(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a, f(a, f(b, f(x, y)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(x, y)))))))
f(a, f(c, f(x, y))) → f(b, f(x, y))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:
F(a, f(a, f(b, f(z0, z1)))) → c1(F(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(z0, z1)))))), F(b, f(a, f(a, f(a, f(z0, z1))))), F(a, f(a, f(a, f(z0, z1)))), F(a, f(a, f(z0, z1))), F(a, f(z0, z1)), F(z0, z1))
F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1))
S tuples:
F(a, f(a, f(b, f(z0, z1)))) → c1(F(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(z0, z1)))))), F(b, f(a, f(a, f(a, f(z0, z1))))), F(a, f(a, f(a, f(z0, z1)))), F(a, f(a, f(z0, z1))), F(a, f(z0, z1)), F(z0, z1))
F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, c2
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(a, f(a, f(b, f(z0, z1)))) → c1(F(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1))))))), F(c, f(b, f(a, f(a, f(a, f(z0, z1)))))), F(b, f(a, f(a, f(a, f(z0, z1))))), F(a, f(a, f(a, f(z0, z1)))), F(a, f(a, f(z0, z1))), F(a, f(z0, z1)), F(z0, z1))
F(a, f(c, f(z0, z1))) → c2(F(b, f(z0, z1)), F(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, f(a, f(b, f(z0, z1)))) → f(b, f(c, f(b, f(a, f(a, f(a, f(z0, z1)))))))
f(a, f(c, f(z0, z1))) → f(b, f(z0, z1))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))