(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
and(not(not(x)), y, not(z)) → and(y, band(x, z), 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(not(not(z0)), z1, not(z2)) → and(z1, band(z0, z2), z0)
Tuples:
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
S tuples:
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
K tuples:none
Defined Rule Symbols:
and
Defined Pair Symbols:
AND
Compound Symbols:
c
(3) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
and(not(not(z0)), z1, not(z2)) → and(z1, band(z0, z2), z0)
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
S tuples:
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
AND
Compound Symbols:
c
(5) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
We considered the (Usable) Rules:none
And the Tuples:
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(AND(x1, x2, x3)) = [4]x1 + [5]x2
POL(band(x1, x2)) = [3]
POL(c(x1)) = x1
POL(not(x1)) = [4] + x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
S tuples:none
K tuples:
AND(not(not(z0)), z1, not(z2)) → c(AND(z1, band(z0, z2), z0))
Defined Rule Symbols:none
Defined Pair Symbols:
AND
Compound Symbols:
c
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)