(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))
Rewrite Strategy: FULL
(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))
S is empty.
Rewrite Strategy: FULL
(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(4) Obligation:
TRS:
Rules:
active(f(x)) → mark(x)
top(active(c)) → top(mark(c))
top(mark(x)) → top(check(x))
check(f(x)) → f(check(x))
check(x) → start(match(f(X), x))
match(f(x), f(y)) → f(match(x, y))
match(X, x) → proper(x)
proper(c) → ok(c)
proper(f(x)) → f(proper(x))
f(ok(x)) → ok(f(x))
start(ok(x)) → found(x)
f(found(x)) → found(f(x))
top(found(x)) → top(active(x))
active(f(x)) → f(active(x))
f(mark(x)) → mark(f(x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
top,
check,
f,
match,
proper,
activeThey will be analysed ascendingly in the following order:
check < top
active < top
f < check
match < check
f < match
f < proper
f < active
proper < match
(6) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
The following defined symbols remain to be analysed:
f, top, check, match, proper, active
They will be analysed ascendingly in the following order:
check < top
active < top
f < check
match < check
f < match
f < proper
f < active
proper < match
(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
f(
gen_mark:c:X:ok:found3_0(
+(
1,
n5_0))) →
*4_0, rt ∈ Ω(n5
0)
Induction Base:
f(gen_mark:c:X:ok:found3_0(+(1, 0)))
Induction Step:
f(gen_mark:c:X:ok:found3_0(+(1, +(n5_0, 1)))) →RΩ(1)
mark(f(gen_mark:c:X:ok:found3_0(+(1, n5_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(8) Complex Obligation (BEST)
(9) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
The following defined symbols remain to be analysed:
proper, top, check, match, active
They will be analysed ascendingly in the following order:
check < top
active < top
match < check
proper < match
(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(11) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
The following defined symbols remain to be analysed:
match, top, check, active
They will be analysed ascendingly in the following order:
check < top
active < top
match < check
(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol match.
(13) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
The following defined symbols remain to be analysed:
check, top, active
They will be analysed ascendingly in the following order:
check < top
active < top
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol check.
(15) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
The following defined symbols remain to be analysed:
active, top
They will be analysed ascendingly in the following order:
active < top
(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(17) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
The following defined symbols remain to be analysed:
top
(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(19) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
No more defined symbols left to analyse.
(20) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
(21) BOUNDS(n^1, INF)
(22) Obligation:
TRS:
Rules:
active(
f(
x)) →
mark(
x)
top(
active(
c)) →
top(
mark(
c))
top(
mark(
x)) →
top(
check(
x))
check(
f(
x)) →
f(
check(
x))
check(
x) →
start(
match(
f(
X),
x))
match(
f(
x),
f(
y)) →
f(
match(
x,
y))
match(
X,
x) →
proper(
x)
proper(
c) →
ok(
c)
proper(
f(
x)) →
f(
proper(
x))
f(
ok(
x)) →
ok(
f(
x))
start(
ok(
x)) →
found(
x)
f(
found(
x)) →
found(
f(
x))
top(
found(
x)) →
top(
active(
x))
active(
f(
x)) →
f(
active(
x))
f(
mark(
x)) →
mark(
f(
x))
Types:
active :: mark:c:X:ok:found → mark:c:X:ok:found
f :: mark:c:X:ok:found → mark:c:X:ok:found
mark :: mark:c:X:ok:found → mark:c:X:ok:found
top :: mark:c:X:ok:found → top
c :: mark:c:X:ok:found
check :: mark:c:X:ok:found → mark:c:X:ok:found
start :: mark:c:X:ok:found → mark:c:X:ok:found
match :: mark:c:X:ok:found → mark:c:X:ok:found → mark:c:X:ok:found
X :: mark:c:X:ok:found
proper :: mark:c:X:ok:found → mark:c:X:ok:found
ok :: mark:c:X:ok:found → mark:c:X:ok:found
found :: mark:c:X:ok:found → mark:c:X:ok:found
hole_mark:c:X:ok:found1_0 :: mark:c:X:ok:found
hole_top2_0 :: top
gen_mark:c:X:ok:found3_0 :: Nat → mark:c:X:ok:found
Lemmas:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:c:X:ok:found3_0(0) ⇔ c
gen_mark:c:X:ok:found3_0(+(x, 1)) ⇔ mark(gen_mark:c:X:ok:found3_0(x))
No more defined symbols left to analyse.
(23) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_mark:c:X:ok:found3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n50)
(24) BOUNDS(n^1, INF)