(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
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
f(mark(X)) →+ mark(f(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(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
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(f(f(X))) → mark(c(f(g(f(X)))))
active(c(X)) → mark(d(X))
active(h(X)) → mark(c(d(X)))
active(f(X)) → f(active(X))
active(h(X)) → h(active(X))
f(mark(X)) → mark(f(X))
h(mark(X)) → mark(h(X))
proper(f(X)) → f(proper(X))
proper(c(X)) → c(proper(X))
proper(g(X)) → g(proper(X))
proper(d(X)) → d(proper(X))
proper(h(X)) → h(proper(X))
f(ok(X)) → ok(f(X))
c(ok(X)) → ok(c(X))
g(ok(X)) → ok(g(X))
d(ok(X)) → ok(d(X))
h(ok(X)) → ok(h(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
c,
f,
g,
d,
h,
proper,
topThey will be analysed ascendingly in the following order:
c < active
f < active
g < active
d < active
h < active
active < top
c < proper
f < proper
g < proper
d < proper
h < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
c, active, f, g, d, h, proper, top
They will be analysed ascendingly in the following order:
c < active
f < active
g < active
d < active
h < active
active < top
c < proper
f < proper
g < proper
d < proper
h < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol c.
(10) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
f, active, g, d, h, proper, top
They will be analysed ascendingly in the following order:
f < active
g < active
d < active
h < active
active < top
f < proper
g < proper
d < 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(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
g, active, d, h, proper, top
They will be analysed ascendingly in the following order:
g < active
d < active
h < active
active < top
g < proper
d < proper
h < proper
proper < top
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol g.
(14) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
d, active, h, proper, top
They will be analysed ascendingly in the following order:
d < active
h < active
active < top
d < proper
h < proper
proper < top
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol d.
(16) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark: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
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol h.
(18) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark: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
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(20) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(22) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
top
(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(24) Obligation:
TRS:
Rules:
active(
f(
f(
X))) →
mark(
c(
f(
g(
f(
X)))))
active(
c(
X)) →
mark(
d(
X))
active(
h(
X)) →
mark(
c(
d(
X)))
active(
f(
X)) →
f(
active(
X))
active(
h(
X)) →
h(
active(
X))
f(
mark(
X)) →
mark(
f(
X))
h(
mark(
X)) →
mark(
h(
X))
proper(
f(
X)) →
f(
proper(
X))
proper(
c(
X)) →
c(
proper(
X))
proper(
g(
X)) →
g(
proper(
X))
proper(
d(
X)) →
d(
proper(
X))
proper(
h(
X)) →
h(
proper(
X))
f(
ok(
X)) →
ok(
f(
X))
c(
ok(
X)) →
ok(
c(
X))
g(
ok(
X)) →
ok(
g(
X))
d(
ok(
X)) →
ok(
d(
X))
h(
ok(
X)) →
ok(
h(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
f :: mark:ok → mark:ok
mark :: mark:ok → mark:ok
c :: mark:ok → mark:ok
g :: mark:ok → mark:ok
d :: mark:ok → mark:ok
h :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
No more defined symbols left to analyse.