(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
f(x, x) → f(g(x), x)
g(x) → s(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(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)
Tuples:
F(z0, z0) → c(F(g(z0), z0), G(z0))
G(z0) → c1
S tuples:
F(z0, z0) → c(F(g(z0), z0), G(z0))
G(z0) → c1
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F, G
Compound Symbols:
c, c1
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
G(z0) → c1
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)
Tuples:
F(z0, z0) → c(F(g(z0), z0), G(z0))
S tuples:
F(z0, z0) → c(F(g(z0), z0), G(z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, z0) → f(g(z0), z0)
g(z0) → s(z0)
Tuples:
F(z0, z0) → c(F(g(z0), z0))
S tuples:
F(z0, z0) → c(F(g(z0), z0))
K tuples:none
Defined Rule Symbols:
f, g
Defined Pair Symbols:
F
Compound Symbols:
c
(7) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
f(z0, z0) → f(g(z0), z0)
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(z0) → s(z0)
Tuples:
F(z0, z0) → c(F(g(z0), z0))
S tuples:
F(z0, z0) → c(F(g(z0), z0))
K tuples:none
Defined Rule Symbols:
g
Defined Pair Symbols:
F
Compound Symbols:
c
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
z0,
z0) →
c(
F(
g(
z0),
z0)) by
F(z0, z0) → c(F(s(z0), z0))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(z0) → s(z0)
Tuples:
F(z0, z0) → c(F(s(z0), z0))
S tuples:
F(z0, z0) → c(F(s(z0), z0))
K tuples:none
Defined Rule Symbols:
g
Defined Pair Symbols:
F
Compound Symbols:
c
(11) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(z0, z0) → c(F(s(z0), z0))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(z0) → s(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
g
Defined Pair Symbols:none
Compound Symbols:none
(13) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(14) BOUNDS(1, 1)