Step
*
1
2
1
1
of Lemma
Long-theorem
1. x : Atom
2. y : Atom
3. ¬(x = y ∈ Atom)
4. a : ℤ
5. b : ℤ
6. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]_d
0(y:=((k ⋅ℤ-rng 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅ℤ-rng 1)*atom(x)
+atom(y)))^(d (i + 1)))
∈ PowerSeries(ℤ-rng))
7. n : ℕ+
8. k : ℕ+
9. upto(k) ∈ bag(ℕk)
⊢ Σ(i∈upto(k)). ((((k - i) ⋅ℤ-rng 1)*atom(x)+atom(y)))^(if (i + 1 =z 0) then 1
if (i + 1 =z 1) then n - 1
else 0
fi )
= (((k)*atom(x)+atom(y)))^(n - 1)
∈ PowerSeries(ℤ-rng)
BY
{ TACTIC:(InstLemma `bag-summation-single-non-zero-no-repeats`
[⌜ℕk⌝;⌜PowerSeries(ℤ-rng)⌝;⌜IntDeq⌝;⌜λx,y. (x*y)⌝;⌜1⌝;⌜upto(k)⌝
; ⌜λ2i.((((k - i) ⋅ℤ-rng 1)*atom(x)+atom(y)))^(if (i + 1 =z 0) then 1
if (i + 1 =z 1) then n - 1
else 0
fi )⌝;⌜0⌝]⋅
THENA Auto
) }
1
1. x : Atom
2. y : Atom
3. ¬(x = y ∈ Atom)
4. a : ℤ
5. b : ℤ
6. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]_d
0(y:=((k ⋅ℤ-rng 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅ℤ-rng 1)*atom(x)
+atom(y)))^(d (i + 1)))
∈ PowerSeries(ℤ-rng))
7. n : ℕ+
8. k : ℕ+
9. upto(k) ∈ bag(ℕk)
⊢ IsMonoid(PowerSeries(ℤ-rng);λx,y. (x*y);1)
2
1. x : Atom
2. y : Atom
3. ¬(x = y ∈ Atom)
4. a : ℤ
5. b : ℤ
6. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]_d
0(y:=((k ⋅ℤ-rng 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅ℤ-rng 1)*atom(x)
+atom(y)))^(d (i + 1)))
∈ PowerSeries(ℤ-rng))
7. n : ℕ+
8. k : ℕ+
9. upto(k) ∈ bag(ℕk)
10. IsMonoid(PowerSeries(ℤ-rng);λx,y. (x*y);1)
⊢ Comm(PowerSeries(ℤ-rng);λx,y. (x*y))
3
1. x : Atom
2. y : Atom
3. ¬(x = y ∈ Atom)
4. a : ℤ
5. b : ℤ
6. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]_d
0(y:=((k ⋅ℤ-rng 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅ℤ-rng 1)*atom(x)
+atom(y)))^(d (i + 1)))
∈ PowerSeries(ℤ-rng))
7. n : ℕ+
8. k : ℕ+
9. upto(k) ∈ bag(ℕk)
10. x@0 : ℕk
11. x@0 ↓∈ upto(k)
⊢ (x@0 = 0 ∈ ℕk)
∨ (((((k - x@0) ⋅ℤ-rng 1)*atom(x)+atom(y)))^(if (x@0 + 1 =z 0) then 1
if (x@0 + 1 =z 1) then n - 1
else 0
fi )
= 1
∈ PowerSeries(ℤ-rng))
4
1. x : Atom
2. y : Atom
3. ¬(x = y ∈ Atom)
4. a : ℤ
5. b : ℤ
6. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]_d
0(y:=((k ⋅ℤ-rng 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅ℤ-rng 1)*atom(x)
+atom(y)))^(d (i + 1)))
∈ PowerSeries(ℤ-rng))
7. n : ℕ+
8. k : ℕ+
9. upto(k) ∈ bag(ℕk)
⊢ bag-no-repeats(ℕk;upto(k))
5
1. x : Atom
2. y : Atom
3. ¬(x = y ∈ Atom)
4. a : ℤ
5. b : ℤ
6. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]_d
0(y:=((k ⋅ℤ-rng 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅ℤ-rng 1)*atom(x)
+atom(y)))^(d (i + 1)))
∈ PowerSeries(ℤ-rng))
7. n : ℕ+
8. k : ℕ+
9. upto(k) ∈ bag(ℕk)
10. bag-no-repeats(ℕk;upto(k))
⊢ 0 ↓∈ upto(k)
6
1. x : Atom
2. y : Atom
3. ¬(x = y ∈ Atom)
4. a : ℤ
5. b : ℤ
6. ∀[d:ℕ ⟶ ℕ]. ∀[k:ℕ].
(Moessner(ℤ-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]_d
0(y:=((k ⋅ℤ-rng 1)*atom(x)+atom(y)))*Π(i∈upto(k)).((((k - i) ⋅ℤ-rng 1)*atom(x)
+atom(y)))^(d (i + 1)))
∈ PowerSeries(ℤ-rng))
7. n : ℕ+
8. k : ℕ+
9. upto(k) ∈ bag(ℕk)
10. Σ(x@0∈upto(k)). ((((k - x@0) ⋅ℤ-rng 1)*atom(x)+atom(y)))^(if (x@0 + 1 =z 0) then 1
if (x@0 + 1 =z 1) then n - 1
else 0
fi )
= ((((k - 0) ⋅ℤ-rng 1)*atom(x)+atom(y)))^(if (0 + 1 =z 0) then 1
if (0 + 1 =z 1) then n - 1
else 0
fi )
∈ PowerSeries(ℤ-rng)
⊢ Σ(i∈upto(k)). ((((k - i) ⋅ℤ-rng 1)*atom(x)+atom(y)))^(if (i + 1 =z 0) then 1
if (i + 1 =z 1) then n - 1
else 0
fi )
= (((k)*atom(x)+atom(y)))^(n - 1)
∈ PowerSeries(ℤ-rng)
Latex:
Latex:
1. x : Atom
2. y : Atom
3. \mneg{}(x = y)
4. a : \mBbbZ{}
5. b : \mBbbZ{}
6. \mforall{}[d:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[k:\mBbbN{}].
(Moessner(\mBbbZ{}-rng;x;y;((a - b)*atom(x)+(b)*atom(y));d;k)
= ([((a - b)*atom(x)+(b)*atom(y))]\_d
0(y:=((k \mcdot{}\mBbbZ{}-rng 1)*atom(x)
+atom(y)))*\mPi{}(i\mmember{}upto(k)).((((k - i) \mcdot{}\mBbbZ{}-rng 1)*atom(x)
+atom(y)))\^{}(d (i + 1))))
7. n : \mBbbN{}\msupplus{}
8. k : \mBbbN{}\msupplus{}
9. upto(k) \mmember{} bag(\mBbbN{}k)
\mvdash{} \mSigma{}(i\mmember{}upto(k)). ((((k - i) \mcdot{}\mBbbZ{}-rng 1)*atom(x)+atom(y)))\^{}(if (i + 1 =\msubz{} 0) then 1
if (i + 1 =\msubz{} 1) then n - 1
else 0
fi )
= (((k)*atom(x)+atom(y)))\^{}(n - 1)
By
Latex:
TACTIC:(InstLemma `bag-summation-single-non-zero-no-repeats`
[\mkleeneopen{}\mBbbN{}k\mkleeneclose{};\mkleeneopen{}PowerSeries(\mBbbZ{}-rng)\mkleeneclose{};\mkleeneopen{}IntDeq\mkleeneclose{};\mkleeneopen{}\mlambda{}x,y. (x*y)\mkleeneclose{};\mkleeneopen{}1\mkleeneclose{};\mkleeneopen{}upto(k)\mkleeneclose{}
; \mkleeneopen{}\mlambda{}\msubtwo{}i.((((k - i) \mcdot{}\mBbbZ{}-rng 1)*atom(x)+atom(y)))\^{}(if (i + 1 =\msubz{} 0) then 1
if (i + 1 =\msubz{} 1) then n - 1
else 0
fi )\mkleeneclose{};\mkleeneopen{}0\mkleeneclose{}]\mcdot{}
THENA Auto
)
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