matita/matita/contribs/lambda_delta/basic_2/reducibility/crf_append.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2/substitution/ldrop_append.ma".
  16 include "basic_2/reducibility/crf.ma".
  17
  18 (* CONTEXT-SENSITIVE REDUCIBLE TERMS ****************************************)
  19
  20 (* Advanved properties ******************************************************)
  21
  22 lemma crf_labst_last: ∀L,T,W. L ⊢ 𝐑⦃T⦄  → ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄.
  23 #L #T #W #H elim H -L -T /2 width=1/
  24 #L #K #V #i #HLK
  25 lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
  26 lapply (ldrop_O1_append_sn_le … HLK … (⋆.ⓛW)) -HLK /2 width=2/ -Hi /2 width=3/
  27 qed.
  28
  29 lemma crf_trf: ∀T,W. ⋆ ⊢ 𝐑⦃T⦄ → ⋆.ⓛW ⊢ 𝐑⦃T⦄.
  30 #T #W #H lapply (crf_labst_last … W H) //
  31 qed.
  32
  33 (* Advanced inversion lemmas ************************************************)
  34
  35 fact crf_inv_labst_last_aux: ∀L1,T,W. L1 ⊢ 𝐑⦃T⦄  →
  36                              ∀L2. L1 = ⋆.ⓛW @@ L2 → L2 ⊢ 𝐑⦃T⦄.
  37 #L1 #T #W #H elim H -L1 -T /2 width=1/ /3 width=1/
  38 [ #L1 #K1 #V1 #i #HLK1 #L2 #H destruct
  39   lapply (ldrop_fwd_ldrop2_length … HLK1)
  40   >append_length >commutative_plus normalize in ⊢ (??% → ?); #H
  41   elim (le_to_or_lt_eq i (|L2|) ?) /2 width=1/ -H #Hi destruct
  42   [ elim (ldrop_O1_lt … Hi) #I2 #K2 #V2 #HLK2
  43     lapply (ldrop_O1_inv_append1_le … HLK1 … HLK2) -HLK1 /2 width=2/ -Hi
  44     normalize #H destruct /2 width=3/
  45   | lapply (ldrop_O1_inv_append1_ge … HLK1 ?) -HLK1 // <minus_n_n #H
  46     lapply (ldrop_inv_refl … H) -H #H destruct
  47   ]
  48 | #a #I #L1 #V #T #HI #_ #IHT #L2 #H destruct /3 width=1/
  49 ]
  50 qed.
  51
  52 lemma crf_inv_labst_last: ∀L,T,W. ⋆.ⓛW @@ L ⊢ 𝐑⦃T⦄  → L ⊢ 𝐑⦃T⦄.
  53 /2 width=4/ qed-.
  54
  55 lemma crf_inv_trf: ∀T,W. ⋆.ⓛW ⊢ 𝐑⦃T⦄  → ⋆ ⊢ 𝐑⦃T⦄.
  56 /2 width=4/ qed-.