+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/congruence".
-
-include "nat/relevant_equations.ma".
-include "nat/primes.ma".
-
-definition S_mod: nat \to nat \to nat \def
-\lambda n,m:nat. (S m) \mod n.
-
-definition congruent: nat \to nat \to nat \to Prop \def
-\lambda n,m,p:nat. mod n p = mod m p.
-
-theorem congruent_n_n: \forall n,p:nat.congruent n n p.
-intros.unfold congruent.reflexivity.
-qed.
-
-theorem transitive_congruent: \forall p:nat. transitive nat
-(\lambda n,m. congruent n m p).
-intros.unfold transitive.unfold congruent.intros.
-whd.apply (trans_eq ? ? (y \mod p)).
-apply H.apply H1.
-qed.
-
-theorem le_to_mod: \forall n,m:nat. n \lt m \to n = n \mod m.
-intros.
-apply (div_mod_spec_to_eq2 n m O n (n/m) (n \mod m)).
-constructor 1.assumption.simplify.reflexivity.
-apply div_mod_spec_div_mod.
-apply (le_to_lt_to_lt O n m).apply le_O_n.assumption.
-qed.
-
-theorem mod_mod : \forall n,p:nat. O<p \to n \mod p = (n \mod p) \mod p.
-intros.
-rewrite > (div_mod (n \mod p) p) in \vdash (? ? % ?).
-rewrite > (eq_div_O ? p).reflexivity.
-(* uffa: hint non lo trova lt vs. le*)
-apply lt_mod_m_m.
-assumption.
-assumption.
-qed.
-
-theorem mod_times_mod : \forall n,m,p:nat. O<p \to O<m \to n \mod p = (n \mod (m*p)) \mod p.
-intros.
-apply (div_mod_spec_to_eq2 n p (n/p) (n \mod p)
-(n/(m*p)*m + (n \mod (m*p)/p))).
-apply div_mod_spec_div_mod.assumption.
-constructor 1.
-apply lt_mod_m_m.assumption.
-rewrite > times_plus_l.
-rewrite > assoc_plus.
-rewrite < div_mod.
-rewrite > assoc_times.
-rewrite < div_mod.
-reflexivity.
-rewrite > (times_n_O O).
-apply lt_times.
-assumption.assumption.assumption.
-qed.
-
-theorem congruent_n_mod_n :
-\forall n,p:nat. O < p \to congruent n (n \mod p) p.
-intros.unfold congruent.
-apply mod_mod.assumption.
-qed.
-
-theorem congruent_n_mod_times :
-\forall n,m,p:nat. O < p \to O < m \to congruent n (n \mod (m*p)) p.
-intros.unfold congruent.
-apply mod_times_mod.assumption.assumption.
-qed.
-
-theorem eq_times_plus_to_congruent: \forall n,m,p,r:nat. O< p \to
-n = r*p+m \to congruent n m p.
-intros.unfold congruent.
-apply (div_mod_spec_to_eq2 n p (div n p) (mod n p) (r +(div m p)) (mod m p)).
-apply div_mod_spec_div_mod.assumption.
-constructor 1.
-apply lt_mod_m_m.assumption.
-rewrite > sym_times.
-rewrite > distr_times_plus.
-rewrite > sym_times.
-rewrite > (sym_times p).
-rewrite > assoc_plus.
-rewrite < div_mod.
-assumption.assumption.
-qed.
-
-theorem divides_to_congruent: \forall n,m,p:nat. O < p \to m \le n \to
-divides p (n - m) \to congruent n m p.
-intros.elim H2.
-apply (eq_times_plus_to_congruent n m p n2).
-assumption.
-rewrite < sym_plus.
-apply minus_to_plus.assumption.
-rewrite > sym_times. assumption.
-qed.
-
-theorem congruent_to_divides: \forall n,m,p:nat.
-O < p \to congruent n m p \to divides p (n - m).
-intros.unfold congruent in H1.
-apply (witness ? ? ((n / p)-(m / p))).
-rewrite > sym_times.
-rewrite > (div_mod n p) in \vdash (? ? % ?).
-rewrite > (div_mod m p) in \vdash (? ? % ?).
-rewrite < (sym_plus (m \mod p)).
-rewrite < H1.
-rewrite < (eq_minus_minus_minus_plus ? (n \mod p)).
-rewrite < minus_plus_m_m.
-apply sym_eq.
-apply times_minus_l.
-assumption.assumption.
-qed.
-
-theorem mod_times: \forall n,m,p:nat.
-O < p \to mod (n*m) p = mod ((mod n p)*(mod m p)) p.
-intros.
-change with (congruent (n*m) ((mod n p)*(mod m p)) p).
-apply (eq_times_plus_to_congruent ? ? p
-((n / p)*p*(m / p) + (n / p)*(m \mod p) + (n \mod p)*(m / p))).
-assumption.
-apply (trans_eq ? ? (((n/p)*p+(n \mod p))*((m/p)*p+(m \mod p)))).
-apply eq_f2.
-apply div_mod.assumption.
-apply div_mod.assumption.
-apply (trans_eq ? ? (((n/p)*p)*((m/p)*p) + (n/p)*p*(m \mod p) +
-(n \mod p)*((m / p)*p) + (n \mod p)*(m \mod p))).
-apply times_plus_plus.
-apply eq_f2.
-rewrite < assoc_times.
-rewrite > (assoc_times (n/p) p (m \mod p)).
-rewrite > (sym_times p (m \mod p)).
-rewrite < (assoc_times (n/p) (m \mod p) p).
-rewrite < times_plus_l.
-rewrite < (assoc_times (n \mod p)).
-rewrite < times_plus_l.
-apply eq_f2.
-apply eq_f2.reflexivity.
-reflexivity.reflexivity.
-reflexivity.
-qed.
-
-theorem congruent_times: \forall n,m,n1,m1,p. O < p \to congruent n n1 p \to
-congruent m m1 p \to congruent (n*m) (n1*m1) p.
-unfold congruent.
-intros.
-rewrite > (mod_times n m p H).
-rewrite > H1.
-rewrite > H2.
-apply sym_eq.
-apply mod_times.assumption.
-qed.
-
-theorem congruent_pi: \forall f:nat \to nat. \forall n,m,p:nat.O < p \to
-congruent (pi n f m) (pi n (\lambda m. mod (f m) p) m) p.
-intros.
-elim n.change with (congruent (f m) (f m \mod p) p).
-apply congruent_n_mod_n.assumption.
-change with (congruent ((f (S n1+m))*(pi n1 f m))
-(((f (S n1+m))\mod p)*(pi n1 (\lambda m.(f m) \mod p) m)) p).
-apply congruent_times.
-assumption.
-apply congruent_n_mod_n.assumption.
-assumption.
-qed.
projects / helm.git / blobdiff