additions and corrections for the article on λδ-2B

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / static / aaa_dec.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/aaa_dec.ma b/matita/matita/contribs/lambdadelta/static_2/static/aaa_dec.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 "static_2/static/aaa_drops.ma".
+include "static_2/static/aaa_aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Main properties **********************************************************)
+
+theorem aaa_dec (G) (L) (T): Decidable (∃A. ⦃G,L⦄ ⊢ T ⁝ A).
+#G #L #T @(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T
+#G0 #L0 #T0 #IH #G #L * * [||| #p * | * ]
+[ #s #HG #HL #HT destruct -IH
+  /3 width=2 by aaa_sort, ex_intro, or_introl/
+| #i #HG #HL #HT destruct
+  elim (drops_F_uni L i) [| * * #I [| #V ] #K ] #HLK
+  [1,2: -IH
+    @or_intror * #A #H
+    elim (aaa_inv_lref_drops … H) -H #J #Y #X #HLY #_ -G -A
+    lapply (drops_mono … HLY … HLK) -L -i #H destruct
+  | elim (IH G K V) -IH [3: /2 width=2 by fqup_lref/ ]
+    [ * /4 width=6 by aaa_lref_drops, ex_intro, or_introl/
+    | #H0 @or_intror * #A #H
+      lapply (aaa_pair_inv_lref … H … HLK) -I -L -i
+      /3 width=2 by ex_intro/
+    ]
+  ]
+| #l #HG #HL #HT destruct -IH
+  @or_intror * #A #H
+  @(aaa_inv_gref … H)
+| #V #T #HG #HL #HT destruct
+  elim (IH G L V) [ * #B #HB | #HnB | // ]
+  [ elim (IH G (L.ⓓV) T) [ * #A #HA | #HnA | // ] ] -IH
+  [ /4 width=2 by aaa_abbr, ex_intro, or_introl/ ]
+  @or_intror * #A #H
+  elim (aaa_inv_abbr … H) -H #B0 #HB0 #HA0
+  /3 width=2 by ex_intro/
+| #W #T #HG #HL #HT destruct
+  elim (IH G L W) [ * #B #HB | #HnB | // ]
+  [ elim (IH G (L.ⓛW) T) [ * #A #HA | #HnA | // ] ] -IH
+  [ /4 width=2 by aaa_abst, ex_intro, or_introl/ ]
+  @or_intror * #A #H
+  elim (aaa_inv_abst … H) -H #B0 #A0 #HB0 #HA0 #H destruct
+  /3 width=2 by ex_intro/
+| #V #T #HG #HL #HT destruct
+  elim (IH G L V) [ * #B #HB | #HnB | // ]
+  [ elim (IH G L T) [ * #X #HX | #HnX | // ] ] -IH
+  [ elim (is_apear_dec B X) [ * #A #H destruct | #HnX ]
+    [ /4 width=4 by aaa_appl, ex_intro, or_introl/ ]
+  ]
+  @or_intror * #A #H
+  [ lapply (aaa_aaa_inv_appl … H HB HX) -G -L -V -T
+  |*: elim (aaa_inv_appl … H) -H #B0 #HB0 #HA0
+  ]
+  /3 width=2 by ex_intro/
+| #U #T #HG #HL #HT destruct
+  elim (IH G L U) [ * #B #HB | #HnB | // ]
+  [ elim (IH G L T) [ * #A #HA | #HnA | // ] ] -IH
+  [ elim (eq_aarity_dec B A) [ #H destruct | #HnA ]
+    [ /4 width=3 by aaa_cast, ex_intro, or_introl/ ]
+  ]
+  @or_intror * #X #H
+  [ elim (aaa_aaa_inv_cast … H HB HA) -G -L -U -T
+  |*: elim (aaa_inv_cast … H) -H #HU #HT
+  ]
+  /3 width=2 by ex_intro/
+]
+qed-.