Step
*
1
of Lemma
fps-compose-mul
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. x : X
6. g : PowerSeries(X;r)
7. f : PowerSeries(X;r)
8. h : PowerSeries(X;r)
9. ∀L:bag(X) List+. (||L|| ≥ 1 )
10. Assoc(|r|;+r)
11. IsMonoid(|r|;+r;0)
12. Comm(|r|;+r)
13. Comm(|r|;*)
14. Assoc(|r|;*)
15. ∀L:bag(X) List+. (Πa ∈ tl(L). f a ∈ |r|)
16. b : bag(X)
⊢ Σ(p∈⋃L∈bag-parts'(eq;b;x).bag-map(λp.<L, p>;bag-partitions(eq;hd(L) + bag-rep(||tl(L)||;x)))). (* (g (fst(snd(p))))
(h (snd(snd(p)))))
*
Πa ∈ tl(fst(p)). f a
= Σ(p∈⋃p∈bag-partitions(eq;b).bag-map(λp1.<p, p1>;bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x)))
((g (hd(fst(snd(p))) + bag-rep(||tl(fst(snd(p)))||;x))) * Πa ∈ tl(fst(snd(p))). f a)
*
((h (hd(snd(snd(p))) + bag-rep(||tl(snd(snd(p)))||;x))) * Πa ∈ tl(snd(snd(p))). f a)
∈ |r|
BY
{ TACTIC:Assert ⌜Σ(p∈⋃p∈bag-partitions(eq;b).bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x))
((g (hd(fst(p)) + bag-rep(||tl(fst(p))||;x))) * Πa ∈ tl(fst(p)). f a)
*
((h (hd(snd(p)) + bag-rep(||tl(snd(p))||;x))) * Πa ∈ tl(snd(p)). f a)
= Σ(p∈⋃p∈bag-partitions(eq;b).bag-map(λp1.<p, p1>;bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x)))
((g (hd(fst(snd(p))) + bag-rep(||tl(fst(snd(p)))||;x))) * Πa ∈ tl(fst(snd(p))). f a)
*
((h (hd(snd(snd(p))) + bag-rep(||tl(snd(snd(p)))||;x))) * Πa ∈ tl(snd(snd(p))). f a)
∈ |r|⌝⋅ }
1
.....assertion.....
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. x : X
6. g : PowerSeries(X;r)
7. f : PowerSeries(X;r)
8. h : PowerSeries(X;r)
9. ∀L:bag(X) List+. (||L|| ≥ 1 )
10. Assoc(|r|;+r)
11. IsMonoid(|r|;+r;0)
12. Comm(|r|;+r)
13. Comm(|r|;*)
14. Assoc(|r|;*)
15. ∀L:bag(X) List+. (Πa ∈ tl(L). f a ∈ |r|)
16. b : bag(X)
⊢ Σ(p∈⋃p∈bag-partitions(eq;b).bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x)). ((g
(hd(fst(p))
+ bag-rep(||tl(fst(p))||;x)))
*
Πa ∈ tl(fst(p)). f a)
*
((h
(hd(snd(p))
+ bag-rep(||tl(snd(p))||;x)))
*
Πa ∈ tl(snd(p)). f a)
= Σ(p∈⋃p∈bag-partitions(eq;b).bag-map(λp1.<p, p1>;bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x)))
((g (hd(fst(snd(p))) + bag-rep(||tl(fst(snd(p)))||;x))) * Πa ∈ tl(fst(snd(p))). f a)
*
((h (hd(snd(snd(p))) + bag-rep(||tl(snd(snd(p)))||;x))) * Πa ∈ tl(snd(snd(p))). f a)
∈ |r|
2
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. x : X
6. g : PowerSeries(X;r)
7. f : PowerSeries(X;r)
8. h : PowerSeries(X;r)
9. ∀L:bag(X) List+. (||L|| ≥ 1 )
10. Assoc(|r|;+r)
11. IsMonoid(|r|;+r;0)
12. Comm(|r|;+r)
13. Comm(|r|;*)
14. Assoc(|r|;*)
15. ∀L:bag(X) List+. (Πa ∈ tl(L). f a ∈ |r|)
16. b : bag(X)
17. Σ(p∈⋃p∈bag-partitions(eq;b).bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x)). ((g
(hd(fst(p))
+ bag-rep(||tl(fst(p))||;x)))
*
Πa ∈ tl(fst(p)). f a)
*
((h
(hd(snd(p))
+ bag-rep(||tl(snd(p))||;x)))
*
Πa ∈ tl(snd(p)). f a)
= Σ(p∈⋃p∈bag-partitions(eq;b).bag-map(λp1.<p, p1>;bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x)))
((g (hd(fst(snd(p))) + bag-rep(||tl(fst(snd(p)))||;x))) * Πa ∈ tl(fst(snd(p))). f a)
*
((h (hd(snd(snd(p))) + bag-rep(||tl(snd(snd(p)))||;x))) * Πa ∈ tl(snd(snd(p))). f a)
∈ |r|
⊢ Σ(p∈⋃L∈bag-parts'(eq;b;x).bag-map(λp.<L, p>;bag-partitions(eq;hd(L) + bag-rep(||tl(L)||;x)))). (* (g (fst(snd(p))))
(h (snd(snd(p)))))
*
Πa ∈ tl(fst(p)). f a
= Σ(p∈⋃p∈bag-partitions(eq;b).bag-map(λp1.<p, p1>;bag-parts'(eq;fst(p);x) × bag-parts'(eq;snd(p);x)))
((g (hd(fst(snd(p))) + bag-rep(||tl(fst(snd(p)))||;x))) * Πa ∈ tl(fst(snd(p))). f a)
*
((h (hd(snd(snd(p))) + bag-rep(||tl(snd(snd(p)))||;x))) * Πa ∈ tl(snd(snd(p))). f a)
∈ |r|
Latex:
Latex:
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. x : X
6. g : PowerSeries(X;r)
7. f : PowerSeries(X;r)
8. h : PowerSeries(X;r)
9. \mforall{}L:bag(X) List\msupplus{}. (||L|| \mgeq{} 1 )
10. Assoc(|r|;+r)
11. IsMonoid(|r|;+r;0)
12. Comm(|r|;+r)
13. Comm(|r|;*)
14. Assoc(|r|;*)
15. \mforall{}L:bag(X) List\msupplus{}. (\mPi{}a \mmember{} tl(L). f a \mmember{} |r|)
16. b : bag(X)
\mvdash{} \mSigma{}(p\mmember{}\mcup{}L\mmember{}bag-parts'(eq;b;x).bag-map(\mlambda{}p.<L, p>bag-partitions(eq;hd(L) + bag-rep(||tl(L)||;x))))
(* (g (fst(snd(p)))) (h (snd(snd(p))))) * \mPi{}a \mmember{} tl(fst(p)). f a
= \mSigma{}(p\mmember{}\mcup{}p\mmember{}bag-partitions(eq;b).
bag-map(\mlambda{}p1.<p, p1>bag-parts'(eq;fst(p);x) \mtimes{} bag-parts'(eq;snd(p);x)))
((g (hd(fst(snd(p))) + bag-rep(||tl(fst(snd(p)))||;x))) * \mPi{}a \mmember{} tl(fst(snd(p))). f a)
*
((h (hd(snd(snd(p))) + bag-rep(||tl(snd(snd(p)))||;x))) * \mPi{}a \mmember{} tl(snd(snd(p))). f a)
By
Latex:
TACTIC:Assert \mkleeneopen{}\mSigma{}(p\mmember{}\mcup{}p\mmember{}bag-partitions(eq;b).bag-parts'(eq;fst(p);x) \mtimes{} bag-parts'(eq;snd(p);x))
((g (hd(fst(p)) + bag-rep(||tl(fst(p))||;x))) * \mPi{}a \mmember{} tl(fst(p)). f a)
*
((h (hd(snd(p)) + bag-rep(||tl(snd(p))||;x))) * \mPi{}a \mmember{} tl(snd(p)). f a)
= \mSigma{}(p\mmember{}\mcup{}p\mmember{}bag-partitions(eq;b).
bag-map(\mlambda{}p1.<p, p1>bag-parts'(eq;fst(p);x) \mtimes{} bag-parts'(eq;snd(p);x)))
((g (hd(fst(snd(p))) + bag-rep(||tl(fst(snd(p)))||;x)))
*
\mPi{}a \mmember{} tl(fst(snd(p))). f a)
*
((h (hd(snd(snd(p))) + bag-rep(||tl(snd(snd(p)))||;x)))
*
\mPi{}a \mmember{} tl(snd(snd(p))). f a)\mkleeneclose{}\mcdot{}
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