(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
and(mark(X1), X2) →+ mark(and(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(and(tt, X)) → mark(X)
active(plus(N, 0')) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
active(and(tt, X)) → mark(X)
active(plus(N, 0')) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
s,
plus,
and,
proper,
topThey will be analysed ascendingly in the following order:
s < active
plus < active
and < active
active < top
s < proper
plus < proper
and < proper
proper < top
(8) Obligation:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
s, active, plus, and, proper, top
They will be analysed ascendingly in the following order:
s < active
plus < active
and < active
active < top
s < proper
plus < proper
and < proper
proper < top
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(
gen_tt:mark:0':ok3_0(
+(
1,
n5_0))) →
*4_0, rt ∈ Ω(n5
0)
Induction Base:
s(gen_tt:mark:0':ok3_0(+(1, 0)))
Induction Step:
s(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:0':ok3_0(+(1, n5_0)))) →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:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
plus, active, and, proper, top
They will be analysed ascendingly in the following order:
plus < active
and < active
active < top
plus < proper
and < proper
proper < top
(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
plus(
gen_tt:mark:0':ok3_0(
+(
1,
n392_0)),
gen_tt:mark:0':ok3_0(
b)) →
*4_0, rt ∈ Ω(n392
0)
Induction Base:
plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
Induction Step:
plus(gen_tt:mark:0':ok3_0(+(1, +(n392_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':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:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3920)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
and, active, proper, top
They will be analysed ascendingly in the following order:
and < active
active < top
and < proper
proper < top
(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
and(
gen_tt:mark:0':ok3_0(
+(
1,
n1700_0)),
gen_tt:mark:0':ok3_0(
b)) →
*4_0, rt ∈ Ω(n1700
0)
Induction Base:
and(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))
Induction Step:
and(gen_tt:mark:0':ok3_0(+(1, +(n1700_0, 1))), gen_tt:mark:0':ok3_0(b)) →RΩ(1)
mark(and(gen_tt:mark:0':ok3_0(+(1, n1700_0)), gen_tt:mark:0':ok3_0(b))) →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:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3920)
and(gen_tt:mark:0':ok3_0(+(1, n1700_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n17000)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':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:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3920)
and(gen_tt:mark:0':ok3_0(+(1, n1700_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n17000)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':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:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3920)
and(gen_tt:mark:0':ok3_0(+(1, n1700_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n17000)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':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:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3920)
and(gen_tt:mark:0':ok3_0(+(1, n1700_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n17000)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(24) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
(25) BOUNDS(n^1, INF)
(26) Obligation:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3920)
and(gen_tt:mark:0':ok3_0(+(1, n1700_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n17000)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(27) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
(28) BOUNDS(n^1, INF)
(29) Obligation:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
plus(gen_tt:mark:0':ok3_0(+(1, n392_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3920)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(30) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
(31) BOUNDS(n^1, INF)
(32) Obligation:
Innermost TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Lemmas:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.
(33) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
s(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
(34) BOUNDS(n^1, INF)