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