Proof of Nuprl Lemma : fps-deriv-mul

Step * 7 7 of Lemma fps-deriv-mul

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. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).

      ((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))

12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))

13. ∀c:|r|. ∀f:PowerSeries(X;r).  ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))

14. b : bag(X)
15. fps-ucont(X;eq;r;f.d(*f)/dx)
16. fps-ucont(X;eq;r;f.((*df/dx)+(d/dx*f)))
17. ∀f,g:PowerSeries(X;r). (d(*(f+g))/dx = (d(*f)/dx+d(*g)/dx) ∈ PowerSeries(X;r))
18. ∀f,g:PowerSeries(X;r).

      (((<b>*d(f+g)/dx)+(d<b>/dx*(f+g))) = (((<b>*df/dx)+(d<b>/dx*f))+((<b>*dg/dx)+(d<b>/dx*g))) ∈ PowerSeries(X;r))

19. ∀c:|r|. ∀f:PowerSeries(X;r). (d(*(c)*f)/dx = (c)*d(*f)/dx ∈ PowerSeries(X;r))

20. ∀c:|r|. ∀f:PowerSeries(X;r).  (((<b>*d(c)*f/dx)+(d<b>/dx*(c)*f)) = (c)*((<b>*df/dx)+(d<b>/dx*f)) ∈ PowerSeries(X;r))

21. b@0 : bag(X)
⊢ d(*<b@0>)/dx = ((*d<b@0>/dx)+(d/dx*<b@0>)) ∈ PowerSeries(X;r)
BY
{ (RenameVar `c' (-1)
 THEN RWW "fps-mul-single fps-deriv-single" 0
 THEN Auto
 THEN (RW (AddrC [3;3] (LemmaC `fps-mul-comm`)) 0 THENA Auto)
 THEN (RWO "fps-scalar-mul-mul.2<" 0 THENA Auto)
 THEN (RWO "fps-mul-single" 0 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. fps-ucont(X;eq;r;f.d(f*g)/dx)
9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g)))
10. ∀f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx) ∈ PowerSeries(X;r))
11. ∀f,g@0:PowerSeries(X;r).

      ((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g))) ∈ PowerSeries(X;r))

12. ∀c:|r|. ∀f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx ∈ PowerSeries(X;r))

13. ∀c:|r|. ∀f:PowerSeries(X;r).  ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))

      (((<b>*d(f+g)/dx)+(d<b>/dx*(f+g))) = (((<b>*df/dx)+(d<b>/dx*f))+((<b>*dg/dx)+(d<b>/dx*g))) ∈ PowerSeries(X;r))

19. ∀c:|r|. ∀f:PowerSeries(X;r). (d(*(c)*f)/dx = (c)*d(*f)/dx ∈ PowerSeries(X;r))

20. ∀c:|r|. ∀f:PowerSeries(X;r).  (((<b>*d(c)*f/dx)+(d<b>/dx*(c)*f)) = (c)*((<b>*df/dx)+(d<b>/dx*f)) ∈ PowerSeries(X;r))

21. c : bag(X)
⊢ (int-to-ring(r;(#x in b + c)))*<bag-drop(eq;b + c;x)>
= ((int-to-ring(r;(#x in c)))*+(int-to-ring(r;(#x in b)))*<c + bag-drop(eq;b;x)>)
∈ 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. fps-ucont(X;eq;r;f.d(f*g)/dx) 9. fps-ucont(X;eq;r;f.((f*dg/dx)+(df/dx*g))) 10. \mforall{}f,g@0:PowerSeries(X;r). (d((f+g@0)*g)/dx = (d(f*g)/dx+d(g@0*g)/dx)) 11. \mforall{}f,g@0:PowerSeries(X;r). ((((f+g@0)*dg/dx)+(d(f+g@0)/dx*g)) = (((f*dg/dx)+(df/dx*g))+((g@0*dg/dx)+(dg@0/dx*g)))) 12. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). (d((c)*f*g)/dx = (c)*d(f*g)/dx) 13. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). ((((c)*f*dg/dx)+(d(c)*f/dx*g)) = (c)*((f*dg/dx)+(df/dx*g))) 14. b : bag(X) 15. fps-ucont(X;eq;r;f.d(*f)/dx) 16. fps-ucont(X;eq;r;f.((*df/dx)+(d/dx*f))) 17. \mforall{}f,g:PowerSeries(X;r). (d(*(f+g))/dx = (d(*f)/dx+d(*g)/dx)) 18. \mforall{}f,g:PowerSeries(X;r). (((*d(f+g)/dx)+(d/dx*(f+g))) = (((*df/dx)+(d/dx*f))+((*dg/dx)+(d/dx*g)))) 19. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). (d(*(c)*f)/dx = (c)*d(*f)/dx) 20. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). (((*d(c)*f/dx)+(d/dx*(c)*f)) = (c)*((*df/dx)+(d/dx*f))) 21. b@0 : bag(X) \mvdash{} d(*<b@0>)/dx = ((*d<b@0>/dx)+(d/dx*<b@0>)) By Latex: (RenameVar `c' (-1) THEN RWW "fps-mul-single fps-deriv-single" 0 THEN Auto THEN (RW (AddrC [3;3] (LemmaC `fps-mul-comm`)) 0 THENA Auto) THEN (RWO "fps-scalar-mul-mul.2<" 0 THENA Auto) THEN (RWO "fps-mul-single" 0 THENA Auto)) Home Index