Upload file for report 9aa7ad0f-2e14-4e6d-bebf-a96311c9dac1

uploads/9aa7ad0f-2e14-4e6d-bebf-a96311c9dac1_Class.v

@@ -0,0 +1,176 @@
+Parameter P Q R : Prop.
+Theorem T' : (P -> Q -> R) -> P -> Q -> R.
+Proof.
+    exact (fun Hpqr Hp Hq => Hpqr Hp Hq).
+Qed.
+
+(*
+; T' : (P Q R) [P Q -> R] P Q -> R 
+(define (T' Hpqr Hp Hq)
+    (Hpqr Hp Hq))
+*)
+
+Check false. 
+
+Check True.
+
+Print True.
+
+Theorem T3 : (not (1 = 2)). 
+Proof.
+    unfold not.
+    intros H.
+    discriminate H.
+Qed.
+
+Definition and_1 (p: P /\ Q) : P :=
+    match p with
+    | conj Hp Hq => Hp
+    end.
+
+(* 
+(define-struct conj [a b])
+(define-contract (And P Q (Struct conj P Q))
+
+(: and_1 (All (P Q) (-> (And P Q) P)))
+(define (and_1 p)
+    (conj-a p)))
+*)
+
+(*Theorem demorgan1 : forall P Q: Prop,
+    not(P \/ Q) <-> not P /\ not Q.
+Proof.
+    intros P Q.
+    split.
+    constructor.
+    - (* intros H.*)
+    refine (fun H => _).
+    unfold not in *.
+    (* constructor *)
+    refine (conj _ _).
+    eauto.
+    eauto.*)
+
+Theorem demorganprop: forall P Q: bool,
+    negb (orb P Q) = andb (negb P) (negb Q).
+Proof.
+    intros P Q.
+    destruct P; destruct Q; reflexivity.
+Qed.
+
+Theorem demorganliteauto : forall P Q : Prop,
+    not (P \/ Q) <-> not P /\ not Q.
+Proof.
+    intros P Q.
+    constructor.
+    - intros. constructor; info_eauto.
+    - intros. destruct H. unfold not. intros. 
+      destruct H1; eauto.
+Qed.
+
+
+Inductive ArithExp : Set :=
+| num : nat -> ArithExp
+| add : ArithExp -> ArithExp -> ArithExp.
+
+Definition a1 := num 1.
+Definition a2 := add (num 2) a1. 
+Definition a4:= match a2 with
+| add sub1 _ => sub1
+| num _ => a1
+ end.
+    
+Eval compute in a4.
+
+Inductive ArithExp1 : Set :=
+| num1 (n : nat)
+| add1 (l r : ArithExp1).
+
+Definition partial := (add (num 3)).
+Definition a3 := partial (num 5).
+
+Eval compute in a3.
+
+Fixpoint eval (e : ArithExp) : nat :=
+    match e with
+    | num n => n
+    | add e1 e2 => Nat.add (eval e1) (eval e2)
+    end.
+
+Eval compute in (eval a3).
+
+Fixpoint const_fold (e : ArithExp) : ArithExp :=
+    match e with
+    | num n => num n
+    | add l r =>
+    match l, r with
+    | num n1, num n2 => num (Nat.add n1 n2)
+    | l', r' => add (const_fold l') (const_fold r')
+    end
+end.
+
+Inductive Color : Set := 
+| red : Color
+| green : Color
+| blue : Color.
+
+Inductive RGB : Set :=
+| mk :nat -> nat -> nat -> RGB.
+
+Definition colorToRGB (c : Color) : RGB :=
+    match c with
+    | red => mk 255 0 0
+    | green => mk 0 255 0
+    | blue => mk 0 0 255
+    end.
+
+Definition RGBtoColor (r: RGB) : Color :=
+    match r with
+    | mk 255 0 0 => red
+    | mk 0 255 0 => green
+    | mk 0 0 255 => blue
+    | _ => red
+    end.
+
+Theorem colorRGB_roundtrip : forall c,
+    RGBtoColor (colorToRGB c) = c.
+Proof.
+    intros c.
+    destruct c; reflexivity.
+Qed.
+
+Fixpoint double (n : nat) : nat :=
+    match n with
+    | 0 => 0
+    | S n' => S (S (double n'))
+    end.
+
+Eval compute in double 10.
+
+Theorem double_succ : forall n,
+  double (S n) = S (S (double n)).   
+Proof.
+    intros n.
+    simpl.
+    reflexivity.
+Qed.
+
+Inductive ev : nat -> Prop :=
+| ev_0 : ev 0
+| ev_SS : forall n, 
+   ev n -> ev (S (S n)). 
+
+Fixpoint ev_pred (n : nat) : Prop :=
+    match n with
+    | 0 => True
+    | S (S n) => ev_pred n'
+    | _ => False
+    end.
+
+Theorem ev_4 : ev 4.
+Proof. 
+    apply ev_SS.
+    apply ev_SS.
+    apply ev_0.
+Qed.
+