arithmetics for λδ

[helm.git] / matita / matita / contribs / lambdadelta / ground / arith / nat_succ.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.ma b/matita/matita/contribs/lambdadelta/ground/arith/nat_succ.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                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground/arith/nat.ma".
+
+(* NON-NEGATIVE INTEGERS ****************************************************)
+
+definition nsucc: nat → nat ≝ λm. match m with
+[ nzero  ⇒ ninj (𝟏)
+| ninj p ⇒ ninj (↑p)
+].
+
+interpretation
+  "successor (non-negative integers"
+  'UpArrow m = (nsucc m).
+
+(* Basic rewrites ***********************************************************)
+
+lemma nsucc_zero: ninj (𝟏) = ↑𝟎.
+// qed.
+
+lemma nsucc_inj (p): ninj (↑p) = ↑(ninj p).
+// qed.
+
+(* Basic eliminations *******************************************************)
+
+(*** nat_ind *)
+lemma nat_ind (Q:predicate …):
+      Q (𝟎) → (∀n. Q n → Q (↑n)) → ∀n. Q n.
+#Q #IH1 #IH2 * //
+#p elim p -p /2 width=1 by/
+qed-.
+
+(*** nat_elim2 *)
+lemma nat_ind_2 (Q:relation2 …):
+      (∀n. Q (𝟎) n) →
+      (∀m. Q (↑m) (𝟎)) →
+      (∀m,n. Q m n → Q (↑m) (↑n)) →
+      ∀m,n. Q m n.
+#Q #IH1 #IH2 #IH3 #m elim m -m [ // ]
+#m #IH #n elim n -n /2 width=1 by/
+qed-.
+
+(* Basic inversions *********************************************************)
+
+(*** injective_S *)
+lemma eq_inv_nsucc_bi: injective … nsucc.
+* [| #p1 ] * [2,4: #p2 ]
+[1,4: <nsucc_zero <nsucc_inj #H destruct
+| <nsucc_inj <nsucc_inj #H destruct //
+| //
+]
+qed-.
+
+lemma eq_inv_nsucc_zero (m): ↑m = 𝟎 → ⊥.
+* [ <nsucc_zero | #p <nsucc_inj ] #H destruct
+qed-.
+
+lemma eq_inv_nzero_succ (m): 𝟎 = ↑m → ⊥.
+* [ <nsucc_zero | #p <nsucc_inj ] #H destruct
+qed-.