(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
c(c(c(a(x1)))) → d(d(x1))
d(b(x1)) → c(c(x1))
b(c(x1)) → b(a(c(x1)))
c(x1) → a(a(x1))
d(x1) → b(c(x1))
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)))) → d(d(z0))
c(z0) → a(a(z0))
d(b(z0)) → c(c(z0))
d(z0) → b(c(z0))
b(c(z0)) → b(a(c(z0)))
Tuples:
C(c(c(a(z0)))) → c1(D(d(z0)), D(z0))
D(b(z0)) → c3(C(c(z0)), C(z0))
D(z0) → c4(B(c(z0)), C(z0))
B(c(z0)) → c5(B(a(c(z0))), C(z0))
S tuples:
C(c(c(a(z0)))) → c1(D(d(z0)), D(z0))
D(b(z0)) → c3(C(c(z0)), C(z0))
D(z0) → c4(B(c(z0)), C(z0))
B(c(z0)) → c5(B(a(c(z0))), C(z0))
K tuples:none
Defined Rule Symbols:
c, d, b
Defined Pair Symbols:
C, D, B
Compound Symbols:
c1, c3, c4, c5
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
C(c(c(a(z0)))) → c1(D(d(z0)), D(z0))
D(b(z0)) → c3(C(c(z0)), C(z0))
B(c(z0)) → c5(B(a(c(z0))), C(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
c(c(c(a(z0)))) → d(d(z0))
c(z0) → a(a(z0))
d(b(z0)) → c(c(z0))
d(z0) → b(c(z0))
b(c(z0)) → b(a(c(z0)))
Tuples:
D(z0) → c4(B(c(z0)), C(z0))
S tuples:
D(z0) → c4(B(c(z0)), C(z0))
K tuples:none
Defined Rule Symbols:
c, d, b
Defined Pair Symbols:
D
Compound Symbols:
c4
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
D(z0) → c4(B(c(z0)), C(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
c(c(c(a(z0)))) → d(d(z0))
c(z0) → a(a(z0))
d(b(z0)) → c(c(z0))
d(z0) → b(c(z0))
b(c(z0)) → b(a(c(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
c, d, b
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))