matita/matita/contribs/lambdadelta/basic_2A/multiple/frees_lift.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "basic_2A/substitution/drop_drop.ma".
  16 include "basic_2A/multiple/frees.ma".
  17
  18 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 lemma frees_dec: ∀L,U,l,i. Decidable (frees l L U i).
  23 #L #U @(f2_ind … rfw … L U) -L -U
  24 #x #IH #L * *
  25 [ -IH /3 width=5 by frees_inv_sort, or_intror/
  26 | #j #Hx #l #i elim (lt_or_eq_or_gt i j) #Hji
  27   [ -x @or_intror #H elim (lt_refl_false i)
  28     lapply (frees_inv_lref_ge … H ?) -L -l /2 width=1 by lt_to_le/
  29   | -x /2 width=1 by or_introl/
  30   | elim (ylt_split j l) #Hli
  31     [ -x @or_intror #H elim (lt_refl_false i)
  32       lapply (frees_inv_lref_skip … H ?) -L //
  33     | elim (lt_or_ge j (|L|)) #Hj
  34       [ elim (drop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
  35         elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, drop_fwd_rfw, or_introl/ ] #HnW
  36         @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -l
  37         lapply (drop_mono … HLY … HLK) -L #H destruct /2 width=1 by/
  38       | -x @or_intror #H elim (lt_refl_false i)
  39         lapply (frees_inv_lref_free … H ?) -l //
  40       ]
  41     ]
  42   ]
  43 | -IH /3 width=5 by frees_inv_gref, or_intror/
  44 | #a #I #W #U #Hx #l #i destruct
  45   elim (IH L W … l i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
  46   elim (IH (L.ⓑ{I}W) U … (⫯l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
  47   @or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
  48 | #I #W #U #Hx #l #i destruct
  49   elim (IH L W … l i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
  50   elim (IH L U … l i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
  51   @or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
  52 ]
  53 qed-.
  54
  55 lemma frees_S: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[yinj l]⦃U⦄ → ∀I,K,W. ⬇[l] L ≡ K.ⓑ{I}W →
  56                (K ⊢ i-l-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯l]⦃U⦄.
  57 #L #U #l #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
  58 * #I #K #W #j #Hlj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
  59 lapply (yle_inv_inj … Hlj) -Hlj #Hlj
  60 elim (le_to_or_lt_eq … Hlj) -Hlj
  61 [ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
  62 | -Hji -HnU #H destruct
  63   lapply (drop_mono … HLK0 … HLK) #H destruct -I
  64   elim HnW0 -L -U -HnW0 //
  65 ]
  66 qed.
  67
  68 (* Note: lemma 1250 *)
  69 lemma frees_bind_dx_O: ∀a,I,L,W,U,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ →
  70                        L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄.
  71 #a #I #L #W #U #i #HU elim (frees_dec L W 0 i)
  72 /4 width=5 by frees_S, frees_bind_dx, frees_bind_sn/
  73 qed.
  74
  75 (* Properties on relocation *************************************************)
  76
  77 lemma frees_lift_ge: ∀K,T,l,i. K ⊢ i ϵ𝐅*[l]⦃T⦄ →
  78                      ∀L,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
  79                      ∀U. ⬆[l0, m0] T ≡ U → l0 ≤ i →
  80                      L ⊢ i+m0 ϵ 𝐅*[l]⦃U⦄.
  81 #K #T #l #i #H elim H -K -T -l -i
  82 [ #K #T #l #i #HnT #L #s #l0 #m0 #_ #U #HTU #Hl0i -K -s
  83   @frees_eq #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
  84 | #I #K #K0 #T #V #l #i #j #Hlj #Hji #HnT #HK0 #HV #IHV #L #s #l0 #m0 #HLK #U #HTU #Hl0i
  85   elim (lt_or_ge j l0) #H1
  86   [ elim (drop_trans_lt … HLK … HK0) // -K #L0 #W #HL0 #HLK0 #HVW
  87     @(frees_be … HL0) -HL0 -HV
  88     /3 width=3 by lt_plus_to_minus_r, lt_to_le_to_lt/
  89     [ #X #HXU >(plus_minus_m_m l0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
  90       elim (lift_div_le … HXU … HTU ?) -U /2 width=2 by monotonic_pred/
  91     | >minus_plus <plus_minus // <minus_plus
  92       /3 width=5 by monotonic_le_minus_l2/
  93     ]
  94   | lapply (drop_trans_ge … HLK … HK0 ?) // -K #HLK0
  95     lapply (drop_inv_gen … HLK0) >commutative_plus -HLK0 #HLK0
  96     @(frees_be … HLK0) -HLK0 -IHV
  97     /2 width=1 by yle_plus_dx1_trans, lt_minus_to_plus/
  98     #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
  99   ]
 100 ]
 101 qed.
 102
 103 (* Inversion lemmas on relocation *******************************************)
 104
 105 lemma frees_inv_lift_be: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
 106                          ∀K,s,l0,m0. ⬇[s, l0, m0+1] L ≡ K →
 107                          ∀T. ⬆[l0, m0+1] T ≡ U → l0 ≤ i → i ≤ l0 + m0 → ⊥.
 108 #L #U #l #i #H elim H -L -U -l -i
 109 [ #L #U #l #i #HnU #K #s #l0 #m0 #_ #T #HTU #Hl0i #Hilm0
 110   elim (lift_split … HTU i m0) -HTU /2 width=2 by/
 111 | #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hl0i #Hilm0
 112   elim (lt_or_ge j l0) #H1
 113   [ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
 114     @(IHW … HKL0 … HVW)
 115     [ /2 width=1 by monotonic_le_minus_l2/
 116     | >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
 117     ]
 118   | elim (lift_split … HTU j m0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
 119   ]
 120 ]
 121 qed-.
 122
 123 lemma frees_inv_lift_ge: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
 124                          ∀K,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
 125                          ∀T. ⬆[l0, m0] T ≡ U → l0 + m0 ≤ i →
 126                          K ⊢ i-m0 ϵ𝐅*[l-yinj m0]⦃T⦄.
 127 #L #U #l #i #H elim H -L -U -l -i
 128 [ #L #U #l #i #HnU #K #s #l0 #m0 #HLK #T #HTU #Hlm0i -L -s
 129   elim (le_inv_plus_l … Hlm0i) -Hlm0i #Hl0im0 #Hm0i @frees_eq #X #HXT -K
 130   elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
 131 | #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hlm0i
 132   elim (lt_or_ge j l0) #H1
 133   [ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
 134     elim (le_inv_plus_l … Hlm0i) #H0 #Hm0i
 135     @(frees_be … H) -H
 136     [ /3 width=1 by yle_plus_dx1_trans, monotonic_yle_minus_dx/
 137     | /2 width=3 by lt_to_le_to_lt/
 138     | #X #HXT elim (lift_trans_ge … HXT … HTU) -T /2 width=2 by/
 139     | lapply (IHW … HKL0 … HVW ?) // -I -K -K0 -L0 -V -W -T -U -s
 140       >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
 141     ]
 142   | elim (lt_or_ge j (l0+m0)) [ >commutative_plus |] #H2
 143     [ -L -I -W lapply (lt_plus_to_minus ???? H2) // -H2 #H2
 144       elim (lift_split … HTU j (m0-1)) -HTU //
 145       [ >minus_minus_associative /2 width=2 by ltn_to_ltO/ <minus_n_n
 146         #X #_ #H elim (HnU … H)
 147       | >commutative_plus /3 width=1 by le_minus_to_plus, monotonic_pred/
 148       ]
 149     | lapply (drop_conf_ge … HLK … HLK0 ?) // -L #HK0
 150       elim (le_inv_plus_l … H2) -H2 #H2 #Hm0j
 151       @(frees_be … HK0)
 152       [ /2 width=1 by monotonic_yle_minus_dx/
 153       | /2 width=1 by monotonic_lt_minus_l/
 154       | #X #HXT elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
 155       | >arith_b1 /2 width=5 by/
 156       ]
 157     ]
 158   ]
 159 ]
 160 qed-.