(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__b → c
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
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, g(z0), z1) → a__f(z1, z1, z1)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__g(b) → c
a__g(z0) → g(z0)
a__b → c
a__b → b
mark(f(z0, z1, z2)) → a__f(z0, z1, z2)
mark(g(z0)) → a__g(mark(z0))
mark(b) → a__b
mark(c) → c
Tuples:
A__F(z0, g(z0), z1) → c1(A__F(z1, z1, z1))
A__F(z0, z1, z2) → c2
A__G(b) → c3
A__G(z0) → c4
A__B → c5
A__B → c6
MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2))
MARK(g(z0)) → c8(A__G(mark(z0)), MARK(z0))
MARK(b) → c9(A__B)
MARK(c) → c10
S tuples:
A__F(z0, g(z0), z1) → c1(A__F(z1, z1, z1))
A__F(z0, z1, z2) → c2
A__G(b) → c3
A__G(z0) → c4
A__B → c5
A__B → c6
MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2))
MARK(g(z0)) → c8(A__G(mark(z0)), MARK(z0))
MARK(b) → c9(A__B)
MARK(c) → c10
K tuples:none
Defined Rule Symbols:
a__f, a__g, a__b, mark
Defined Pair Symbols:
A__F, A__G, A__B, MARK
Compound Symbols:
c1, c2, c3, c4, c5, c6, c7, c8, c9, c10
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 9 trailing nodes:
A__F(z0, z1, z2) → c2
A__B → c5
A__G(z0) → c4
A__G(b) → c3
MARK(b) → c9(A__B)
MARK(c) → c10
A__F(z0, g(z0), z1) → c1(A__F(z1, z1, z1))
A__B → c6
MARK(f(z0, z1, z2)) → c7(A__F(z0, z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__f(z0, g(z0), z1) → a__f(z1, z1, z1)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__g(b) → c
a__g(z0) → g(z0)
a__b → c
a__b → b
mark(f(z0, z1, z2)) → a__f(z0, z1, z2)
mark(g(z0)) → a__g(mark(z0))
mark(b) → a__b
mark(c) → c
Tuples:
MARK(g(z0)) → c8(A__G(mark(z0)), MARK(z0))
S tuples:
MARK(g(z0)) → c8(A__G(mark(z0)), MARK(z0))
K tuples:none
Defined Rule Symbols:
a__f, a__g, a__b, mark
Defined Pair Symbols:
MARK
Compound Symbols:
c8
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a__f(z0, g(z0), z1) → a__f(z1, z1, z1)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__g(b) → c
a__g(z0) → g(z0)
a__b → c
a__b → b
mark(f(z0, z1, z2)) → a__f(z0, z1, z2)
mark(g(z0)) → a__g(mark(z0))
mark(b) → a__b
mark(c) → c
Tuples:
MARK(g(z0)) → c8(MARK(z0))
S tuples:
MARK(g(z0)) → c8(MARK(z0))
K tuples:none
Defined Rule Symbols:
a__f, a__g, a__b, mark
Defined Pair Symbols:
MARK
Compound Symbols:
c8
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
a__f(z0, g(z0), z1) → a__f(z1, z1, z1)
a__f(z0, z1, z2) → f(z0, z1, z2)
a__g(b) → c
a__g(z0) → g(z0)
a__b → c
a__b → b
mark(f(z0, z1, z2)) → a__f(z0, z1, z2)
mark(g(z0)) → a__g(mark(z0))
mark(b) → a__b
mark(c) → c
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
MARK(g(z0)) → c8(MARK(z0))
S tuples:
MARK(g(z0)) → c8(MARK(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
MARK
Compound Symbols:
c8
(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(g(z0)) → c8(MARK(z0))
We considered the (Usable) Rules:none
And the Tuples:
MARK(g(z0)) → c8(MARK(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(MARK(x1)) = [5]x1
POL(c8(x1)) = x1
POL(g(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
MARK(g(z0)) → c8(MARK(z0))
S tuples:none
K tuples:
MARK(g(z0)) → c8(MARK(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
MARK
Compound Symbols:
c8
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)