(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:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))
Rewrite Strategy: FULL
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 1th argument of plus: times
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
(2) 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:
times(X, s(Y)) → plus(X, times(Y, X))
Rewrite Strategy: FULL
(3) RcToIrcProof (BOTH BOUNDS(ID, ID) transformation)
Converted rc-obligation to irc-obligation.
As the TRS does not nest defined symbols, we have rc = irc.
(4) 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:
times(X, s(Y)) → plus(X, times(Y, X))
Rewrite Strategy: INNERMOST
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
times(z0, s(z1)) → plus(z0, times(z1, z0))
Tuples:
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
S tuples:
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
K tuples:none
Defined Rule Symbols:
times
Defined Pair Symbols:
TIMES
Compound Symbols:
c
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
times(z0, s(z1)) → plus(z0, times(z1, z0))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
S tuples:
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
TIMES
Compound Symbols:
c
(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
We considered the (Usable) Rules:none
And the Tuples:
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(TIMES(x1, x2)) = x1 + x2
POL(c(x1)) = x1
POL(s(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
S tuples:none
K tuples:
TIMES(z0, s(z1)) → c(TIMES(z1, z0))
Defined Rule Symbols:none
Defined Pair Symbols:
TIMES
Compound Symbols:
c
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)