Step
*
1
2
1
2
1
1
of Lemma
bag-moebius-property1
1. T : Type
2. valueall-type(T)
3. eq : EqDecider(T)
4. b : T List
5. ¬(b = {} ∈ bag(T))
6. Comm(ℤ;λx,y. (x + y)) ∧ IsMonoid(ℤ;λx,y. (x + y);0)
7. ||b|| ≥ 1
⊢ ((Σ(x∈[x∈[x∈sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|0 <z (#hd(b) in x)]). bag-moebius(eq;x)
+ Σ(x∈[x∈[x∈sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|¬b0 <z (#hd(b) in x)]). bag-moebius(eq;x))
+ 0)
= 0
∈ ℤ
BY
{ (InstLemma `bag-summation-minus` [⌜bag(T)⌝;⌜ℤ-rng⌝;
⌜[x∈[x∈sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|¬b0 <z (#hd(b) in x)]⌝;⌜λ2x.bag-moebius(eq;x)⌝]⋅
THENA Auto
) }
1
1. T : Type
2. valueall-type(T)
3. eq : EqDecider(T)
4. b : T List
5. ¬(b = {} ∈ bag(T))
6. Comm(ℤ;λx,y. (x + y)) ∧ IsMonoid(ℤ;λx,y. (x + y);0)
7. ||b|| ≥ 1
8. Σ(x∈[x∈[x∈sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|¬b0 <z (#hd(b) in x)]). -ℤ-rng bag-moebius(eq;x)
= (-ℤ-rng Σ(x∈[x∈[x∈sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|¬b0 <z (#hd(b) in x)]). bag-moebius(eq;x))
∈ |ℤ-rng|
⊢ ((Σ(x∈[x∈[x∈sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|0 <z (#hd(b) in x)]). bag-moebius(eq;x)
+ Σ(x∈[x∈[x∈sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|¬b0 <z (#hd(b) in x)]). bag-moebius(eq;x))
+ 0)
= 0
∈ ℤ
Latex:
Latex:
1. T : Type
2. valueall-type(T)
3. eq : EqDecider(T)
4. b : T List
5. \mneg{}(b = \{\})
6. Comm(\mBbbZ{};\mlambda{}x,y. (x + y)) \mwedge{} IsMonoid(\mBbbZ{};\mlambda{}x,y. (x + y);0)
7. ||b|| \mgeq{} 1
\mvdash{} ((\mSigma{}(x\mmember{}[x\mmember{}[x\mmember{}sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|0 <z (\#hd(b) in x)]). bag-moebius(eq;x)
+ \mSigma{}(x\mmember{}[x\mmember{}[x\mmember{}sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|\mneg{}\msubb{}0 <z (\#hd(b) in x)]). bag-moebius(eq;x))
+ 0)
= 0
By
Latex:
(InstLemma `bag-summation-minus` [\mkleeneopen{}bag(T)\mkleeneclose{};\mkleeneopen{}\mBbbZ{}-rng\mkleeneclose{};
\mkleeneopen{}[x\mmember{}[x\mmember{}sub-bags(eq;b)|bag-has-no-repeats(eq;x)]|\mneg{}\msubb{}0 <z (\#hd(b) in x)]\mkleeneclose{};\mkleeneopen{}\mlambda{}\msubtwo{}x.bag-moebius(eq;x)\mkleeneclose{}]\mcdot{}
THENA Auto
)
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