(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:
g(0, f(x, x)) → x
g(x, s(y)) → g(f(x, y), 0)
g(s(x), y) → g(f(x, y), 0)
g(f(x, y), 0) → f(g(x, 0), g(y, 0))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(0, f(z0, z0)) → z0
g(z0, s(z1)) → g(f(z0, z1), 0)
g(s(z0), z1) → g(f(z0, z1), 0)
g(f(z0, z1), 0) → f(g(z0, 0), g(z1, 0))
Tuples:
G(0, f(z0, z0)) → c
G(z0, s(z1)) → c1(G(f(z0, z1), 0))
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
S tuples:
G(0, f(z0, z0)) → c
G(z0, s(z1)) → c1(G(f(z0, z1), 0))
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:
g
Defined Pair Symbols:
G
Compound Symbols:
c, c1, c2, c3
(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)
Removed 1 leading nodes:
G(z0, s(z1)) → c1(G(f(z0, z1), 0))
Removed 1 trailing nodes:
G(0, f(z0, z0)) → c
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
g(0, f(z0, z0)) → z0
g(z0, s(z1)) → g(f(z0, z1), 0)
g(s(z0), z1) → g(f(z0, z1), 0)
g(f(z0, z1), 0) → f(g(z0, 0), g(z1, 0))
Tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
S tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:
g
Defined Pair Symbols:
G
Compound Symbols:
c2, c3
(5) CdtUsableRulesProof (EQUIVALENT transformation)
The following rules are not usable and were removed:
g(0, f(z0, z0)) → z0
g(z0, s(z1)) → g(f(z0, z1), 0)
g(s(z0), z1) → g(f(z0, z1), 0)
g(f(z0, z1), 0) → f(g(z0, 0), g(z1, 0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
S tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
K tuples:none
Defined Rule Symbols:none
Defined Pair Symbols:
G
Compound Symbols:
c2, c3
(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
G(s(z0), z1) → c2(G(f(z0, z1), 0))
We considered the (Usable) Rules:none
And the Tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(G(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(f(x1, x2)) = x1 + x2
POL(s(x1)) = [1] + x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
S tuples:
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
K tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
Defined Rule Symbols:none
Defined Pair Symbols:
G
Compound Symbols:
c2, c3
(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
We considered the (Usable) Rules:none
And the Tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(0) = 0
POL(G(x1, x2)) = x1 + x2
POL(c2(x1)) = x1
POL(c3(x1, x2)) = x1 + x2
POL(f(x1, x2)) = [1] + x1 + x2
POL(s(x1)) = [1] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:none
Tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
S tuples:none
K tuples:
G(s(z0), z1) → c2(G(f(z0, z1), 0))
G(f(z0, z1), 0) → c3(G(z0, 0), G(z1, 0))
Defined Rule Symbols:none
Defined Pair Symbols:
G
Compound Symbols:
c2, c3
(11) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(12) BOUNDS(1, 1)