matita/matita/contribs/lambdadelta/basic_2/substitution/cpss.ma

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  14
  15 include "basic_2/relocation/ldrop_append.ma".
  16 include "basic_2/substitution/lsubr.ma".
  17
  18 (* CONTEXT-SENSITIVE PARALLEL SUBSTITUTION FOR TERMS ************************)
  19
  20 inductive cpss: lenv → relation term ≝
  21 | cpss_atom : ∀I,L. cpss L (⓪{I}) (⓪{I})
  22 | cpss_delta: ∀L,K,V,V2,W2,i.
  23               ⇩[0, i] L ≡ K. ⓓV → cpss K V V2 →
  24               ⇧[0, i + 1] V2 ≡ W2 → cpss L (#i) W2
  25 | cpss_bind : ∀a,I,L,V1,V2,T1,T2.
  26               cpss L V1 V2 → cpss (L. ⓑ{I} V1) T1 T2 →
  27               cpss L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
  28 | cpss_flat : ∀I,L,V1,V2,T1,T2.
  29               cpss L V1 V2 → cpss L T1 T2 →
  30               cpss L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
  31 .
  32
  33 interpretation "context-sensitive parallel substitution (term)"
  34    'PSubstStar L T1 T2 = (cpss L T1 T2).
  35
  36 (* Basic properties *********************************************************)
  37
  38 lemma cpss_lsubr_trans: lsubr_trans … cpss.
  39 #L1 #T1 #T2 #H elim H -L1 -T1 -T2
  40 [ //
  41 | #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
  42   elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
  43 | /4 width=1/
  44 | /3 width=1/
  45 ]
  46 qed-.
  47
  48 (* Basic_1: was by definition: subst1_refl *)
  49 lemma cpss_refl: ∀T,L. L ⊢ T ▶* T.
  50 #T elim T -T //
  51 #I elim I -I /2 width=1/
  52 qed.
  53
  54 (* Basic_1: was only: subst1_ex *)
  55 lemma cpss_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
  56                    ∃∃T2,T. L ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
  57 #K #V #T1 elim T1 -T1
  58 [ * #i #L #d #HLK /2 width=4/
  59   elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
  60   destruct
  61   elim (lift_total V 0 (i+1)) #W #HVW
  62   elim (lift_split … HVW i i) // /3 width=6/
  63 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
  64   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
  65   [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
  66   | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
  67   ]
  68 ]
  69 qed-.
  70
  71 lemma cpss_append: l_appendable_sn … cpss.
  72 #K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/
  73 #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
  74 lapply (ldrop_fwd_ldrop2_length … HK0) #H
  75 @(cpss_delta … (L@@K0) V1 … HVW2) //
  76 @(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
  77 qed.
  78
  79 (* Basic inversion lemmas ***************************************************)
  80
  81 fact cpss_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ∀I. T1 = ⓪{I} →
  82                          T2 = ⓪{I} ∨
  83                          ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
  84                                      K ⊢ V ▶* V2 &
  85                                      ⇧[O, i + 1] V2 ≡ T2 &
  86                                      I = LRef i.
  87 #L #T1 #T2 * -L -T1 -T2
  88 [ #I #L #J #H destruct /2 width=1/
  89 | #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #I #H destruct /3 width=8/
  90 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
  91 | #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
  92 ]
  93 qed-.
  94
  95 lemma cpss_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ▶* T2 →
  96                       T2 = ⓪{I} ∨
  97                       ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
  98                                   K ⊢ V ▶* V2 &
  99                                   ⇧[O, i + 1] V2 ≡ T2 &
 100                                   I = LRef i.
 101 /2 width=3 by cpss_inv_atom1_aux/ qed-.
 102
 103 (* Basic_1: was only: subst1_gen_sort *)
 104 lemma cpss_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ▶* T2 → T2 = ⋆k.
 105 #L #T2 #k #H
 106 elim (cpss_inv_atom1 … H) -H //
 107 * #K #V #V2 #i #_ #_ #_ #H destruct
 108 qed-.
 109
 110 (* Basic_1: was only: subst1_gen_lref *)
 111 lemma cpss_inv_lref1: ∀L,T2,i. L ⊢ #i ▶* T2 →
 112                       T2 = #i ∨
 113                       ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
 114                                 K ⊢ V ▶* V2 &
 115                                 ⇧[O, i + 1] V2 ≡ T2.
 116 #L #T2 #i #H
 117 elim (cpss_inv_atom1 … H) -H /2 width=1/
 118 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
 119 qed-.
 120
 121 lemma cpss_inv_gref1: ∀L,T2,p. L ⊢ §p ▶* T2 → T2 = §p.
 122 #L #T2 #p #H
 123 elim (cpss_inv_atom1 … H) -H //
 124 * #K #V #V2 #i #_ #_ #_ #H destruct
 125 qed-.
 126
 127 fact cpss_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
 128                          ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
 129                          ∃∃V2,T2. L ⊢ V1 ▶* V2 &
 130                                   L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
 131                                   U2 = ⓑ{a,I} V2. T2.
 132 #L #U1 #U2 * -L -U1 -U2
 133 [ #I #L #b #J #W1 #U1 #H destruct
 134 | #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
 135 | #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5/
 136 | #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
 137 ]
 138 qed-.
 139
 140 lemma cpss_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* U2 →
 141                       ∃∃V2,T2. L ⊢ V1 ▶* V2 &
 142                                L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
 143                                U2 = ⓑ{a,I} V2. T2.
 144 /2 width=3 by cpss_inv_bind1_aux/ qed-.
 145
 146 fact cpss_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
 147                          ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
 148                          ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
 149                                   U2 =  ⓕ{I} V2. T2.
 150 #L #U1 #U2 * -L -U1 -U2
 151 [ #I #L #J #W1 #U1 #H destruct
 152 | #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
 153 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
 154 | #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5/
 155 ]
 156 qed-.
 157
 158 lemma cpss_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* U2 →
 159                       ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
 160                                U2 =  ⓕ{I} V2. T2.
 161 /2 width=3 by cpss_inv_flat1_aux/ qed-.
 162
 163 (* Basic forward lemmas *****************************************************)
 164
 165 lemma cpss_fwd_tw: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ♯{T1} ≤ ♯{T2}.
 166 #L #T1 #T2 #H elim H -L -T1 -T2 normalize
 167 /3 width=1 by monotonic_le_plus_l, le_plus/ (**) (* auto is too slow without trace *)
 168 qed-.
 169
 170 lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
 171                        ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
 172 #L1 @(lenv_ind_dx … L1) -L1 normalize
 173 [ #L #T1 #T #HT1
 174   @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
 175 | #I #L1 #V1 #IH #L #T1 #X
 176   >shift_append_assoc normalize #H
 177   elim (cpss_inv_bind1 … H) -H
 178   #V0 #T0 #_ #HT10 #H destruct
 179   elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
 180   >append_length >HL12 -HL12
 181   @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
 182 ]
 183 qed-.
 184
 185 (* Basic_1: removed theorems 27:
 186             subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
 187             subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
 188             subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
 189             subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
 190             subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
 191             subst0_confluence_lift subst0_tlt
 192             subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
 193             subst1_gen_lift_eq subst1_confluence_neq
 194 *)