lambdadelta

[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 d203f0da4a3990a38ec9243932984e27f628354e..537a97d1e309fc967a9dbb194de36c19d02f491c 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/equivalence/lsubss.ma".
+include "basic_2/notation/relations/crsubeqv_4.ma".
 include "basic_2/dynamic/snv.ma".
 
 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
@@ -21,10 +21,10 @@ include "basic_2/dynamic/snv.ma".
 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,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)
+               lsubsv h g (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubsv_abbr: ∀L1,L2,W,V,W1,V2,l. ⦃h, L1⦄ ⊢ ⓝW.V ¡[g] → ⦃h, L2⦄ ⊢ W ¡[g] →
+               ⦃h, L1⦄ ⊢ V •[g] ⦃l+1, W1⦄ → ⦃h, L2⦄ ⊢ W •[g] ⦃l, V2⦄ →
+               lsubsv h g L1 L2 → lsubsv h g (L1.ⓓⓝW.V) (L2.ⓛW)
 .
 
 interpretation
@@ -37,7 +37,7 @@ fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 = ⋆ → L
 #h #g #L1 #L2 * -L1 -L2
 [ //
 | #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V1 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
@@ -45,30 +45,32 @@ lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ¡⊑[g] L2 → L2 = ⋆.
 /2 width=5 by lsubsv_inv_atom1_aux/ qed-.
 
 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,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.
+                           ∀I,K1,X. L1 = K1.ⓑ{I}X →
+                           (∃∃K2. h ⊢ K1 ¡⊑[g] K2 & L2 = K2.ⓑ{I}X) ∨
+                           ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
+                                             ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
+                                             h ⊢ K1 ¡⊑[g] K2 &
+                                             I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 #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 #V2 #W1 #W2 #l #HV1 #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K1 #U1 #H destruct /3 width=11/
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
+| #L1 #L2 #W #V #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K1 #X #H destruct /3 width=12/
 ]
 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,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.
+lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,X. h ⊢ K1.ⓑ{I}X ¡⊑[g] L2 →
+                        (∃∃K2. h ⊢ K1 ¡⊑[g] K2 & L2 = K2.ⓑ{I}X) ∨
+                        ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
+                                          ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
+                                          h ⊢ K1 ¡⊑[g] K2 &
+                                          I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
 /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 #V2 #W1 #W2 #l #_ #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
@@ -76,37 +78,29 @@ lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ¡⊑[g] ⋆ → L1 = ⋆.
 /2 width=5 by lsubsv_inv_atom2_aux/ qed-.
 
 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,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.
+                           ∀I,K2,W. L2 = K2.ⓑ{I}W →
+                           (∃∃K1. h ⊢ K1 ¡⊑[g] K2 & L1 = K1.ⓑ{I}W) ∨
+                           ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
+                                           ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
+                                           h ⊢ K1 ¡⊑[g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
 #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 #V2 #W1 #W2 #l #HV #HVW1 #HW12 #HW2 #HWV2 #HL12 #J #K2 #U2 #H destruct /3 width=11/
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
+| #L1 #L2 #W #V #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K2 #U #H destruct /3 width=10/
 ]
 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,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.
+lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 ¡⊑[g] K2.ⓑ{I}W →
+                        (∃∃K1. h ⊢ K1 ¡⊑[g] K2 & L1 = K1.ⓑ{I}W) ∨
+                        ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[g] & ⦃h, K2⦄ ⊢ W ¡[g] &
+                                        ⦃h, K1⦄ ⊢ V •[g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[g] ⦃l, V2⦄ &
+                                        h ⊢ K1 ¡⊑[g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
 /2 width=3 by lsubsv_inv_pair2_aux/ qed-.
 
 (* Basic_forward lemmas *****************************************************)
 
-lemma lsubsv_fwd_lsubss: ∀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_lsubr1: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 ⊑ L2.
-/3 width=3 by lsubsv_fwd_lsubss, lsubss_fwd_lsubr1/
-qed-.
-
-lemma lsubsv_fwd_lsubr2: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 ⊑ L2.
-/3 width=3 by lsubsv_fwd_lsubss, lsubss_fwd_lsubr2/
+lemma lsubsv_fwd_lsubx: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[g] L2 → L1 ⓝ⊑ L2.
+#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
 qed-.
 
 (* Basic properties *********************************************************)
@@ -117,5 +111,5 @@ 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_lsubss, lsubss_cprs_trans/
+/3 width=5 by lsubsv_fwd_lsubx, lsubx_cprs_trans/
 qed-.