(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)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
h(mark(X)) →+ mark(h(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative 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)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
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)
active(h(X)) → h(active(X))
active(g(X1, X2)) → g(active(X1), X2)
active(f(X1, X2)) → f(active(X1), X2)
h(mark(X)) → mark(h(X))
g(mark(X1), X2) → mark(g(X1, X2))
f(mark(X1), X2) → mark(f(X1, X2))
proper(h(X)) → h(proper(X))
proper(g(X1, X2)) → g(proper(X1), proper(X2))
proper(a) → ok(a)
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(b) → ok(b)
h(ok(X)) → ok(h(X))
g(ok(X1), ok(X2)) → ok(g(X1, X2))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
g,
f,
h,
proper,
topThey will be analysed ascendingly in the following order:
g < active
f < active
h < active
active < top
g < proper
f < proper
h < proper
proper < top
(8) Obligation:
TRS:
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)
active(
h(
X)) →
h(
active(
X))
active(
g(
X1,
X2)) →
g(
active(
X1),
X2)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
h(
mark(
X)) →
mark(
h(
X))
g(
mark(
X1),
X2) →
mark(
g(
X1,
X2))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
h(
X)) →
h(
proper(
X))
proper(
g(
X1,
X2)) →
g(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
b) →
ok(
b)
h(
ok(
X)) →
ok(
h(
X))
g(
ok(
X1),
ok(
X2)) →
ok(
g(
X1,
X2))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))
The following defined symbols remain to be analysed:
g, active, f, h, proper, top
They will be analysed ascendingly in the following order:
g < active
f < active
h < active
active < top
g < proper
f < proper
h < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol g.
(10) Obligation:
TRS:
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)
active(
h(
X)) →
h(
active(
X))
active(
g(
X1,
X2)) →
g(
active(
X1),
X2)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
h(
mark(
X)) →
mark(
h(
X))
g(
mark(
X1),
X2) →
mark(
g(
X1,
X2))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
h(
X)) →
h(
proper(
X))
proper(
g(
X1,
X2)) →
g(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
b) →
ok(
b)
h(
ok(
X)) →
ok(
h(
X))
g(
ok(
X1),
ok(
X2)) →
ok(
g(
X1,
X2))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))
The following defined symbols remain to be analysed:
f, active, h, proper, top
They will be analysed ascendingly in the following order:
f < active
h < active
active < top
f < proper
h < proper
proper < top
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol f.
(12) Obligation:
TRS:
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)
active(
h(
X)) →
h(
active(
X))
active(
g(
X1,
X2)) →
g(
active(
X1),
X2)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
h(
mark(
X)) →
mark(
h(
X))
g(
mark(
X1),
X2) →
mark(
g(
X1,
X2))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
h(
X)) →
h(
proper(
X))
proper(
g(
X1,
X2)) →
g(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
b) →
ok(
b)
h(
ok(
X)) →
ok(
h(
X))
g(
ok(
X1),
ok(
X2)) →
ok(
g(
X1,
X2))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))
The following defined symbols remain to be analysed:
h, active, proper, top
They will be analysed ascendingly in the following order:
h < active
active < top
h < proper
proper < top
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol h.
(14) Obligation:
TRS:
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)
active(
h(
X)) →
h(
active(
X))
active(
g(
X1,
X2)) →
g(
active(
X1),
X2)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
h(
mark(
X)) →
mark(
h(
X))
g(
mark(
X1),
X2) →
mark(
g(
X1,
X2))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
h(
X)) →
h(
proper(
X))
proper(
g(
X1,
X2)) →
g(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
b) →
ok(
b)
h(
ok(
X)) →
ok(
h(
X))
g(
ok(
X1),
ok(
X2)) →
ok(
g(
X1,
X2))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))
The following defined symbols remain to be analysed:
active, proper, top
They will be analysed ascendingly in the following order:
active < top
proper < top
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(16) Obligation:
TRS:
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)
active(
h(
X)) →
h(
active(
X))
active(
g(
X1,
X2)) →
g(
active(
X1),
X2)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
h(
mark(
X)) →
mark(
h(
X))
g(
mark(
X1),
X2) →
mark(
g(
X1,
X2))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
h(
X)) →
h(
proper(
X))
proper(
g(
X1,
X2)) →
g(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
b) →
ok(
b)
h(
ok(
X)) →
ok(
h(
X))
g(
ok(
X1),
ok(
X2)) →
ok(
g(
X1,
X2))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(18) Obligation:
TRS:
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)
active(
h(
X)) →
h(
active(
X))
active(
g(
X1,
X2)) →
g(
active(
X1),
X2)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
h(
mark(
X)) →
mark(
h(
X))
g(
mark(
X1),
X2) →
mark(
g(
X1,
X2))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
h(
X)) →
h(
proper(
X))
proper(
g(
X1,
X2)) →
g(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
b) →
ok(
b)
h(
ok(
X)) →
ok(
h(
X))
g(
ok(
X1),
ok(
X2)) →
ok(
g(
X1,
X2))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))
The following defined symbols remain to be analysed:
top
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(20) Obligation:
TRS:
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)
active(
h(
X)) →
h(
active(
X))
active(
g(
X1,
X2)) →
g(
active(
X1),
X2)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
h(
mark(
X)) →
mark(
h(
X))
g(
mark(
X1),
X2) →
mark(
g(
X1,
X2))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
h(
X)) →
h(
proper(
X))
proper(
g(
X1,
X2)) →
g(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
b) →
ok(
b)
h(
ok(
X)) →
ok(
h(
X))
g(
ok(
X1),
ok(
X2)) →
ok(
g(
X1,
X2))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:a:b:ok → mark:a:b:ok
h :: mark:a:b:ok → mark:a:b:ok
mark :: mark:a:b:ok → mark:a:b:ok
g :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
a :: mark:a:b:ok
f :: mark:a:b:ok → mark:a:b:ok → mark:a:b:ok
b :: mark:a:b:ok
proper :: mark:a:b:ok → mark:a:b:ok
ok :: mark:a:b:ok → mark:a:b:ok
top :: mark:a:b:ok → top
hole_mark:a:b:ok1_0 :: mark:a:b:ok
hole_top2_0 :: top
gen_mark:a:b:ok3_0 :: Nat → mark:a:b:ok
Generator Equations:
gen_mark:a:b:ok3_0(0) ⇔ a
gen_mark:a:b:ok3_0(+(x, 1)) ⇔ mark(gen_mark:a:b:ok3_0(x))
No more defined symbols left to analyse.