(* new script *)
-include "While/syntax.ma".
-
+include "While/syntax.ma".
+include "arithmetics/div_and_mod.ma".
(* state *)
-definition state ≝ \
\ No newline at end of file
+definition state ≝ Var → nat.
+
+definition eqvar ≝ λv1,v2.
+ match v1 with
+ [Id n => match v2 with [Id m => eqb n m]].
+
+definition update : state → Var → nat → state ≝ λs,v,a,v1.
+ if eqvar v1 v then a else s v1.
+
+(* Semantics of Arithmetic expressions *)
+\ 5pre class="smallmargin" style="display: inline;"\ 6let rec evalA a s ≝
+ match a with
+ [ Const n => n
+ | Aid v => s v
+ | Add a b => evalA a s+evalA b s
+ | Sub a b => evalA a s-evalA b s
+ | Mul a b => evalA a s*evalA b s
+ | Div a b => div (evalA a s) (evalA b s)
+ | Mod a b => mod (evalA a s) (evalA b s)
+ ]
+ .
+
+example exA1: evalA ((Const 2)+(Const 3)*(Aid (Id 2))) (λx.\ 5span style="text-decoration: underline;"\ 61\ 5/span\ 6) = \ 5span style="text-decoration: underline;"\ 65\ 5/span\ 6.
+normalize // qed
+
+let rec evalB b s ≝
+ match b with
+ [ BFalse => false
+ | BTtrue => true
+ | BNot b => ¬(evalB b s)
+ | BAnd b1 b2 => evalB b1 s ∧ evalB b2 s
+ | BOr b1 b2 => evalB b1 s ∨ evalB b2 s \ 5pre class="smallmargin" style="display: inline;"\ 6\ 5/pre\ 6
+ | Eq a1 a2 => eqb (evalA a1 s) (evalA a2 s)
+ | LE a1 a2 => leb (evalA a1 s) (evalA a2 s)
+ ].
+
+inductive Cmd_sem: Cmd → state → state\ 5pre class="smallmargin" style="display: inline;"\ 6→ Prop\ 5/pre\ 6≝
+ | Skip_sem : \forall s. Cmd_sem Skip s s
+ | Assign_sem : \forall s,v,a. Cmd_sem (Assign v a) s (update s v (evalA a s))
+ | Seq_sem : \forall c1,c2,s1,s2,s3. Cmd_sem c1 s1 s2 \to Cmd_sem c2 s2 s3 \to Cmd_sem (Seq c1 c2) s1 s3
+ | If_sem_true : \forall b,ct,cf,s,st. evalB b s = true \to Cmd_sem ct s st \to Cmd_sem (If b ct cf) s st
+ | If_sem_false : \forall b,ct,cf,s,sf. evalB b s = false \to Cmd_sem cf s sf \to Cmd_sem (If b ct cf) s sf
+ | While_true : \forall b,c,s,s1,s2. evalB b s = true \to Cmd_sem c s s1 \to Cmd_sem (While b c) s1 s2 \to Cmd_sem (While b c) s s2
+ | While_false : \forall b,c,s. evalB b s = false \to Cmd_sem (While b c) s s
+.\ 5/pre\ 6
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