(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:
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: INNERMOST
(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 (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
(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2]
transitions:
c0(0) → 0
00() → 0
a0(0, 0) → 1
b0(0, 0) → 2
c1(0) → 4
01() → 6
a1(6, 0) → 5
a1(4, 5) → 3
c1(3) → 2
0 → 1
6 → 5
4 → 3
(4) BOUNDS(1, n^1)
(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:
B(z0, z1) → c2(A(c(z1), a(0, z0)), A(0, z0))
A(z0, z1) → c1
(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)