(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a, x) → f(b, f(c, x))
f(a, f(b, x)) → f(b, f(a, x))
f(d, f(c, x)) → f(d, f(a, x))
f(a, f(c, x)) → f(c, f(a, x))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, z0) → f(b, f(c, z0))
f(a, f(b, z0)) → f(b, f(a, z0))
f(d, f(c, z0)) → f(d, f(a, z0))
f(a, f(c, z0)) → f(c, f(a, z0))
Tuples:
F(a, z0) → c1(F(b, f(c, z0)), F(c, z0))
F(a, f(b, z0)) → c2(F(b, f(a, z0)), F(a, z0))
F(d, f(c, z0)) → c3(F(d, f(a, z0)), F(a, z0))
F(a, f(c, z0)) → c4(F(c, f(a, z0)), F(a, z0))
S tuples:
F(a, z0) → c1(F(b, f(c, z0)), F(c, z0))
F(a, f(b, z0)) → c2(F(b, f(a, z0)), F(a, z0))
F(d, f(c, z0)) → c3(F(d, f(a, z0)), F(a, z0))
F(a, f(c, z0)) → c4(F(c, f(a, z0)), F(a, z0))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1, c2, c3, c4
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(a, f(b, z0)) → c2(F(b, f(a, z0)), F(a, z0))
F(d, f(c, z0)) → c3(F(d, f(a, z0)), F(a, z0))
F(a, f(c, z0)) → c4(F(c, f(a, z0)), F(a, z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, z0) → f(b, f(c, z0))
f(a, f(b, z0)) → f(b, f(a, z0))
f(d, f(c, z0)) → f(d, f(a, z0))
f(a, f(c, z0)) → f(c, f(a, z0))
Tuples:
F(a, z0) → c1(F(b, f(c, z0)), F(c, z0))
S tuples:
F(a, z0) → c1(F(b, f(c, z0)), F(c, z0))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c1
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(a, z0) → c1(F(b, f(c, z0)), F(c, z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a, z0) → f(b, f(c, z0))
f(a, f(b, z0)) → f(b, f(a, z0))
f(d, f(c, z0)) → f(d, f(a, z0))
f(a, f(c, z0)) → f(c, f(a, z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))