(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^2).
The TRS R consists of the following rules:
and(tt, X) → activate(X)
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))
x(N, 0) → 0
x(N, s(M)) → plus(x(N, M), N)
activate(X) → X
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
and(tt, z0) → activate(z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
activate(z0) → z0
Tuples:
AND(tt, z0) → c(ACTIVATE(z0))
PLUS(z0, 0) → c1
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, 0) → c3
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) → c5
S tuples:
AND(tt, z0) → c(ACTIVATE(z0))
PLUS(z0, 0) → c1
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, 0) → c3
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
ACTIVATE(z0) → c5
K tuples:none
Defined Rule Symbols:
and, plus, x, activate
Defined Pair Symbols:
AND, PLUS, X, ACTIVATE
Compound Symbols:
c, c1, c2, c3, c4, c5
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
X(z0, 0) → c3
PLUS(z0, 0) → c1
AND(tt, z0) → c(ACTIVATE(z0))
ACTIVATE(z0) → c5
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
and(tt, z0) → activate(z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
activate(z0) → z0
Tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
Defined Rule Symbols:
and, plus, x, activate
Defined Pair Symbols:
PLUS, X
Compound Symbols:
c2, c4
(5) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
and(tt, z0) → activate(z0)
activate(z0) → z0
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:none
Defined Rule Symbols:
x, plus
Defined Pair Symbols:
PLUS, X
Compound Symbols:
c2, c4
(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(PLUS(x1, x2)) = x2
POL(X(x1, x2)) = [2]x1·x2
POL(c2(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(plus(x1, x2)) = [2]x22
POL(s(x1)) = [2] + x1
POL(x(x1, x2)) = 0
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
K tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
Defined Rule Symbols:
x, plus
Defined Pair Symbols:
PLUS, X
Compound Symbols:
c2, c4
(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.
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(PLUS(x1, x2)) = 0
POL(X(x1, x2)) = x2
POL(c2(x1)) = x1
POL(c4(x1, x2)) = x1 + x2
POL(plus(x1, x2)) = 0
POL(s(x1)) = [1] + x1
POL(x(x1, x2)) = 0
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
x(z0, 0) → 0
x(z0, s(z1)) → plus(x(z0, z1), z0)
plus(z0, 0) → z0
plus(z0, s(z1)) → s(plus(z0, z1))
Tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
S tuples:none
K tuples:
PLUS(z0, s(z1)) → c2(PLUS(z0, z1))
X(z0, s(z1)) → c4(PLUS(x(z0, z1), z0), X(z0, z1))
Defined Rule Symbols:
x, plus
Defined Pair Symbols:
PLUS, X
Compound Symbols:
c2, c4
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)