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