Step
*
5
1
1
of Lemma
fps-deriv-compose
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. b : bag(X)
⊢ d<b>(x:=g)/dx = ((int-to-ring(r;(#x in b)))*<bag-drop(eq;b;x)>(x:=g)*dg/dx) ∈ PowerSeries(X;r)
BY
{ ((RWO "fps-compose-single-general" 0 THENA Auto) THEN (GenConcl ⌜(g-(g[{}])*1) = h ∈ PowerSeries(X;r)⌝⋅ THENA Auto)) }
1
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. b : bag(X)
9. h : PowerSeries(X;r)
10. (g-(g[{}])*1) = h ∈ PowerSeries(X;r)
⊢ d(<(b|¬x)>*(h)^(#((b|x))))/dx
= ((int-to-ring(r;(#x in b)))*(<(bag-drop(eq;b;x)|¬x)>*(h)^(#((bag-drop(eq;b;x)|x))))*dg/dx)
∈ PowerSeries(X;r)
Latex:
Latex:
1. X : Type
2. valueall-type(X)
3. eq : EqDecider(X)
4. r : CRng
5. f : PowerSeries(X;r)
6. g : PowerSeries(X;r)
7. x : X
8. b : bag(X)
\mvdash{} d<b>(x:=g)/dx = ((int-to-ring(r;(\#x in b)))*<bag-drop(eq;b;x)>(x:=g)*dg/dx)
By
Latex:
((RWO "fps-compose-single-general" 0 THENA Auto) THEN (GenConcl \mkleeneopen{}(g-(g[\{\}])*1) = h\mkleeneclose{}\mcdot{} THENA Auto))
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