active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Heuristically decided to analyse the following defined symbols:
active', U12', s', plus', U11', proper', top'
They will be analysed ascendingly in the following order:
U12' < active'
s' < active'
plus' < active'
U11' < active'
active' < top'
U12' < proper'
s' < proper'
plus' < proper'
U11' < proper'
proper' < top'
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))
The following defined symbols remain to be analysed:
U12', active', s', plus', U11', proper', top'
They will be analysed ascendingly in the following order:
U12' < active'
s' < active'
plus' < active'
U11' < active'
active' < top'
U12' < proper'
s' < proper'
plus' < proper'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
Induction Base:
U12'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))
Induction Step:
U12'(_gen_tt':mark':0':ok'3(+(1, +(_$n6, 1))), _gen_tt':mark':0':ok'3(_b1018), _gen_tt':mark':0':ok'3(_c1019)) →RΩ(1)
mark'(U12'(_gen_tt':mark':0':ok'3(+(1, _$n6)), _gen_tt':mark':0':ok'3(_b1018), _gen_tt':mark':0':ok'3(_c1019))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))
The following defined symbols remain to be analysed:
s', active', plus', U11', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
plus' < active'
U11' < active'
active' < top'
s' < proper'
plus' < proper'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_tt':mark':0':ok'3(+(1, _n2826))) → _*4, rt ∈ Ω(n2826)
Induction Base:
s'(_gen_tt':mark':0':ok'3(+(1, 0)))
Induction Step:
s'(_gen_tt':mark':0':ok'3(+(1, +(_$n2827, 1)))) →RΩ(1)
mark'(s'(_gen_tt':mark':0':ok'3(+(1, _$n2827)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n2826))) → _*4, rt ∈ Ω(n2826)
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))
The following defined symbols remain to be analysed:
plus', active', U11', proper', top'
They will be analysed ascendingly in the following order:
plus' < active'
U11' < active'
active' < top'
plus' < proper'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n4283)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4283)
Induction Base:
plus'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))
Induction Step:
plus'(_gen_tt':mark':0':ok'3(+(1, +(_$n4284, 1))), _gen_tt':mark':0':ok'3(_b5752)) →RΩ(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n4284)), _gen_tt':mark':0':ok'3(_b5752))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n2826))) → _*4, rt ∈ Ω(n2826)
plus'(_gen_tt':mark':0':ok'3(+(1, _n4283)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4283)
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))
The following defined symbols remain to be analysed:
U11', active', proper', top'
They will be analysed ascendingly in the following order:
U11' < active'
active' < top'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
U11'(_gen_tt':mark':0':ok'3(+(1, _n7191)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n7191)
Induction Base:
U11'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))
Induction Step:
U11'(_gen_tt':mark':0':ok'3(+(1, +(_$n7192, 1))), _gen_tt':mark':0':ok'3(_b9662), _gen_tt':mark':0':ok'3(_c9663)) →RΩ(1)
mark'(U11'(_gen_tt':mark':0':ok'3(+(1, _$n7192)), _gen_tt':mark':0':ok'3(_b9662), _gen_tt':mark':0':ok'3(_c9663))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n2826))) → _*4, rt ∈ Ω(n2826)
plus'(_gen_tt':mark':0':ok'3(+(1, _n4283)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4283)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7191)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n7191)
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(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'
Could not prove a rewrite lemma for the defined symbol active'.
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n2826))) → _*4, rt ∈ Ω(n2826)
plus'(_gen_tt':mark':0':ok'3(+(1, _n4283)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4283)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7191)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n7191)
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))
The following defined symbols remain to be analysed:
proper', top'
They will be analysed ascendingly in the following order:
proper' < top'
Could not prove a rewrite lemma for the defined symbol proper'.
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n2826))) → _*4, rt ∈ Ω(n2826)
plus'(_gen_tt':mark':0':ok'3(+(1, _n4283)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4283)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7191)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n7191)
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))
The following defined symbols remain to be analysed:
top'
Could not prove a rewrite lemma for the defined symbol top'.
Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'
Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n2826))) → _*4, rt ∈ Ω(n2826)
plus'(_gen_tt':mark':0':ok'3(+(1, _n4283)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4283)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7191)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n7191)
Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n5)