le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
inc(0) → 0
inc(s(x)) → s(inc(x))
minus(0, y) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
log(x) → log2(x, 0)
log2(x, y) → if(le(x, 0), le(x, s(0)), x, inc(y))
if(true, b, x, y) → log_undefined
if(false, b, x, y) → if2(b, x, y)
if2(true, x, s(y)) → y
if2(false, x, y) → log2(quot(x, s(s(0))), y)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Infered types.
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Types:
le' :: 0':s':log_undefined' → 0':s':log_undefined' → true':false'
0' :: 0':s':log_undefined'
true' :: true':false'
s' :: 0':s':log_undefined' → 0':s':log_undefined'
false' :: true':false'
inc' :: 0':s':log_undefined' → 0':s':log_undefined'
minus' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
quot' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log' :: 0':s':log_undefined' → 0':s':log_undefined'
log2' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
if' :: true':false' → true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log_undefined' :: 0':s':log_undefined'
if2' :: true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
_hole_true':false'1 :: true':false'
_hole_0':s':log_undefined'2 :: 0':s':log_undefined'
_gen_0':s':log_undefined'3 :: Nat → 0':s':log_undefined'
Heuristically decided to analyse the following defined symbols:
le', inc', minus', quot', log2'
They will be analysed ascendingly in the following order:
le' < log2'
inc' < log2'
minus' < quot'
quot' < log2'
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Types:
le' :: 0':s':log_undefined' → 0':s':log_undefined' → true':false'
0' :: 0':s':log_undefined'
true' :: true':false'
s' :: 0':s':log_undefined' → 0':s':log_undefined'
false' :: true':false'
inc' :: 0':s':log_undefined' → 0':s':log_undefined'
minus' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
quot' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log' :: 0':s':log_undefined' → 0':s':log_undefined'
log2' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
if' :: true':false' → true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log_undefined' :: 0':s':log_undefined'
if2' :: true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
_hole_true':false'1 :: true':false'
_hole_0':s':log_undefined'2 :: 0':s':log_undefined'
_gen_0':s':log_undefined'3 :: Nat → 0':s':log_undefined'
Generator Equations:
_gen_0':s':log_undefined'3(0) ⇔ 0'
_gen_0':s':log_undefined'3(+(x, 1)) ⇔ s'(_gen_0':s':log_undefined'3(x))
The following defined symbols remain to be analysed:
le', inc', minus', quot', log2'
They will be analysed ascendingly in the following order:
le' < log2'
inc' < log2'
minus' < quot'
quot' < log2'
Proved the following rewrite lemma:
le'(_gen_0':s':log_undefined'3(_n5), _gen_0':s':log_undefined'3(_n5)) → true', rt ∈ Ω(1 + n5)
Induction Base:
le'(_gen_0':s':log_undefined'3(0), _gen_0':s':log_undefined'3(0)) →RΩ(1)
true'
Induction Step:
le'(_gen_0':s':log_undefined'3(+(_$n6, 1)), _gen_0':s':log_undefined'3(+(_$n6, 1))) →RΩ(1)
le'(_gen_0':s':log_undefined'3(_$n6), _gen_0':s':log_undefined'3(_$n6)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Types:
le' :: 0':s':log_undefined' → 0':s':log_undefined' → true':false'
0' :: 0':s':log_undefined'
true' :: true':false'
s' :: 0':s':log_undefined' → 0':s':log_undefined'
false' :: true':false'
inc' :: 0':s':log_undefined' → 0':s':log_undefined'
minus' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
quot' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log' :: 0':s':log_undefined' → 0':s':log_undefined'
log2' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
if' :: true':false' → true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log_undefined' :: 0':s':log_undefined'
if2' :: true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
_hole_true':false'1 :: true':false'
_hole_0':s':log_undefined'2 :: 0':s':log_undefined'
_gen_0':s':log_undefined'3 :: Nat → 0':s':log_undefined'
Lemmas:
le'(_gen_0':s':log_undefined'3(_n5), _gen_0':s':log_undefined'3(_n5)) → true', rt ∈ Ω(1 + n5)
Generator Equations:
_gen_0':s':log_undefined'3(0) ⇔ 0'
_gen_0':s':log_undefined'3(+(x, 1)) ⇔ s'(_gen_0':s':log_undefined'3(x))
The following defined symbols remain to be analysed:
inc', minus', quot', log2'
They will be analysed ascendingly in the following order:
inc' < log2'
minus' < quot'
quot' < log2'
Proved the following rewrite lemma:
inc'(_gen_0':s':log_undefined'3(_n815)) → _gen_0':s':log_undefined'3(_n815), rt ∈ Ω(1 + n815)
Induction Base:
inc'(_gen_0':s':log_undefined'3(0)) →RΩ(1)
0'
Induction Step:
inc'(_gen_0':s':log_undefined'3(+(_$n816, 1))) →RΩ(1)
s'(inc'(_gen_0':s':log_undefined'3(_$n816))) →IH
s'(_gen_0':s':log_undefined'3(_$n816))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Types:
le' :: 0':s':log_undefined' → 0':s':log_undefined' → true':false'
0' :: 0':s':log_undefined'
true' :: true':false'
s' :: 0':s':log_undefined' → 0':s':log_undefined'
false' :: true':false'
inc' :: 0':s':log_undefined' → 0':s':log_undefined'
minus' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
quot' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log' :: 0':s':log_undefined' → 0':s':log_undefined'
log2' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
if' :: true':false' → true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log_undefined' :: 0':s':log_undefined'
if2' :: true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
_hole_true':false'1 :: true':false'
_hole_0':s':log_undefined'2 :: 0':s':log_undefined'
_gen_0':s':log_undefined'3 :: Nat → 0':s':log_undefined'
Lemmas:
le'(_gen_0':s':log_undefined'3(_n5), _gen_0':s':log_undefined'3(_n5)) → true', rt ∈ Ω(1 + n5)
inc'(_gen_0':s':log_undefined'3(_n815)) → _gen_0':s':log_undefined'3(_n815), rt ∈ Ω(1 + n815)
Generator Equations:
_gen_0':s':log_undefined'3(0) ⇔ 0'
_gen_0':s':log_undefined'3(+(x, 1)) ⇔ s'(_gen_0':s':log_undefined'3(x))
The following defined symbols remain to be analysed:
minus', quot', log2'
They will be analysed ascendingly in the following order:
minus' < quot'
quot' < log2'
Proved the following rewrite lemma:
minus'(_gen_0':s':log_undefined'3(_n1404), _gen_0':s':log_undefined'3(_n1404)) → _gen_0':s':log_undefined'3(0), rt ∈ Ω(1 + n1404)
Induction Base:
minus'(_gen_0':s':log_undefined'3(0), _gen_0':s':log_undefined'3(0)) →RΩ(1)
0'
Induction Step:
minus'(_gen_0':s':log_undefined'3(+(_$n1405, 1)), _gen_0':s':log_undefined'3(+(_$n1405, 1))) →RΩ(1)
minus'(_gen_0':s':log_undefined'3(_$n1405), _gen_0':s':log_undefined'3(_$n1405)) →IH
_gen_0':s':log_undefined'3(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Types:
le' :: 0':s':log_undefined' → 0':s':log_undefined' → true':false'
0' :: 0':s':log_undefined'
true' :: true':false'
s' :: 0':s':log_undefined' → 0':s':log_undefined'
false' :: true':false'
inc' :: 0':s':log_undefined' → 0':s':log_undefined'
minus' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
quot' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log' :: 0':s':log_undefined' → 0':s':log_undefined'
log2' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
if' :: true':false' → true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log_undefined' :: 0':s':log_undefined'
if2' :: true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
_hole_true':false'1 :: true':false'
_hole_0':s':log_undefined'2 :: 0':s':log_undefined'
_gen_0':s':log_undefined'3 :: Nat → 0':s':log_undefined'
Lemmas:
le'(_gen_0':s':log_undefined'3(_n5), _gen_0':s':log_undefined'3(_n5)) → true', rt ∈ Ω(1 + n5)
inc'(_gen_0':s':log_undefined'3(_n815)) → _gen_0':s':log_undefined'3(_n815), rt ∈ Ω(1 + n815)
minus'(_gen_0':s':log_undefined'3(_n1404), _gen_0':s':log_undefined'3(_n1404)) → _gen_0':s':log_undefined'3(0), rt ∈ Ω(1 + n1404)
Generator Equations:
_gen_0':s':log_undefined'3(0) ⇔ 0'
_gen_0':s':log_undefined'3(+(x, 1)) ⇔ s'(_gen_0':s':log_undefined'3(x))
The following defined symbols remain to be analysed:
quot', log2'
They will be analysed ascendingly in the following order:
quot' < log2'
Could not prove a rewrite lemma for the defined symbol quot'.
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Types:
le' :: 0':s':log_undefined' → 0':s':log_undefined' → true':false'
0' :: 0':s':log_undefined'
true' :: true':false'
s' :: 0':s':log_undefined' → 0':s':log_undefined'
false' :: true':false'
inc' :: 0':s':log_undefined' → 0':s':log_undefined'
minus' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
quot' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log' :: 0':s':log_undefined' → 0':s':log_undefined'
log2' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
if' :: true':false' → true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log_undefined' :: 0':s':log_undefined'
if2' :: true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
_hole_true':false'1 :: true':false'
_hole_0':s':log_undefined'2 :: 0':s':log_undefined'
_gen_0':s':log_undefined'3 :: Nat → 0':s':log_undefined'
Lemmas:
le'(_gen_0':s':log_undefined'3(_n5), _gen_0':s':log_undefined'3(_n5)) → true', rt ∈ Ω(1 + n5)
inc'(_gen_0':s':log_undefined'3(_n815)) → _gen_0':s':log_undefined'3(_n815), rt ∈ Ω(1 + n815)
minus'(_gen_0':s':log_undefined'3(_n1404), _gen_0':s':log_undefined'3(_n1404)) → _gen_0':s':log_undefined'3(0), rt ∈ Ω(1 + n1404)
Generator Equations:
_gen_0':s':log_undefined'3(0) ⇔ 0'
_gen_0':s':log_undefined'3(+(x, 1)) ⇔ s'(_gen_0':s':log_undefined'3(x))
The following defined symbols remain to be analysed:
log2'
Could not prove a rewrite lemma for the defined symbol log2'.
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
inc'(0') → 0'
inc'(s'(x)) → s'(inc'(x))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
log'(x) → log2'(x, 0')
log2'(x, y) → if'(le'(x, 0'), le'(x, s'(0')), x, inc'(y))
if'(true', b, x, y) → log_undefined'
if'(false', b, x, y) → if2'(b, x, y)
if2'(true', x, s'(y)) → y
if2'(false', x, y) → log2'(quot'(x, s'(s'(0'))), y)
Types:
le' :: 0':s':log_undefined' → 0':s':log_undefined' → true':false'
0' :: 0':s':log_undefined'
true' :: true':false'
s' :: 0':s':log_undefined' → 0':s':log_undefined'
false' :: true':false'
inc' :: 0':s':log_undefined' → 0':s':log_undefined'
minus' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
quot' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log' :: 0':s':log_undefined' → 0':s':log_undefined'
log2' :: 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
if' :: true':false' → true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
log_undefined' :: 0':s':log_undefined'
if2' :: true':false' → 0':s':log_undefined' → 0':s':log_undefined' → 0':s':log_undefined'
_hole_true':false'1 :: true':false'
_hole_0':s':log_undefined'2 :: 0':s':log_undefined'
_gen_0':s':log_undefined'3 :: Nat → 0':s':log_undefined'
Lemmas:
le'(_gen_0':s':log_undefined'3(_n5), _gen_0':s':log_undefined'3(_n5)) → true', rt ∈ Ω(1 + n5)
inc'(_gen_0':s':log_undefined'3(_n815)) → _gen_0':s':log_undefined'3(_n815), rt ∈ Ω(1 + n815)
minus'(_gen_0':s':log_undefined'3(_n1404), _gen_0':s':log_undefined'3(_n1404)) → _gen_0':s':log_undefined'3(0), rt ∈ Ω(1 + n1404)
Generator Equations:
_gen_0':s':log_undefined'3(0) ⇔ 0'
_gen_0':s':log_undefined'3(+(x, 1)) ⇔ s'(_gen_0':s':log_undefined'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
le'(_gen_0':s':log_undefined'3(_n5), _gen_0':s':log_undefined'3(_n5)) → true', rt ∈ Ω(1 + n5)