assembly.ma splitted into many files

[helm.git] / helm / software / matita / library / assembly / extra.ma

diff --git a/helm/software/matita/library/assembly/extra.ma b/helm/software/matita/library/assembly/extra.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/assembly/extra".
+
+include "nat/div_and_mod.ma".
+include "nat/primes.ma".
+include "list/list.ma".
+
+axiom mod_plus: ∀a,b,m. (a + b) \mod m = (a \mod m + b \mod m) \mod m.
+axiom mod_mod: ∀a,n,m. n∣m → a \mod n = a \mod n \mod m.
+axiom eq_mod_times_n_m_m_O: ∀n,m. O < m → n * m \mod m = O.
+axiom eq_mod_to_eq_plus_mod: ∀a,b,c,m. a \mod m = b \mod m → (a+c) \mod m = (b+c) \mod m.
+axiom eq_mod_times_times_mod: ∀a,b,n,m. m = a*n → (a*b) \mod m = a * (b \mod n).
+
+inductive cartesian_product (A,B: Type) : Type ≝
+ couple: ∀a:A.∀b:B. cartesian_product A B.
+
+lemma le_to_lt: ∀n,m. n ≤ m → n < S m.
+ intros;
+ autobatch.
+qed.
+
+alias num (instance 0) = "natural number".
+definition nat_of_bool ≝
+ λb. match b with [ true ⇒ 1 | false ⇒ 0 ].
+
+theorem lt_trans: ∀x,y,z. x < y → y < z → x < z.
+ unfold lt;
+ intros;
+ autobatch.
+qed.
+
+lemma leq_m_n_to_eq_div_n_m_S: ∀n,m:nat. 0 < m → m ≤ n → ∃z. n/m = S z.
+ intros;
+ unfold div;
+ apply (ex_intro ? ? (div_aux (pred n) (n-m) (pred m)));
+ cut (∃w.m = S w);
+  [ elim Hcut;
+    rewrite > H2;
+    rewrite > H2 in H1;
+    clear Hcut; clear H2; clear H; (*clear m;*)
+    simplify;
+    unfold in ⊢ (? ? % ?);
+    cut (∃z.n = S z);
+     [ elim Hcut; clear Hcut;
+       rewrite > H in H1;
+       rewrite > H; clear m;
+       change in ⊢ (? ? % ?)  with
+        (match leb (S a1) a with
+         [ true ⇒ O
+         | false ⇒ S (div_aux a1 ((S a1) - S a) a)]);
+       cut (S a1 ≰ a);
+        [ apply (leb_elim (S a1) a);
+           [ intro;
+             elim (Hcut H2)
+           | intro;
+             simplify;
+             reflexivity
+           ]
+        | intro;
+          autobatch
+        ]
+     | elim H1; autobatch
+     ]
+  | autobatch
+  ].
+qed.
+
+axiom daemon: False.