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