(**************************************************************************)
-(* ___ *)
+(* ___ *)
(* ||M|| *)
(* ||A|| A project by Andrea Asperti *)
(* ||T|| *)
set "baseuri" "cic:/matita/nat/div_and_mod".
include "nat/minus.ma".
-include "nat/orders_op.ma".
-include "nat/compare.ma".
let rec mod_aux p m n: nat \def
match (leb m n) with
| false \Rightarrow
match p with
[O \Rightarrow m
- |(S q) \Rightarrow mod_aux q (minus m (S n)) n]].
+ |(S q) \Rightarrow mod_aux q (m-(S n)) n]].
definition mod : nat \to nat \to nat \def
\lambda n,m.
[O \Rightarrow m
| (S p) \Rightarrow mod_aux n n p].
+interpretation "natural remainder" 'module x y =
+ (cic:/matita/nat/div_and_mod/mod.con x y).
+
let rec div_aux p m n : nat \def
match (leb m n) with
[ true \Rightarrow O
| false \Rightarrow
match p with
[O \Rightarrow O
- |(S q) \Rightarrow S (div_aux q (minus m (S n)) n)]].
+ |(S q) \Rightarrow S (div_aux q (m-(S n)) n)]].
definition div : nat \to nat \to nat \def
\lambda n,m.
[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).
+
theorem le_mod_aux_m_m:
-\forall p,n,m. (le n p) \to (le (mod_aux p n m) m).
+\forall p,n,m. n \leq p \to (mod_aux p n m) \leq m.
intro.elim p.
-apply le_n_O_elim n H (\lambda n.(le (mod_aux O n m) m)).
+apply (le_n_O_elim n H (\lambda n.(mod_aux O n m) \leq m)).
simplify.apply le_O_n.
simplify.
-apply leb_elim n1 m.
+apply (leb_elim n1 m).
simplify.intro.assumption.
simplify.intro.apply H.
-cut (le n1 (S n)) \to (le (minus n1 (S m)) n).
+cut (n1 \leq (S n) \to n1-(S m) \leq n).
apply Hcut.assumption.
elim n1.
simplify.apply le_O_n.
-simplify.apply trans_le ? n2 n.
+simplify.apply (trans_le ? n2 n).
apply le_minus_m.apply le_S_S_to_le.assumption.
qed.
-theorem lt_mod_m_m: \forall n,m. (lt O m) \to (lt (mod n m) m).
+theorem lt_mod_m_m: \forall n,m. O < m \to (n \mod m) < m.
intros 2.elim m.apply False_ind.
-apply not_le_Sn_O O H.
-simplify.apply le_S_S.apply le_mod_aux_m_m.
+apply (not_le_Sn_O O H).
+simplify.unfold lt.apply le_S_S.apply le_mod_aux_m_m.
apply le_n.
qed.
theorem div_aux_mod_aux: \forall p,n,m:nat.
-(n=plus (times (div_aux p n m) (S m)) (mod_aux p n m)).
+(n=(div_aux p n m)*(S m) + (mod_aux p n m)).
intro.elim p.
-simplify.elim leb n m.
+simplify.elim (leb n m).
simplify.apply refl_eq.
simplify.apply refl_eq.
simplify.
-apply leb_elim n1 m.
+apply (leb_elim n1 m).
simplify.intro.apply refl_eq.
simplify.intro.
rewrite > assoc_plus.
-elim (H (minus n1 (S m)) m).
-change with (n1=plus (S m) (minus n1 (S m))).
+elim (H (n1-(S m)) m).
+change with (n1=(S m)+(n1-(S m))).
rewrite < sym_plus.
apply plus_minus_m_m.
-change with lt m n1.
+change with (m < n1).
apply not_le_to_lt.exact H1.
qed.
-theorem div_mod: \forall n,m:nat.
-(lt O m) \to n=plus (times (div n m) m) (mod n m).
+theorem div_mod: \forall n,m:nat. O < m \to n=(n / m)*m+(n \mod m).
intros 2.elim m.elim (not_le_Sn_O O H).
simplify.
apply div_aux_mod_aux.
qed.
inductive div_mod_spec (n,m,q,r:nat) : Prop \def
-div_mod_spec_intro:
-(lt r m) \to n=plus (times q m) r \to (div_mod_spec n m q r).
+div_mod_spec_intro: r < m \to n=q*m+r \to (div_mod_spec n m q r).
(*
definition div_mod_spec : nat \to nat \to nat \to nat \to Prop \def
-\lambda n,m,q,r:nat.(And (lt r m) n=plus (times q m) r).
+\lambda n,m,q,r:nat.r < m \land n=q*m+r).
*)
-theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to Not (m=O).
-intros 4.simplify.intros.elim H.absurd le (S r) O.
+theorem div_mod_spec_to_not_eq_O: \forall n,m,q,r.(div_mod_spec n m q r) \to m \neq O.
+intros 4.unfold Not.intros.elim H.absurd (le (S r) O).
rewrite < H1.assumption.
-exact not_le_Sn_O r.
+exact (not_le_Sn_O r).
qed.
theorem div_mod_spec_div_mod:
-\forall n,m. (lt O m) \to (div_mod_spec n m (div n m) (mod n m)).
+\forall n,m. O < m \to (div_mod_spec n m (n / m) (n \mod m)).
intros.
apply div_mod_spec_intro.
apply lt_mod_m_m.assumption.
(div_mod_spec a b q r) \to (div_mod_spec a b q1 r1) \to
(eq nat q q1).
intros.elim H.elim H1.
-apply nat_compare_elim q q1.intro.
+apply (nat_compare_elim q q1).intro.
apply False_ind.
-cut eq nat (plus (times (minus q1 q) b) r1) r.
-cut le b (plus (times (minus q1 q) b) r1).
-cut le b r.
-apply lt_to_not_le r b H2 Hcut2.
+cut (eq nat ((q1-q)*b+r1) r).
+cut (b \leq (q1-q)*b+r1).
+cut (b \leq r).
+apply (lt_to_not_le r b H2 Hcut2).
elim Hcut.assumption.
-apply trans_le ? (times (minus q1 q) b) ?.
+apply (trans_le ? ((q1-q)*b)).
apply le_times_n.
apply le_SO_minus.exact H6.
rewrite < sym_plus.
apply le_plus_n.
rewrite < sym_times.
rewrite > distr_times_minus.
-(* ATTENZIONE ALL' ORDINAMENTO DEI GOALS *)
-rewrite > plus_minus ? ? ? ?.
+rewrite > plus_minus.
rewrite > sym_times.
rewrite < H5.
rewrite < sym_times.
apply plus_to_minus.
-apply eq_plus_to_le ? ? ? H3.
apply H3.
apply le_times_r.
apply lt_to_le.
(* the following case is symmetric *)
intro.
apply False_ind.
-cut eq nat (plus (times (minus q q1) b) r) r1.
-cut le b (plus (times (minus q q1) b) r).
-cut le b r1.
-apply lt_to_not_le r1 b H4 Hcut2.
+cut (eq nat ((q-q1)*b+r) r1).
+cut (b \leq (q-q1)*b+r).
+cut (b \leq r1).
+apply (lt_to_not_le r1 b H4 Hcut2).
elim Hcut.assumption.
-apply trans_le ? (times (minus q q1) b) ?.
+apply (trans_le ? ((q-q1)*b)).
apply le_times_n.
apply le_SO_minus.exact H6.
rewrite < sym_plus.
apply le_plus_n.
rewrite < sym_times.
rewrite > distr_times_minus.
-rewrite > plus_minus ? ? ? ?.
+rewrite > plus_minus.
rewrite > sym_times.
rewrite < H3.
rewrite < sym_times.
apply plus_to_minus.
-apply eq_plus_to_le ? ? ? H5.
apply H5.
apply le_times_r.
apply lt_to_le.
apply H6.
qed.
+
+theorem div_mod_spec_to_eq2 :\forall a,b,q,r,q1,r1.
+(div_mod_spec a b q r) \to (div_mod_spec a b q1 r1) \to
+(eq nat r r1).
+intros.elim H.elim H1.
+apply (inj_plus_r (q*b)).
+rewrite < H3.
+rewrite > (div_mod_spec_to_eq a b q r q1 r1 H H1).
+assumption.
+qed.
+
+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.
+qed.
+
+(* some properties of div and mod *)
+theorem div_times: \forall n,m:nat. ((S n)*m) / (S n) = m.
+intros.
+apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O).
+goal 15. (* ?11 is closed with the following tactics *)
+apply div_mod_spec_div_mod.
+unfold lt.apply le_S_S.apply le_O_n.
+apply div_mod_spec_times.
+qed.
+
+theorem div_n_n: \forall n:nat. O < n \to n / n = S O.
+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.
+qed.
+
+theorem eq_div_O: \forall n,m. n < m \to n / m = O.
+intros.
+apply (div_mod_spec_to_eq n m (n/m) (n \mod m) O n).
+apply div_mod_spec_div_mod.
+apply (le_to_lt_to_lt O n m).
+apply le_O_n.assumption.
+constructor 1.assumption.reflexivity.
+qed.
+
+theorem mod_n_n: \forall n:nat. O < n \to n \mod n = 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.
+qed.
+
+theorem mod_S: \forall n,m:nat. O < m \to S (n \mod m) < m \to
+((S n) \mod m) = S (n \mod m).
+intros.
+apply (div_mod_spec_to_eq2 (S n) m ((S n) / m) ((S n) \mod m) (n / m) (S (n \mod m))).
+apply div_mod_spec_div_mod.assumption.
+constructor 1.assumption.rewrite < plus_n_Sm.
+apply eq_f.
+apply div_mod.
+assumption.
+qed.
+
+theorem mod_O_n: \forall n:nat.O \mod n = O.
+intro.elim n.simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+theorem lt_to_eq_mod:\forall n,m:nat. n < m \to n \mod m = n.
+intros.
+apply (div_mod_spec_to_eq2 n m (n/m) (n \mod m) O n).
+apply div_mod_spec_div_mod.
+apply (le_to_lt_to_lt O n m).apply le_O_n.assumption.
+constructor 1.
+assumption.reflexivity.
+qed.
+
+(* injectivity *)
+theorem injective_times_r: \forall n:nat.injective nat nat (\lambda m:nat.(S n)*m).
+change with (\forall n,p,q:nat.(S n)*p = (S n)*q \to p=q).
+intros.
+rewrite < (div_times n).
+rewrite < (div_times n q).
+apply eq_f2.assumption.
+reflexivity.
+qed.
+
+variant inj_times_r : \forall n,p,q:nat.(S n)*p = (S n)*q \to p=q \def
+injective_times_r.
+
+theorem lt_O_to_injective_times_r: \forall n:nat. O < n \to injective nat nat (\lambda m:nat.n*m).
+change with (\forall n. O < n \to \forall p,q:nat.n*p = n*q \to p=q).
+intros 4.
+apply (lt_O_n_elim n H).intros.
+apply (inj_times_r m).assumption.
+qed.
+
+variant inj_times_r1:\forall n. O < n \to \forall p,q:nat.n*p = n*q \to p=q
+\def lt_O_to_injective_times_r.
+
+theorem injective_times_l: \forall n:nat.injective nat nat (\lambda m:nat.m*(S n)).
+change with (\forall n,p,q:nat.p*(S n) = q*(S n) \to p=q).
+intros.
+apply (inj_times_r n p q).
+rewrite < sym_times.
+rewrite < (sym_times q).
+assumption.
+qed.
+
+variant inj_times_l : \forall n,p,q:nat. p*(S n) = q*(S n) \to p=q \def
+injective_times_l.
+
+theorem lt_O_to_injective_times_l: \forall n:nat. O < n \to injective nat nat (\lambda m:nat.m*n).
+change with (\forall n. O < n \to \forall p,q:nat.p*n = q*n \to p=q).
+intros 4.
+apply (lt_O_n_elim n H).intros.
+apply (inj_times_l m).assumption.
+qed.
+
+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.
projects / helm.git / blobdiff