X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2FWhile%2Fsemantics.ma;h=ee4362b71ccd197ce70a660c9f8c9b95a27b0d39;hb=84b38ac86f1f92b91ae8913cd0dbcb5c3485dc3a;hp=ec11cf2ec002457254c6a2094ce13bbd0a1a5155;hpb=e0671ce496760cec78ec904747af76ed55fd03a2;p=helm.git

diff --git a/weblib/While/semantics.ma b/weblib/While/semantics.ma
index ec11cf2ec..ee4362b71 100644
--- a/weblib/While/semantics.ma
+++ b/weblib/While/semantics.ma
@@ -1,53 +1,52 @@
 (* new script *)
 
 include "While/syntax.ma". 
-include "arithmetics/div_and_mod.ma. 
-
+include "arithmetics/div_and_mod.ma". 
 
 (* state *)
 
-definition state ≝ a href="cic:/matita/While/syntax/Var.ind(1,0,0)"Var/a → a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"nat/a.
+definition state ≝ Var → nat.
 
-(* Semantics of Arithmetic expressions *)
+definition eqvar ≝ λv1,v2.
+  match v1 with
+  [Id n => match v2 with [Id m => eqb n m]].
 
-pre class="smallmargin" style="display: inline;"let rec eval_Aexp a s ≝
+definition update : state → Var → nat → state ≝ λs,v,a,v1. 
+  if eqvar v1 v then a else s v1.
+
+(* Semantics of Arithmetic expressions *)
+pre class="smallmargin" style="display: inline;"let rec evalA a s ≝
   match a with 
-  | Const n => n
+  [ Const n => n
   | Aid v => s v
-  | Add a b => a+b
-  | Sub a b => a-b
-  | Mul a b => a*b
-  | Div a b => div a b
-  | Mod a b => mod a b
+  | 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) 
+  ]
  .
 
-interpretation "Aexp plus" 'plus x y = (Add x y).
-interpretation "Aexp minus" 'plus x y = (Sub x y).
-interpretation "Aexp times" 'times x y = (Mul x y).
-
-(* Boolean expressions *)
-inductive Bexp : Type[0] ≝
-  | BFalse : Bexp
-  | BTtrue : Bexp
-  | BNot : Bexp → Bexp
-  | BAnd : Bexp → Bexp → Bexp
-  | BOr : Bexp → Bexp → Bexp
-  | Eq : a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"Aexp/a → a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"Aexp/a → Bexp
-  | LE : a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"Aexp/a → a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"Aexp/a → Bexp
-.
-
-interpretation "Bexp not" 'not x = (BNot x).
-interpretation "Bexp and" 'and x y = (BAnd x y).
-interpretation "Bexp or" 'or x y = (BOr x y).
-interpretation "Bexp 'less or equal to'" 'leq x y = (LE x y).
-
-(* Commands *)
-inductive Cmd : Type[0] ≝
-  | Skip : Cmd
-  | Assign : Var → a href="cic:/matita/While/syntax/Aexp.ind(1,0,0)"Aexp/a → Cmd
-  | Seq : Cmd → Cmd → Cmd
-  | If : a href="cic:/matita/While/syntax/Bexp.ind(1,0,0)"Bexp/a → Cmd → Cmd → Cmd
-  | While : a href="cic:/matita/While/syntax/Bexp.ind(1,0,0)"Bexp/a → Cmd → Cmd
-.
-
-/pre 
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+example exA1: evalA ((Const 2)+(Const 3)*(Aid (Id 2))) (λx.span style="text-decoration: underline;"1/span) = span style="text-decoration: underline;"5/span.
+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 pre class="smallmargin" style="display: inline;"/pre  
+  | 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 → statepre class="smallmargin" style="display: inline;"→ Prop/pre≝
+  | 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  
+./pre
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