(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:
b(y, z) → f(c(c(y, z, z), a, a))
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 0th argument of c: c
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
b(b(z, y), a) → z
c(f(z), f(c(a, x, a)), y) → c(f(b(x, z)), c(z, y, a), a)
(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:
b(y, z) → f(c(c(y, z, z), a, a))
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:
b(y, z) → f(c(c(y, z, z), a, a))
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(z0, z1) → f(c(c(z0, z1, z1), a, a))
Tuples:
B(z0, z1) → c1
S tuples:
B(z0, z1) → c1
K tuples:none
Defined Rule Symbols:
b
Defined Pair Symbols:
B
Compound Symbols:
c1
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
B(z0, z1) → c1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
b(z0, z1) → f(c(c(z0, z1, z1), a, a))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
b
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)