matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma

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  14
  15 include "ground_2/relocation/rtmap_id.ma".
  16 include "basic_2/notation/relations/relationstar_4.ma".
  17 include "basic_2/grammar/ceq.ma".
  18 include "basic_2/relocation/lexs.ma".
  19 include "basic_2/static/frees.ma".
  20
  21 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
  22
  23 definition lfxs (R) (T): relation lenv ≝
  24                 λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2.
  25
  26 interpretation "generic extension on referred entries (local environment)"
  27    'RelationStar R T L1 L2 = (lfxs R T L1 L2).
  28
  29 definition R_confluent_lfxs: relation4 (relation3 lenv term term)
  30                                        (relation3 lenv term term) … ≝
  31                              λR1,R2,RP1,RP2.
  32                              ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
  33                              ∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 →
  34                              ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
  35
  36 (* Basic properties ***********************************************************)
  37
  38 lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
  39 /3 width=3 by lexs_atom, frees_atom, ex2_intro/
  40 qed.
  41
  42 lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
  43                  L1 ⦻*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2.
  44 #R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/
  45 qed.
  46
  47 lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 →
  48                  R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2.
  49 #R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
  50 qed.
  51
  52 lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
  53                  L1 ⦻*[R, #i] L2 → L1.ⓑ{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2.
  54 #R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
  55 qed.
  56
  57 lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
  58                  L1 ⦻*[R, §l] L2 → L1.ⓑ{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2.
  59 #R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/
  60 qed.
  61
  62 lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
  63                          L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 →
  64                          ∀V2. R L1 V V2 →
  65                          L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2.
  66 #R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
  67 /3 width=5 by lexs_pair_repl, ex2_intro/
  68 qed-.
  69
  70 lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
  71                ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2.
  72 #R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/
  73 qed-.
  74
  75 lemma pippo: ∀R1,R2,RP1,RP2. R_confluent_lfxs R1 R2 RP1 RP2 →
  76              lexs_confluent R1 R2 RP1 cfull RP2 cfull.
  77 #R1 #R2 #RP1 #RP2 #HR #f #L0 #T0 #T1 #HT01 #T2 #HT02 #L1 #HL01 #L2 #HL02
  78
  79 (* Basic inversion lemmas ***************************************************)
  80
  81 lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
  82 #R #I #Y2 * /2 width=4 by lexs_inv_atom1/
  83 qed-.
  84
  85 lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
  86 #R #I #Y1 * /2 width=4 by lexs_inv_atom2/
  87 qed-.
  88
  89 lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
  90                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
  91                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
  92                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
  93 #R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2
  94 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
  95 | lapply (frees_inv_sort … H1) -H1 #Hf
  96   elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
  97   elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
  98   /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/
  99 ]
 100 qed-.
 101
 102 lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
 103                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
 104                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 105                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
 106 #R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
 107 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
 108 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
 109   /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
 110 ]
 111 qed-.
 112
 113 lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
 114                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
 115                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
 116                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
 117 #R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
 118 [ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
 119 | #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
 120   /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
 121 ]
 122 qed-.
 123
 124 lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
 125                      (Y1 = ⋆ ∧ Y2 = ⋆) ∨
 126                      ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
 127                                       Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
 128 #R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
 129 [ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
 130 | lapply (frees_inv_gref … H1) -H1 #Hf
 131   elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
 132   elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
 133   /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
 134 ]
 135 qed-.
 136
 137 lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
 138                      L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
 139 #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
 140 /6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
 141 qed-.
 142
 143 lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
 144                      L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
 145 #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
 146 /5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
 147 qed-.
 148
 149 (* Advanced inversion lemmas ************************************************)
 150
 151 lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
 152                              ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
 153 #R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
 154 [ #H destruct
 155 | #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 156 ]
 157 qed-.
 158
 159 lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
 160                              ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
 161 #R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
 162 [ #_ #H destruct
 163 | #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 164 ]
 165 qed-.
 166
 167 lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
 168                              ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 169                                       Y2 = L2.ⓑ{I}V2.
 170 #R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
 171 [ #H destruct
 172 | #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
 173   /2 width=5 by ex3_2_intro/
 174 ]
 175 qed-.
 176
 177 lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
 178                              ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
 179                                       Y1 = L1.ⓑ{I}V1.
 180 #R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
 181 [ #_ #H destruct
 182 | #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
 183   /2 width=5 by ex3_2_intro/
 184 ]
 185 qed-.
 186
 187 lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
 188                              ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
 189 #R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
 190 [ #H destruct
 191 | #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 192 ]
 193 qed-.
 194
 195 lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
 196                              ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
 197 #R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
 198 [ #_ #H destruct
 199 | #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 200 ]
 201 qed-.
 202
 203 lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
 204                              ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
 205 #R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
 206 [ #H destruct
 207 | #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 208 ]
 209 qed-.
 210
 211 lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
 212                              ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
 213 #R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
 214 [ #_ #H destruct
 215 | #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
 216 ]
 217 qed-.
 218
 219 (* Basic forward lemmas *****************************************************)
 220
 221 lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
 222 #R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
 223 /4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
 224 qed-.
 225
 226 lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
 227                         R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
 228 #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
 229 qed-.
 230
 231 lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
 232 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 233 qed-.
 234
 235 lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
 236 #R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
 237 qed-.
 238
 239 lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
 240 #R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
 241 qed-.
 242
 243 (* Basic_2A1: removed theorems 24:
 244               llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
 245               llpx_sn_bind llpx_sn_flat
 246               llpx_sn_inv_bind llpx_sn_inv_flat
 247               llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
 248               llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
 249               llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
 250               llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
 251 *)