X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2Flibrary_auto%2Fauto%2Fnat%2Fdiv_and_mod.ma;fp=matita%2Fcontribs%2Flibrary_auto%2Fauto%2Fnat%2Fdiv_and_mod.ma;h=bbb3d49b1c32a13721e9bc3b996d628aedd9f6a4;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1 diff --git a/matita/contribs/library_auto/auto/nat/div_and_mod.ma b/matita/contribs/library_auto/auto/nat/div_and_mod.ma new file mode 100644 index 000000000..bbb3d49b1 --- /dev/null +++ b/matita/contribs/library_auto/auto/nat/div_and_mod.ma @@ -0,0 +1,425 @@ +(**************************************************************************) +(* ___ *) +(* ||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/library_autobatch/nat/div_and_mod". + +include "datatypes/constructors.ma". +include "auto/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 m +| (S p) \Rightarrow mod_aux n n p]. + +interpretation "natural remainder" 'module x y = + (cic:/matita/library_autobatch/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/library_autobatch/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)). + autobatch + (*simplify. + apply le_O_n*) +| simplify. + apply (leb_elim n1 m);simplify;intro + [ assumption + | apply H. + cut (n1 \leq (S n) \to n1-(S m) \leq n) + [ autobatch + (*apply Hcut. + assumption*) + | elim n1;simplify;autobatch + (*[ apply le_O_n. + | 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. + autobatch + (*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);autobatch + (*simplify;apply refl_eq.*) +| apply (leb_elim n1 m);simplify;intro + [ apply refl_eq + | rewrite > assoc_plus. + elim (H (n1-(S m)) m). + change with (n1=(S m)+(n1-(S m))). + rewrite < sym_plus. + autobatch + (*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. + +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);autobatch. +(*[ 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. +autobatch. +(*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));autobatch + (*[ 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;autobatch + (*[ 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 *) + autobatch + (*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));autobatch + (*[ 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;autobatch + (*[ 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. +autobatch. +(*constructor 1 +[ unfold lt. + apply le_S_S. + apply le_O_n +| rewrite < plus_n_O. + rewrite < sym_times. + reflexivity +]*) +qed. + + +(*il corpo del seguente teorema non e' stato strutturato *) +(* 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.autobatch. +| skip +| autobatch +] +(*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);autobatch. +(*[ 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);autobatch. +(*[ 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);autobatch. +(*[ 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))) +[ autobatch + (*apply div_mod_spec_div_mod. + assumption*) +| constructor 1 + [ assumption + | rewrite < plus_n_Sm. + autobatch + (*apply eq_f. + apply div_mod. + assumption*) + ] +] +qed. + +theorem mod_O_n: \forall n:nat.O \mod n = O. +intro. +elim n;autobatch. + (*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);autobatch. +(*[ 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). +autobatch. +(*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. +autobatch. +(*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. +autobatch. +(*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. +autobatch. +(*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.