matita/matita/contribs/lambdadelta/basic_2/etc_new/drops/drops_append.etc

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  14
  15 include "basic_2/grammar/lenv_append.ma".
  16 include "basic_2/substitution/drop.ma".
  17
  18 (* DROPPING *****************************************************************)
  19
  20 (* Properties on append for local environments ******************************)
  21
  22 fact drop_O1_append_sn_le_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
  23                                l = 0 → m ≤ |L1| →
  24                                ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2.
  25 #L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m //
  26 [ #l #m #_ #_ #H >(yle_inv_O2 … H) -m //
  27 | /4 width=1 by drop_drop, yle_inv_succ/
  28 | #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #H elim (ysucc_inv_O_dx … H)
  29 ]
  30 qed-.
  31
  32 lemma drop_O1_append_sn_le: ∀L1,L2,s,m. ⬇[s, yinj 0, m] L1 ≡ L2 → m ≤ |L1| →
  33                             ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2.
  34 /2 width=3 by drop_O1_append_sn_le_aux/ qed.
  35
  36 (* Inversion lemmas on append for local environments ************************)
  37
  38 lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K →
  39                               ∀m0. |L2| + m0 = m → ⬇[s, 0, m0] L1 ≡ K.
  40 #K #L1 #L2 elim L2 -L2
  41 [ #s #m #H #m0 >yplus_O1 #H0 destruct //
  42 | #L2 #I #V #IHL2 #s #m #H #m0 >yplus_succ1
  43   #H0 elim (drop_inv_O1_pair1 … H) -H * #Hm #HL12 destruct
  44   [ elim (ysucc_inv_O_dx … Hm)
  45   | /2 width=3 by/
  46   ]
  47 ]
  48 qed-.
  49
  50 lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K → m ≤ |L2| →
  51                               ∀K2. ⬇[s, 0, m] L2 ≡ K2 → K = L1 @@ K2.
  52 #K #L1 #L2 elim L2 -L2
  53 [ #s #m #H1 #H2 #K2 #H3 lapply (yle_inv_O2 … H2) -H2
  54   #H2 elim (drop_inv_atom1 … H3) -H3 #H3 #_ destruct
  55   >(drop_inv_O2 … H1) -H1 //
  56 | #L2 #I #V #IHL2 #s #m @(ynat_ind … m) -m [ -IHL2 || -IHL2 ]
  57   [ #H1 #_ #K2 #H2
  58     lapply (drop_inv_O2 … H1) -H1 #H1
  59     lapply (drop_inv_O2 … H2) -H2 #H2 destruct //
  60   | /3 width=7 by drop_inv_drop1, yle_inv_succ/
  61   | #_ #H lapply (yle_inv_Y1 … H) -H
  62     #H elim (ylt_yle_false (|L2.ⓑ{I}V|) (∞)) //
  63   ]
  64 ]
  65 qed-.