(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(S(x), x2) → f(x2, x)
f(0, x2) → 0
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(S(z0), z1) → f(z1, z0)
f(0, z0) → 0
Tuples:
F(S(z0), z1) → c(F(z1, z0))
F(0, z0) → c1
S tuples:
F(S(z0), z1) → c(F(z1, z0))
F(0, z0) → c1
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(0, z0) → c1
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(S(z0), z1) → f(z1, z0)
f(0, z0) → 0
Tuples:
F(S(z0), z1) → c(F(z1, z0))
S tuples:
F(S(z0), z1) → c(F(z1, z0))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
f(S(z0), z1) → f(z1, z0)
f(0, z0) → 0
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
F(S(z0), z1) → c(F(z1, z0))
S tuples:
F(S(z0), z1) → c(F(z1, z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
F
Compound Symbols:
c
(7) CdtRuleRemovalProof (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(S(z0), z1) → c(F(z1, z0))
We considered the (Usable) Rules:none
And the Tuples:
F(S(z0), z1) → c(F(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F(x1, x2)) = [2]x1 + [2]x2
POL(S(x1)) = [1] + x1
POL(c(x1)) = x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
F(S(z0), z1) → c(F(z1, z0))
S tuples:none
K tuples:
F(S(z0), z1) → c(F(z1, z0))
Defined Rule Symbols:none
Defined Pair Symbols:
F
Compound Symbols:
c
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))