(0) 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:
w(r(x)) → r(w(x))
b(r(x)) → r(b(x))
b(w(x)) → w(b(x))
Rewrite Strategy: INNERMOST
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
b(w(x)) → w(b(x))
(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:
b(r(x)) → r(b(x))
w(r(x)) → r(w(x))
Rewrite Strategy: INNERMOST
(3) CpxTrsMatchBoundsProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.
Start state: 3
Accept states: [4]
Transitions:
3→4[b_1|0, w_1|0]
3→5[r_1|1]
3→6[r_1|1]
4→4[r_1|0]
5→4[b_1|1]
5→5[r_1|1]
6→4[w_1|1]
6→6[r_1|1]
(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:
b(r(z0)) → r(b(z0))
w(r(z0)) → r(w(z0))
Tuples:
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
S tuples:
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
K tuples:none
Defined Rule Symbols:
b, w
Defined Pair Symbols:
B, W
Compound Symbols:
c, c1
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
b(r(z0)) → r(b(z0))
w(r(z0)) → r(w(z0))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
S tuples:
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
B, W
Compound Symbols:
c, c1
(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
We considered the (Usable) Rules:none
And the Tuples:
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(B(x1)) = x12
POL(W(x1)) = x1
POL(c(x1)) = x1
POL(c1(x1)) = x1
POL(r(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
S tuples:none
K tuples:
B(r(z0)) → c(B(z0))
W(r(z0)) → c1(W(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
B, W
Compound Symbols:
c, c1
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)