(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0, 1, X) → h(X, X)
h(0, X) → f(0, X, X)
g(X, Y) → X
g(X, Y) → Y
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → h(z0, z0)
h(0, z0) → f(0, z0, z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:
F(0, 1, z0) → c(H(z0, z0))
H(0, z0) → c1(F(0, z0, z0))
S tuples:
F(0, 1, z0) → c(H(z0, z0))
H(0, z0) → c1(F(0, z0, z0))
K tuples:none
Defined Rule Symbols:
f, h, g
Defined Pair Symbols:
F, H
Compound Symbols:
c, c1
(3) CdtInstantiationProof (BOTH BOUNDS(ID, ID) transformation)
Use instantiation to replace
F(
0,
1,
z0) →
c(
H(
z0,
z0)) by
F(0, 1, 1) → c(H(1, 1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → h(z0, z0)
h(0, z0) → f(0, z0, z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:
H(0, z0) → c1(F(0, z0, z0))
F(0, 1, 1) → c(H(1, 1))
S tuples:
H(0, z0) → c1(F(0, z0, z0))
F(0, 1, 1) → c(H(1, 1))
K tuples:none
Defined Rule Symbols:
f, h, g
Defined Pair Symbols:
H, F
Compound Symbols:
c1, c
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(0, 1, 1) → c(H(1, 1))
H(0, z0) → c1(F(0, z0, z0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → h(z0, z0)
h(0, z0) → f(0, z0, z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, h, g
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))