Decreasing degrees of Qispan> 's
in ∃x ∈ ]a, b[, P =
0 and Q1 > 0 and ... Qk
> 0
We replace it by the following disjonction:
-
(a = 0 and ∃x, (P
without axn
= 0 and Q1
> 0 and ... Qk
> 0))
This part is now simplier because the degree of P
without axn is strictly
lower than the degree of P. We apply again this step to the new formula.
-
or (a ≠ 0 and
∃x,
(P = 0 and T1
> 0 and ... Tk
> 0))
where each Ti is obtained by the following
algorithm:
Ti
:= Qi
while(degree(Ti) ≥
degree(P))
Ti := a2.Ti -
a.b.P.xdegree(Qi)-degree(P)
where
:
- P = axdegree(P)
+...
- Ti = bxdegree(Ti)
+...
endWhile
At the end, we have
degreex(Ti)
<
degreex(P).
x + 1 = 0 and x2
> 0
is replaced by x + 1 = 0 and x2 -
x(x + 1) > 0, that is to say, x + 1 = 0 and -x > 0.
Then
x + 1 = 0 and -x > 0 is replaced by x + 1 = 0 and -x -
(-1)(x +
1) > 0, that is to say, x + 1 = 0 and 1 > 0.
We can then
evaluate 1 > 0, (it is true) and the initial formula is equivalent to x + 1 = 0.