(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), 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:
f(f(z0, z1, z2), z3, f(z0, z1, z4)) → f(z0, z1, f(z2, z3, z4))
f(z0, z1, z1) → z1
f(z0, z1, g(z1)) → z0
f(z0, z0, z1) → z0
f(g(z0), z0, z1) → z1
Tuples:
F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(z0, z0, z1) → c3
F(g(z0), z0, z1) → c4
S tuples:
F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(z0, z0, z1) → c3
F(g(z0), z0, z1) → c4
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1, c2, c3, c4
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
F(z0, z1, z1) → c1
F(z0, z0, z1) → c3
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c4
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, z1, z2), z3, f(z0, z1, z4)) → f(z0, z1, f(z2, z3, z4))
f(z0, z1, z1) → z1
f(z0, z1, g(z1)) → z0
f(z0, z0, z1) → z0
f(g(z0), z0, z1) → z1
Tuples:
F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))
S tuples:
F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c
(5) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
F(f(z0, z1, z2), z3, f(z0, z1, z4)) → c(F(z0, z1, f(z2, z3, z4)), F(z2, z3, z4))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(f(z0, z1, z2), z3, f(z0, z1, z4)) → f(z0, z1, f(z2, z3, z4))
f(z0, z1, z1) → z1
f(z0, z1, g(z1)) → z0
f(z0, z0, z1) → z0
f(g(z0), z0, z1) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)