(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0, 1, X) → f(g(X, X), 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) → f(g(z0, z0), z0, z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:
F(0, 1, z0) → c(F(g(z0, z0), z0, z0), G(z0, z0))
S tuples:
F(0, 1, z0) → c(F(g(z0, z0), z0, z0), G(z0, z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c
(3) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(g(z0, z0), z0, z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:
F(0, 1, z0) → c(F(g(z0, z0), z0, z0))
S tuples:
F(0, 1, z0) → c(F(g(z0, z0), z0, z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
0,
1,
z0) →
c(
F(
g(
z0,
z0),
z0,
z0)) by
F(0, 1, z0) → c(F(z0, z0, z0))
F(0, 1, x0) → c
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(g(z0, z0), z0, z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:
F(0, 1, z0) → c(F(z0, z0, z0))
F(0, 1, x0) → c
S tuples:
F(0, 1, z0) → c(F(z0, z0, z0))
F(0, 1, x0) → c
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c, c
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing nodes:
F(0, 1, z0) → c(F(z0, z0, z0))
F(0, 1, x0) → c
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(g(z0, z0), z0, z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))