matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "ground_2/ynat/ynat_plus.ma".
  16 include "basic_2/notation/relations/freestar_4.ma".
  17 include "basic_2/substitution/lift_neg.ma".
  18 include "basic_2/substitution/drop.ma".
  19
  20 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
  21
  22 inductive frees: relation4 ynat lenv term ynat ≝
  23 | frees_eq: ∀L,U,l,i. (∀T. ⬆[i, 1] T ≡ U → ⊥) → frees l L U i
  24 | frees_be: ∀I,L,K,U,W,l,i,j. l ≤ yinj j → yinj j < i →
  25             (∀T. ⬆[j, 1] T ≡ U → ⊥) → ⬇[j]L ≡ K.ⓑ{I}W →
  26             frees 0 K W (⫰(i-j)) → frees l L U i.
  27
  28 interpretation
  29    "context-sensitive free variables (term)"
  30    'FreeStar L i l U = (frees l L U i).
  31
  32 definition frees_trans: predicate (relation3 lenv term term) ≝
  33                         λR. ∀L,U1,U2,i. R L U1 U2 → L ⊢ i ϵ 𝐅*[0]⦃U2⦄ → L ⊢ i ϵ 𝐅*[0]⦃U1⦄.
  34
  35 (* Basic inversion lemmas ***************************************************)
  36
  37 lemma frees_inv: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
  38                  (∀T. ⬆[i, 1] T ≡ U → ⊥) ∨
  39                  ∃∃I,K,W,j. l ≤ yinj j & j < i & (∀T. ⬆[j, 1] T ≡ U → ⊥) &
  40                             ⬇[j]L ≡ K.ⓑ{I}W & K ⊢ ⫰(i-j) ϵ 𝐅*[yinj 0]⦃W⦄.
  41 #L #U #l #i * -L -U -l -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
  42 qed-.
  43
  44 lemma frees_inv_sort: ∀L,l,i,k. L ⊢ i ϵ 𝐅*[l]⦃⋆k⦄ → ⊥.
  45 #L #l #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
  46 qed-.
  47
  48 lemma frees_inv_gref: ∀L,l,i,p. L ⊢ i ϵ 𝐅*[l]⦃§p⦄ → ⊥.
  49 #L #l #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
  50 qed-.
  51
  52 lemma frees_inv_lref: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ →
  53                       yinj j = i ∨
  54                       ∃∃I,K,W. l ≤ yinj j & yinj j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ ⫰(i-j) ϵ 𝐅*[yinj 0]⦃W⦄.
  55 #L #l #x #i #H elim (frees_inv … H) -H
  56 [ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
  57 | * #I #K #W #j #Hlj #Hji #Hnx #HLK #HW
  58   lapply (nlift_inv_lref_be_SO … Hnx) -Hnx #H
  59   lapply (yinj_inj … H) -H #H destruct
  60   /3 width=5 by ex4_3_intro, or_intror/
  61 ]
  62 qed-.
  63
  64 lemma frees_inv_lref_free: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → |L| ≤ j → yinj j = i.
  65 #L #l #j #i #H #Hj elim (frees_inv_lref … H) -H //
  66 * #I #K #W #_ #_ #HLK lapply (drop_fwd_length_lt2 … HLK) -I
  67 #H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
  68 qed-.
  69
  70 lemma frees_inv_lref_skip: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → yinj j < l → yinj j = i.
  71 #L #l #j #i #H #Hjl elim (frees_inv_lref … H) -H //
  72 * #I #K #W #Hlj elim (ylt_yle_false … Hlj) -Hlj //
  73 qed-.
  74
  75 lemma frees_inv_lref_ge: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → i ≤ j → yinj j = i.
  76 #L #l #j #i #H #Hij elim (frees_inv_lref … H) -H //
  77 * #I #K #W #_ #Hji elim (ylt_yle_false … Hji Hij)
  78 qed-.
  79
  80 lemma frees_inv_lref_lt: ∀L,l,j,i.L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → j < i →
  81                          ∃∃I,K,W. l ≤ yinj j & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ ⫰(i-j) ϵ 𝐅*[yinj 0]⦃W⦄.
  82 #L #l #j #i #H #Hji elim (frees_inv_lref … H) -H
  83 [ #H elim (ylt_yle_false … Hji) //
  84 | * /2 width=5 by ex3_3_intro/
  85 ]
  86 qed-.
  87
  88 lemma frees_inv_bind: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄ →
  89                       L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[⫯l]⦃U⦄ .
  90 #a #J #L #V #U #l #i #H elim (frees_inv … H) -H
  91 [ #HnX elim (nlift_inv_bind … HnX) -HnX
  92   /4 width=2 by frees_eq, or_intror, or_introl/
  93 | * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
  94   [ /4 width=9 by frees_be, or_introl/
  95   | #HnT @or_intror @(frees_be … HnT) -HnT
  96     [4: lapply (yle_succ … Hlj) // (**)
  97     |5: lapply (ylt_succ … Hji) // (**)
  98     |6: /2 width=4 by drop_drop/
  99     |7: <yminus_succ in HW; // (**)
 100     |*: skip
 101     ]
 102   ]
 103 ]
 104 qed-.
 105
 106 lemma frees_inv_flat: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄ →
 107                       L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[l]⦃U⦄ .
 108 #J #L #V #U #l #i #H elim (frees_inv … H) -H
 109 [ #HnX elim (nlift_inv_flat … HnX) -HnX
 110   /4 width=2 by frees_eq, or_intror, or_introl/
 111 | * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
 112   /4 width=9 by frees_be, or_intror, or_introl/
 113 ]
 114 qed-.
 115
 116 (* Basic properties *********************************************************)
 117
 118 lemma frees_lref_eq: ∀L,l,i. L ⊢ i ϵ 𝐅*[l]⦃#i⦄.
 119 /4 width=7 by frees_eq, lift_inv_lref2_be, ylt_inj/ qed.
 120
 121 lemma frees_lref_be: ∀I,L,K,W,l,i,j. l ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
 122                      K ⊢ ⫰(i-j) ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[l]⦃#j⦄.
 123 /4 width=9 by frees_be, lift_inv_lref2_be, ylt_inj/ qed.
 124
 125 lemma frees_bind_sn: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
 126                      L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
 127 #a #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
 128 /4 width=9 by frees_be, frees_eq, nlift_bind_sn/
 129 qed.
 130
 131 lemma frees_bind_dx: ∀a,I,L,W,U,l,i. L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[⫯l]⦃U⦄ →
 132                      L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
 133 #a #J #L #V #U #l #i #H elim (frees_inv … H) -H
 134 [ /4 width=9 by frees_eq, nlift_bind_dx/
 135 | * #I #K #W #j #Hlj elim (yle_inv_succ1 … Hlj) -Hlj #Hlj
 136   #Hj <Hj >yminus_succ
 137   lapply (ylt_O … Hj) -Hj #Hj #H
 138   lapply (ylt_inv_succ … H) -H #Hji #HnU #HLK #HW
 139   @(frees_be … Hlj Hji … HW) -HW -Hlj -Hji (**) (* explicit constructor *)
 140   [2: #X #H lapply (nlift_bind_dx … H) /2 width=2 by/ (**)
 141   |3: lapply (drop_inv_drop1_lt … HLK ?) -HLK //
 142   |*: skip
 143 ]
 144 qed.
 145
 146 lemma frees_flat_sn: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
 147                      L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
 148 #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
 149 /4 width=9 by frees_be, frees_eq, nlift_flat_sn/
 150 qed.
 151
 152 lemma frees_flat_dx: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
 153                      L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
 154 #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
 155 /4 width=9 by frees_be, frees_eq, nlift_flat_dx/
 156 qed.
 157
 158 lemma frees_weak: ∀L,U,l1,i. L ⊢ i ϵ 𝐅*[l1]⦃U⦄ →
 159                   ∀l2. l2 ≤ l1 → L ⊢ i ϵ 𝐅*[l2]⦃U⦄.
 160 #L #U #l1 #i #H elim H -L -U -l1 -i
 161 /3 width=9 by frees_be, frees_eq, yle_trans/
 162 qed-.
 163
 164 (* Advanced inversion lemmas ************************************************)
 165
 166 lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
 167                         L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[0]⦃U⦄ .
 168 #a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
 169 /3 width=3 by frees_weak, or_intror, or_introl/
 170 qed-.