(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(g(x), s(0), y) → f(g(s(0)), y, g(x))
g(s(x)) → s(g(x))
g(0) → 0
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), s(0), z1) → f(g(s(0)), z1, g(z0))
g(s(z0)) → s(g(z0))
g(0) → 0
Tuples:
F(g(z0), s(0), z1) → c(F(g(s(0)), z1, g(z0)), G(s(0)), G(z0))
G(s(z0)) → c1(G(z0))
S tuples:
F(g(z0), s(0), z1) → c(F(g(s(0)), z1, g(z0)), G(s(0)), G(z0))
G(s(z0)) → c1(G(z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F, G
Compound Symbols:
c, c1
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(g(z0), s(0), z1) → c(F(g(s(0)), z1, g(z0)), G(s(0)), G(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), s(0), z1) → f(g(s(0)), z1, g(z0))
g(s(z0)) → s(g(z0))
g(0) → 0
Tuples:
G(s(z0)) → c1(G(z0))
S tuples:
G(s(z0)) → c1(G(z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
G
Compound Symbols:
c1
(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.
G(s(z0)) → c1(G(z0))
We considered the (Usable) Rules:none
And the Tuples:
G(s(z0)) → c1(G(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(G(x1)) = [5]x1
POL(c1(x1)) = x1
POL(s(x1)) = [1] + x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), s(0), z1) → f(g(s(0)), z1, g(z0))
g(s(z0)) → s(g(z0))
g(0) → 0
Tuples:
G(s(z0)) → c1(G(z0))
S tuples:none
K tuples:
G(s(z0)) → c1(G(z0))
Defined Rule Symbols:
f, g
Defined Pair Symbols:
G
Compound Symbols:
c1
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))