(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^2).
The TRS R consists of the following rules:
h(z, e(x)) → h(c(z), d(z, x))
d(z, g(0, 0)) → e(0)
d(z, g(x, y)) → g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) → g(d(c(z), g(x, y)), d(z, g(x, y)))
g(e(x), e(y)) → e(g(x, y))
Rewrite Strategy: INNERMOST
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The following defined symbols can occur below the 1th argument of h: d
The following defined symbols can occur below the 1th argument of d: d
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
d(z, g(0, 0)) → e(0)
d(z, g(x, y)) → g(e(x), d(z, y))
d(c(z), g(g(x, y), 0)) → g(d(c(z), g(x, y)), d(z, g(x, y)))
(2) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^2).
The TRS R consists of the following rules:
h(z, e(x)) → h(c(z), d(z, x))
g(e(x), e(y)) → e(g(x, y))
Rewrite Strategy: INNERMOST
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, e(z1)) → h(c(z0), d(z0, z1))
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:
H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)))
G(e(z0), e(z1)) → c2(G(z0, z1))
S tuples:
H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)))
G(e(z0), e(z1)) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:
h, g
Defined Pair Symbols:
H, G
Compound Symbols:
c1, c2
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
H(z0, e(z1)) → c1(H(c(z0), d(z0, z1)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
h(z0, e(z1)) → h(c(z0), d(z0, z1))
g(e(z0), e(z1)) → e(g(z0, z1))
Tuples:
G(e(z0), e(z1)) → c2(G(z0, z1))
S tuples:
G(e(z0), e(z1)) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:
h, g
Defined Pair Symbols:
G
Compound Symbols:
c2
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
h(z0, e(z1)) → h(c(z0), d(z0, z1))
g(e(z0), e(z1)) → e(g(z0, z1))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(e(z0), e(z1)) → c2(G(z0, z1))
S tuples:
G(e(z0), e(z1)) → c2(G(z0, z1))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
G
Compound Symbols:
c2
(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
G(e(z0), e(z1)) → c2(G(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
G(e(z0), e(z1)) → c2(G(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(G(x1, x2)) = [2]x22
POL(c2(x1)) = x1
POL(e(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(e(z0), e(z1)) → c2(G(z0, z1))
S tuples:none
K tuples:
G(e(z0), e(z1)) → c2(G(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
G
Compound Symbols:
c2
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)