(0) 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(c(c(a, y, a), b(x, z), a)) → b(y, f(c(f(a), z, z)))
f(b(b(x, f(y)), z)) → c(z, x, f(b(b(f(a), y), y)))
c(b(a, a), b(y, z), x) → b(a, b(z, z))
Rewrite Strategy: INNERMOST
(1) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)
The following rules are not reachable from basic terms in the dependency graph and can be removed:
f(c(c(a, y, a), b(x, z), a)) → b(y, f(c(f(a), z, z)))
f(b(b(x, f(y)), z)) → c(z, x, f(b(b(f(a), y), y)))
(2) 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:
c(b(a, a), b(y, z), x) → b(a, b(z, z))
Rewrite Strategy: INNERMOST
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
c(b(a, a), b(z0, z1), z2) → b(a, b(z1, z1))
Tuples:
C(b(a, a), b(z0, z1), z2) → c1
S tuples:
C(b(a, a), b(z0, z1), z2) → c1
K tuples:none
Defined Rule Symbols:
c
Defined Pair Symbols:
C
Compound Symbols:
c1
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
C(b(a, a), b(z0, z1), z2) → c1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
c(b(a, a), b(z0, z1), z2) → b(a, b(z1, z1))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
c
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)