[isabelle] nonnegative quadratic polynomial



Dear all,


Please I need to prove that for any nonnegative quadratic polynomial
equation the discriminant will be nonpositive. I found a proposition in a
text book bellow :
https://books.google.co.uk/books?id=TPE0fXGnYtMC&pg=PA81&lpg=PA81&dq=if+f(x)+%3E0+then+b%5E2-4ac%3C0&source=bl&ots=eUAdjNvJpT&sig=KwJxcX_5RqIsCL4wJ3nrfvfZe9g&hl=en&sa=X&ved=0ahUKEwjlxpjm2ZjNAhWMLsAKHX2vAs8Q6AEILzAE#v=onepage&q&f=false

 but for this one, the equality hold iff x=-b/2*a and I don't want to put
this as an assumption as I need this lemma in another work. the lemma I'm
trying is:

lemma
  fixes a b c x :: real
  assumes  "a > 0"
  and "âx. a*x^2 + b*x + câ0 "
shows " discrim a b c â0"

I proved that for strictly positive equation as bellow:

lemma
  fixes a b c x :: real
  assumes  "a > 0"
  and "âx. (a*(x)^2 + b*x + c) >0 "
shows " discrim a b c â0"
using assms by (metis discriminant_pos_ex less_le not_less)

but for nonnegative equation (the first lemma) I couldn't and way. please
could any one help me in this because I spend alot of time trying to prove
it but unfortunately I failed.

Omar



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