(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(x1) → x1
a(x1) → b(x1)
b(x1) → x1
b(a(c(x1))) → c(c(a(a(b(x1)))))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(z0) → b(z0)
b(z0) → z0
b(a(c(z0))) → c(c(a(a(b(z0)))))
Tuples:
A(z0) → c2(B(z0))
B(a(c(z0))) → c4(A(a(b(z0))), A(b(z0)), B(z0))
S tuples:
A(z0) → c2(B(z0))
B(a(c(z0))) → c4(A(a(b(z0))), A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c2, c4
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
B(a(c(z0))) → c4(A(a(b(z0))), A(b(z0)), B(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(z0) → b(z0)
b(z0) → z0
b(a(c(z0))) → c(c(a(a(b(z0)))))
Tuples:
A(z0) → c2(B(z0))
S tuples:
A(z0) → c2(B(z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A
Compound Symbols:
c2
(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
A(z0) → c2(B(z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0) → z0
a(z0) → b(z0)
b(z0) → z0
b(a(c(z0))) → c(c(a(a(b(z0)))))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))