renaming in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / static / rex_rex.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/rex_rex.ma b/matita/matita/contribs/lambdadelta/basic_2/static/rex_rex.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 "basic_2/relocation/sex_sex.ma".
+include "basic_2/static/frees_fqup.ma".
+include "basic_2/static/rex.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R, T] L2 →
+                     ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R, cfull, f] L2.
+#R #L1 #L2 #T * /3 width=6 by frees_mono, sex_eq_repl_back/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: llpx_sn_dec *)
+lemma rex_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+               ∀L1,L2,T. Decidable (L1 ⪤[R, T] L2).
+#R #HR #L1 #L2 #T
+elim (frees_total L1 T) #f #Hf
+elim (sex_dec (cext2 R) cfull … L1 L2 f)
+/4 width=3 by rex_inv_frees, cfull_dec, ext2_dec, ex2_intro, or_intror, or_introl/
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
+theorem rex_bind: ∀R,p,I,L1,L2,V1,V2,T.
+                  L1 ⪤[R, V1] L2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2 →
+                  L1 ⪤[R, ⓑ{p,I}V1.T] L2.
+#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
+lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind, sex_join, isfin_tl, ex2_intro/
+qed.
+
+(* Basic_2A1: llpx_sn_flat *)
+theorem rex_flat: ∀R,I,L1,L2,V,T.
+                  L1 ⪤[R, V] L2 → L1 ⪤[R, T] L2 →
+                  L1 ⪤[R, ⓕ{I}V.T] L2.
+#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
+/3 width=7 by frees_fwd_isfin, frees_flat, sex_join, ex2_intro/
+qed.
+
+theorem rex_bind_void: ∀R,p,I,L1,L2,V,T.
+                       L1 ⪤[R, V] L2 → L1.ⓧ ⪤[R, T] L2.ⓧ →
+                       L1 ⪤[R, ⓑ{p,I}V.T] L2.
+#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
+lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind_void, sex_join, isfin_tl, ex2_intro/
+qed.
+
+(* Negated inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
+lemma rnex_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                     ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
+                     (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R, T] L2.ⓑ{I}V → ⊥).
+#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
+/4 width=2 by rex_bind, or_intror, or_introl/
+qed-.
+
+(* Basic_2A1: uses: nllpx_sn_inv_flat *)
+lemma rnex_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                     ∀I,L1,L2,V,T. (L1 ⪤[R, ⓕ{I}V.T] L2 → ⊥) →
+                     (L1 ⪤[R, V] L2 → ⊥) ∨ (L1 ⪤[R, T] L2 → ⊥).
+#R #HR #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
+/4 width=1 by rex_flat, or_intror, or_introl/
+qed-.
+
+lemma rnex_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                          ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
+                          (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R, T] L2.ⓧ → ⊥).
+#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
+/4 width=2 by rex_bind_void, or_intror, or_introl/
+qed-.