renaming

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
index 04b4044be31d18c4d09b5e431f6d0eabab1c2aca..89b84566be4e1e6aadced8a7c63dd302c36dbf69 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
@@ -13,23 +13,77 @@
 (**************************************************************************)
 
 include "basic_2/relocation/lexs_lexs.ma".
+include "basic_2/static/frees_fqup.ma".
 include "basic_2/static/lfxs.ma".
 
 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
 
-(* Main properties ******************************************************)
+(* Advanced inversion lemmas ************************************************)
 
-theorem lfxs_bind: ∀R,I,L1,L2,V1,V2,T,p.
-                   L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 →
-                   L1 ⦻*[R, ⓑ{p,I}V1.T] L2.
-#R #I #L1 #L2 #V1 #V2 #T #p * #f1 #HV #Hf1 * #f2 #HT #Hf2
-elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2))
+lemma lfxs_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, lexs_eq_repl_back/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: llpx_sn_dec *)
+lemma lfxs_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 (lexs_dec (cext2 R) cfull … L1 L2 f)
+/4 width=3 by lfxs_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 lfxs_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 (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
 /3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
 qed.
 
+(* Basic_2A1: llpx_sn_flat *)
 theorem lfxs_flat: ∀R,I,L1,L2,V,T.
-                   L1 ⦻*[R, V] L2 â\86\92 L1 ⦻*[R, T] L2 →
-                   L1 ⦻*[R, ⓕ{I}V.T] L2.
+                   L1 ⪤*[R, V] L2 â\86\92 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, lexs_join, ex2_intro/
 qed.
+
+theorem lfxs_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 (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind_void, lexs_join, isfin_tl, ex2_intro/
+qed.
+
+(* Negated inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
+lemma lfnxs_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 (lfxs_dec … HR L1 L2 V)
+/4 width=2 by lfxs_bind, or_intror, or_introl/
+qed-.
+
+(* Basic_2A1: uses: nllpx_sn_inv_flat *)
+lemma lfnxs_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 (lfxs_dec … HR L1 L2 V)
+/4 width=1 by lfxs_flat, or_intror, or_introl/
+qed-.
+
+lemma lfnxs_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 (lfxs_dec … HR L1 L2 V)
+/4 width=2 by lfxs_bind_void, or_intror, or_introl/
+qed-.