(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(0, 1, x) → f(h(x), h(x), x)
h(0) → 0
h(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(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
S tuples:
F(0, 1, z0) → c(F(h(z0), h(z0), z0), H(z0), H(z0))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(3) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, z0) → c(F(h(z0), h(z0), z0))
S tuples:
F(0, 1, z0) → c(F(h(z0), h(z0), z0))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
0,
1,
z0) →
c(
F(
h(
z0),
h(
z0),
z0)) by
F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:
F(0, 1, 0) → c(F(h(0), 0, 0))
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(7) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 4 dangling nodes:
F(0, 1, 0) → c(F(h(0), 0, 0))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:
F(0, 1, g(z0, z1)) → c(F(h(g(z0, z1)), z1, g(z0, z1)))
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(9) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
0,
1,
g(
z0,
z1)) →
c(
F(
h(
g(
z0,
z1)),
z1,
g(
z0,
z1))) by
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
S tuples:
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(11) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 3 dangling nodes:
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:
F(0, 1, 0) → c(F(0, h(0), 0))
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
0,
1,
0) →
c(
F(
0,
h(
0),
0)) by
F(0, 1, 0) → c(F(0, 0, 0))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))
S tuples:
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
F(0, 1, 0) → c(F(0, 0, 0))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(15) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 2 dangling nodes:
F(0, 1, 0) → c(F(0, 0, 0))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
S tuples:
F(0, 1, g(z0, z1)) → c(F(z1, h(g(z0, z1)), g(z0, z1)))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
0,
1,
g(
z0,
z1)) →
c(
F(
z1,
h(
g(
z0,
z1)),
g(
z0,
z1))) by
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
S tuples:
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:
F
Compound Symbols:
c
(19) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)
Removed 1 of 1 dangling nodes:
F(0, 1, g(z0, z1)) → c(F(z1, z1, g(z0, z1)))
(20) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(0, 1, z0) → f(h(z0), h(z0), z0)
h(0) → 0
h(g(z0, z1)) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f, h
Defined Pair Symbols:none
Compound Symbols:none
(21) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(22) BOUNDS(O(1), O(1))