- lambdadelta: third recursive part of preservation finally proved!

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / dynamic / lsubsv.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
index 9979efb4ec9b9dd1a575e6eff34b28837468e1b0..fcd0deb2986ef173d87c578c549ce1286a122488 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/equivalence/lsubse.ma".
 include "basic_2/dynamic/snv.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
@@ -21,8 +22,8 @@ inductive lsubsv (h:sh) (g:sd h): relation lenv ≝
 | lsubsv_atom: lsubsv h g (⋆) (⋆)
 | lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 →
                lsubsv h g (L1. ⓑ{I} V) (L2. ⓑ{I} V)
-| lsubsv_abbr: ∀L1,L2,V1,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → ⦃h, L1⦄ ⊢ V1 •[g, l+1] W1 →
-               L1 ⊢ W2 ⬌* W1 → ⦃h, L2⦄ ⊩  W2 :[g] →
+| lsubsv_abbr: ∀L1,L2,V1,V2,W1,W2,l. ⦃h, L1⦄ ⊩ V1 :[g] → ⦃h, L1⦄ ⊢ V1 •[g, l+1] W1 →
+               L1 ⊢ W1 ⬌* W2 → ⦃h, L2⦄ ⊩ W2 :[g] → ⦃h, L2⦄ ⊢ W2 •[g, l] V2 →
                lsubsv h g L1 L2 → lsubsv h g (L1. ⓓV1) (L2. ⓛW2)
 .
 
@@ -36,7 +37,7 @@ fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 = ⋆ →
 #h #g #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V1 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
@@ -46,28 +47,28 @@ lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ⊩:⊑[g] L2 → L2 = ⋆.
 fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
                            ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
                            (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
-                           ∃∃K2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
-                                         K1 ⊢ W2 ⬌* W1 & ⦃h, K2⦄ ⊩  W2 :[g] &
-                                         h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+                           ∃∃K2,V2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
+                                            K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+                                            h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
 #h #g #L1 #L2 * -L1 -L2
 [ #J #K1 #U1 #H destruct
 | #I #L1 #L2 #V #HL12 #J #K1 #U1 #H destruct /3 width=3/
-| #L1 #L2 #V1 #W1 #W2 #l #HV1 #HVW1 #HW21 #HW2 #HL12 #J #K1 #U1 #H destruct /3 width=9/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV1 #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=11/
 ]
 qed-.
 
 lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,V1. h ⊢ K1. ⓑ{I} V1 ⊩:⊑[g] L2 →
                         (∃∃K2. h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓑ{I} V1) ∨
-                        ∃∃K2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
-                                      K1 ⊢ W2 ⬌* W1 & ⦃h, K2⦄ ⊩ W2 :[g] &
-                                      h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
+                        ∃∃K2,V2,W1,W2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
+                                         K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+                                         h ⊢ K1 ⊩:⊑[g] K2 & L2 = K2. ⓛW2 & I = Abbr.
 /2 width=3 by lsubsv_inv_pair1_aux/ qed-.
 
 fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
 #h #g #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V1 #W1 #W2 #l #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
@@ -77,31 +78,35 @@ lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ⊩:⊑[g] ⋆ → L1 = ⋆.
 fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
                            ∀I,K2,W2. L2 = K2. ⓑ{I} W2 →
                            (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
-                           ∃∃K1,W1,V1,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
-                                         K1 ⊢ W2 ⬌* W1 & ⦃h, K2⦄ ⊩ W2 :[g] &
-                                         h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+                           ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
+                                            K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+                                            h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
 #h #g #L1 #L2 * -L1 -L2
 [ #J #K2 #U2 #H destruct
 | #I #L1 #L2 #V #HL12 #J #K2 #U2 #H destruct /3 width=3/
-| #L1 #L2 #V1 #W1 #W2 #l #HV #HVW1 #HW21 #HW2 #HL12 #J #K2 #U2 #H destruct /3 width=9/
+| #L1 #L2 #V1 #V2 #W1 #W2 #l #HV #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/
 ]
 qed-.
 
 lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W2. h ⊢ L1 ⊩:⊑[g] K2. ⓑ{I} W2 →
                         (∃∃K1. h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓑ{I} W2) ∨
-                        ∃∃K1,W1,V1,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
-                                      K1 ⊢ W2 ⬌* W1 & ⦃h, K2⦄ ⊩ W2 :[g] &
-                                      h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
+                        ∃∃K1,W1,V1,V2,l. ⦃h, K1⦄ ⊩ V1 :[g] & ⦃h, K1⦄ ⊢ V1 •[g,l+1] W1 &
+                                         K1 ⊢ W1 ⬌* W2 & ⦃h, K2⦄ ⊩ W2 :[g] & ⦃h, K2⦄ ⊢ W2 •[g,l] V2 &
+                                         h ⊢ K1 ⊩:⊑[g] K2 & L1 = K1. ⓓV1 & I = Abst.
 /2 width=3 by lsubsv_inv_pair2_aux/ qed-.
 
 (* Basic_forward lemmas *****************************************************)
 
+lemma lsubsv_fwd_lsubse: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → h ⊢ L1 ⊢•⊑[g] L2.
+#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/ /2 width=6/
+qed-.
+
 lemma lsubsv_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L1|] L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+/3 width=3 by lsubsv_fwd_lsubse, lsubse_fwd_lsubs1/
 qed-.
 
 lemma lsubsv_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 → L1 ≼[0, |L2|] L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+/3 width=3 by lsubsv_fwd_lsubse, lsubse_fwd_lsubs2/
 qed-.
 
 (* Basic properties *********************************************************)
@@ -112,5 +117,5 @@ qed.
 
 lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ⊩:⊑[g] L2 →
                          ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
-/3 width=5 by lsubsv_fwd_lsubs2, cprs_lsubs_trans/
+/3 width=5 by lsubsv_fwd_lsubse, lsubse_cprs_trans/
 qed-.