(**************************************************************************)
-(* ___ *)
+(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
| S : nat \to nat.
definition pred: nat \to nat \def
-\lambda n:nat. match n with
-[ O \Rightarrow O
-| (S p) \Rightarrow p ].
+ \lambda n:nat. match n with
+ [ O \Rightarrow O
+ | (S p) \Rightarrow p ].
theorem pred_Sn : \forall n:nat.n=(pred (S n)).
-intros. reflexivity.
+ intros. reflexivity.
qed.
theorem injective_S : injective nat nat S.
-unfold injective.
-intros.
-rewrite > pred_Sn.
-rewrite > (pred_Sn y).
-apply eq_f. assumption.
+ unfold injective.
+ intros.
+ rewrite > pred_Sn.
+ rewrite > (pred_Sn y).
+ apply eq_f. assumption.
qed.
-theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m
-\def injective_S.
+theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m \def
+ injective_S.
theorem not_eq_S : \forall n,m:nat.
-\lnot n=m \to S n \neq S m.
-intros. unfold Not. intros.
-apply H. apply injective_S. assumption.
+ \lnot n=m \to S n \neq S m.
+ intros. unfold Not. intros.
+ apply H. apply injective_S. assumption.
qed.
definition not_zero : nat \to Prop \def
-\lambda n: nat.
+ \lambda n: nat.
match n with
[ O \Rightarrow False
| (S p) \Rightarrow True ].
theorem not_eq_O_S : \forall n:nat. O \neq S n.
-intros. unfold Not. intros.
-cut (not_zero O).
-exact Hcut.
-rewrite > H.exact I.
+ intros. unfold Not. intros.
+ cut (not_zero O).
+ exact Hcut.
+ rewrite > H.exact I.
qed.
theorem not_eq_n_Sn : \forall n:nat. n \neq S n.
-intros.elim n.
-apply not_eq_O_S.
-apply not_eq_S.assumption.
+ intros.elim n.
+ apply not_eq_O_S.
+ apply not_eq_S.assumption.
qed.
theorem nat_case:
-\forall n:nat.\forall P:nat \to Prop.
-P O \to (\forall m:nat. P (S m)) \to P n.
-intros.elim n.assumption.apply H1.
+ \forall n:nat.\forall P:nat \to Prop.
+ P O \to (\forall m:nat. P (S m)) \to P n.
+intros.elim n
+ [ assumption
+ | apply H1 ]
qed.
theorem nat_case1:
-\forall n:nat.\forall P:nat \to Prop.
-(n=O \to P O) \to (\forall m:nat. (n=(S m) \to P (S m))) \to P n.
-intros 2.elim n.
-apply H.reflexivity.
-apply H2.reflexivity.
+ \forall n:nat.\forall P:nat \to Prop.
+ (n=O \to P O) \to (\forall m:nat. (n=(S m) \to P (S m))) \to P n.
+intros 2; elim n
+ [ apply H;reflexivity
+ | apply H2;reflexivity ]
qed.
theorem nat_elim2 :
-\forall R:nat \to nat \to Prop.
-(\forall n:nat. R O n) \to
-(\forall n:nat. R (S n) O) \to
-(\forall n,m:nat. R n m \to R (S n) (S m)) \to \forall n,m:nat. R n m.
-intros 5.elim n.
-apply H.
-apply (nat_case m).apply H1.
-intros.apply H2. apply H3.
+ \forall R:nat \to nat \to Prop.
+ (\forall n:nat. R O n) \to
+ (\forall n:nat. R (S n) O) \to
+ (\forall n,m:nat. R n m \to R (S n) (S m)) \to \forall n,m:nat. R n m.
+intros 5;elim n
+ [ apply H
+ | apply (nat_case m)
+ [ apply H1
+ | intro; apply H2; apply H3 ] ]
qed.
theorem decidable_eq_nat : \forall n,m:nat.decidable (n=m).
-intros.unfold decidable.
-apply (nat_elim2 (\lambda n,m.(Or (n=m) ((n=m) \to False)))).
-intro.elim n1.
-left.reflexivity.
-right.apply not_eq_O_S.
-intro.right.intro.
-apply (not_eq_O_S n1).
-apply sym_eq.assumption.
-intros.elim H.
-left.apply eq_f. assumption.
-right.intro.apply H1.apply inj_S.assumption.
+ intros.unfold decidable.
+ apply (nat_elim2 (\lambda n,m.(Or (n=m) ((n=m) \to False))))
+ [ intro; elim n1
+ [ left; reflexivity
+ | right; apply not_eq_O_S ]
+ | intro; right; intro; apply (not_eq_O_S n1); apply sym_eq; assumption
+ | intros; elim H
+ [ left; apply eq_f; assumption
+ | right; intro; apply H1; apply inj_S; assumption ] ]
qed.
-