update in static_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / toeq.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/toeq.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/toeq.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/notation/relations/topiso_2.ma".
+include "static_2/syntax/term.ma".
+
+(* SORT-IRRELEVANT OUTER EQUIVALENCE FOR TERMS ******************************)
+
+(* Basic_2A1: includes: tsts_atom tsts_pair *)
+inductive toeq: relation term ≝
+| toeq_sort: ∀s1,s2. toeq (⋆s1) (⋆s2)
+| toeq_lref: ∀i. toeq (#i) (#i)
+| toeq_gref: ∀l. toeq (§l) (§l)
+| toeq_pair: ∀I,V1,V2,T1,T2. toeq (②{I}V1.T1) (②{I}V2.T2)
+.
+
+interpretation
+  "sort-irrelevant outer equivalence (term)"
+  'TopIso T1 T2 = (toeq T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact toeq_inv_sort1_aux: ∀X,Y. X ⩳ Y → ∀s1. X = ⋆s1 →
+                         ∃s2. Y = ⋆s2.
+#X #Y * -X -Y
+[ #s1 #s2 #s #H destruct /2 width=2 by ex_intro/
+| #i #s #H destruct
+| #l #s #H destruct
+| #I #V1 #V2 #T1 #T2 #s #H destruct
+]
+qed-.
+
+(* Basic_1: was just: iso_gen_sort *)
+lemma toeq_inv_sort1: ∀Y,s1. ⋆s1 ⩳ Y →
+                      ∃s2. Y = ⋆s2.
+/2 width=4 by toeq_inv_sort1_aux/ qed-.
+
+fact toeq_inv_lref1_aux: ∀X,Y. X ⩳ Y → ∀i. X = #i → Y = #i.
+#X #Y * -X -Y //
+[ #s1 #s2 #j #H destruct
+| #I #V1 #V2 #T1 #T2 #j #H destruct
+]
+qed-.
+
+(* Basic_1: was: iso_gen_lref *)
+lemma toeq_inv_lref1: ∀Y,i. #i ⩳ Y → Y = #i.
+/2 width=5 by toeq_inv_lref1_aux/ qed-.
+
+fact toeq_inv_gref1_aux: ∀X,Y. X ⩳ Y → ∀l. X = §l → Y = §l.
+#X #Y * -X -Y //
+[ #s1 #s2 #k #H destruct
+| #I #V1 #V2 #T1 #T2 #k #H destruct
+]
+qed-.
+
+lemma toeq_inv_gref1: ∀Y,l. §l ⩳ Y → Y = §l.
+/2 width=5 by toeq_inv_gref1_aux/ qed-.
+
+fact toeq_inv_pair1_aux: ∀T1,T2. T1 ⩳ T2 →
+                         ∀J,W1,U1. T1 = ②{J}W1.U1 →
+                         ∃∃W2,U2. T2 = ②{J}W2.U2.
+#T1 #T2 * -T1 -T2
+[ #s1 #s2 #J #W1 #U1 #H destruct
+| #i #J #W1 #U1 #H destruct
+| #l #J #W1 #U1 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_1: was: iso_gen_head *)
+(* Basic_2A1: was: tsts_inv_pair1 *)
+lemma toeq_inv_pair1: ∀J,W1,U1,T2. ②{J}W1.U1 ⩳ T2 →
+                      ∃∃W2,U2. T2 = ②{J}W2. U2.
+/2 width=7 by toeq_inv_pair1_aux/ qed-.
+
+fact toeq_inv_pair2_aux: ∀T1,T2. T1 ⩳ T2 →
+                         ∀J,W2,U2. T2 = ②{J}W2.U2 →
+                         ∃∃W1,U1. T1 = ②{J}W1.U1.
+#T1 #T2 * -T1 -T2
+[ #s1 #s2 #J #W2 #U2 #H destruct
+| #i #J #W2 #U2 #H destruct
+| #l #J #W2 #U2 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was: tsts_inv_pair2 *)
+lemma toeq_inv_pair2: ∀J,T1,W2,U2. T1 ⩳ ②{J}W2.U2 →
+                      ∃∃W1,U1. T1 = ②{J}W1.U1.
+/2 width=7 by toeq_inv_pair2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma toeq_inv_pair: ∀I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ⩳ ②{I2}V2.T2 →
+                     I1 = I2.
+#I1 #I2 #V1 #V2 #T1 #T2 #H elim (toeq_inv_pair1 … H) -H
+#V0 #T0 #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: iso_refl *)
+(* Basic_2A1: was: tsts_refl *)
+lemma toeq_refl: reflexive … toeq.
+* //
+* /2 width=1 by toeq_lref, toeq_gref/
+qed.
+
+(* Basic_2A1: was: tsts_sym *)
+lemma toeq_sym: symmetric … toeq.
+#T1 #T2 * -T1 -T2 /2 width=3 by toeq_sort/
+qed-.
+
+(* Basic_2A1: was: tsts_dec *)
+lemma toeq_dec: ∀T1,T2. Decidable (T1 ⩳ T2).
+* [ * #s1 | #I1 #V1 #T1 ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ /3 width=1 by toeq_sort, or_introl/
+|2,3,13:
+  @or_intror #H
+  elim (toeq_inv_sort1 … H) -H #x #H destruct
+|4,6,14:
+  @or_intror #H
+  lapply (toeq_inv_lref1 … H) -H #H destruct
+|5:
+  elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+  @or_intror #H
+  lapply (toeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+  @or_intror #H
+  lapply (toeq_inv_gref1 … H) -H #H destruct
+|9:
+  elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+  @or_intror #H
+  lapply (toeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+  @or_intror #H
+  elim (toeq_inv_pair1 … H) -H #X1 #X2 #H destruct
+|16:
+  elim (eq_item2_dec I1 I2) #HI12 destruct
+  [ /3 width=1 by toeq_pair, or_introl/ ]
+  @or_intror #H
+  lapply (toeq_inv_pair … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 2:
+              tsts_inv_atom1 tsts_inv_atom2
+*)