defining new format for SMT queries

coq/SMTArith.v

+(*
+  Formalization of SMT Arithmetic for FloVer. Original author: Joachim Bard.
+*)
+
+Require Import Coq.Reals.Reals Coq.QArith.Qreals.
+
+Open Scope R.
+
+Inductive SMTArith :=
+| ConstQ (r: Q) : SMTArith
+| VarQ (x: nat) : SMTArith
+| UMinusQ (e: SMTArith) : SMTArith
+| PlusQ (e1 e2: SMTArith) : SMTArith
+| MinusQ (e1 e2: SMTArith) : SMTArith
+| TimesQ (e1 e2: SMTArith) : SMTArith
+| DivQ (e1 e2: SMTArith) : SMTArith.
+
+
+Inductive SMTLogic :=
+| LessQ (e1 e2: SMTArith) : SMTLogic
+| LessEqQ (e1 e2: SMTArith) : SMTLogic
+| EqualsQ (e1 e2: SMTArith) : SMTLogic
+| NotQ (q: SMTLogic) : SMTLogic
+| AndQ (q1 q2: SMTLogic) : SMTLogic
+| OrQ (q1 q2: SMTLogic) : SMTLogic.
+
+Fixpoint evalArith (f: nat -> R) (e: SMTArith) : R :=
+  match e with
+  | ConstQ r => Q2R r
+  | VarQ x => f x
+  | UMinusQ e => - (evalArith f e)
+  | PlusQ e1 e2 => (evalArith f e1) + (evalArith f e2)
+  | MinusQ e1 e2 => (evalArith f e1) - (evalArith f e2)
+  | TimesQ e1 e2 => (evalArith f e1) * (evalArith f e2)
+  | DivQ e1 e2 => (evalArith f e1) / (evalArith f e2)
+  end.
+
+Fixpoint evalLogic (f: nat -> R) (q: SMTLogic) : Prop :=
+  match q with
+  | LessQ q1 q2 => (evalArith f q1) < (evalArith f q2)
+  | LessEqQ q1 q2 => (evalArith f q1) <= (evalArith f q2)
+  | EqualsQ q1 q2 => (evalArith f q1) = (evalArith f q2)
+  | NotQ q => ~ (evalLogic f q)
+  | AndQ q1 q2 => (evalLogic f q1) /\ (evalLogic f q2)
+  | OrQ q1 q2 => (evalLogic f q1) \/ (evalLogic f q2)
+  end.