End CGSSolve.
Section CGWithOne.
Variable Done : D.
Variable Done_pos : Dle Done Dz = false.
Variable
Done_min_pos : forall d : D, Dle d Dz = false -> Dle Done d = true.
(*s Defining the negation of a formula : $\neg x \leq y+d \Leftrightarrow x>y+d \Leftrightarrow x \geq y+d+1$ *)
Fixpoint CGSnot (fs : CGSForm) : CGSForm :=
match fs with
| CGSleq x y d => CGSleq y x (Dneg (Dplus d Done))
| CGSand f0 f1 => CGSor (CGSnot f0) (CGSnot f1)
| CGSor f0 f1 => CGSand (CGSnot f0) (CGSnot f1)
end.

Lemma Dle_lt_1 :
forall d d' : D, Dle d' d = false -> Dle d (Dplus d' (Dneg Done)) = true.