matita/matita/contribs/lambdadelta/basic_2/etc_new/frees/frees_lift.etc

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  14
  15 include "ground_2/ynat/ynat_max.ma".
  16 include "basic_2/substitution/drop_drop.ma".
  17 include "basic_2/multiple/frees.ma".
  18
  19 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma frees_dec: ∀L,U,l,i. Decidable (frees l L U i).
  24 #L #U @(f2_ind … rfw … L U) -L -U
  25 #x #IH #L * *
  26 [ -IH /3 width=5 by frees_inv_sort, or_intror/
  27 | #j #Hx #l #i elim (ylt_split_eq i j) #Hji
  28   [ -x @or_intror #H elim (ylt_yle_false … Hji)
  29     lapply (frees_inv_lref_ge … H ?) -L -l /2 width=1 by ylt_fwd_le/
  30   | -x /2 width=1 by or_introl/
  31   | elim (ylt_split j l) #Hli
  32     [ -x @or_intror #H elim (ylt_yle_false … Hji)
  33       lapply (frees_inv_lref_skip … H ?) -L //
  34     | elim (lt_or_ge j (|L|)) #Hj
  35       [ elim (drop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
  36         elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, drop_fwd_rfw, or_introl/ ] #HnW
  37         @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -l
  38         lapply (drop_mono … HLY … HLK) -L #H destruct /2 width=1 by/
  39       | -x @or_intror #H elim (ylt_yle_false … Hji)
  40         lapply (frees_inv_lref_free … H ?) -l //
  41       ]
  42     ]
  43   ]
  44 | -IH /3 width=5 by frees_inv_gref, or_intror/
  45 | #a #I #W #U #Hx #l #i destruct
  46   elim (IH L W … l i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
  47   elim (IH (L.ⓑ{I}W) U … (⫯l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
  48   @or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
  49 | #I #W #U #Hx #l #i destruct
  50   elim (IH L W … l i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
  51   elim (IH L U … l i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
  52   @or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
  53 ]
  54 qed-.
  55
  56 lemma frees_S: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[yinj l]⦃U⦄ → ∀I,K,W. ⬇[l] L ≡ K.ⓑ{I}W →
  57                (K ⊢ ⫰(i-l) ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯l]⦃U⦄.
  58 #L #U #l #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
  59 * #I #K #W #j #Hlj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
  60 lapply (yle_inv_inj … Hlj) -Hlj #Hlj
  61 elim (le_to_or_lt_eq … Hlj) -Hlj
  62 [ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
  63 | -Hji -HnU #H destruct
  64   lapply (drop_mono … HLK0 … HLK) #H destruct -I
  65   elim HnW0 -L -U -HnW0 //
  66 ]
  67 qed.
  68
  69 (* Note: lemma 1250 *)
  70 lemma frees_bind_dx_O: ∀a,I,L,W,U,i. L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[0]⦃U⦄ →
  71                        L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄.
  72 #a #I #L #W #U #i #HU elim (frees_dec L W 0 i)
  73 /4 width=5 by frees_S, frees_bind_dx, frees_bind_sn/
  74 qed.