- we set up the support for the "bt-reduction" of Automath literature

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

diff --git a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv.ma b/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv.ma
index 23a93550cc90cc02168f9e063d97fcd4cf968775..2a234c2267ce4ff63fcead7177798e8e9966cd17 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/dynamic/snv.ma
@@ -21,12 +21,12 @@ include "basic_2/equivalence/cpcs.ma".
 inductive snv (h:sh) (g:sd h): lenv → predicate term ≝
 | snv_sort: ∀L,k. snv h g L (⋆k)
 | 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: ∀I,L,V,T. snv h g L V → snv h g (L.ⓑ{I}V) T → snv h g L (ⓑ{I}V.T)
-| snv_appl: ∀L,V,W,W0,T,U,l. snv h g L V → snv h g L T →
+| 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 →
-            â¦\83h, Lâ¦\84 â\8a¢ T â\9e¸*[g] â\93\9bW0.U → snv h g L (ⓐV.T)
+            â¦\83h, Lâ¦\84 â\8a¢ T â\80¢â\9e¡*[g] â\93\9b{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 ⊢ W ⬌* U → snv h g L (ⓝW.T)
 .
 
 interpretation "stratified native validity (term)"
@@ -34,36 +34,68 @@ interpretation "stratified native validity (term)"
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma snv_inv_bind_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀I,V,T. X = ⓑ{I}V.T →
-                        ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ T :[g].
+fact snv_inv_lref_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀i. X = #i →
+                       ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊩ V :[g].
 #h #g #L #X * -L -X
-[ #L #k #I #V #T #H destruct
-| #I0 #L #K #V0 #i #_ #_ #I #V #T #H destruct
-| #I0 #L #V0 #T0 #HV0 #HT0 #I #V #T #H destruct /2 width=1/
-| #L #V0 #W0 #W00 #T0 #U0 #l #_ #_ #_ #_ #_ #I #V #T #H destruct
-| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #I #V #T #H destruct
+[ #L #k #i #H destruct
+| #I #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5/
+| #a #I #L #V #T #_ #_ #i #H destruct
+| #a #L #V #W #W0 #T #U #l #_ #_ #_ #_ #_ #i #H destruct
+| #L #W #T #U #l #_ #_ #_ #_ #i #H destruct
 ]
 qed.
 
-lemma snv_inv_bind: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊩ ⓑ{I}V.T :[g] →
-                        ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ T :[g].
+lemma snv_inv_lref: ∀h,g,L,i. ⦃h, L⦄ ⊩ #i :[g] →
+                    ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊩ V :[g].
 /2 width=3/ qed-.
 
-lemma snv_inv_appl_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀V,T. X = ⓐV.T →
-                        ∃∃W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ T :[g] &
-                                    ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 &
-                                    ⦃h, L⦄ ⊢ T ➸*[g] ⓛW0.U.
+fact snv_inv_bind_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
+                       ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ 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
+| #b #I0 #L #V0 #T0 #HV0 #HT0 #a #I #V #T #H destruct /2 width=1/
+| #b #L #V0 #W0 #W00 #T0 #U0 #l #_ #_ #_ #_ #_ #a #I #V #T #H destruct
+| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #a #I #V #T #H destruct
+]
+qed.
+
+lemma snv_inv_bind: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊩ ⓑ{a,I}V.T :[g] →
+                        ⦃h, L⦄ ⊩ V :[g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊩ T :[g].
+/2 width=4/ qed-.
+
+fact snv_inv_appl_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀V,T. X = ⓐV.T →
+                       ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ 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
 | #I #L #K #V0 #i #_ #_ #V #T #H destruct
-| #I #L #V0 #T0 #_ #_ #V #T #H destruct
-| #L #V0 #W0 #W00 #T0 #U0 #l #HV0 #HT0 #HVW0 #HW00 #HTU0 #V #T #H destruct /2 width=7/
+| #a #I #L #V0 #T0 #_ #_ #V #T #H destruct
+| #a #L #V0 #W0 #W00 #T0 #U0 #l #HV0 #HT0 #HVW0 #HW00 #HTU0 #V #T #H destruct /2 width=8/
 | #L #W0 #T0 #U0 #l #_ #_ #_ #_ #V #T #H destruct
 ]
 qed.
 
 lemma snv_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊩ ⓐV.T :[g] →
-                    ∃∃W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ T :[g] &
+                    ∃∃a,W,W0,U,l. ⦃h, L⦄ ⊩ V :[g] & ⦃h, L⦄ ⊩ T :[g] &
                                 ⦃h, L⦄ ⊢ V •[g, l + 1] W & L ⊢ W ➡* W0 &
-                                ⦃h, L⦄ ⊢ T ➸*[g] ⓛW0.U.
+                                ⦃h, L⦄ ⊢ T •➡*[g] ⓛ{a}W0.U.
+/2 width=3/ qed-.
+
+fact snv_inv_cast_aux: ∀h,g,L,X. ⦃h, L⦄ ⊩ X :[g] → ∀W,T. X = ⓝW.T →
+                       ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] & 
+                              L ⊢ W ⬌* U & ⦃h, L⦄ ⊢ T •[g, l + 1] U.
+#h #g #L #X * -L -X
+[ #L #k #W #T #H destruct
+| #I #L #K #V #i #_ #_ #W #T #H destruct
+| #a #I #L #V #T0 #_ #_ #W #T #H destruct
+| #a #L #V #W0 #W00 #T0 #U #l #_ #_ #_ #_ #_ #W #T #H destruct
+| #L #W0 #T0 #U0 #l #HW0 #HT0 #HTU0 #HWU0 #W #T #H destruct /2 width=4/
+]
+qed.
+
+lemma snv_inv_cast: ∀h,g,L,W,T. ⦃h, L⦄ ⊩ ⓝW.T :[g] →
+                    ∃∃U,l. ⦃h, L⦄ ⊩ W :[g] & ⦃h, L⦄ ⊩ T :[g] & 
+                           L ⊢ W ⬌* U & ⦃h, L⦄ ⊢ T •[g, l + 1] U.
 /2 width=3/ qed-.