(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(1) → f(g(1))
f(f(x)) → f(x)
g(0) → g(f(0))
g(g(x)) → g(x)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(1) → f(g(1))
f(f(z0)) → f(z0)
g(0) → g(f(0))
g(g(z0)) → g(z0)
Tuples:
F(1) → c(F(g(1)), G(1))
F(f(z0)) → c1(F(z0))
G(0) → c2(G(f(0)), F(0))
G(g(z0)) → c3(G(z0))
S tuples:
F(1) → c(F(g(1)), G(1))
F(f(z0)) → c1(F(z0))
G(0) → c2(G(f(0)), F(0))
G(g(z0)) → c3(G(z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F, G
Compound Symbols:
c, c1, c2, c3
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(f(z0)) → c1(F(z0))
G(g(z0)) → c3(G(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(1) → f(g(1))
f(f(z0)) → f(z0)
g(0) → g(f(0))
g(g(z0)) → g(z0)
Tuples:
F(1) → c(F(g(1)), G(1))
G(0) → c2(G(f(0)), F(0))
S tuples:
F(1) → c(F(g(1)), G(1))
G(0) → c2(G(f(0)), F(0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F, G
Compound Symbols:
c, c2
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(1) → c(F(g(1)), G(1))
G(0) → c2(G(f(0)), F(0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(1) → f(g(1))
f(f(z0)) → f(z0)
g(0) → g(f(0))
g(g(z0)) → g(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))