- main proposition on lsx finally proved!

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / reduction / cpx.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
index bc5044b6dfb379e0cc12fb09ab6b5bc38213b52a..9fd1ccf480bbda238bed577bf1db8904076ee76b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
@@ -23,8 +23,8 @@ inductive cpx (h) (g): relation4 genv lenv term term ≝
 | cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
 | cpx_sort : ∀G,L,k,l. deg h g k (l+1) → cpx h g G L (⋆k) (⋆(next h k))
 | cpx_delta: ∀I,G,L,K,V,V2,W2,i.
-             ⇩[0, i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
-             ⇧[0, i + 1] V2 ≡ W2 → cpx h g G L (#i) W2
+             ⇩[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
+             ⇧[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
 | cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
              cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
              cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
@@ -78,7 +78,7 @@ lemma cpx_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
 #h #g * /2 width=1 by cpx_bind, cpx_flat/
 qed.
 
-lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
+lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,d. ⇩[d] L ≡ (K.ⓑ{I}V) →
                   ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & ⇧[d, 1] T ≡ T2.
 #h #g #I #G #K #V #T1 elim T1 -T1
 [ * #i #L #d /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
@@ -88,28 +88,19 @@ lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
   elim (lift_split … HVW i i) /3 width=7 by cpx_delta, ex2_2_intro/
 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
-  [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /3 width=9 by cpx_bind, ldrop_ldrop, lift_bind, ex2_2_intro/
+  [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /3 width=9 by cpx_bind, ldrop_drop, lift_bind, ex2_2_intro/
   | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, ex2_2_intro/
   ]
 ]
 qed-.
 
-lemma cpx_append: ∀h,g,G. l_appendable_sn … (cpx h g G).
-#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-/2 width=3 by cpx_sort, cpx_bind, cpx_flat, cpx_zeta, cpx_tau, cpx_ti, cpx_beta, cpx_theta/
-#I #G #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
-lapply (ldrop_fwd_length_lt2 … HK0) #H
-@(cpx_delta … I … (L@@K0) V1 … HVW2) // 
-@(ldrop_O1_append_sn_le … HK0) /2 width=2 by lt_to_le/ (**) (* /3/ does not work *)
-qed.
-
 (* Basic inversion lemmas ***************************************************)
 
 fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
                         ∨∨ T2 = ⓪{J}
                          | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
-                         | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
-                                         ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
+                         | ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                         ⇧[O, i+1] V2 ≡ T2 & J = LRef i.
 #G #h #g #L #T1 #T2 * -L -T1 -T2
 [ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
 | #G #L #k #l #Hkl #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
@@ -127,8 +118,8 @@ qed-.
 lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
                      ∨∨ T2 = ⓪{J}
                       | ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
-                      | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
-                                      ⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
+                      | ∃∃I,K,V,V2,i. ⇩[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                      ⇧[O, i+1] V2 ≡ T2 & J = LRef i.
 /2 width=3 by cpx_inv_atom1_aux/ qed-.
 
 lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
@@ -142,8 +133,8 @@ qed-.
 
 lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
                      T2 = #i ∨
-                     ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
-                                 ⇧[O, i + 1] V2 ≡ T2.
+                     ∃∃I,K,V,V2. ⇩[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                                 ⇧[O, i+1] V2 ≡ T2.
 #h #g #G #L #T2 #i #H
 elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
 [ #k #l #_ #_ #H destruct
@@ -311,20 +302,3 @@ elim (cpx_inv_bind1 … H) -H *
 | #T2 #_ #_ #H destruct
 ]
 qed-.
-
-lemma cpx_fwd_shift1: ∀h,g,G,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡[h, g] T →
-                      ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#h #g #G #L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #L #T1 #T #HT1
-  @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #L #T1 #X
-  >shift_append_assoc normalize #H
-  elim (cpx_inv_bind1 … H) -H *
-  [ #V0 #T0 #_ #HT10 #H destruct
-    elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
-    >append_length >HL12 -HL12
-    @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] /2 width=3 by refl, trans_eq/ (**) (* explicit constructor *)
-  | #T #_ #_ #H destruct
-  ]
-]
-qed-.