(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(x, y) → c(a(c(y), a(0, x)))
a(y, x) → y
a(y, c(b(a(0, x), 0))) → b(a(c(b(0, y)), x), 0)
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 1th argument of a: a
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
a(y, c(b(a(0, x), 0))) → b(a(c(b(0, y)), x), 0)
(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:
a(y, x) → y
b(x, y) → c(a(c(y), a(0, x)))
Rewrite Strategy: FULL
(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS is a non-duplicating overlay system, 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:
a(y, x) → y
b(x, y) → c(a(c(y), a(0, x)))
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0, z1) → z0
b(z0, z1) → c(a(c(z1), a(0, z0)))
Tuples:
A(z0, z1) → c1
B(z0, z1) → c2(A(c(z1), a(0, z0)), A(0, z0))
S tuples:
A(z0, z1) → c1
B(z0, z1) → c2(A(c(z1), a(0, z0)), A(0, z0))
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:
A, B
Compound Symbols:
c1, c2
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
A(z0, z1) → c1
B(z0, z1) → c2(A(c(z1), a(0, z0)), A(0, z0))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(z0, z1) → z0
b(z0, z1) → c(a(c(z1), a(0, z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a, b
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)