(0) Obligation:
The Runtime Complexity (full) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))
Rewrite Strategy: FULL
(1) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS does not nest defined symbols, we have rc = irc.
(2) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))
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 2.
The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1]
transitions:
not0(0) → 0
or0(0, 0) → 0
implies0(0, 0) → 1
or1(0, 0) → 1
or1(0, 0) → 2
implies1(0, 2) → 1
implies1(0, 0) → 3
or1(0, 3) → 1
or1(0, 2) → 1
or1(0, 0) → 3
implies1(0, 2) → 3
or1(0, 3) → 3
implies2(0, 0) → 4
or2(0, 4) → 1
or1(0, 2) → 3
or2(0, 4) → 3
or1(0, 0) → 4
implies1(0, 2) → 4
or1(0, 3) → 4
or1(0, 2) → 4
or2(0, 4) → 4
(4) BOUNDS(1, n^1)