milestone uupdate in basic_2

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_ind.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_ind.ma
index 2e655049e9d864e5c6ebb1662b402c5ac8882671..4de1ae8019b368e26b172f24c38685620de5978d 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_ind.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/nta_ind.ma
@@ -77,3 +77,107 @@ lemma nta_ind_rest_cnv (h) (Q:relation4 …):
   /4 width=4 by nta_cast/
 ]
 qed-.
+
+lemma nta_ind_ext_cnv_mixed (h) (Q:relation4 …):
+      (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
+      (∀G,K,V,W,U.
+        ⦃G,K⦄ ⊢ V :*[h] W → ⬆*[1] W ≘ U →
+        Q G K V W → Q G (K.ⓓV) (#0) U
+      ) →
+      (∀G,K,W,U. ⦃G,K⦄ ⊢ W !*[h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+      (∀I,G,K,W,U,i.
+        ⦃G,K⦄ ⊢ #i :*[h] W → ⬆*[1] W ≘ U →
+        Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
+      ) →
+      (∀p,I,G,K,V,T,U.
+        ⦃G,K⦄ ⊢ V !*[h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h] U →
+        Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
+      ) →
+      (∀p,G,L,V,W,T,U.
+        ⦃G,L⦄ ⊢ V :*[h] W → ⦃G,L⦄ ⊢ T :*[h] ⓛ{p}W.U →
+        Q G L V W → Q G L T (ⓛ{p}W.U) → Q G L (ⓐV.T) (ⓐV.ⓛ{p}W.U)
+      ) →
+      (∀G,L,V,T,U.
+        ⦃G,L⦄ ⊢ T :*[h] U → ⦃G,L⦄ ⊢ ⓐV.U !*[h] →
+        Q G L T U → Q G L (ⓐV.T) (ⓐV.U)
+      ) →
+      (∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U → Q G L (ⓝU.T) U
+      ) →
+      (∀G,L,T,U1,U2.
+        ⦃G,L⦄ ⊢ T :*[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 !*[h] →
+        Q G L T U1 → Q G L T U2
+      ) →
+      ∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T
+@(fqup_wf_ind_eq (Ⓣ) … G L T) -G -L -T #G0 #L0 #T0 #IH #G #L * * [|||| * ]
+[ #s #HG #HL #HT #X #H destruct -IH
+  elim (nta_inv_sort_sn … H) -H #HUX #HX
+  /2 width=4 by/
+| * [| #i ] #HG #HL #HT #X #H destruct
+  [ elim (nta_inv_lref_sn_drops_cnv … H) -H *
+    [ #K #V #W #U #H #HVW #HWU #HUX #HX
+      lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+      /5 width=7 by nta_ldef, fqu_fqup, fqu_lref_O/
+    | #K #W #U #H #HW #HWU #HUX #HX
+      lapply (drops_fwd_isid … H ?) -H [ // ] #H destruct
+      /3 width=4 by nta_ldec_cnv/
+    ]
+  | elim (nta_inv_lref_sn … H) -H #I #K #T #U #HT #HTU #HUX #HX #H destruct
+    /5 width=7 by nta_lref, fqu_fqup/
+  ]
+| #l #HG #HL #HT #U #H destruct -IH
+  elim (nta_inv_gref_sn … H)
+| #p #I #V #T #HG #HL #HT #X #H destruct
+  elim (nta_inv_bind_sn_cnv … H) -H #U #HV #HTU #HUX #HX
+  /4 width=5 by nta_bind_cnv/
+| #V #T #HG #HL #HT #X #H destruct
+  elim (nta_inv_pure_sn_cnv … H) -H *
+  [ #p #W #U #HVW #HTU #HUX #HX
+    /4 width=9 by nta_appl/
+  | #U #HTU #HVU #HUX #HX
+    /4 width=6 by nta_pure_cnv/
+  ]
+| #U #T #HG #HL #HT #X #H destruct
+  elim (nta_inv_cast_sn … H) -H #HTU #HUX #HX
+  /4 width=4 by nta_cast/
+]
+qed-.
+
+lemma nta_ind_ext_cnv (h) (Q:relation4 …):
+      (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
+      (∀G,K,V,W,U.
+        ⦃G,K⦄ ⊢ V :*[h] W → ⬆*[1] W ≘ U →
+        Q G K V W → Q G (K.ⓓV) (#0) U
+      ) →
+      (∀G,K,W,U. ⦃G,K⦄ ⊢ W !*[h] → ⬆*[1] W ≘ U → Q G (K.ⓛW) (#0) U) →
+      (∀I,G,K,W,U,i.
+        ⦃G,K⦄ ⊢ #i :*[h] W → ⬆*[1] W ≘ U →
+        Q G K (#i) W → Q G (K.ⓘ{I}) (#↑i) U
+      ) →
+      (∀p,I,G,K,V,T,U.
+        ⦃G,K⦄ ⊢ V !*[h] → ⦃G,K.ⓑ{I}V⦄ ⊢ T :*[h] U →
+        Q G (K.ⓑ{I}V) T U → Q G K (ⓑ{p,I}V.T) (ⓑ{p,I}V.U)
+      ) →
+      (∀p,G,K,V,W,T,U.
+        ⦃G,K⦄ ⊢ V :*[h] W → ⦃G,K.ⓛW⦄ ⊢ T :*[h] U →
+        Q G K V W → Q G (K.ⓛW) T U → Q G K (ⓐV.ⓛ{p}W.T) (ⓐV.ⓛ{p}W.U)
+      ) →
+      (∀G,L,V,T,U.
+        ⦃G,L⦄ ⊢ T :*[h] U → ⦃G,L⦄ ⊢ ⓐV.U !*[h] →
+        Q G L T U → Q G L (ⓐV.T) (ⓐV.U)
+      ) →
+      (∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U → Q G L (ⓝU.T) U
+      ) →
+      (∀G,L,T,U1,U2.
+        ⦃G,L⦄ ⊢ T :*[h] U1 → ⦃G,L⦄ ⊢ U1 ⬌*[h] U2 → ⦃G,L⦄ ⊢ U2 !*[h] →
+        Q G L T U1 → Q G L T U2
+      ) →
+      ∀G,L,T,U. ⦃G,L⦄ ⊢ T :*[h] U → Q G L T U.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T #U #H
+@(nta_ind_ext_cnv_mixed … IH1 IH2 IH3 IH4 IH5 … IH7 IH8 IH9 … H) -G -L -T -U -IH1 -IH2 -IH3 -IH4 -IH5 -IH6 -IH8 -IH9
+#p #G #L #V #W #T #U #HVW #HTU #_ #IHTU
+lapply (nta_fwd_cnv_dx … HTU) #H
+elim (cnv_inv_bind … H) -H #_ #HU
+elim (cnv_nta_sn … HU) -HU #X #HUX
+/4 width=2 by nta_beta, nta_fwd_cnv_sn/
+qed-.