(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(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
__(mark(X1), X2) →+ mark(__(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
__,
and,
isNePal,
proper,
topThey will be analysed ascendingly in the following order:
__ < active
and < active
isNePal < active
active < top
__ < proper
and < proper
isNePal < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
The following defined symbols remain to be analysed:
__, active, and, isNePal, proper, top
They will be analysed ascendingly in the following order:
__ < active
and < active
isNePal < active
active < top
__ < proper
and < proper
isNePal < proper
proper < top
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
__(
gen_mark:nil:tt:ok3_0(
+(
1,
n5_0)),
gen_mark:nil:tt:ok3_0(
b)) →
*4_0, rt ∈ Ω(n5
0)
Induction Base:
__(gen_mark:nil:tt:ok3_0(+(1, 0)), gen_mark:nil:tt:ok3_0(b))
Induction Step:
__(gen_mark:nil:tt:ok3_0(+(1, +(n5_0, 1))), gen_mark:nil:tt:ok3_0(b)) →RΩ(1)
mark(__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
The following defined symbols remain to be analysed:
and, active, isNePal, proper, top
They will be analysed ascendingly in the following order:
and < active
isNePal < active
active < top
and < proper
isNePal < proper
proper < top
(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
and(
gen_mark:nil:tt:ok3_0(
+(
1,
n1017_0)),
gen_mark:nil:tt:ok3_0(
b)) →
*4_0, rt ∈ Ω(n1017
0)
Induction Base:
and(gen_mark:nil:tt:ok3_0(+(1, 0)), gen_mark:nil:tt:ok3_0(b))
Induction Step:
and(gen_mark:nil:tt:ok3_0(+(1, +(n1017_0, 1))), gen_mark:nil:tt:ok3_0(b)) →RΩ(1)
mark(and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n10170)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
The following defined symbols remain to be analysed:
isNePal, active, proper, top
They will be analysed ascendingly in the following order:
isNePal < active
active < top
isNePal < proper
proper < top
(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
isNePal(
gen_mark:nil:tt:ok3_0(
+(
1,
n2134_0))) →
*4_0, rt ∈ Ω(n2134
0)
Induction Base:
isNePal(gen_mark:nil:tt:ok3_0(+(1, 0)))
Induction Step:
isNePal(gen_mark:nil:tt:ok3_0(+(1, +(n2134_0, 1)))) →RΩ(1)
mark(isNePal(gen_mark:nil:tt:ok3_0(+(1, n2134_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(16) Complex Obligation (BEST)
(17) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n10170)
isNePal(gen_mark:nil:tt:ok3_0(+(1, n2134_0))) → *4_0, rt ∈ Ω(n21340)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt: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
(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(19) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n10170)
isNePal(gen_mark:nil:tt:ok3_0(+(1, n2134_0))) → *4_0, rt ∈ Ω(n21340)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(21) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n10170)
isNePal(gen_mark:nil:tt:ok3_0(+(1, n2134_0))) → *4_0, rt ∈ Ω(n21340)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
The following defined symbols remain to be analysed:
top
(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(23) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n10170)
isNePal(gen_mark:nil:tt:ok3_0(+(1, n2134_0))) → *4_0, rt ∈ Ω(n21340)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
No more defined symbols left to analyse.
(24) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(25) BOUNDS(n^1, INF)
(26) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n10170)
isNePal(gen_mark:nil:tt:ok3_0(+(1, n2134_0))) → *4_0, rt ∈ Ω(n21340)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
No more defined symbols left to analyse.
(27) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(28) BOUNDS(n^1, INF)
(29) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
and(gen_mark:nil:tt:ok3_0(+(1, n1017_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n10170)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
No more defined symbols left to analyse.
(30) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(31) BOUNDS(n^1, INF)
(32) Obligation:
TRS:
Rules:
active(
__(
__(
X,
Y),
Z)) →
mark(
__(
X,
__(
Y,
Z)))
active(
__(
X,
nil)) →
mark(
X)
active(
__(
nil,
X)) →
mark(
X)
active(
and(
tt,
X)) →
mark(
X)
active(
isNePal(
__(
I,
__(
P,
I)))) →
mark(
tt)
active(
__(
X1,
X2)) →
__(
active(
X1),
X2)
active(
__(
X1,
X2)) →
__(
X1,
active(
X2))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
isNePal(
X)) →
isNePal(
active(
X))
__(
mark(
X1),
X2) →
mark(
__(
X1,
X2))
__(
X1,
mark(
X2)) →
mark(
__(
X1,
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
isNePal(
mark(
X)) →
mark(
isNePal(
X))
proper(
__(
X1,
X2)) →
__(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
isNePal(
X)) →
isNePal(
proper(
X))
__(
ok(
X1),
ok(
X2)) →
ok(
__(
X1,
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
isNePal(
ok(
X)) →
ok(
isNePal(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok
Lemmas:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))
No more defined symbols left to analyse.
(33) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
__(gen_mark:nil:tt:ok3_0(+(1, n5_0)), gen_mark:nil:tt:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(34) BOUNDS(n^1, INF)