(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
ack_in(0, n) → ack_out(s(n))
ack_in(s(m), 0) → u11(ack_in(m, s(0)))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
ack_in(s(s(m36_3)), 0) →+ u11(u21(ack_in(s(m36_3), 0), m36_3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [m36_3 / s(m36_3)].
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:
ack_in(0', n) → ack_out(s(n))
ack_in(s(m), 0') → u11(ack_in(m, s(0')))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
ack_in(0', n) → ack_out(s(n))
ack_in(s(m), 0') → u11(ack_in(m, s(0')))
u11(ack_out(n)) → ack_out(n)
ack_in(s(m), s(n)) → u21(ack_in(s(m), n), m)
u21(ack_out(n), m) → u22(ack_in(m, n))
u22(ack_out(n)) → ack_out(n)
Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
ack_in,
u21They will be analysed ascendingly in the following order:
ack_in = u21
(8) Obligation:
TRS:
Rules:
ack_in(
0',
n) →
ack_out(
s(
n))
ack_in(
s(
m),
0') →
u11(
ack_in(
m,
s(
0')))
u11(
ack_out(
n)) →
ack_out(
n)
ack_in(
s(
m),
s(
n)) →
u21(
ack_in(
s(
m),
n),
m)
u21(
ack_out(
n),
m) →
u22(
ack_in(
m,
n))
u22(
ack_out(
n)) →
ack_out(
n)
Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
The following defined symbols remain to be analysed:
u21, ack_in
They will be analysed ascendingly in the following order:
ack_in = u21
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol u21.
(10) Obligation:
TRS:
Rules:
ack_in(
0',
n) →
ack_out(
s(
n))
ack_in(
s(
m),
0') →
u11(
ack_in(
m,
s(
0')))
u11(
ack_out(
n)) →
ack_out(
n)
ack_in(
s(
m),
s(
n)) →
u21(
ack_in(
s(
m),
n),
m)
u21(
ack_out(
n),
m) →
u22(
ack_in(
m,
n))
u22(
ack_out(
n)) →
ack_out(
n)
Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
The following defined symbols remain to be analysed:
ack_in
They will be analysed ascendingly in the following order:
ack_in = u21
(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
ack_in(
gen_0':s3_0(
1),
gen_0':s3_0(
+(
1,
n4249_0))) →
*4_0, rt ∈ Ω(n4249
0)
Induction Base:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, 0)))
Induction Step:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, +(n4249_0, 1)))) →RΩ(1)
u21(ack_in(s(gen_0':s3_0(0)), gen_0':s3_0(+(1, n4249_0))), gen_0':s3_0(0)) →IH
u21(*4_0, gen_0':s3_0(0))
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(12) Complex Obligation (BEST)
(13) Obligation:
TRS:
Rules:
ack_in(
0',
n) →
ack_out(
s(
n))
ack_in(
s(
m),
0') →
u11(
ack_in(
m,
s(
0')))
u11(
ack_out(
n)) →
ack_out(
n)
ack_in(
s(
m),
s(
n)) →
u21(
ack_in(
s(
m),
n),
m)
u21(
ack_out(
n),
m) →
u22(
ack_in(
m,
n))
u22(
ack_out(
n)) →
ack_out(
n)
Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)
Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
The following defined symbols remain to be analysed:
u21
They will be analysed ascendingly in the following order:
ack_in = u21
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol u21.
(15) Obligation:
TRS:
Rules:
ack_in(
0',
n) →
ack_out(
s(
n))
ack_in(
s(
m),
0') →
u11(
ack_in(
m,
s(
0')))
u11(
ack_out(
n)) →
ack_out(
n)
ack_in(
s(
m),
s(
n)) →
u21(
ack_in(
s(
m),
n),
m)
u21(
ack_out(
n),
m) →
u22(
ack_in(
m,
n))
u22(
ack_out(
n)) →
ack_out(
n)
Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)
Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
No more defined symbols left to analyse.
(16) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)
(17) BOUNDS(n^1, INF)
(18) Obligation:
TRS:
Rules:
ack_in(
0',
n) →
ack_out(
s(
n))
ack_in(
s(
m),
0') →
u11(
ack_in(
m,
s(
0')))
u11(
ack_out(
n)) →
ack_out(
n)
ack_in(
s(
m),
s(
n)) →
u21(
ack_in(
s(
m),
n),
m)
u21(
ack_out(
n),
m) →
u22(
ack_in(
m,
n))
u22(
ack_out(
n)) →
ack_out(
n)
Types:
ack_in :: 0':s → 0':s → ack_out
0' :: 0':s
ack_out :: 0':s → ack_out
s :: 0':s → 0':s
u11 :: ack_out → ack_out
u21 :: ack_out → 0':s → ack_out
u22 :: ack_out → ack_out
hole_ack_out1_0 :: ack_out
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)
Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
No more defined symbols left to analyse.
(19) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
ack_in(gen_0':s3_0(1), gen_0':s3_0(+(1, n4249_0))) → *4_0, rt ∈ Ω(n42490)
(20) BOUNDS(n^1, INF)