(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
U11(tt, s(M521_3), N) →+ s(U11(tt, M521_3, activate(activate(N))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [M521_3 / s(M521_3)].
The result substitution is [N / activate(activate(N))].
(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:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X
Types:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
plus,
xThey will be analysed ascendingly in the following order:
plus < x
(8) Obligation:
TRS:
Rules:
U11(
tt,
M,
N) →
U12(
tt,
activate(
M),
activate(
N))
U12(
tt,
M,
N) →
s(
plus(
activate(
N),
activate(
M)))
U21(
tt,
M,
N) →
U22(
tt,
activate(
M),
activate(
N))
U22(
tt,
M,
N) →
plus(
x(
activate(
N),
activate(
M)),
activate(
N))
plus(
N,
0') →
Nplus(
N,
s(
M)) →
U11(
tt,
M,
N)
x(
N,
0') →
0'x(
N,
s(
M)) →
U21(
tt,
M,
N)
activate(
X) →
XTypes:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'
Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))
The following defined symbols remain to be analysed:
plus, x
They will be analysed ascendingly in the following order:
plus < x
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
plus(
gen_s:0'3_0(
a),
gen_s:0'3_0(
n5_0)) →
gen_s:0'3_0(
+(
n5_0,
a)), rt ∈ Ω(1 + n5
0)
Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) →RΩ(1)
gen_s:0'3_0(a)
Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) →RΩ(1)
U11(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) →RΩ(1)
U12(tt, activate(gen_s:0'3_0(n5_0)), activate(gen_s:0'3_0(a))) →RΩ(1)
U12(tt, gen_s:0'3_0(n5_0), activate(gen_s:0'3_0(a))) →RΩ(1)
U12(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) →RΩ(1)
s(plus(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n5_0)))) →RΩ(1)
s(plus(gen_s:0'3_0(a), activate(gen_s:0'3_0(n5_0)))) →RΩ(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0))) →IH
s(gen_s:0'3_0(+(a, c6_0)))
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
U11(
tt,
M,
N) →
U12(
tt,
activate(
M),
activate(
N))
U12(
tt,
M,
N) →
s(
plus(
activate(
N),
activate(
M)))
U21(
tt,
M,
N) →
U22(
tt,
activate(
M),
activate(
N))
U22(
tt,
M,
N) →
plus(
x(
activate(
N),
activate(
M)),
activate(
N))
plus(
N,
0') →
Nplus(
N,
s(
M)) →
U11(
tt,
M,
N)
x(
N,
0') →
0'x(
N,
s(
M)) →
U21(
tt,
M,
N)
activate(
X) →
XTypes:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'
Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n50)
Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))
The following defined symbols remain to be analysed:
x
(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
x(
gen_s:0'3_0(
a),
gen_s:0'3_0(
n580_0)) →
gen_s:0'3_0(
*(
n580_0,
a)), rt ∈ Ω(1 + a·n580
0 + n580
0)
Induction Base:
x(gen_s:0'3_0(a), gen_s:0'3_0(0)) →RΩ(1)
0'
Induction Step:
x(gen_s:0'3_0(a), gen_s:0'3_0(+(n580_0, 1))) →RΩ(1)
U21(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) →RΩ(1)
U22(tt, activate(gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) →RΩ(1)
U22(tt, gen_s:0'3_0(n580_0), activate(gen_s:0'3_0(a))) →RΩ(1)
U22(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) →RΩ(1)
plus(x(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) →RΩ(1)
plus(x(gen_s:0'3_0(a), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) →RΩ(1)
plus(x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) →IH
plus(gen_s:0'3_0(*(c581_0, a)), activate(gen_s:0'3_0(a))) →RΩ(1)
plus(gen_s:0'3_0(*(n580_0, a)), gen_s:0'3_0(a)) →LΩ(1 + a)
gen_s:0'3_0(+(a, *(n580_0, a)))
We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
Rules:
U11(
tt,
M,
N) →
U12(
tt,
activate(
M),
activate(
N))
U12(
tt,
M,
N) →
s(
plus(
activate(
N),
activate(
M)))
U21(
tt,
M,
N) →
U22(
tt,
activate(
M),
activate(
N))
U22(
tt,
M,
N) →
plus(
x(
activate(
N),
activate(
M)),
activate(
N))
plus(
N,
0') →
Nplus(
N,
s(
M)) →
U11(
tt,
M,
N)
x(
N,
0') →
0'x(
N,
s(
M)) →
U21(
tt,
M,
N)
activate(
X) →
XTypes:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'
Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n50)
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n5800 + n5800)
Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))
No more defined symbols left to analyse.
(15) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n2) was proven with the following lemma:
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n5800 + n5800)
(16) BOUNDS(n^2, INF)
(17) Obligation:
TRS:
Rules:
U11(
tt,
M,
N) →
U12(
tt,
activate(
M),
activate(
N))
U12(
tt,
M,
N) →
s(
plus(
activate(
N),
activate(
M)))
U21(
tt,
M,
N) →
U22(
tt,
activate(
M),
activate(
N))
U22(
tt,
M,
N) →
plus(
x(
activate(
N),
activate(
M)),
activate(
N))
plus(
N,
0') →
Nplus(
N,
s(
M)) →
U11(
tt,
M,
N)
x(
N,
0') →
0'x(
N,
s(
M)) →
U21(
tt,
M,
N)
activate(
X) →
XTypes:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'
Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n50)
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n5800 + n5800)
Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))
No more defined symbols left to analyse.
(18) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n2) was proven with the following lemma:
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n5800 + n5800)
(19) BOUNDS(n^2, INF)
(20) Obligation:
TRS:
Rules:
U11(
tt,
M,
N) →
U12(
tt,
activate(
M),
activate(
N))
U12(
tt,
M,
N) →
s(
plus(
activate(
N),
activate(
M)))
U21(
tt,
M,
N) →
U22(
tt,
activate(
M),
activate(
N))
U22(
tt,
M,
N) →
plus(
x(
activate(
N),
activate(
M)),
activate(
N))
plus(
N,
0') →
Nplus(
N,
s(
M)) →
U11(
tt,
M,
N)
x(
N,
0') →
0'x(
N,
s(
M)) →
U21(
tt,
M,
N)
activate(
X) →
XTypes:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'
Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n50)
Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))
No more defined symbols left to analyse.
(21) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n50)
(22) BOUNDS(n^1, INF)