(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a) → g(h(a))
h(g(x)) → g(h(f(x)))
k(x, h(x), a) → h(x)
k(f(x), y, x) → f(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(a) → g(h(a))
h(g(z0)) → g(h(f(z0)))
k(z0, h(z0), a) → h(z0)
k(f(z0), z1, z0) → f(z0)
Tuples:
F(a) → c(H(a))
H(g(z0)) → c1(H(f(z0)), F(z0))
K(z0, h(z0), a) → c2(H(z0))
K(f(z0), z1, z0) → c3(F(z0))
S tuples:
F(a) → c(H(a))
H(g(z0)) → c1(H(f(z0)), F(z0))
K(z0, h(z0), a) → c2(H(z0))
K(f(z0), z1, z0) → c3(F(z0))
K tuples:none
Defined Rule Symbols:
f, h, k
Defined Pair Symbols:
F, H, K
Compound Symbols:
c, c1, c2, c3
(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
K(z0, h(z0), a) → c2(H(z0))
Removed 2 trailing nodes:
F(a) → c(H(a))
K(f(z0), z1, z0) → c3(F(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a) → g(h(a))
h(g(z0)) → g(h(f(z0)))
k(z0, h(z0), a) → h(z0)
k(f(z0), z1, z0) → f(z0)
Tuples:
H(g(z0)) → c1(H(f(z0)), F(z0))
S tuples:
H(g(z0)) → c1(H(f(z0)), F(z0))
K tuples:none
Defined Rule Symbols:
f, h, k
Defined Pair Symbols:
H
Compound Symbols:
c1
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a) → g(h(a))
h(g(z0)) → g(h(f(z0)))
k(z0, h(z0), a) → h(z0)
k(f(z0), z1, z0) → f(z0)
Tuples:
H(g(z0)) → c1(H(f(z0)))
S tuples:
H(g(z0)) → c1(H(f(z0)))
K tuples:none
Defined Rule Symbols:
f, h, k
Defined Pair Symbols:
H
Compound Symbols:
c1
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
h(g(z0)) → g(h(f(z0)))
k(z0, h(z0), a) → h(z0)
k(f(z0), z1, z0) → f(z0)
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a) → g(h(a))
Tuples:
H(g(z0)) → c1(H(f(z0)))
S tuples:
H(g(z0)) → c1(H(f(z0)))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
H
Compound Symbols:
c1
(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.
H(g(z0)) → c1(H(f(z0)))
We considered the (Usable) Rules:
f(a) → g(h(a))
And the Tuples:
H(g(z0)) → c1(H(f(z0)))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(H(x1)) = [4]x1
POL(a) = [4]
POL(c1(x1)) = x1
POL(f(x1)) = x1
POL(g(x1)) = [2] + x1
POL(h(x1)) = 0
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(a) → g(h(a))
Tuples:
H(g(z0)) → c1(H(f(z0)))
S tuples:none
K tuples:
H(g(z0)) → c1(H(f(z0)))
Defined Rule Symbols:
f
Defined Pair Symbols:
H
Compound Symbols:
c1
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)