(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
and(tt, X) → activate(X)
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X
Rewrite Strategy: INNERMOST
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
plus(N, s(M)) →+ s(plus(N, M))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [M / s(M)].
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:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
and(tt, X) → activate(X)
plus(N, 0') → N
plus(N, s(M)) → s(plus(N, M))
activate(X) → X
Types:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
plus
(8) Obligation:
Innermost TRS:
Rules:
and(
tt,
X) →
activate(
X)
plus(
N,
0') →
Nplus(
N,
s(
M)) →
s(
plus(
N,
M))
activate(
X) →
XTypes:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s
Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
The following defined symbols remain to be analysed:
plus
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
plus(
gen_0':s4_0(
a),
gen_0':s4_0(
n6_0)) →
gen_0':s4_0(
+(
n6_0,
a)), rt ∈ Ω(1 + n6
0)
Induction Base:
plus(gen_0':s4_0(a), gen_0':s4_0(0)) →RΩ(1)
gen_0':s4_0(a)
Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n6_0, 1))) →RΩ(1)
s(plus(gen_0':s4_0(a), gen_0':s4_0(n6_0))) →IH
s(gen_0':s4_0(+(a, c7_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:
and(
tt,
X) →
activate(
X)
plus(
N,
0') →
Nplus(
N,
s(
M)) →
s(
plus(
N,
M))
activate(
X) →
XTypes:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s
Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)
Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
No more defined symbols left to analyse.
(12) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)
(13) BOUNDS(n^1, INF)
(14) Obligation:
Innermost TRS:
Rules:
and(
tt,
X) →
activate(
X)
plus(
N,
0') →
Nplus(
N,
s(
M)) →
s(
plus(
N,
M))
activate(
X) →
XTypes:
and :: tt → and:activate → and:activate
tt :: tt
activate :: and:activate → and:activate
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
hole_and:activate1_0 :: and:activate
hole_tt2_0 :: tt
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s
Lemmas:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)
Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
No more defined symbols left to analyse.
(15) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n6_0)) → gen_0':s4_0(+(n6_0, a)), rt ∈ Ω(1 + n60)
(16) BOUNDS(n^1, INF)