(0) Obligation:
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)
pred(s(x)) → x
minus(x, 0) → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)
Rewrite Strategy: INNERMOST
(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(2) Obligation:
Runtime Complexity Relative 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)
pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0', y) → y
gcd(s(x), 0') → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)
S is empty.
Rewrite Strategy: INNERMOST
(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(4) Obligation:
Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
pred(s(x)) → x
minus(x, 0') → x
minus(x, s(y)) → pred(minus(x, y))
gcd(0', y) → y
gcd(s(x), 0') → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, x, y) → gcd(minus(x, y), y)
if_gcd(false, x, y) → gcd(minus(y, x), x)
Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
pred :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s
if_gcd :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
le,
minus,
gcdThey will be analysed ascendingly in the following order:
le < gcd
minus < gcd
(6) Obligation:
Innermost TRS:
Rules:
le(
0',
y) →
truele(
s(
x),
0') →
falsele(
s(
x),
s(
y)) →
le(
x,
y)
pred(
s(
x)) →
xminus(
x,
0') →
xminus(
x,
s(
y)) →
pred(
minus(
x,
y))
gcd(
0',
y) →
ygcd(
s(
x),
0') →
s(
x)
gcd(
s(
x),
s(
y)) →
if_gcd(
le(
y,
x),
s(
x),
s(
y))
if_gcd(
true,
x,
y) →
gcd(
minus(
x,
y),
y)
if_gcd(
false,
x,
y) →
gcd(
minus(
y,
x),
x)
Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
pred :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s
if_gcd :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
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:
le, minus, gcd
They will be analysed ascendingly in the following order:
le < gcd
minus < gcd
(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
le(
gen_0':s3_0(
n5_0),
gen_0':s3_0(
n5_0)) →
true, rt ∈ Ω(1 + n5
0)
Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
true
Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →IH
true
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(8) Complex Obligation (BEST)
(9) Obligation:
Innermost TRS:
Rules:
le(
0',
y) →
truele(
s(
x),
0') →
falsele(
s(
x),
s(
y)) →
le(
x,
y)
pred(
s(
x)) →
xminus(
x,
0') →
xminus(
x,
s(
y)) →
pred(
minus(
x,
y))
gcd(
0',
y) →
ygcd(
s(
x),
0') →
s(
x)
gcd(
s(
x),
s(
y)) →
if_gcd(
le(
y,
x),
s(
x),
s(
y))
if_gcd(
true,
x,
y) →
gcd(
minus(
x,
y),
y)
if_gcd(
false,
x,
y) →
gcd(
minus(
y,
x),
x)
Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
pred :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s
if_gcd :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
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:
minus, gcd
They will be analysed ascendingly in the following order:
minus < gcd
(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
minus(
gen_0':s3_0(
a),
gen_0':s3_0(
+(
1,
n330_0))) →
*4_0, rt ∈ Ω(n330
0)
Induction Base:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, 0)))
Induction Step:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n330_0, 1)))) →RΩ(1)
pred(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n330_0)))) →IH
pred(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(11) Complex Obligation (BEST)
(12) Obligation:
Innermost TRS:
Rules:
le(
0',
y) →
truele(
s(
x),
0') →
falsele(
s(
x),
s(
y)) →
le(
x,
y)
pred(
s(
x)) →
xminus(
x,
0') →
xminus(
x,
s(
y)) →
pred(
minus(
x,
y))
gcd(
0',
y) →
ygcd(
s(
x),
0') →
s(
x)
gcd(
s(
x),
s(
y)) →
if_gcd(
le(
y,
x),
s(
x),
s(
y))
if_gcd(
true,
x,
y) →
gcd(
minus(
x,
y),
y)
if_gcd(
false,
x,
y) →
gcd(
minus(
y,
x),
x)
Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
pred :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s
if_gcd :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n330_0))) → *4_0, rt ∈ Ω(n3300)
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:
gcd
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol gcd.
(14) Obligation:
Innermost TRS:
Rules:
le(
0',
y) →
truele(
s(
x),
0') →
falsele(
s(
x),
s(
y)) →
le(
x,
y)
pred(
s(
x)) →
xminus(
x,
0') →
xminus(
x,
s(
y)) →
pred(
minus(
x,
y))
gcd(
0',
y) →
ygcd(
s(
x),
0') →
s(
x)
gcd(
s(
x),
s(
y)) →
if_gcd(
le(
y,
x),
s(
x),
s(
y))
if_gcd(
true,
x,
y) →
gcd(
minus(
x,
y),
y)
if_gcd(
false,
x,
y) →
gcd(
minus(
y,
x),
x)
Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
pred :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s
if_gcd :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n330_0))) → *4_0, rt ∈ Ω(n3300)
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.
(15) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
(16) BOUNDS(n^1, INF)
(17) Obligation:
Innermost TRS:
Rules:
le(
0',
y) →
truele(
s(
x),
0') →
falsele(
s(
x),
s(
y)) →
le(
x,
y)
pred(
s(
x)) →
xminus(
x,
0') →
xminus(
x,
s(
y)) →
pred(
minus(
x,
y))
gcd(
0',
y) →
ygcd(
s(
x),
0') →
s(
x)
gcd(
s(
x),
s(
y)) →
if_gcd(
le(
y,
x),
s(
x),
s(
y))
if_gcd(
true,
x,
y) →
gcd(
minus(
x,
y),
y)
if_gcd(
false,
x,
y) →
gcd(
minus(
y,
x),
x)
Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
pred :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s
if_gcd :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n330_0))) → *4_0, rt ∈ Ω(n3300)
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.
(18) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
(19) BOUNDS(n^1, INF)
(20) Obligation:
Innermost TRS:
Rules:
le(
0',
y) →
truele(
s(
x),
0') →
falsele(
s(
x),
s(
y)) →
le(
x,
y)
pred(
s(
x)) →
xminus(
x,
0') →
xminus(
x,
s(
y)) →
pred(
minus(
x,
y))
gcd(
0',
y) →
ygcd(
s(
x),
0') →
s(
x)
gcd(
s(
x),
s(
y)) →
if_gcd(
le(
y,
x),
s(
x),
s(
y))
if_gcd(
true,
x,
y) →
gcd(
minus(
x,
y),
y)
if_gcd(
false,
x,
y) →
gcd(
minus(
y,
x),
x)
Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
pred :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
gcd :: 0':s → 0':s → 0':s
if_gcd :: true:false → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s
Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
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.
(21) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → true, rt ∈ Ω(1 + n50)
(22) BOUNDS(n^1, INF)