X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fdiv_and_mod.ma;h=e9831f82ad1ec5cc01decf9e920f9e80518c3f64;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=4c43d33bd9d1d57ba4b283016c850f0cb0bf7a29;hpb=633474751ddf1074947ff0d324fb1aca2293eff8;p=helm.git diff --git a/helm/matita/library/nat/div_and_mod.ma b/helm/matita/library/nat/div_and_mod.ma index 4c43d33bd..e9831f82a 100644 --- a/helm/matita/library/nat/div_and_mod.ma +++ b/helm/matita/library/nat/div_and_mod.ma @@ -1,5 +1,5 @@ (**************************************************************************) -(* ___ *) +(* ___ *) (* ||M|| *) (* ||A|| A project by Andrea Asperti *) (* ||T|| *) @@ -15,8 +15,6 @@ 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 @@ -24,7 +22,7 @@ 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. @@ -32,13 +30,16 @@ match m with [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. @@ -46,73 +47,74 @@ 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. (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. @@ -123,27 +125,25 @@ 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 (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. @@ -153,26 +153,146 @@ intros.assumption. (* 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.