(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:
a__f(X, X) → a__f(a, b)
a__b → a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__b → b
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__f(z0, z0) → a__f(a, b)
a__f(z0, z1) → f(z0, z1)
a__b → a
a__b → b
mark(f(z0, z1)) → a__f(mark(z0), z1)
mark(b) → a__b
mark(a) → a
Tuples:
A__F(z0, z0) → c(A__F(a, b))
A__F(z0, z1) → c1
A__B → c2
A__B → c3
MARK(f(z0, z1)) → c4(A__F(mark(z0), z1), MARK(z0))
MARK(b) → c5(A__B)
MARK(a) → c6
S tuples:
A__F(z0, z0) → c(A__F(a, b))
A__F(z0, z1) → c1
A__B → c2
A__B → c3
MARK(f(z0, z1)) → c4(A__F(mark(z0), z1), MARK(z0))
MARK(b) → c5(A__B)
MARK(a) → c6
K tuples:none
Defined Rule Symbols:
a__f, a__b, mark
Defined Pair Symbols:
A__F, A__B, MARK
Compound Symbols:
c, c1, c2, c3, c4, c5, c6
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
MARK(a) → c6
A__B → c3
A__F(z0, z1) → c1
A__F(z0, z0) → c(A__F(a, b))
A__B → c2
MARK(b) → c5(A__B)
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__f(z0, z0) → a__f(a, b)
a__f(z0, z1) → f(z0, z1)
a__b → a
a__b → b
mark(f(z0, z1)) → a__f(mark(z0), z1)
mark(b) → a__b
mark(a) → a
Tuples:
MARK(f(z0, z1)) → c4(A__F(mark(z0), z1), MARK(z0))
S tuples:
MARK(f(z0, z1)) → c4(A__F(mark(z0), z1), MARK(z0))
K tuples:none
Defined Rule Symbols:
a__f, a__b, mark
Defined Pair Symbols:
MARK
Compound Symbols:
c4
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__f(z0, z0) → a__f(a, b)
a__f(z0, z1) → f(z0, z1)
a__b → a
a__b → b
mark(f(z0, z1)) → a__f(mark(z0), z1)
mark(b) → a__b
mark(a) → a
Tuples:
MARK(f(z0, z1)) → c4(MARK(z0))
S tuples:
MARK(f(z0, z1)) → c4(MARK(z0))
K tuples:none
Defined Rule Symbols:
a__f, a__b, mark
Defined Pair Symbols:
MARK
Compound Symbols:
c4
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
a__f(z0, z0) → a__f(a, b)
a__f(z0, z1) → f(z0, z1)
a__b → a
a__b → b
mark(f(z0, z1)) → a__f(mark(z0), z1)
mark(b) → a__b
mark(a) → a
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
MARK(f(z0, z1)) → c4(MARK(z0))
S tuples:
MARK(f(z0, z1)) → c4(MARK(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
MARK
Compound Symbols:
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.
MARK(f(z0, z1)) → c4(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:
MARK(f(z0, z1)) → c4(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(MARK(x1)) = x1
POL(c4(x1)) = x1
POL(f(x1, x2)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
MARK(f(z0, z1)) → c4(MARK(z0))
S tuples:none
K tuples:
MARK(f(z0, z1)) → c4(MARK(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
MARK
Compound Symbols:
c4
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)