(0) Obligation:
The Runtime Complexity (full) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
f(f(X)) → f(a(b(f(X))))
f(a(g(X))) → b(X)
b(X) → a(X)
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(f(X)) → f(a(b(f(X))))
(2) Obligation:
The Runtime Complexity (full) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
f(a(g(X))) → b(X)
b(X) → a(X)
Rewrite Strategy: FULL
(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS does not nest defined symbols, we have rc = irc.
(4) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
f(a(g(X))) → b(X)
b(X) → a(X)
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:
F(a(g(z0))) → c(B(z0))
B(z0) → c1
S tuples:
F(a(g(z0))) → c(B(z0))
B(z0) → c1
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:
F, B
Compound Symbols:
c, c1
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(a(g(z0))) → c(B(z0))
B(z0) → c1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a(g(z0))) → b(z0)
b(z0) → a(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, b
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)