End CGDist1.
(*s Definition of concrete formulas : [(CGleq x y d)] means $x\leq y+d$, [(CGeq x y d)] means $x=y+d$ *)
Inductive CGForm : Set :=
| CGleq : ad -> ad -> D -> CGForm
| CGeq : ad -> ad -> D -> CGForm
| CGand : CGForm -> CGForm -> CGForm
| CGor : CGForm -> CGForm -> CGForm
| CGimp : CGForm -> CGForm -> CGForm
| CGnot : CGForm -> CGForm.
(*s Interpretation of concrete formulas as proposition *)
Fixpoint CGeval (rho : ad -> D) (f : CGForm) {struct f} : Prop :=
match f with
| CGleq x y d => Dle (rho x) (Dplus (rho y) d) = true
| CGeq x y d => rho x = Dplus (rho y) d
| CGand f0 f1 => CGeval rho f0 /\ CGeval rho f1
| CGor f0 f1 => CGeval rho f0 \/ CGeval rho f1
| CGimp f0 f1 => CGeval rho f0 -> CGeval rho f1
| CGnot f0 => ~ CGeval rho f0
end.
(*s Decidability of satisfaction *)

Lemma CGeval_dec :
forall (f : CGForm) (rho : ad -> D), {CGeval rho f} + {~ CGeval rho f}.