matita/matita/contribs/lambdadelta/basic_2/i_static/lfxss.ma

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  14
  15 include "basic_2/notation/relations/relationstarstar_4.ma".
  16 include "basic_2/static/lfxs.ma".
  17
  18 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
  19
  20 definition lfxss (R) (T): relation lenv ≝ TC … (lfxs R T).
  21
  22 interpretation "tc of generic extension on referred entries (local environment)"
  23    'RelationStarStar R T L1 L2 = (lfxss R T L1 L2).
  24
  25 (* Basic properties ***********************************************************)
  26
  27 lemma lfxss_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆.
  28 /2 width=1 by inj/ qed.
  29
  30 lemma lfxss_sort: ∀R,I,L1,L2,V1,V2,s.
  31                   L1 ⦻**[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻**[R, ⋆s] L2.ⓑ{I}V2.
  32 #R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
  33 /3 width=4 by lfxs_sort, step, inj/
  34 qed.
  35
  36 lemma lfxss_lref: ∀R,I,L1,L2,V1,V2,i.
  37                   L1 ⦻**[R, #i] L2 → L1.ⓑ{I}V1 ⦻**[R, #⫯i] L2.ⓑ{I}V2.
  38 #R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
  39 /3 width=4 by lfxs_lref, step, inj/
  40 qed.
  41
  42 lemma lfxss_gref: ∀R,I,L1,L2,V1,V2,l.
  43                   L1 ⦻**[R, §l] L2 → L1.ⓑ{I}V1 ⦻**[R, §l] L2.ⓑ{I}V2.
  44 #R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
  45 /3 width=4 by lfxs_gref, step, inj/
  46 qed.
  47
  48 lemma lfxss_sym: ∀R. lexs_frees_confluent R cfull →
  49                  (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
  50                  ∀T. symmetric … (lfxss R T).
  51 #R #H1R #H2R #T #L1 #L2 #H elim H -L2
  52 /4 width=3 by lfxs_sym, TC_strap, inj/
  53 qed-.
  54
  55 lemma lfxss_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
  56                 ∀L1,L2,T. L1 ⦻**[R1, T] L2 → L1 ⦻**[R2, T] L2.
  57 #R1 #R2 #HR #L1 #L2 #T #H elim H -L2
  58 /4 width=5 by lfxs_co, step, inj/
  59 qed-.
  60 (*
  61 (* Basic inversion lemmas ***************************************************)
  62
  63 lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
  64 #R #I #Y2 * /2 width=4 by lexs_inv_atom1/
  65 qed-.
  66
  67 lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
  68 #R #I #Y1 * /2 width=4 by lexs_inv_atom2/
  69 qed-.
  70
  71 lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
  72                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  73                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
  74                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  75 #R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2
  76 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
  77 | lapply (frees_inv_sort … H1) -H1 #Hf
  78   elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
  79   elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
  80   /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/
  81 ]
  82 qed-.
  83
  84 lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
  85                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  86                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
  87                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  88 #R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
  89 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
  90 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
  91   /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
  92 ]
  93 qed-.
  94
  95 lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
  96                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  97                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
  98                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  99 #R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
 100 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
 101 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
 102   /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
 103 ]
 104 qed-.
 105
 106 lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
 107                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
 108                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
 109                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
 110 #R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
 111 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
 112 | lapply (frees_inv_gref … H1) -H1 #Hf
 113   elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
 114   elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
 115   /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
 116 ]
 117 qed-.
 118
 119 lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
 120                      L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
 121 #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
 122 /6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
 123 qed-.
 124
 125 lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
 126                      L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
 127 #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
 128 /5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
 129 qed-.
 130
 131 (* Advanced inversion lemmas ************************************************)
 132
 133 lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
 134                              ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
 135 #R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
 136 [ #H destruct
 137 | #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 138 ]
 139 qed-.
 140
 141 lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
 142                              ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
 143 #R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
 144 [ #_ #H destruct
 145 | #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 146 ]
 147 qed-.
 148
 149 lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
 150                              ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 151                                       Y2 = L2.ⓑ{I}V2.
 152 #R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
 153 [ #H destruct
 154 | #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
 155   /2 width=5 by ex3_2_intro/
 156 ]
 157 qed-.
 158
 159 lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
 160                              ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 161                                       Y1 = L1.ⓑ{I}V1.
 162 #R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
 163 [ #_ #H destruct
 164 | #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
 165   /2 width=5 by ex3_2_intro/
 166 ]
 167 qed-.
 168
 169 lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
 170                              ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
 171 #R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
 172 [ #H destruct
 173 | #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 174 ]
 175 qed-.
 176
 177 lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
 178                              ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
 179 #R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
 180 [ #_ #H destruct
 181 | #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 182 ]
 183 qed-.
 184
 185 lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
 186                              ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
 187 #R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
 188 [ #H destruct
 189 | #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 190 ]
 191 qed-.
 192
 193 lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
 194                              ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
 195 #R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
 196 [ #_ #H destruct
 197 | #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 198 ]
 199 qed-.
 200
 201 (* Basic forward lemmas *****************************************************)
 202
 203 lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
 204 #R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
 205 /4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
 206 qed-.
 207
 208 lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
 209                         R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
 210 #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
 211 qed-.
 212
 213 lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
 214 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 215 qed-.
 216
 217 lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
 218 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 219 qed-.
 220
 221 lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
 222 #R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
 223 qed-.
 224
 225 (* Basic_2A1: removed theorems 24:
 226               llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
 227               llpx_sn_bind llpx_sn_flat
 228               llpx_sn_inv_bind llpx_sn_inv_flat
 229               llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
 230               llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
 231               llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
 232               llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
 233 *)
 234 *)