(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(h(X)) → mark(g(X, X))
active(g(a, X)) → mark(f(b, X))
active(f(X, X)) → mark(h(a))
active(a) → mark(b)
mark(h(X)) → active(h(mark(X)))
mark(g(X1, X2)) → active(g(mark(X1), X2))
mark(a) → active(a)
mark(f(X1, X2)) → active(f(mark(X1), X2))
mark(b) → active(b)
h(mark(X)) → h(X)
h(active(X)) → h(X)
g(mark(X1), X2) → g(X1, X2)
g(X1, mark(X2)) → g(X1, X2)
g(active(X1), X2) → g(X1, X2)
g(X1, active(X2)) → g(X1, X2)
f(mark(X1), X2) → f(X1, X2)
f(X1, mark(X2)) → f(X1, X2)
f(active(X1), X2) → f(X1, X2)
f(X1, active(X2)) → f(X1, X2)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
mark(h(z0)) → active(h(mark(z0)))
mark(g(z0, z1)) → active(g(mark(z0), z1))
mark(a) → active(a)
mark(f(z0, z1)) → active(f(mark(z0), z1))
mark(b) → active(b)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
g(mark(z0), z1) → g(z0, z1)
g(z0, mark(z1)) → g(z0, z1)
g(active(z0), z1) → g(z0, z1)
g(z0, active(z1)) → g(z0, z1)
f(mark(z0), z1) → f(z0, z1)
f(z0, mark(z1)) → f(z0, z1)
f(active(z0), z1) → f(z0, z1)
f(z0, active(z1)) → f(z0, z1)
Tuples:
ACTIVE(h(z0)) → c(MARK(g(z0, z0)), G(z0, z0))
ACTIVE(g(a, z0)) → c1(MARK(f(b, z0)), F(b, z0))
ACTIVE(f(z0, z0)) → c2(MARK(h(a)), H(a))
ACTIVE(a) → c3(MARK(b))
MARK(h(z0)) → c4(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(g(z0, z1)) → c5(ACTIVE(g(mark(z0), z1)), G(mark(z0), z1), MARK(z0))
MARK(a) → c6(ACTIVE(a))
MARK(f(z0, z1)) → c7(ACTIVE(f(mark(z0), z1)), F(mark(z0), z1), MARK(z0))
MARK(b) → c8(ACTIVE(b))
H(mark(z0)) → c9(H(z0))
H(active(z0)) → c10(H(z0))
G(mark(z0), z1) → c11(G(z0, z1))
G(z0, mark(z1)) → c12(G(z0, z1))
G(active(z0), z1) → c13(G(z0, z1))
G(z0, active(z1)) → c14(G(z0, z1))
F(mark(z0), z1) → c15(F(z0, z1))
F(z0, mark(z1)) → c16(F(z0, z1))
F(active(z0), z1) → c17(F(z0, z1))
F(z0, active(z1)) → c18(F(z0, z1))
S tuples:
ACTIVE(h(z0)) → c(MARK(g(z0, z0)), G(z0, z0))
ACTIVE(g(a, z0)) → c1(MARK(f(b, z0)), F(b, z0))
ACTIVE(f(z0, z0)) → c2(MARK(h(a)), H(a))
ACTIVE(a) → c3(MARK(b))
MARK(h(z0)) → c4(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(g(z0, z1)) → c5(ACTIVE(g(mark(z0), z1)), G(mark(z0), z1), MARK(z0))
MARK(a) → c6(ACTIVE(a))
MARK(f(z0, z1)) → c7(ACTIVE(f(mark(z0), z1)), F(mark(z0), z1), MARK(z0))
MARK(b) → c8(ACTIVE(b))
H(mark(z0)) → c9(H(z0))
H(active(z0)) → c10(H(z0))
G(mark(z0), z1) → c11(G(z0, z1))
G(z0, mark(z1)) → c12(G(z0, z1))
G(active(z0), z1) → c13(G(z0, z1))
G(z0, active(z1)) → c14(G(z0, z1))
F(mark(z0), z1) → c15(F(z0, z1))
F(z0, mark(z1)) → c16(F(z0, z1))
F(active(z0), z1) → c17(F(z0, z1))
F(z0, active(z1)) → c18(F(z0, z1))
K tuples:none
Defined Rule Symbols:
active, mark, h, g, f
Defined Pair Symbols:
ACTIVE, MARK, H, G, F
Compound Symbols:
c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
ACTIVE(h(z0)) → c(MARK(g(z0, z0)), G(z0, z0))
ACTIVE(g(a, z0)) → c1(MARK(f(b, z0)), F(b, z0))
ACTIVE(f(z0, z0)) → c2(MARK(h(a)), H(a))
MARK(h(z0)) → c4(ACTIVE(h(mark(z0))), H(mark(z0)), MARK(z0))
MARK(g(z0, z1)) → c5(ACTIVE(g(mark(z0), z1)), G(mark(z0), z1), MARK(z0))
MARK(f(z0, z1)) → c7(ACTIVE(f(mark(z0), z1)), F(mark(z0), z1), MARK(z0))
H(mark(z0)) → c9(H(z0))
H(active(z0)) → c10(H(z0))
G(mark(z0), z1) → c11(G(z0, z1))
G(z0, mark(z1)) → c12(G(z0, z1))
G(active(z0), z1) → c13(G(z0, z1))
G(z0, active(z1)) → c14(G(z0, z1))
F(mark(z0), z1) → c15(F(z0, z1))
F(z0, mark(z1)) → c16(F(z0, z1))
F(active(z0), z1) → c17(F(z0, z1))
F(z0, active(z1)) → c18(F(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
mark(h(z0)) → active(h(mark(z0)))
mark(g(z0, z1)) → active(g(mark(z0), z1))
mark(a) → active(a)
mark(f(z0, z1)) → active(f(mark(z0), z1))
mark(b) → active(b)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
g(mark(z0), z1) → g(z0, z1)
g(z0, mark(z1)) → g(z0, z1)
g(active(z0), z1) → g(z0, z1)
g(z0, active(z1)) → g(z0, z1)
f(mark(z0), z1) → f(z0, z1)
f(z0, mark(z1)) → f(z0, z1)
f(active(z0), z1) → f(z0, z1)
f(z0, active(z1)) → f(z0, z1)
Tuples:
ACTIVE(a) → c3(MARK(b))
MARK(a) → c6(ACTIVE(a))
MARK(b) → c8(ACTIVE(b))
S tuples:
ACTIVE(a) → c3(MARK(b))
MARK(a) → c6(ACTIVE(a))
MARK(b) → c8(ACTIVE(b))
K tuples:none
Defined Rule Symbols:
active, mark, h, g, f
Defined Pair Symbols:
ACTIVE, MARK
Compound Symbols:
c3, c6, c8
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
MARK(a) → c6(ACTIVE(a))
ACTIVE(a) → c3(MARK(b))
MARK(b) → c8(ACTIVE(b))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
active(h(z0)) → mark(g(z0, z0))
active(g(a, z0)) → mark(f(b, z0))
active(f(z0, z0)) → mark(h(a))
active(a) → mark(b)
mark(h(z0)) → active(h(mark(z0)))
mark(g(z0, z1)) → active(g(mark(z0), z1))
mark(a) → active(a)
mark(f(z0, z1)) → active(f(mark(z0), z1))
mark(b) → active(b)
h(mark(z0)) → h(z0)
h(active(z0)) → h(z0)
g(mark(z0), z1) → g(z0, z1)
g(z0, mark(z1)) → g(z0, z1)
g(active(z0), z1) → g(z0, z1)
g(z0, active(z1)) → g(z0, z1)
f(mark(z0), z1) → f(z0, z1)
f(z0, mark(z1)) → f(z0, z1)
f(active(z0), z1) → f(z0, z1)
f(z0, active(z1)) → f(z0, z1)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
active, mark, h, g, f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))