(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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
f(mark(X1), X2) →+ mark(f(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(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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(f(g(X), Y)) → mark(f(X, f(g(X), Y)))
active(f(X1, X2)) → f(active(X1), X2)
active(g(X)) → g(active(X))
f(mark(X1), X2) → mark(f(X1, X2))
g(mark(X)) → mark(g(X))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
f(ok(X1), ok(X2)) → ok(f(X1, X2))
g(ok(X)) → ok(g(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:ok
g :: mark:ok → mark:ok
mark :: 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,
f,
g,
proper,
topThey will be analysed ascendingly in the following order:
f < active
g < active
active < top
f < proper
g < proper
proper < top
(8) Obligation:
Innermost TRS:
Rules:
active(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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, proper, top
They will be analysed ascendingly in the following order:
f < active
g < active
active < top
f < proper
g < proper
proper < top
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
f(
gen_mark:ok3_0(
+(
1,
n5_0)),
gen_mark:ok3_0(
b)) →
*4_0, rt ∈ Ω(n5
0)
Induction Base:
f(gen_mark:ok3_0(+(1, 0)), gen_mark:ok3_0(b))
Induction Step:
f(gen_mark:ok3_0(+(1, +(n5_0, 1))), gen_mark:ok3_0(b)) →RΩ(1)
mark(f(gen_mark:ok3_0(+(1, n5_0)), gen_mark: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:
Innermost TRS:
Rules:
active(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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
Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
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, proper, top
They will be analysed ascendingly in the following order:
g < active
active < top
g < proper
proper < top
(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
g(
gen_mark:ok3_0(
+(
1,
n728_0))) →
*4_0, rt ∈ Ω(n728
0)
Induction Base:
g(gen_mark:ok3_0(+(1, 0)))
Induction Step:
g(gen_mark:ok3_0(+(1, +(n728_0, 1)))) →RΩ(1)
mark(g(gen_mark:ok3_0(+(1, n728_0)))) →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(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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
Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
g(gen_mark:ok3_0(+(1, n728_0))) → *4_0, rt ∈ Ω(n7280)
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
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(16) Obligation:
Innermost TRS:
Rules:
active(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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
Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
g(gen_mark:ok3_0(+(1, n728_0))) → *4_0, rt ∈ Ω(n7280)
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
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(18) Obligation:
Innermost TRS:
Rules:
active(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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
Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
g(gen_mark:ok3_0(+(1, n728_0))) → *4_0, rt ∈ Ω(n7280)
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
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(20) Obligation:
Innermost TRS:
Rules:
active(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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
Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
g(gen_mark:ok3_0(+(1, n728_0))) → *4_0, rt ∈ Ω(n7280)
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.
(21) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(22) BOUNDS(n^1, INF)
(23) Obligation:
Innermost TRS:
Rules:
active(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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
Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
g(gen_mark:ok3_0(+(1, n728_0))) → *4_0, rt ∈ Ω(n7280)
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.
(24) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(25) BOUNDS(n^1, INF)
(26) Obligation:
Innermost TRS:
Rules:
active(
f(
g(
X),
Y)) →
mark(
f(
X,
f(
g(
X),
Y)))
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
g(
ok(
X)) →
ok(
g(
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:ok
g :: mark:ok → mark:ok
mark :: 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
Lemmas:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
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.
(27) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_mark:ok3_0(+(1, n5_0)), gen_mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(28) BOUNDS(n^1, INF)