X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;fp=matita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;h=538515a8cc806003287340ef9d1299b4625a17a0;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/library/nat/div_and_mod.ma b/matita/library/nat/div_and_mod.ma new file mode 100644 index 000000000..538515a8c --- /dev/null +++ b/matita/library/nat/div_and_mod.ma @@ -0,0 +1,397 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| A.Asperti, C.Sacerdoti Coen, *) +(* ||A|| E.Tassi, S.Zacchiroli *) +(* \ / *) +(* \ / Matita is distributed under the terms of the *) +(* v GNU Lesser General Public License Version 2.1 *) +(* *) +(**************************************************************************) + +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 +| false \Rightarrow + match p with + [O \Rightarrow m + |(S q) \Rightarrow mod_aux q (m-(S n)) n]]. + +definition mod : nat \to nat \to nat \def +\lambda n,m. +match m with +[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). + +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 (m-(S n)) n)]]. + +definition div : nat \to nat \to nat \def +\lambda n,m. +match m 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). + +theorem le_mod_aux_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.(mod_aux O n m) \leq m)). +simplify.apply le_O_n. +simplify. +apply (leb_elim n1 m). +simplify.intro.assumption. +simplify.intro.apply H. +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). +apply le_minus_m.apply le_S_S_to_le.assumption. +qed. + +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.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=(div_aux p n m)*(S m) + (mod_aux p n m)). +intro.elim p. +simplify.elim (leb n m). +simplify.apply refl_eq. +simplify.apply refl_eq. +simplify. +apply (leb_elim n1 m). +simplify.intro.apply refl_eq. +simplify.intro. +rewrite > assoc_plus. +elim (H (n1-(S m)) m). +change with (n1=(S m)+(n1-(S m))). +rewrite < sym_plus. +apply plus_minus_m_m. +change with (m < n1). +apply not_le_to_lt.exact H1. +qed. + +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. + +theorem eq_times_div_minus_mod: +\forall a,b:nat. O \lt b \to +(a /b)*b = a - (a \mod b). +intros. +rewrite > (div_mod a b) in \vdash (? ? ? (? % ?)) +[ apply (minus_plus_m_m (times (div a b) b) (mod a b)) +| assumption +] +qed. + +inductive div_mod_spec (n,m,q,r:nat) : Prop \def +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.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 m \neq O. +intros 4.unfold Not.intros.elim H.absurd (le (S r) O). +rewrite < H1.assumption. +exact (not_le_Sn_O r). +qed. + +theorem div_mod_spec_div_mod: +\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. +apply div_mod.assumption. +qed. + +theorem div_mod_spec_to_eq :\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 q q1). +intros.elim H.elim H1. +apply (nat_compare_elim q q1).intro. +apply False_ind. +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 ? ((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. +rewrite > plus_minus. +rewrite > sym_times. +rewrite < H5. +rewrite < sym_times. +apply plus_to_minus. +apply H3. +apply le_times_r. +apply lt_to_le. +apply H6. +(* eq case *) +intros.assumption. +(* the following case is symmetric *) +intro. +apply False_ind. +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 ? ((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 > sym_times. +rewrite < H3. +rewrite < sym_times. +apply plus_to_minus. +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. + +lemma div_plus_times: \forall m,q,r:nat. r < m \to (q*m+r)/ m = q. +intros. +apply (div_mod_spec_to_eq (q*m+r) m ? ((q*m+r) \mod m) ? r) + [apply div_mod_spec_div_mod. + apply (le_to_lt_to_lt ? r) + [apply le_O_n|assumption] + |apply div_mod_spec_intro[assumption|reflexivity] + ] +qed. + +lemma mod_plus_times: \forall m,q,r:nat. r < m \to (q*m+r) \mod m = r. +intros. +apply (div_mod_spec_to_eq2 (q*m+r) m ((q*m+r)/ m) ((q*m+r) \mod m) q r) + [apply div_mod_spec_div_mod. + apply (le_to_lt_to_lt ? r) + [apply le_O_n|assumption] + |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. +apply (div_mod_spec_to_eq ((S n)*m) (S n) ? ? ? O); +[2: apply div_mod_spec_div_mod. + unfold lt.apply le_S_S.apply le_O_n. +| skip +| apply div_mod_spec_times +] +qed. + +(*a simple variant of div_times theorem *) +theorem lt_O_to_div_times: \forall a,b:nat. O \lt b \to +a*b/b = a. +intros. +rewrite > sym_times. +rewrite > (S_pred b H). +apply div_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. + +theorem mod_SO: \forall n:nat. mod n (S O) = O. +intro. +apply sym_eq. +apply le_n_O_to_eq. +apply le_S_S_to_le. +apply lt_mod_m_m. +apply le_n. +qed. + +theorem div_SO: \forall n:nat. div n (S O) = n. +intro. +rewrite > (div_mod ? (S O)) in \vdash (? ? ? %) + [rewrite > mod_SO. + rewrite < plus_n_O. + apply times_n_SO + |apply le_n + ] +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) sym_plus. + rewrite < H1 in ⊢ (? ? ? (? ? %)). + rewrite < plus_n_Sm. + apply eq_f. + apply div_mod. + assumption + ] + ] +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). +simplify. +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)). +simplify. +intros. +apply (inj_times_r n x y). +rewrite < sym_times. +rewrite < (sym_times y). +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). +simplify. +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. + + +(* n_divides computes the pair (div,mod) *) + +(* p is just an upper bound, acc is an accumulator *) +let rec n_divides_aux p n m acc \def + match n \mod m with + [ O \Rightarrow + match p with + [ O \Rightarrow pair nat nat acc n + | (S p) \Rightarrow n_divides_aux p (n / m) m (S acc)] + | (S a) \Rightarrow pair nat nat acc n]. + +(* n_divides n m = if m divides n q times, with remainder r *) +definition n_divides \def \lambda n,m:nat.n_divides_aux n n m O. +