(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
not(true) → false
not(false) → true
evenodd(x, 0) → not(evenodd(x, s(0)))
evenodd(0, s(0)) → false
evenodd(s(x), s(0)) → evenodd(x, 0)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:
NOT(true) → c
NOT(false) → c1
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(0, s(0)) → c3
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
S tuples:
NOT(true) → c
NOT(false) → c1
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(0, s(0)) → c3
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
K tuples:none
Defined Rule Symbols:
not, evenodd
Defined Pair Symbols:
NOT, EVENODD
Compound Symbols:
c, c1, c2, c3, c4
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
NOT(true) → c
NOT(false) → c1
EVENODD(0, s(0)) → c3
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
S tuples:
EVENODD(z0, 0) → c2(NOT(evenodd(z0, s(0))), EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
K tuples:none
Defined Rule Symbols:
not, evenodd
Defined Pair Symbols:
EVENODD
Compound Symbols:
c2, c4
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
Tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:none
Defined Rule Symbols:
not, evenodd
Defined Pair Symbols:
EVENODD
Compound Symbols:
c4, c2
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
not(true) → false
not(false) → true
evenodd(z0, 0) → not(evenodd(z0, s(0)))
evenodd(0, s(0)) → false
evenodd(s(z0), s(0)) → evenodd(z0, 0)
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
EVENODD
Compound Symbols:
c4, c2
(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
We considered the (Usable) Rules:none
And the Tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(EVENODD(x1, x2)) = x1
POL(c2(x1)) = x1
POL(c4(x1)) = x1
POL(s(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
S tuples:
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
K tuples:
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Defined Rule Symbols:none
Defined Pair Symbols:
EVENODD
Compound Symbols:
c4, c2
(11) CdtKnowledgeProof (EQUIVALENT transformation)
The following tuples could be moved from S to K by knowledge propagation:
EVENODD(z0, 0) → c2(EVENODD(z0, s(0)))
EVENODD(s(z0), s(0)) → c4(EVENODD(z0, 0))
Now S is empty
(12) BOUNDS(O(1), O(1))