(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/div_and_mod".
-
include "datatypes/constructors.ma".
include "nat/minus.ma".
-
let rec mod_aux p m n: nat \def
match (leb m n) with
[ true \Rightarrow m
[O \Rightarrow n
| (S p) \Rightarrow mod_aux n n p].
-interpretation "natural remainder" 'module x y =
- (cic:/matita/nat/div_and_mod/mod.con x y).
+interpretation "natural remainder" 'module x y = (mod x y).
let rec div_aux p m n : nat \def
match (leb m n) with
[O \Rightarrow S n
| (S p) \Rightarrow div_aux n n p].
-interpretation "natural divide" 'divide x y =
- (cic:/matita/nat/div_and_mod/div.con x y).
+interpretation "natural divide" 'divide x y = (div x y).
theorem le_mod_aux_m_m:
\forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
rewrite < sym_times.
rewrite > distr_times_minus.
rewrite > plus_minus.
+lapply(plus_to_minus ??? H3); demodulate all.
+(*
rewrite > sym_times.
rewrite < H5.
rewrite < sym_times.
apply plus_to_minus.
apply H3.
+*)
apply le_times_r.
apply lt_to_le.
apply H6.
theorem div_mod_spec_times : \forall n,m:nat.div_mod_spec ((S n)*m) (S n) m O.
intros.constructor 1.
-unfold lt.apply le_S_S.apply le_O_n.
-rewrite < plus_n_O.rewrite < sym_times.reflexivity.
+unfold lt.apply le_S_S.apply le_O_n. demodulate. reflexivity.
+(*rewrite < plus_n_O.rewrite < sym_times.reflexivity.*)
qed.
lemma div_plus_times: \forall m,q,r:nat. r < m \to (q*m+r)/ m = q.
|apply div_mod_spec_intro[assumption|reflexivity]
]
qed.
+
(* some properties of div and mod *)
theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
intros.
intros.
apply (div_mod_spec_to_eq n n (n / n) (n \mod n) (S O) O).
apply div_mod_spec_div_mod.assumption.
-constructor 1.assumption.
-rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.
+constructor 1.assumption. demodulate. reflexivity. (*
+rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.*)
qed.
theorem eq_div_O: \forall n,m. n < m \to n / m = O.
intros.
apply (div_mod_spec_to_eq2 n n (n / n) (n \mod n) (S O) O).
apply div_mod_spec_div_mod.assumption.
-constructor 1.assumption.
-rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.
+constructor 1.assumption. demodulate. reflexivity.(*
+rewrite < plus_n_O.simplify.rewrite < plus_n_O.reflexivity.*)
qed.
theorem mod_S: \forall n,m:nat. O < m \to S (n \mod m) < m \to
]
qed.
-theorem le_div: \forall n,m. O < n \to m/n \le m.
-intros.
-rewrite > (div_mod m n) in \vdash (? ? %)
- [apply (trans_le ? (m/n*n))
- [rewrite > times_n_SO in \vdash (? % ?).
- apply le_times
- [apply le_n|assumption]
- |apply le_plus_n_r
- ]
- |assumption
- ]
-qed.
-
theorem or_div_mod: \forall n,q. O < q \to
((S (n \mod q)=q) \land S n = (S (div n q)) * q \lor
((S (n \mod q)<q) \land S n= (div n q) * q + S (n\mod q))).
variant inj_times_l1:\forall n. O < n \to \forall p,q:nat.p*n = q*n \to p=q
\def lt_O_to_injective_times_l.
+
(* n_divides computes the pair (div,mod) *)
(* p is just an upper bound, acc is an accumulator *)