From 4421f312e9614e608d0ddf6fa029c761f87ceb45 Mon Sep 17 00:00:00 2001 From: Andrea Asperti Date: Tue, 13 Mar 2007 10:53:33 +0000 Subject: [PATCH] new version of div_and_mod --- .../matita/library/nat/div_and_mod_new.ma | 360 ++++++++++++++++++ 1 file changed, 360 insertions(+) create mode 100644 helm/software/matita/library/nat/div_and_mod_new.ma diff --git a/helm/software/matita/library/nat/div_and_mod_new.ma b/helm/software/matita/library/nat/div_and_mod_new.ma new file mode 100644 index 000000000..325244f7a --- /dev/null +++ b/helm/software/matita/library/nat/div_and_mod_new.ma @@ -0,0 +1,360 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +set "baseuri" "cic:/matita/nat/div_and_mod". + +include "datatypes/constructors.ma". +include "nat/minus.ma". + +let rec mod_aux t m n: nat \def +match (leb (S m) n) with +[ true \Rightarrow m +| false \Rightarrow + match t with + [O \Rightarrow m (* if t is large enough this case never happens *) + |(S t1) \Rightarrow mod_aux t1 (m-n) n + ] +]. + +definition mod: nat \to nat \to nat \def +\lambda m,n.mod_aux m m n. + +interpretation "natural remainder" 'module x y = + (cic:/matita/nat/div_and_mod/mod.con x y). + +lemma O_to_mod_aux: \forall m,n. mod_aux O m n = m. +intros. +simplify.elim (leb (S m) n);reflexivity. +qed. + +lemma lt_to_mod_aux: \forall t,m,n. m < n \to mod_aux (S t) m n = m. +intros. +change with +( match (leb (S m) n) with + [ true \Rightarrow m | false \Rightarrow mod_aux t (m-n) n] = m). +rewrite > (le_to_leb_true ? ? H). +reflexivity. +qed. + +lemma le_to_mod_aux: \forall t,m,n. n \le m \to +mod_aux (S t) m n = mod_aux t (m-n) n. +intros. +change with +(match (leb (S m) n) with +[ true \Rightarrow m | false \Rightarrow mod_aux t (m-n) n] = mod_aux t (m-n) n). +apply (leb_elim (S m) n);intro + [apply False_ind.apply (le_to_not_lt ? ? H).apply H1 + |reflexivity + ] +qed. + +let rec div_aux p m n : nat \def +match (leb (S m) n) with +[ true \Rightarrow O +| false \Rightarrow + match p with + [O \Rightarrow O + |(S q) \Rightarrow S (div_aux q (m-n) n)]]. + +definition div : nat \to nat \to nat \def +\lambda n,m.div_aux n n m. + +interpretation "natural divide" 'divide x y = + (cic:/matita/nat/div_and_mod/div.con x y). + +theorem lt_mod_aux_m_m: +\forall n. O < n \to \forall t,m. m \leq t \to (mod_aux t m n) < n. +intros 3. +elim t + [rewrite > O_to_mod_aux. + apply (le_n_O_elim ? H1). + assumption + |apply (leb_elim (S m) n);intros + [rewrite > lt_to_mod_aux[assumption|assumption] + |rewrite > le_to_mod_aux + [apply H1. + apply le_plus_to_minus. + apply (trans_le ? ? ? H2). + apply (lt_O_n_elim ? H).intro. + rewrite < plus_n_Sm. + apply le_S_S. + apply le_plus_n_r + |apply not_lt_to_le. + assumption + ] + ] + ] +qed. + +theorem lt_mod_m_m: \forall n,m. O < m \to (n \mod m) < m. +intros.unfold mod. +apply lt_mod_aux_m_m[assumption|apply le_n] +qed. + +lemma mod_aux_O: \forall p,n:nat. mod_aux p n O = n. +intros. +elim p + [reflexivity + |simplify.rewrite < minus_n_O.assumption + ] +qed. + +theorem div_aux_mod_aux: \forall m,p,n:nat. +(n=(div_aux p n m)*m + (mod_aux p n m)). +intro. +apply (nat_case m) + [intros.rewrite < times_n_O.simplify.apply sym_eq.apply mod_aux_O + |intros 2.elim p + [simplify.elim (leb n m1);reflexivity + |simplify.apply (leb_elim n1 m1);intro + [reflexivity + |simplify. + rewrite > assoc_plus. + rewrite < (H (n1-(S m1))). + change with (n1=(S m1)+(n1-(S m1))). + rewrite < sym_plus. + apply plus_minus_m_m. + change with (m1 < 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.apply (div_aux_mod_aux m n n). +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.auto. +(* +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 q = q1. +intros.elim H.elim H1. +apply (nat_compare_elim q q1);intro + [apply False_ind. + cut ((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 + ] + |auto depth=4. 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 + [auto. + (* + rewrite > sym_times. + rewrite < H5. + rewrite < sym_times. + apply plus_to_minus. + apply H3 + *) + |auto. + (* + apply le_times_r. + apply lt_to_le. + apply H6 + *) + ] + ] +(* eq case *) + |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. + +(* 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). +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. -- 2.39.2