matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/cpss.etc

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   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 notation "hvbox( L ⊢ break term 46 T1 ▶* break term 46 T2 )"
  16    non associative with precedence 45
  17    for @{ 'PSubstStar $L $T1 $T2 }.
  18
  19 include "basic_2/grammar/cl_shift.ma".
  20 include "basic_2/relocation/ldrop_append.ma".
  21 include "basic_2/substitution/lsubr.ma".
  22
  23 (* CONTEXT-SENSITIVE PARALLEL SUBSTITUTION FOR TERMS ************************)
  24
  25 inductive cpss: lenv → relation term ≝
  26 | cpss_atom : ∀I,L. cpss L (⓪{I}) (⓪{I})
  27 | cpss_delta: ∀L,K,V,V2,W2,i.
  28               ⇩[0, i] L ≡ K. ⓓV → cpss K V V2 →
  29               ⇧[0, i + 1] V2 ≡ W2 → cpss L (#i) W2
  30 | cpss_bind : ∀a,I,L,V1,V2,T1,T2.
  31               cpss L V1 V2 → cpss (L. ⓑ{I} V1) T1 T2 →
  32               cpss L (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
  33 | cpss_flat : ∀I,L,V1,V2,T1,T2.
  34               cpss L V1 V2 → cpss L T1 T2 →
  35               cpss L (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
  36 .
  37
  38 interpretation "context-sensitive parallel substitution (term)"
  39    'PSubstStar L T1 T2 = (cpss L T1 T2).
  40
  41 (* Basic properties *********************************************************)
  42
  43 lemma cpss_lsubr_trans: lsub_trans … cpss lsubr.
  44 #L1 #T1 #T2 #H elim H -L1 -T1 -T2
  45 [ //
  46 | #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
  47   elim (lsubr_fwd_ldrop2_abbr … HL12 … HLK1) -HL12 -HLK1 /3 width=6/
  48 | /4 width=1/
  49 | /3 width=1/
  50 ]
  51 qed-.
  52
  53 (* Basic_1: was by definition: subst1_refl *)
  54 lemma cpss_refl: ∀T,L. L ⊢ T ▶* T.
  55 #T elim T -T //
  56 #I elim I -I /2 width=1/
  57 qed.
  58
  59 (* Basic_1: was only: subst1_ex *)
  60 lemma cpss_delift: ∀K,V,T1,L,d. ⇩[0, d] L ≡ (K. ⓓV) →
  61                    ∃∃T2,T. L ⊢ T1 ▶* T2 & ⇧[d, 1] T ≡ T2.
  62 #K #V #T1 elim T1 -T1
  63 [ * #i #L #d #HLK /2 width=4/
  64   elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
  65   destruct
  66   elim (lift_total V 0 (i+1)) #W #HVW
  67   elim (lift_split … HVW i i) // /3 width=6/
  68 | * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
  69   elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
  70   [ elim (IHU1 (L. ⓑ{I} W1) (d+1)) -IHU1 /2 width=1/ -HLK /3 width=9/
  71   | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8/
  72   ]
  73 ]
  74 qed-.
  75
  76 lemma cpss_append: l_appendable_sn … cpss.
  77 #K #T1 #T2 #H elim H -K -T1 -T2 // /2 width=1/
  78 #K #K0 #V1 #V2 #W2 #i #HK0 #_ #HVW2 #IHV12 #L
  79 lapply (ldrop_fwd_length_lt2 … HK0) #H
  80 @(cpss_delta … (L@@K0) V1 … HVW2) //
  81 @(ldrop_O1_append_sn_le … HK0) /2 width=2/ (**) (* /3/ does not work *)
  82 qed.
  83
  84 (* Basic inversion lemmas ***************************************************)
  85
  86 fact cpss_inv_atom1_aux: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ∀I. T1 = ⓪{I} →
  87                          T2 = ⓪{I} ∨
  88                          ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
  89                                      K ⊢ V ▶* V2 &
  90                                      ⇧[O, i + 1] V2 ≡ T2 &
  91                                      I = LRef i.
  92 #L #T1 #T2 * -L -T1 -T2
  93 [ #I #L #J #H destruct /2 width=1/
  94 | #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #I #H destruct /3 width=8/
  95 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
  96 | #I #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
  97 ]
  98 qed-.
  99
 100 lemma cpss_inv_atom1: ∀I,L,T2. L ⊢ ⓪{I} ▶* T2 →
 101                       T2 = ⓪{I} ∨
 102                       ∃∃K,V,V2,i. ⇩[O, i] L ≡ K. ⓓV &
 103                                   K ⊢ V ▶* V2 &
 104                                   ⇧[O, i + 1] V2 ≡ T2 &
 105                                   I = LRef i.
 106 /2 width=3 by cpss_inv_atom1_aux/ qed-.
 107
 108 (* Basic_1: was only: subst1_gen_sort *)
 109 lemma cpss_inv_sort1: ∀L,T2,k. L ⊢ ⋆k ▶* T2 → T2 = ⋆k.
 110 #L #T2 #k #H
 111 elim (cpss_inv_atom1 … H) -H //
 112 * #K #V #V2 #i #_ #_ #_ #H destruct
 113 qed-.
 114
 115 (* Basic_1: was only: subst1_gen_lref *)
 116 lemma cpss_inv_lref1: ∀L,T2,i. L ⊢ #i ▶* T2 →
 117                       T2 = #i ∨
 118                       ∃∃K,V,V2. ⇩[O, i] L ≡ K. ⓓV &
 119                                 K ⊢ V ▶* V2 &
 120                                 ⇧[O, i + 1] V2 ≡ T2.
 121 #L #T2 #i #H
 122 elim (cpss_inv_atom1 … H) -H /2 width=1/
 123 * #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=6/
 124 qed-.
 125
 126 lemma cpss_inv_gref1: ∀L,T2,p. L ⊢ §p ▶* T2 → T2 = §p.
 127 #L #T2 #p #H
 128 elim (cpss_inv_atom1 … H) -H //
 129 * #K #V #V2 #i #_ #_ #_ #H destruct
 130 qed-.
 131
 132 fact cpss_inv_bind1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
 133                          ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
 134                          ∃∃V2,T2. L ⊢ V1 ▶* V2 &
 135                                   L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
 136                                   U2 = ⓑ{a,I} V2. T2.
 137 #L #U1 #U2 * -L -U1 -U2
 138 [ #I #L #b #J #W1 #U1 #H destruct
 139 | #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W1 #U1 #H destruct
 140 | #a #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5/
 141 | #I #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W1 #U1 #H destruct
 142 ]
 143 qed-.
 144
 145 lemma cpss_inv_bind1: ∀a,I,L,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* U2 →
 146                       ∃∃V2,T2. L ⊢ V1 ▶* V2 &
 147                                L. ⓑ{I} V1 ⊢ T1 ▶* T2 &
 148                                U2 = ⓑ{a,I} V2. T2.
 149 /2 width=3 by cpss_inv_bind1_aux/ qed-.
 150
 151 fact cpss_inv_flat1_aux: ∀L,U1,U2. L ⊢ U1 ▶* U2 →
 152                          ∀I,V1,T1. U1 = ⓕ{I} V1. T1 →
 153                          ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
 154                                   U2 =  ⓕ{I} V2. T2.
 155 #L #U1 #U2 * -L -U1 -U2
 156 [ #I #L #J #W1 #U1 #H destruct
 157 | #L #K #V #V2 #W2 #i #_ #_ #_ #J #W1 #U1 #H destruct
 158 | #a #I #L #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
 159 | #I #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5/
 160 ]
 161 qed-.
 162
 163 lemma cpss_inv_flat1: ∀I,L,V1,T1,U2. L ⊢ ⓕ{I} V1. T1 ▶* U2 →
 164                       ∃∃V2,T2. L ⊢ V1 ▶* V2 & L ⊢ T1 ▶* T2 &
 165                                U2 =  ⓕ{I} V2. T2.
 166 /2 width=3 by cpss_inv_flat1_aux/ qed-.
 167
 168 (* Basic forward lemmas *****************************************************)
 169
 170 lemma cpss_fwd_tw: ∀L,T1,T2. L ⊢ T1 ▶* T2 → ♯{T1} ≤ ♯{T2}.
 171 #L #T1 #T2 #H elim H -L -T1 -T2 normalize
 172 /3 width=1 by monotonic_le_plus_l, le_plus/ (**) (* auto is too slow without trace *)
 173 qed-.
 174
 175 lemma cpss_fwd_shift1: ∀L1,L,T1,T. L ⊢ L1 @@ T1 ▶* T →
 176                        ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
 177 #L1 @(lenv_ind_dx … L1) -L1 normalize
 178 [ #L #T1 #T #HT1
 179   @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
 180 | #I #L1 #V1 #IH #L #T1 #X
 181   >shift_append_assoc normalize #H
 182   elim (cpss_inv_bind1 … H) -H
 183   #V0 #T0 #_ #HT10 #H destruct
 184   elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
 185   >append_length >HL12 -HL12
 186   @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
 187 ]
 188 qed-.
 189
 190 (* Basic_1: removed theorems 27:
 191             subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
 192             subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
 193             subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
 194             subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
 195             subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
 196             subst0_confluence_lift subst0_tlt
 197             subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
 198             subst1_gen_lift_eq subst1_confluence_neq
 199 *)