The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(1)).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(c(a(z0, z1)))) → b(c(c(c(c(z1)))), z0)
c(c(b(c(z0), 0))) → a(0, c(c(a(z0, 0))))
c(c(a(a(z0, 0), z1))) → c(z0)

Tuples:

C(c(c(a(z0, z1)))) → c1(C(c(c(c(z1)))), C(c(c(z1))), C(c(z1)), C(z1))
C(c(b(c(z0), 0))) → c2(C(c(a(z0, 0))), C(a(z0, 0)))
C(c(a(a(z0, 0), z1))) → c3(C(z0))

S tuples:

C(c(c(a(z0, z1)))) → c1(C(c(c(c(z1)))), C(c(c(z1))), C(c(z1)), C(z1))
C(c(b(c(z0), 0))) → c2(C(c(a(z0, 0))), C(a(z0, 0)))
C(c(a(a(z0, 0), z1))) → c3(C(z0))

K tuples:none
Defined Rule Symbols:

c

Defined Pair Symbols:

C

Compound Symbols:

c1, c2, c3

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

C(c(c(a(z0, z1)))) → c1(C(c(c(c(z1)))), C(c(c(z1))), C(c(z1)), C(z1))
C(c(b(c(z0), 0))) → c2(C(c(a(z0, 0))), C(a(z0, 0)))
C(c(a(a(z0, 0), z1))) → c3(C(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

c(c(c(a(z0, z1)))) → b(c(c(c(c(z1)))), z0)
c(c(b(c(z0), 0))) → a(0, c(c(a(z0, 0))))
c(c(a(a(z0, 0), z1))) → c(z0)

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

c

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))