syntactic components detached from basic_2 become static_2

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

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/req.ma b/matita/matita/contribs/lambdadelta/static_2/static/req.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/ideqsn_3.ma".
+include "static_2/static/rex.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
+
+(* Basic_2A1: was: lleq *)
+definition req: relation3 term lenv lenv ≝
+                rex ceq.
+
+interpretation
+   "syntactic equivalence on referred entries (local environment)"
+   'IdEqSn T L1 L2 = (req T L1 L2).
+
+(* Note: "req_transitive R" is equivalent to "rex_transitive ceq R R" *)
+(* Basic_2A1: uses: lleq_transitive *)
+definition req_transitive: predicate (relation3 lenv term term) ≝
+           λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma req_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 →
+                    ∧∧ L1 ≡[V] L2 & L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
+/2 width=2 by rex_inv_bind/ qed-.
+
+lemma req_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 →
+                    ∧∧ L1 ≡[V] L2 & L1 ≡[T] L2.
+/2 width=2 by rex_inv_flat/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma req_inv_zero_pair_sn: ∀I,L2,K1,V. K1.ⓑ{I}V ≡[#0] L2 →
+                            ∃∃K2. K1 ≡[V] K2 & L2 = K2.ⓑ{I}V.
+#I #L2 #K1 #V #H
+elim (rex_inv_zero_pair_sn … H) -H #K2 #X #HK12 #HX #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma req_inv_zero_pair_dx: ∀I,L1,K2,V. L1 ≡[#0] K2.ⓑ{I}V →
+                            ∃∃K1. K1 ≡[V] K2 & L1 = K1.ⓑ{I}V.
+#I #L1 #K2 #V #H
+elim (rex_inv_zero_pair_dx … H) -H #K1 #X #HK12 #HX #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma req_inv_lref_bind_sn: ∀I1,K1,L2,i. K1.ⓘ{I1} ≡[#↑i] L2 →
+                            ∃∃I2,K2. K1 ≡[#i] K2 & L2 = K2.ⓘ{I2}.
+/2 width=2 by rex_inv_lref_bind_sn/ qed-.
+
+lemma req_inv_lref_bind_dx: ∀I2,K2,L1,i. L1 ≡[#↑i] K2.ⓘ{I2} →
+                            ∃∃I1,K1. K1 ≡[#i] K2 & L1 = K1.ⓘ{I1}.
+/2 width=2 by rex_inv_lref_bind_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_2A1: was: llpx_sn_lrefl *)
+(* Basic_2A1: this should have been lleq_fwd_llpx_sn *)
+lemma req_fwd_rex: ∀R. c_reflexive … R →
+                   ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R, T] L2.
+#R #HR #L1 #L2 #T * #f #Hf #HL12
+/4 width=7 by sex_co, cext2_co, ex2_intro/
+qed-.
+
+(* Basic_properties *********************************************************)
+
+lemma frees_req_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
+                      ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
+#f #L1 #T #H elim H -f -L1 -T
+[ /2 width=3 by frees_sort/
+| #f #i #Hf #L2 #H2
+  >(rex_inv_atom_sn … H2) -L2
+  /2 width=1 by frees_atom/
+| #f #I #L1 #V1 #_ #IH #Y #H2
+  elim (req_inv_zero_pair_sn … H2) -H2 #L2 #HL12 #H destruct
+  /3 width=1 by frees_pair/
+| #f #I #L1 #Hf #Y #H2
+  elim (rex_inv_zero_unit_sn … H2) -H2 #g #L2 #_ #_ #H destruct
+  /2 width=1 by frees_unit/
+| #f #I #L1 #i #_ #IH #Y #H2
+  elim (req_inv_lref_bind_sn … H2) -H2 #J #L2 #HL12 #H destruct
+  /3 width=1 by frees_lref/
+| /2 width=1 by frees_gref/
+| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
+  elim (req_inv_bind … H2) -H2 /3 width=5 by frees_bind/
+| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
+  elim (req_inv_flat … H2) -H2 /3 width=5 by frees_flat/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 10:
+              lleq_ind lleq_fwd_lref
+              lleq_fwd_drop_sn lleq_fwd_drop_dx
+              lleq_skip lleq_lref lleq_free
+              lleq_Y lleq_ge_up lleq_ge
+               
+*)