milestone update in ground_2 and basic_2A

[helm.git] / matita / matita / contribs / lambdadelta / ground_2A / ynat / ynat_le.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma b/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.ma
deleted file mode 100644 (file)
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--- a/matita/matita/contribs/lambdadelta/ground_2A/ynat/ynat_le.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_2A/ynat/ynat_succ.ma".
-
-(* NATURAL NUMBERS WITH INFINITY ********************************************)
-
-(* order relation *)
-inductive yle: relation ynat ≝
-| yle_inj: ∀m,n. m ≤ n → yle m n
-| yle_Y  : ∀m. yle m (∞)
-.
-
-interpretation "ynat 'less or equal to'" 'leq x y = (yle x y).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact yle_inv_inj2_aux: ∀x,y. x ≤ y → ∀n. y = yinj n →
-                       ∃∃m. m ≤ n & x = yinj m.
-#x #y * -x -y
-[ #x #y #Hxy #n #Hy destruct /2 width=3 by ex2_intro/
-| #x #n #Hy destruct
-]
-qed-.
-
-lemma yle_inv_inj2: ∀x,n. x ≤ yinj n → ∃∃m. m ≤ n & x = yinj m.
-/2 width=3 by yle_inv_inj2_aux/ qed-.
-
-lemma yle_inv_inj: ∀m,n. yinj m ≤ yinj n → m ≤ n.
-#m #n #H elim (yle_inv_inj2 … H) -H
-#x #Hxn #H destruct //
-qed-.
-
-fact yle_inv_O2_aux: ∀m:ynat. ∀x:ynat. m ≤ x → x = 0 → m = 0.
-#m #x * -m -x
-[ #m #n #Hmn #H destruct /3 width=1 by le_n_O_to_eq, eq_f/
-| #m #H destruct
-] 
-qed-.
-
-lemma yle_inv_O2: ∀m:ynat. m ≤ 0 → m = 0.
-/2 width =3 by yle_inv_O2_aux/ qed-.
-
-fact yle_inv_Y1_aux: ∀x,n. x ≤ n → x = ∞ → n = ∞.
-#x #n * -x -n //
-#x #n #_ #H destruct
-qed-.
-
-lemma yle_inv_Y1: ∀n. ∞ ≤ n → n = ∞.
-/2 width=3 by yle_inv_Y1_aux/ qed-.
-
-(* Inversion lemmas on successor ********************************************)
-
-fact yle_inv_succ1_aux: ∀x,y. x ≤ y → ∀m. x = ⫯m → m ≤ ⫰y ∧ ⫯⫰y = y.
-#x #y * -x -y
-[ #x #y #Hxy #m #H elim (ysucc_inv_inj_sn … H) -H
-  #n #H1 #H2 destruct elim (le_inv_S1 … Hxy) -Hxy
-  #m #Hnm #H destruct /3 width=1 by yle_inj, conj/
-| #x #y #H destruct /2 width=1 by yle_Y, conj/
-]
-qed-.
-
-lemma yle_inv_succ1: ∀m,y. ⫯m ≤ y → m ≤ ⫰y ∧ ⫯⫰y = y.
-/2 width=3 by yle_inv_succ1_aux/ qed-.
-
-lemma yle_inv_succ: ∀m,n. ⫯m ≤ ⫯n → m ≤ n.
-#m #n #H elim (yle_inv_succ1 … H) -H //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma le_O1: ∀n:ynat. 0 ≤ n.
-* /2 width=1 by yle_inj/
-qed.
-
-lemma yle_refl: reflexive … yle.
-* /2 width=1 by le_n, yle_inj/
-qed.
-
-lemma yle_split: ∀x,y:ynat. x ≤ y ∨ y ≤ x.
-* /2 width=1 by or_intror/
-#x * /2 width=1 by or_introl/
-#y elim (le_or_ge x y) /3 width=1 by yle_inj, or_introl, or_intror/
-qed-.
-
-(* Properties on predecessor ************************************************)
-
-lemma yle_pred_sn: ∀m,n. m ≤ n → ⫰m ≤ n.
-#m #n * -m -n /3 width=3 by transitive_le, yle_inj/
-qed.
-
-lemma yle_refl_pred_sn: ∀x. ⫰x ≤ x.
-/2 width=1 by yle_refl, yle_pred_sn/ qed.
-
-lemma yle_pred: ∀m,n. m ≤ n → ⫰m ≤ ⫰n.
-#m #n * -m -n /3 width=1 by yle_inj, monotonic_pred/
-qed.
-
-(* Properties on successor **************************************************)
-
-lemma yle_succ: ∀m,n. m ≤ n → ⫯m ≤ ⫯n.
-#m #n * -m -n /3 width=1 by yle_inj, le_S_S/
-qed.
-
-lemma yle_succ_dx: ∀m,n. m ≤ n → m ≤ ⫯n.
-#m #n * -m -n /3 width=1 by le_S, yle_inj/
-qed.
-
-lemma yle_refl_S_dx: ∀x. x ≤ ⫯x.
-/2 width=1 by yle_succ_dx/ qed.
-
-lemma yle_refl_SP_dx: ∀x. x ≤ ⫯⫰x.
-* // * //
-qed. 
-
-(* Main properties **********************************************************)
-
-theorem yle_trans: Transitive … yle.
-#x #y * -x -y
-[ #x #y #Hxy * //
-  #z #H lapply (yle_inv_inj … H) -H
-  /3 width=3 by transitive_le, yle_inj/ (**) (* full auto too slow *)
-| #x #z #H lapply (yle_inv_Y1 … H) //
-]
-qed-.