renaming

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma
index 99b1ece949863003c3d475e1bdfb47e2f7127c28..76adfa38b376b35601af4a65037f1bf2ca3620bc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_drops.ma
@@ -19,25 +19,30 @@ include "basic_2/static/lfxs.ma".
 
 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
 
-definition dedropable_sn: predicate (relation3 lenv term term) ≝
-                          λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 →
-                          ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≡ U →
-                          ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
+definition f_dedropable_sn: predicate (relation3 lenv term term) ≝
+                            λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
+                            ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
+                            ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
 
-definition dropable_sn: predicate (relation3 lenv term term) ≝
-                        λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ →
-                        ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≡ U →
-                        ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≡ K2.
+definition f_dropable_sn: predicate (relation3 lenv term term) ≝
+                          λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+                          ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
+                          ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
 
-definition dropable_dx: predicate (relation3 lenv term term) ≝
-                        λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
-                        ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U →
-                        ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⪤*[R, T] K2.
+definition f_dropable_dx: predicate (relation3 lenv term term) ≝
+                          λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
+                          ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+                          ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
+
+definition f_transitive_next: relation3 … ≝ λR1,R2,R3.
+                              ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
+                              ∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
+                              lexs_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
 
 (* Properties with generic slicing for local environments *******************)
 
 lemma lfxs_liftable_dedropable_sn: ∀R. (∀L. reflexive ? (R L)) →
-                                   d_liftable2_sn … lifts R → dedropable_sn R.
+                                   d_liftable2_sn … lifts R → f_dedropable_sn R.
 #R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
 elim (frees_total L1 U) #f2 #Hf2
 lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
@@ -45,11 +50,24 @@ elim (lexs_liftable_co_dedropable_sn … HLK1 … HK12 … Hf) -f1 -K1
 /3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, ex3_intro, ex2_intro/
 qed-.
 
+lemma lfxs_trans_next: ∀R1,R2,R3. lfxs_transitive R1 R2 R3 → f_transitive_next R1 R2 R3.
+#R1 #R2 #R3 #HR #f #L1 #T #Hf #g #I1 #K1 #n #HLK #Hgf #I #H
+generalize in match HLK; -HLK elim H -I1 -I
+[ #I #_ #L2 #_ #I2 #H
+  lapply (ext2_inv_unit_sn … H) -H #H destruct
+  /2 width=1 by ext2_unit/
+| #I #V1 #V #HV1 #HLK1 #L2 #HL12 #I2 #H
+  elim (ext2_inv_pair_sn … H) -H #V2 #HV2 #H destruct
+  elim (frees_inv_drops_next … Hf … HLK1 … Hgf) -f -HLK1 #f #Hf #Hfg
+  /5 width=5 by ext2_pair, sle_lexs_trans, ex2_intro/
+]
+qed.
+
 (* Inversion lemmas with generic slicing for local environments *************)
 
 (* Basic_2A1: uses: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
 (* Basic_2A1: was: llpx_sn_drop_conf_O *)
-lemma lfxs_dropable_sn: ∀R. dropable_sn R.
+lemma lfxs_dropable_sn: ∀R. f_dropable_sn R.
 #R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
 elim (frees_total K1 T) #f1 #Hf1
 lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
@@ -59,7 +77,7 @@ qed-.
 
 (* Basic_2A1: was: llpx_sn_drop_trans_O *)
 (* Note: the proof might be simplified *)
-lemma lfxs_dropable_dx: ∀R. dropable_dx R.
+lemma lfxs_dropable_dx: ∀R. f_dropable_dx R.
 #R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
 elim (drops_isuni_ex … H1f L1) #K1 #HLK1
 elim (frees_total K1 T) #f1 #Hf1
@@ -70,36 +88,36 @@ qed-.
 
 (* Basic_2A1: uses: llpx_sn_inv_lift_O *)
 lemma lfxs_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤*[R, U] L2 → ∀b,f. 𝐔⦃f⦄ → 
-                         â\88\80K1,K2. â¬\87*[b, f] L1 â\89¡ K1 â\86\92 â¬\87*[b, f] L2 â\89¡ K2 →
-                         â\88\80T. â¬\86*[f] T â\89¡ U → K1 ⪤*[R, T] K2.
+                         â\88\80K1,K2. â¬\87*[b, f] L1 â\89\98 K1 â\86\92 â¬\87*[b, f] L2 â\89\98 K2 →
+                         â\88\80T. â¬\86*[f] T â\89\98 U → K1 ⪤*[R, T] K2.
 #R #L1 #L2 #U #HL12 #b #f #Hf #K1 #K2 #HLK1 #HLK2 #T #HTU
 elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
 lapply (drops_mono … HY … HLK2) -b -f -L2 #H destruct //
 qed-.
 
-lemma lfxs_inv_lref_pair_sn: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K1,V1. â¬\87*[i] L1 â\89¡ K1.ⓑ{I}V1 →
-                             â\88\83â\88\83K2,V2. â¬\87*[i] L2 â\89¡ K2.ⓑ{I}V2 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
+lemma lfxs_inv_lref_pair_sn: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K1,V1. â¬\87*[i] L1 â\89\98 K1.ⓑ{I}V1 →
+                             â\88\83â\88\83K2,V2. â¬\87*[i] L2 â\89\98 K2.ⓑ{I}V2 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
 #R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
 #Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY
 #K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma lfxs_inv_lref_pair_dx: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K2,V2. â¬\87*[i] L2 â\89¡ K2.ⓑ{I}V2 →
-                             â\88\83â\88\83K1,V1. â¬\87*[i] L1 â\89¡ K1.ⓑ{I}V1 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
+lemma lfxs_inv_lref_pair_dx: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K2,V2. â¬\87*[i] L2 â\89\98 K2.ⓑ{I}V2 →
+                             â\88\83â\88\83K1,V1. â¬\87*[i] L1 â\89\98 K1.ⓑ{I}V1 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
 #R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
 #Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY
 #K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma lfxs_inv_lref_unit_sn: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K1. â¬\87*[i] L1 â\89¡ K1.ⓤ{I} →
-                             â\88\83â\88\83f,K2. â¬\87*[i] L2 â\89¡ K2.ⓤ{I} & K1 ⪤*[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+lemma lfxs_inv_lref_unit_sn: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K1. â¬\87*[i] L1 â\89\98 K1.ⓤ{I} →
+                             â\88\83â\88\83f,K2. â¬\87*[i] L2 â\89\98 K2.ⓤ{I} & K1 ⪤*[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
 #R #L1 #L2 #i #HL12 #I #K1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
 #Y #HY #HLK2 elim (lfxs_inv_zero_unit_sn … HY) -HY
 #f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
 qed-.
 
-lemma lfxs_inv_lref_unit_dx: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K2. â¬\87*[i] L2 â\89¡ K2.ⓤ{I} →
-                             â\88\83â\88\83f,K1. â¬\87*[i] L1 â\89¡ K1.ⓤ{I} & K1 ⪤*[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+lemma lfxs_inv_lref_unit_dx: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 â\86\92 â\88\80I,K2. â¬\87*[i] L2 â\89\98 K2.ⓤ{I} →
+                             â\88\83â\88\83f,K1. â¬\87*[i] L1 â\89\98 K1.ⓤ{I} & K1 ⪤*[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
 #R #L1 #L2 #i #HL12 #I #K2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
 #Y #HLK1 #HY elim (lfxs_inv_zero_unit_dx … HY) -HY
 #f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/