(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL
(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(4) Obligation:
TRS:
Rules:
active(f(X, X)) → mark(f(a, b))
active(b) → mark(a)
active(f(X1, X2)) → f(active(X1), X2)
f(mark(X1), X2) → mark(f(X1, X2))
proper(f(X1, X2)) → f(proper(X1), proper(X2))
proper(a) → ok(a)
proper(b) → ok(b)
f(ok(X1), ok(X2)) → ok(f(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: a:b:mark:ok → a:b:mark:ok
f :: a:b:mark:ok → a:b:mark:ok → a:b:mark:ok
mark :: a:b:mark:ok → a:b:mark:ok
a :: a:b:mark:ok
b :: a:b:mark:ok
proper :: a:b:mark:ok → a:b:mark:ok
ok :: a:b:mark:ok → a:b:mark:ok
top :: a:b:mark:ok → top
hole_a:b:mark:ok1_0 :: a:b:mark:ok
hole_top2_0 :: top
gen_a:b:mark:ok3_0 :: Nat → a:b:mark:ok
(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
f,
proper,
topThey will be analysed ascendingly in the following order:
f < active
active < top
f < proper
proper < top
(6) Obligation:
TRS:
Rules:
active(
f(
X,
X)) →
mark(
f(
a,
b))
active(
b) →
mark(
a)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
b) →
ok(
b)
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:b:mark:ok → a:b:mark:ok
f :: a:b:mark:ok → a:b:mark:ok → a:b:mark:ok
mark :: a:b:mark:ok → a:b:mark:ok
a :: a:b:mark:ok
b :: a:b:mark:ok
proper :: a:b:mark:ok → a:b:mark:ok
ok :: a:b:mark:ok → a:b:mark:ok
top :: a:b:mark:ok → top
hole_a:b:mark:ok1_0 :: a:b:mark:ok
hole_top2_0 :: top
gen_a:b:mark:ok3_0 :: Nat → a:b:mark:ok
Generator Equations:
gen_a:b:mark:ok3_0(0) ⇔ a
gen_a:b:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:ok3_0(x))
The following defined symbols remain to be analysed:
f, active, proper, top
They will be analysed ascendingly in the following order:
f < active
active < top
f < proper
proper < top
(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
f(
gen_a:b:mark:ok3_0(
+(
1,
n5_0)),
gen_a:b:mark:ok3_0(
b)) →
*4_0, rt ∈ Ω(n5
0)
Induction Base:
f(gen_a:b:mark:ok3_0(+(1, 0)), gen_a:b:mark:ok3_0(b))
Induction Step:
f(gen_a:b:mark:ok3_0(+(1, +(n5_0, 1))), gen_a:b:mark:ok3_0(b)) →RΩ(1)
mark(f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b))) →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,
X)) →
mark(
f(
a,
b))
active(
b) →
mark(
a)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
b) →
ok(
b)
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:b:mark:ok → a:b:mark:ok
f :: a:b:mark:ok → a:b:mark:ok → a:b:mark:ok
mark :: a:b:mark:ok → a:b:mark:ok
a :: a:b:mark:ok
b :: a:b:mark:ok
proper :: a:b:mark:ok → a:b:mark:ok
ok :: a:b:mark:ok → a:b:mark:ok
top :: a:b:mark:ok → top
hole_a:b:mark:ok1_0 :: a:b:mark:ok
hole_top2_0 :: top
gen_a:b:mark:ok3_0 :: Nat → a:b:mark:ok
Lemmas:
f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_a:b:mark:ok3_0(0) ⇔ a
gen_a:b:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:b: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
(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(11) Obligation:
TRS:
Rules:
active(
f(
X,
X)) →
mark(
f(
a,
b))
active(
b) →
mark(
a)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
b) →
ok(
b)
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:b:mark:ok → a:b:mark:ok
f :: a:b:mark:ok → a:b:mark:ok → a:b:mark:ok
mark :: a:b:mark:ok → a:b:mark:ok
a :: a:b:mark:ok
b :: a:b:mark:ok
proper :: a:b:mark:ok → a:b:mark:ok
ok :: a:b:mark:ok → a:b:mark:ok
top :: a:b:mark:ok → top
hole_a:b:mark:ok1_0 :: a:b:mark:ok
hole_top2_0 :: top
gen_a:b:mark:ok3_0 :: Nat → a:b:mark:ok
Lemmas:
f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_a:b:mark:ok3_0(0) ⇔ a
gen_a:b:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:b: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
(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(13) Obligation:
TRS:
Rules:
active(
f(
X,
X)) →
mark(
f(
a,
b))
active(
b) →
mark(
a)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
b) →
ok(
b)
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:b:mark:ok → a:b:mark:ok
f :: a:b:mark:ok → a:b:mark:ok → a:b:mark:ok
mark :: a:b:mark:ok → a:b:mark:ok
a :: a:b:mark:ok
b :: a:b:mark:ok
proper :: a:b:mark:ok → a:b:mark:ok
ok :: a:b:mark:ok → a:b:mark:ok
top :: a:b:mark:ok → top
hole_a:b:mark:ok1_0 :: a:b:mark:ok
hole_top2_0 :: top
gen_a:b:mark:ok3_0 :: Nat → a:b:mark:ok
Lemmas:
f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_a:b:mark:ok3_0(0) ⇔ a
gen_a:b:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:ok3_0(x))
The following defined symbols remain to be analysed:
top
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(15) Obligation:
TRS:
Rules:
active(
f(
X,
X)) →
mark(
f(
a,
b))
active(
b) →
mark(
a)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
b) →
ok(
b)
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:b:mark:ok → a:b:mark:ok
f :: a:b:mark:ok → a:b:mark:ok → a:b:mark:ok
mark :: a:b:mark:ok → a:b:mark:ok
a :: a:b:mark:ok
b :: a:b:mark:ok
proper :: a:b:mark:ok → a:b:mark:ok
ok :: a:b:mark:ok → a:b:mark:ok
top :: a:b:mark:ok → top
hole_a:b:mark:ok1_0 :: a:b:mark:ok
hole_top2_0 :: top
gen_a:b:mark:ok3_0 :: Nat → a:b:mark:ok
Lemmas:
f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_a:b:mark:ok3_0(0) ⇔ a
gen_a:b:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:ok3_0(x))
No more defined symbols left to analyse.
(16) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(17) BOUNDS(n^1, INF)
(18) Obligation:
TRS:
Rules:
active(
f(
X,
X)) →
mark(
f(
a,
b))
active(
b) →
mark(
a)
active(
f(
X1,
X2)) →
f(
active(
X1),
X2)
f(
mark(
X1),
X2) →
mark(
f(
X1,
X2))
proper(
f(
X1,
X2)) →
f(
proper(
X1),
proper(
X2))
proper(
a) →
ok(
a)
proper(
b) →
ok(
b)
f(
ok(
X1),
ok(
X2)) →
ok(
f(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: a:b:mark:ok → a:b:mark:ok
f :: a:b:mark:ok → a:b:mark:ok → a:b:mark:ok
mark :: a:b:mark:ok → a:b:mark:ok
a :: a:b:mark:ok
b :: a:b:mark:ok
proper :: a:b:mark:ok → a:b:mark:ok
ok :: a:b:mark:ok → a:b:mark:ok
top :: a:b:mark:ok → top
hole_a:b:mark:ok1_0 :: a:b:mark:ok
hole_top2_0 :: top
gen_a:b:mark:ok3_0 :: Nat → a:b:mark:ok
Lemmas:
f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_a:b:mark:ok3_0(0) ⇔ a
gen_a:b:mark:ok3_0(+(x, 1)) ⇔ mark(gen_a:b:mark:ok3_0(x))
No more defined symbols left to analyse.
(19) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_a:b:mark:ok3_0(+(1, n5_0)), gen_a:b:mark:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(20) BOUNDS(n^1, INF)