update in basic_2

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fsle.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fsle.ma
index cc7218982c531e63403457063615dcccef4cb256..011748dbfd44b20e4c68115c2a9645b2151ce0ca 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fsle.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_fsle.ma
@@ -13,21 +13,24 @@
 (**************************************************************************)
 
 include "basic_2/relocation/lexs_length.ma".
-include "basic_2/static/frees_drops.ma".
 include "basic_2/static/fsle_fsle.ma".
+include "basic_2/static/lfxs_drops.ma".
 include "basic_2/static/lfxs_lfxs.ma".
 
 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
 
-definition R_fsle_compatible: predicate (relation3 …) ≝ λRN.
+definition R_fsge_compatible: predicate (relation3 …) ≝ λRN.
                               ∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄.
 
-definition lfxs_fsle_compatible: predicate (relation3 …) ≝ λRN.
+definition lfxs_fsge_compatible: predicate (relation3 …) ≝ λRN.
                                  ∀L1,L2,T. L1 ⪤*[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄.
 
+definition lfxs_fsle_compatible: predicate (relation3 …) ≝ λRN.
+                                 ∀L1,L2,T. L1 ⪤*[RN, T] L2 → ⦃L1, T⦄ ⊆ ⦃L2, T⦄.
+
 (* Basic inversions with free variables inclusion for restricted closures ***)
 
-lemma frees_lexs_conf: ∀R. lfxs_fsle_compatible R →
+lemma frees_lexs_conf: ∀R. lfxs_fsge_compatible R →
                        ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
                        ∀L2. L1 ⪤*[cext2 R, cfull, f1] L2 →
                        ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
@@ -39,14 +42,14 @@ qed-.
 (* Properties with free variables inclusion for restricted closures *********)
 
 (* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
-lemma fsle_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
+lemma fsge_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
                        ∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
 #R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
 elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
 /4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/
 qed-.
 
-lemma lfxs_sym: ∀R. lfxs_fsle_compatible R →
+lemma lfxs_sym: ∀R. lfxs_fsge_compatible R →
                 (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
                 ∀T. symmetric … (lfxs R T).
 #R #H1R #H2R #T #L1 #L2
@@ -56,7 +59,7 @@ elim (frees_lexs_conf … Hf1 … HL12) -Hf1 //
 qed-.
 
 lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                          lfxs_fsle_compatible R1 →
+                          lfxs_fsge_compatible R1 →
                           ∀L1,L2,V. L1 ⪤*[R1, V] L2 → ∀I,T.
                           ∃∃L. L1 ⪤*[R1, ②{I}V.T] L & L ⪤*[R2, V] L2.
 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
@@ -75,7 +78,7 @@ elim (frees_lexs_conf … Hf … H) -Hf -H
 qed-.
 
 lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                          lfxs_fsle_compatible R1 →
+                          lfxs_fsge_compatible R1 →
                           ∀L1,L2,T. L1 ⪤*[R1, T] L2 → ∀I,V.
                           ∃∃L. L1 ⪤*[R1, ⓕ{I}V.T] L & L ⪤*[R2, T] L2.
 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
@@ -91,7 +94,7 @@ elim (frees_lexs_conf … Hf … H) -Hf -H
 qed-.
 
 lemma lfxs_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                          lfxs_fsle_compatible R1 →
+                          lfxs_fsge_compatible R1 →
                           ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤*[R1, T] L2 → ∀p.
                           ∃∃L,V. L1 ⪤*[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤*[R2, T] L2 & R1 L1 V1 V.
 #R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
@@ -111,7 +114,7 @@ elim (frees_lexs_conf … Hf … H0) -Hf -H0
 qed-.
 
 lemma lfxs_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                               lfxs_fsle_compatible R1 →
+                               lfxs_fsge_compatible R1 →
                                ∀L1,L2,T. L1.ⓧ ⪤*[R1, T] L2 → ∀p,I,V.
                                ∃∃L. L1 ⪤*[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤*[R2, T] L2.
 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
@@ -133,8 +136,8 @@ qed-.
 (* Main properties with free variables inclusion for restricted closures ****)
 
 theorem lfxs_conf: ∀R1,R2.
-                   lfxs_fsle_compatible R1 →
-                   lfxs_fsle_compatible R2 →
+                   lfxs_fsge_compatible R1 →
+                   lfxs_fsge_compatible R2 →
                    R_confluent2_lfxs R1 R2 R1 R2 →
                    ∀T. confluent2 … (lfxs R1 T) (lfxs R2 T).
 #R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
@@ -160,3 +163,14 @@ elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
   ]
 ]
 qed-.
+
+theorem lfxs_trans_fsle: ∀R1,R2,R3.
+                         lfxs_fsle_compatible R1 → lfxs_transitive_next R1 R2 R3 →
+                         ∀L1,L,T. L1 ⪤*[R1, T] L →
+                         ∀L2. L ⪤*[R2, T] L2 → L1 ⪤*[R3, T] L2.
+#R1 #R2 #R3 #H1R #H2R #L1 #L #T #H
+lapply (H1R … H) -H1R #H0
+cases H -H #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
+lapply (fsle_inv_frees_eq … H0 … Hf1 … Hf2) -H0 -Hf2
+/4 width=14 by lexs_trans_gen, lexs_fwd_length, sle_lexs_trans, ex2_intro/
+qed-.