(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(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))
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(c(f(x)))
f(f(x)) → f(d(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:
g(c(0)) → g(d(1))
g(d(x)) → x
g(c(x)) → x
g(c(1)) → g(d(0))
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:
g(c(0)) → g(d(1))
g(d(x)) → x
g(c(x)) → x
g(c(1)) → g(d(0))
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(c(0)) → g(d(1))
g(d(z0)) → z0
g(c(z0)) → z0
g(c(1)) → g(d(0))
Tuples:
G(c(0)) → c1(G(d(1)))
G(d(z0)) → c2
G(c(z0)) → c3
G(c(1)) → c4(G(d(0)))
S tuples:
G(c(0)) → c1(G(d(1)))
G(d(z0)) → c2
G(c(z0)) → c3
G(c(1)) → c4(G(d(0)))
K tuples:none
Defined Rule Symbols:
g
Defined Pair Symbols:
G
Compound Symbols:
c1, c2, c3, c4
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
G(d(z0)) → c2
G(c(1)) → c4(G(d(0)))
G(c(0)) → c1(G(d(1)))
G(c(z0)) → c3
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(c(0)) → g(d(1))
g(d(z0)) → z0
g(c(z0)) → z0
g(c(1)) → g(d(0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
g
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)