notational change for snv and lsubsv: inverted "!" used for now

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
index e39a5764af20f9c251efe461baa87bb2d06cbec1..69906c437d728e0c44632196478bc3a26fae3571 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
@@ -22,10 +22,10 @@ inductive snv (h:sh) (g:sd h): lenv → predicate term ≝
 | snv_lref: ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → snv h g K V → snv h g L (#i)
 | snv_bind: ∀a,I,L,V,T. snv h g L V → snv h g (L.ⓑ{I}V) T → snv h g L (ⓑ{a,I}V.T)
 | snv_appl: ∀a,L,V,W,W0,T,U,l. snv h g L V → snv h g L T →
-            ⦃h, L⦄ ⊢ V •[g, l + 1] W → L ⊢ W ➡* W0 →
+            ⦃h, L⦄ ⊢ V •[g] ⦃l+1, W⦄ → L ⊢ W ➡* W0 →
             ⦃h, L⦄ ⊢ T •*➡*[g] ⓛ{a}W0.U → snv h g L (ⓐV.T)
 | snv_cast: ∀L,W,T,U,l. snv h g L W → snv h g L T →
-            ⦃h, L⦄ ⊢ T •[g, l + 1] U → L ⊢ U ⬌* W → snv h g L (ⓝW.T)
+            ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → L ⊢ U ⬌* W → snv h g L (ⓝW.T)
 .
 
 interpretation "stratified native validity (term)"
@@ -33,8 +33,8 @@ interpretation "stratified native validity (term)"
 
 (* Basic inversion lemmas ***************************************************)
 
-fact snv_inv_lref_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a© X :[g] → ∀i. X = #i →
-                       â\88\83â\88\83I,K,V. â\87©[0, i] L â\89¡ K.â\93\91{I}V & â¦\83h, Kâ¦\84 â\8a© V :[g].
+fact snv_inv_lref_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a¢ X ¡[g] → ∀i. X = #i →
+                       â\88\83â\88\83I,K,V. â\87©[0, i] L â\89¡ K.â\93\91{I}V & â¦\83h, Kâ¦\84 â\8a¢ V ¡[g].
 #h #g #L #X * -L -X
 [ #L #k #i #H destruct
 | #I #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5/
@@ -44,11 +44,11 @@ fact snv_inv_lref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀i. X = #i →
 ]
 qed.
 
-lemma snv_inv_lref: â\88\80h,g,L,i. â¦\83h, Lâ¦\84 â\8a© #i :[g] →
-                    â\88\83â\88\83I,K,V. â\87©[0, i] L â\89¡ K.â\93\91{I}V & â¦\83h, Kâ¦\84 â\8a© V :[g].
+lemma snv_inv_lref: â\88\80h,g,L,i. â¦\83h, Lâ¦\84 â\8a¢ #i ¡[g] →
+                    â\88\83â\88\83I,K,V. â\87©[0, i] L â\89¡ K.â\93\91{I}V & â¦\83h, Kâ¦\84 â\8a¢ V ¡[g].
 /2 width=3/ qed-.
 
-fact snv_inv_gref_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a© X :[g] → ∀p. X = §p → ⊥.
+fact snv_inv_gref_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a¢ X ¡[g] → ∀p. X = §p → ⊥.
 #h #g #L #X * -L -X
 [ #L #k #p #H destruct
 | #I #L #K #V #i #_ #_ #p #H destruct
@@ -58,11 +58,11 @@ fact snv_inv_gref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀p. X = §p → 
 ]
 qed.
 
-lemma snv_inv_gref: â\88\80h,g,L,p. â¦\83h, Lâ¦\84 â\8a© §p :[g] → ⊥.
+lemma snv_inv_gref: â\88\80h,g,L,p. â¦\83h, Lâ¦\84 â\8a¢ §p ¡[g] → ⊥.
 /2 width=7/ qed-.
 
-fact snv_inv_bind_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a© X :[g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
-                       â¦\83h, Lâ¦\84 â\8a© V :[g] â\88§ â¦\83h, L.â\93\91{I}Vâ¦\84 â\8a© T :[g].
+fact snv_inv_bind_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a¢ X ¡[g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
+                       â¦\83h, Lâ¦\84 â\8a¢ V ¡[g] â\88§ â¦\83h, L.â\93\91{I}Vâ¦\84 â\8a¢ T ¡[g].
 #h #g #L #X * -L -X
 [ #L #k #a #I #V #T #H destruct
 | #I0 #L #K #V0 #i #_ #_ #a #I #V #T #H destruct
@@ -72,13 +72,13 @@ fact snv_inv_bind_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀a,I,V,T. X = ⓑ
 ]
 qed.
 
-lemma snv_inv_bind: â\88\80h,g,a,I,L,V,T. â¦\83h, Lâ¦\84 â\8a© â\93\91{a,I}V.T :[g] →
-                        â¦\83h, Lâ¦\84 â\8a© V :[g] â\88§ â¦\83h, L.â\93\91{I}Vâ¦\84 â\8a© T :[g].
+lemma snv_inv_bind: â\88\80h,g,a,I,L,V,T. â¦\83h, Lâ¦\84 â\8a¢ â\93\91{a,I}V.T ¡[g] →
+                        â¦\83h, Lâ¦\84 â\8a¢ V ¡[g] â\88§ â¦\83h, L.â\93\91{I}Vâ¦\84 â\8a¢ T ¡[g].
 /2 width=4/ qed-.
 
-fact snv_inv_appl_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a© X :[g] → ∀V,T. X = ⓐV.T →
-                       â\88\83â\88\83a,W,W0,U,l. â¦\83h, Lâ¦\84 â\8a© V :[g] & â¦\83h, Lâ¦\84 â\8a© T :[g] &
-                                   ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 &
+fact snv_inv_appl_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a¢ X ¡[g] → ∀V,T. X = ⓐV.T →
+                       â\88\83â\88\83a,W,W0,U,l. â¦\83h, Lâ¦\84 â\8a¢ V ¡[g] & â¦\83h, Lâ¦\84 â\8a¢ T ¡[g] &
+                                   ⦃h, L⦄ ⊢ V •[g] ⦃l+1, W⦄ & L ⊢ W ➡* W0 &
                                    ⦃h, L⦄ ⊢ T •*➡*[g] ⓛ{a}W0.U.
 #h #g #L #X * -L -X
 [ #L #k #V #T #H destruct
@@ -89,15 +89,15 @@ fact snv_inv_appl_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀V,T. X = ⓐV.T
 ]
 qed.
 
-lemma snv_inv_appl: â\88\80h,g,L,V,T. â¦\83h, Lâ¦\84 â\8a© â\93\90V.T :[g] →
-                    â\88\83â\88\83a,W,W0,U,l. â¦\83h, Lâ¦\84 â\8a© V :[g] & â¦\83h, Lâ¦\84 â\8a© T :[g] &
-                                ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 &
+lemma snv_inv_appl: â\88\80h,g,L,V,T. â¦\83h, Lâ¦\84 â\8a¢ â\93\90V.T ¡[g] →
+                    â\88\83â\88\83a,W,W0,U,l. â¦\83h, Lâ¦\84 â\8a¢ V ¡[g] & â¦\83h, Lâ¦\84 â\8a¢ T ¡[g] &
+                                ⦃h, L⦄ ⊢ V •[g] ⦃l+1, W⦄ & L ⊢ W ➡* W0 &
                                 ⦃h, L⦄ ⊢ T •*➡*[g] ⓛ{a}W0.U.
 /2 width=3/ qed-.
 
-fact snv_inv_cast_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a© X :[g] → ∀W,T. X = ⓝW.T →
-                       â\88\83â\88\83U,l. â¦\83h, Lâ¦\84 â\8a© W :[g] & â¦\83h, Lâ¦\84 â\8a© T :[g] & 
-                              ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W.
+fact snv_inv_cast_aux: â\88\80h,g,L,X. â¦\83h, Lâ¦\84 â\8a¢ X ¡[g] → ∀W,T. X = ⓝW.T →
+                       â\88\83â\88\83U,l. â¦\83h, Lâ¦\84 â\8a¢ W ¡[g] & â¦\83h, Lâ¦\84 â\8a¢ T ¡[g] &
+                              ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ & L ⊢ U ⬌* W.
 #h #g #L #X * -L -X
 [ #L #k #W #T #H destruct
 | #I #L #K #V #i #_ #_ #W #T #H destruct
@@ -107,7 +107,22 @@ fact snv_inv_cast_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀W,T. X = ⓝW.T
 ]
 qed.
 
-lemma snv_inv_cast: â\88\80h,g,L,W,T. â¦\83h, Lâ¦\84 â\8a© â\93\9dW.T :[g] →
-                    â\88\83â\88\83U,l. â¦\83h, Lâ¦\84 â\8a© W :[g] & â¦\83h, Lâ¦\84 â\8a© T :[g] &
-                           ⦃h, L⦄ ⊢ T •[g, l + 1] U & L ⊢ U ⬌* W.
+lemma snv_inv_cast: â\88\80h,g,L,W,T. â¦\83h, Lâ¦\84 â\8a¢ â\93\9dW.T ¡[g] →
+                    â\88\83â\88\83U,l. â¦\83h, Lâ¦\84 â\8a¢ W ¡[g] & â¦\83h, Lâ¦\84 â\8a¢ T ¡[g] &
+                           ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ & L ⊢ U ⬌* W.
 /2 width=3/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma snv_fwd_ssta: ∀h,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] → ∃∃l,U. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄.
+#h #g #L #T #H elim H -L -T
+[ #L #k elim (deg_total h g k) /3 width=3/
+| * #L #K #V #i #HLK #_ * #l0 #W #HVW
+  [ elim (lift_total W 0 (i+1)) /3 width=8/
+  | elim (lift_total V 0 (i+1)) /3 width=8/
+  ]
+| #a #I #L #V #T #_ #_ #_ * /3 width=3/
+| #a #L #V #W #W1 #T0 #T1 #l #_ #_ #_ #_ #_ #_ * /3 width=3/
+| #L #W #T #U #l #_ #_ #HTU #_ #_ #_ /3 width=3/ (**) (* auto fails without the last #_ *)
+]
+qed-.