(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:
f(g(X), Y) → f(X, n__f(n__g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
g(X) → n__g(X)
activate(n__f(X1, X2)) → f(activate(X1), X2)
activate(n__g(X)) → g(activate(X))
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:
f(g(z0), z1) → f(z0, n__f(n__g(z0), activate(z1)))
f(z0, z1) → n__f(z0, z1)
g(z0) → n__g(z0)
activate(n__f(z0, z1)) → f(activate(z0), z1)
activate(n__g(z0)) → g(activate(z0))
activate(z0) → z0
Tuples:
F(g(z0), z1) → c(F(z0, n__f(n__g(z0), activate(z1))), ACTIVATE(z1))
F(z0, z1) → c1
G(z0) → c2
ACTIVATE(n__f(z0, z1)) → c3(F(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(G(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c5
S tuples:
F(g(z0), z1) → c(F(z0, n__f(n__g(z0), activate(z1))), ACTIVATE(z1))
F(z0, z1) → c1
G(z0) → c2
ACTIVATE(n__f(z0, z1)) → c3(F(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(G(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c5
K tuples:none
Defined Rule Symbols:
f, g, activate
Defined Pair Symbols:
F, G, ACTIVATE
Compound Symbols:
c, c1, c2, c3, c4, c5
(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
F(g(z0), z1) → c(F(z0, n__f(n__g(z0), activate(z1))), ACTIVATE(z1))
Removed 3 trailing nodes:
G(z0) → c2
F(z0, z1) → c1
ACTIVATE(z0) → c5
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), z1) → f(z0, n__f(n__g(z0), activate(z1)))
f(z0, z1) → n__f(z0, z1)
g(z0) → n__g(z0)
activate(n__f(z0, z1)) → f(activate(z0), z1)
activate(n__g(z0)) → g(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__f(z0, z1)) → c3(F(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(G(activate(z0)), ACTIVATE(z0))
S tuples:
ACTIVATE(n__f(z0, z1)) → c3(F(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(G(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
f, g, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3, c4
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(g(z0), z1) → f(z0, n__f(n__g(z0), activate(z1)))
f(z0, z1) → n__f(z0, z1)
g(z0) → n__g(z0)
activate(n__f(z0, z1)) → f(activate(z0), z1)
activate(n__g(z0)) → g(activate(z0))
activate(z0) → z0
Tuples:
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
S tuples:
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:
f, g, activate
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3, c4
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
f(g(z0), z1) → f(z0, n__f(n__g(z0), activate(z1)))
f(z0, z1) → n__f(z0, z1)
g(z0) → n__g(z0)
activate(n__f(z0, z1)) → f(activate(z0), z1)
activate(n__g(z0)) → g(activate(z0))
activate(z0) → z0
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
S tuples:
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3, 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.
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(ACTIVATE(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(n__f(x1, x2)) = [1] + x1
POL(n__g(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
S tuples:none
K tuples:
ACTIVATE(n__f(z0, z1)) → c3(ACTIVATE(z0))
ACTIVATE(n__g(z0)) → c4(ACTIVATE(z0))
Defined Rule Symbols:none
Defined Pair Symbols:
ACTIVATE
Compound Symbols:
c3, c4
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)