z0, z1), s(z2)) → c6(REM(z0, z2))
Defined Rule Symbols:
norm, f, rem
Defined Pair Symbols:
NORM, F, REM
Compound Symbols:
c1, c3, c6
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(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
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(10) BOUNDS(O(1), O(1))
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