ground_2 released and permanently renamed as ground

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / ynat / ynat_succ.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma b/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
deleted file mode 100644 (file)
index d0c7b3f..0000000
--- a/matita/matita/contribs/lambdadelta/ground_2/ynat/ynat_succ.ma
+++ /dev/null
@@ -1,97 +0,0 @@
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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_2/ynat/ynat_pred.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* the successor function *)
-definition ysucc: ynat → ynat ≝ λm. match m with
-[ yinj m ⇒ ↑m
-| Y      ⇒ Y
-].
-
-interpretation "ynat successor" 'UpArrow m = (ysucc m).
-
-lemma ysucc_inj: ∀m:nat. ↑(yinj m) = yinj (↑m).
-// qed.
-
-lemma ysucc_Y: ↑(∞) = ∞.
-// qed.
-
-(* Properties ***************************************************************)
-
-lemma ypred_succ: ∀m. ↓↑m = m.
-* // qed.
-
-lemma ynat_cases: ∀n:ynat. n = 0 ∨ ∃m:ynat. n = ↑m.
-*
-[ * /2 width=1 by or_introl/
-  #n @or_intror @(ex_intro … n) // (**) (* explicit constructor *)
-| @or_intror @(ex_intro … (∞)) // (**) (* explicit constructor *)
-]
-qed-.
-
-lemma ysucc_iter_Y: ∀m. ysucc^m (∞) = ∞.
-#m elim m -m //
-#m #IHm whd in ⊢ (??%?); >IHm //
-qed.
-
-(* Inversion lemmas *********************************************************)
-
-lemma ysucc_inv_inj: ∀m,n. ↑m = ↑n → m = n.
-#m #n #H <(ypred_succ m) <(ypred_succ n) //
-qed-.
-
-lemma ysucc_inv_refl: ∀m. ↑m = m → m = ∞.
-* //
-#m #H lapply (yinj_inj … H) -H (**) (* destruct lemma needed *)
-#H elim (lt_refl_false m) //
-qed-.
-
-lemma ysucc_inv_inj_sn: ∀m2,n1. yinj m2 = ↑n1 →
-                        ∃∃m1. n1 = yinj m1 & m2 = S m1.
-#m2 * normalize
-[ #n1 #H destruct /2 width=3 by ex2_intro/
-| #H destruct
-]
-qed-.
-
-lemma ysucc_inv_inj_dx: ∀m2,n1. ↑n1 = yinj m2  →
-                        ∃∃m1. n1 = yinj m1 & m2 = S m1.
-/2 width=1 by ysucc_inv_inj_sn/ qed-.
-
-lemma ysucc_inv_Y_sn: ∀m. ∞ = ↑m → m = ∞.
-* // normalize
-#m #H destruct
-qed-.
-
-lemma ysucc_inv_Y_dx: ∀m. ↑m = ∞ → m = ∞.
-/2 width=1 by ysucc_inv_Y_sn/ qed-.
-
-lemma ysucc_inv_O_sn: ∀m. yinj 0 = ↑m → ⊥. (**) (* explicit coercion *)
-#m #H elim (ysucc_inv_inj_sn … H) -H
-#n #_ #H destruct
-qed-.
-
-lemma ysucc_inv_O_dx: ∀m:ynat. ↑m = 0 → ⊥.
-/2 width=2 by ysucc_inv_O_sn/ qed-.
-
-(* Eliminators **************************************************************)
-
-lemma ynat_ind: ∀R:predicate ynat.
-                R 0 → (∀n:nat. R n → R (↑n)) → R (∞) →
-                ∀x. R x.
-#R #H1 #H2 #H3 * // #n elim n -n /2 width=1 by/
-qed-.