Step
*
7
7
1
2
1
1
of Lemma
fps-deriv-mul
.....equality.....
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. (#x in b) ≠ 0
16. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
17. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
18. ∀f,g:PowerSeries(X;r). (d(<b>*(f+g))/dx = (d(<b>*f)/dx+d(<b>*g)/dx) ∈ PowerSeries(X;r))
19. ∀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))
20. ∀c:|r|. ∀f:PowerSeries(X;r). (d(<b>*(c)*f)/dx = (c)*d(<b>*f)/dx ∈ PowerSeries(X;r))
21. ∀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))
22. c : bag(X)
23. 0 < (#x in c)
⊢ (b + bag-drop(eq;c;x)) = (bag-drop(eq;b;x) + c) ∈ bag(X)
BY
{ (((InstLemma `bag-drop-append` [⌜X⌝;⌜eq⌝;⌜x⌝;⌜b⌝;⌜c⌝]⋅ THENA Auto) THEN SplitOnHypITE -1 THEN Auto)
THEN (InstLemma `bag-drop-append` [⌜X⌝;⌜eq⌝;⌜x⌝;⌜c⌝;⌜b⌝]⋅ THENA Auto)
THEN SplitOnHypITE -1
THEN Auto) }
1
.....falsecase.....
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. (#x in b) ≠ 0
16. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
17. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
18. ∀f,g:PowerSeries(X;r). (d(<b>*(f+g))/dx = (d(<b>*f)/dx+d(<b>*g)/dx) ∈ PowerSeries(X;r))
19. ∀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))
20. ∀c:|r|. ∀f:PowerSeries(X;r). (d(<b>*(c)*f)/dx = (c)*d(<b>*f)/dx ∈ PowerSeries(X;r))
21. ∀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))
22. c : bag(X)
23. 0 < (#x in c)
24. bag-drop(eq;b + c;x) = (bag-drop(eq;b;x) + c) ∈ bag(X)
25. ¬((#x in b) = 0 ∈ ℤ)
26. bag-drop(eq;c + b;x) = (bag-drop(eq;c;x) + b) ∈ bag(X)
27. ¬((#x in c) = 0 ∈ ℤ)
⊢ (b + bag-drop(eq;c;x)) = (bag-drop(eq;b;x) + c) ∈ bag(X)
Latex:
Latex:
.....equality.....
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. (\#x in b) \mneq{} 0
16. fps-ucont(X;eq;r;f.d(<b>*f)/dx)
17. fps-ucont(X;eq;r;f.((<b>*df/dx)+(d<b>/dx*f)))
18. \mforall{}f,g:PowerSeries(X;r). (d(<b>*(f+g))/dx = (d(<b>*f)/dx+d(<b>*g)/dx))
19. \mforall{}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))))
20. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). (d(<b>*(c)*f)/dx = (c)*d(<b>*f)/dx)
21. \mforall{}c:|r|. \mforall{}f:PowerSeries(X;r).
(((<b>*d(c)*f/dx)+(d<b>/dx*(c)*f)) = (c)*((<b>*df/dx)+(d<b>/dx*f)))
22. c : bag(X)
23. 0 < (\#x in c)
\mvdash{} (b + bag-drop(eq;c;x)) = (bag-drop(eq;b;x) + c)
By
Latex:
(((InstLemma `bag-drop-append` [\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{}]\mcdot{} THENA Auto) THEN SplitOnHypITE -1 THEN Auto)
THEN (InstLemma `bag-drop-append` [\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{};\mkleeneopen{}x\mkleeneclose{};\mkleeneopen{}c\mkleeneclose{};\mkleeneopen{}b\mkleeneclose{}]\mcdot{} THENA Auto)
THEN SplitOnHypITE -1
THEN Auto)
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