(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, n^1).
The TRS R consists of the following rules:
+(X, 0) → X
+(X, s(Y)) → s(+(X, Y))
f(0, s(0), X) → f(X, +(X, X), X)
g(X, Y) → X
g(X, Y) → Y
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
f(0, s(0), z0) → f(z0, +(z0, z0), z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:
+'(z0, 0) → c
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), z0) → c2(F(z0, +(z0, z0), z0), +'(z0, z0))
G(z0, z1) → c3
G(z0, z1) → c4
S tuples:
+'(z0, 0) → c
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), z0) → c2(F(z0, +(z0, z0), z0), +'(z0, z0))
G(z0, z1) → c3
G(z0, z1) → c4
K tuples:none
Defined Rule Symbols:
+, f, g
Defined Pair Symbols:
+', F, G
Compound Symbols:
c, c1, c2, c3, c4
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
G(z0, z1) → c3
+'(z0, 0) → c
G(z0, z1) → c4
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
f(0, s(0), z0) → f(z0, +(z0, z0), z0)
g(z0, z1) → z0
g(z0, z1) → z1
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), z0) → c2(F(z0, +(z0, z0), z0), +'(z0, z0))
S tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), z0) → c2(F(z0, +(z0, z0), z0), +'(z0, z0))
K tuples:none
Defined Rule Symbols:
+, f, g
Defined Pair Symbols:
+', F
Compound Symbols:
c1, c2
(5) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
f(0, s(0), z0) → f(z0, +(z0, z0), z0)
g(z0, z1) → z0
g(z0, z1) → z1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), z0) → c2(F(z0, +(z0, z0), z0), +'(z0, z0))
S tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), z0) → c2(F(z0, +(z0, z0), z0), +'(z0, z0))
K tuples:none
Defined Rule Symbols:
+
Defined Pair Symbols:
+', F
Compound Symbols:
c1, c2
(7) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
F(
0,
s(
0),
z0) →
c2(
F(
z0,
+(
z0,
z0),
z0),
+'(
z0,
z0)) by
F(0, s(0), 0) → c2(F(0, 0, 0), +'(0, 0))
F(0, s(0), s(z1)) → c2(F(s(z1), s(+(s(z1), z1)), s(z1)), +'(s(z1), s(z1)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), 0) → c2(F(0, 0, 0), +'(0, 0))
F(0, s(0), s(z1)) → c2(F(s(z1), s(+(s(z1), z1)), s(z1)), +'(s(z1), s(z1)))
S tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), 0) → c2(F(0, 0, 0), +'(0, 0))
F(0, s(0), s(z1)) → c2(F(s(z1), s(+(s(z1), z1)), s(z1)), +'(s(z1), s(z1)))
K tuples:none
Defined Rule Symbols:
+
Defined Pair Symbols:
+', F
Compound Symbols:
c1, c2
(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
F(0, s(0), 0) → c2(F(0, 0, 0), +'(0, 0))
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), s(z1)) → c2(F(s(z1), s(+(s(z1), z1)), s(z1)), +'(s(z1), s(z1)))
S tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), s(z1)) → c2(F(s(z1), s(+(s(z1), z1)), s(z1)), +'(s(z1), s(z1)))
K tuples:none
Defined Rule Symbols:
+
Defined Pair Symbols:
+', F
Compound Symbols:
c1, c2
(11) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), s(z1)) → c2(+'(s(z1), s(z1)))
S tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
F(0, s(0), s(z1)) → c2(+'(s(z1), s(z1)))
K tuples:none
Defined Rule Symbols:
+
Defined Pair Symbols:
+', F
Compound Symbols:
c1, c2
(13) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
F(0, s(0), s(z1)) → c2(+'(s(z1), s(z1)))
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
S tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
K tuples:none
Defined Rule Symbols:
+
Defined Pair Symbols:
+'
Compound Symbols:
c1
(15) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
+(z0, 0) → z0
+(z0, s(z1)) → s(+(z0, z1))
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
S tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
+'
Compound Symbols:
c1
(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
+'(z0, s(z1)) → c1(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(+'(x1, x2)) = x2
POL(c1(x1)) = x1
POL(s(x1)) = [1] + x1
(18) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
S tuples:none
K tuples:
+'(z0, s(z1)) → c1(+'(z0, z1))
Defined Rule Symbols:none
Defined Pair Symbols:
+'
Compound Symbols:
c1
(19) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(20) BOUNDS(1, 1)