matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma

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  11 (*        v         GNU General Public License Version 2                  *)
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  14
  15 include "basic_2/computation/lprs_cprs.ma".
  16 include "basic_2/conversion/cpc_cpc.ma".
  17 include "basic_2/equivalence/cpcs_cprs.ma".
  18
  19 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
  20
  21 (* Advanced inversion lemmas ************************************************)
  22
  23 lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
  24                      ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
  25 #G #L #T1 #T2 #H @(cpcs_ind … H) -T2
  26 [ /3 width=3 by ex2_intro/
  27 | #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
  28   [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
  29   | /3 width=5 by cprs_strap2, ex2_intro/
  30   ]
  31 ]
  32 qed-.
  33
  34 (* Basic_1: was: pc3_gen_sort *)
  35 lemma cpcs_inv_sort: ∀G,L,s1,s2. ⦃G, L⦄ ⊢ ⋆s1 ⬌* ⋆s2 → s1 = s2.
  36 #G #L #s1 #s2 #H elim (cpcs_inv_cprs … H) -H
  37 #T #H1 >(cprs_inv_sort1 … H1) -T #H2
  38 lapply (cprs_inv_sort1 … H2) -L #H destruct //
  39 qed-.
  40
  41 lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
  42                       ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
  43 #a #G #L #W1 #T1 #T #H
  44 elim (cpcs_inv_cprs … H) -H #X #H1 #H2
  45 elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
  46 /3 width=6 by cprs_bind, ex2_2_intro/
  47 qed-.
  48
  49 lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
  50                       ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
  51 /3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
  52
  53 (* Basic_1: was: pc3_gen_sort_abst *)
  54 lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,s. ⦃G, L⦄ ⊢ ⋆s ⬌* ⓛ{a}W.T → ⊥.
  55 #a #G #L #W #T #s #H
  56 elim (cpcs_inv_cprs … H) -H #X #H1
  57 >(cprs_inv_sort1 … H1) -X #H2
  58 elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
  59 qed-.
  60
  61 (* Basic_1: was: pc3_gen_lift *)
  62 lemma cpcs_inv_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≡ K →
  63                      ∀T1,U1. ⬆[l, k] T1 ≡ U1 → ∀T2,U2. ⬆[l, k] T2 ≡ U2 →
  64                      ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
  65 #G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
  66 elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
  67 elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
  68 elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU
  69 >(lift_inj … HXU … HTU) -X -U -l -k /2 width=3 by cprs_div/
  70 qed-.
  71
  72 (* Advanced properties ******************************************************)
  73
  74 lemma lpr_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
  75                       ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
  76 #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
  77 /4 width=5 by cprs_div, lpr_cprs_trans/
  78 qed-.
  79
  80 lemma lprs_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
  81                        ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
  82 #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
  83 /4 width=5 by cprs_div, lprs_cprs_trans/
  84 qed-.
  85
  86 lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  87 #G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
  88 /2 width=3 by cpr_cprs_div/
  89 qed-.
  90
  91 lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
  92 #G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
  93 /2 width=3 by cprs_cpr_div/
  94 qed-.
  95
  96 lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
  97 #G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
  98 /2 width=3 by cprs_div/
  99 qed-.
 100
 101 lemma lprs_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
 102                       ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 103 #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1
 104 /2 width=3 by cprs_div/
 105 qed-.
 106
 107 (* Basic_1: was: pc3_wcpr0_t *)
 108 (* Basic_1: note: pc3_wcpr0_t should be renamed *)
 109 (* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *)
 110 lemma lpr_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
 111                      ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 112 #G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1
 113 /2 width=3 by cprs_div/
 114 qed-.
 115
 116 (* Basic_1: was only: pc3_pr0_pr2_t *)
 117 (* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
 118 lemma lpr_cpr_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
 119                     ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 120 /3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
 121
 122 (* Basic_1: was only: pc3_thin_dx *)
 123 lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
 124                  ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
 125 #G #L #V1 #V2 #HV12 #T1 #T2 #HT12
 126 elim (cpcs_inv_cprs … HV12) -HV12
 127 elim (cpcs_inv_cprs … HT12) -HT12
 128 /3 width=5 by cprs_flat, cprs_div/
 129 qed.
 130
 131 lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
 132                             ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
 133 /3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
 134
 135 lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
 136                     ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
 137 #a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
 138 /3 width=5 by cprs_div, cprs_bind/
 139 qed.
 140
 141 lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
 142 #a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
 143 /3 width=5 by cprs_div, cprs_bind/
 144 qed.
 145
 146 lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
 147                         ∀L2. L2 ⫃ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 148 #G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
 149 /3 width=5 by cprs_div, lsubr_cprs_trans/
 150 qed-.
 151
 152 (* Basic_1: was: pc3_lift *)
 153 lemma cpcs_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≡ K →
 154                  ∀T1,U1. ⬆[l, k] T1 ≡ U1 → ∀T2,U2. ⬆[l, k] T2 ≡ U2 →
 155                  ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
 156 #G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
 157 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
 158 elim (lift_total T l k) /3 width=12 by cprs_div, cprs_lift/
 159 qed.
 160
 161 lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
 162                   ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
 163 #G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
 164
 165 (* More inversion lemmas ****************************************************)
 166
 167 (* Note: there must be a proof suitable for llpr *)
 168 lemma cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
 169                         ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
 170 #a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
 171 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
 172 elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
 173 elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
 174 lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
 175 lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
 176 /4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
 177 qed-.
 178
 179 lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
 180                         ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
 181 #a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
 182 #HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
 183 qed-.
 184
 185 (* Main properties **********************************************************)
 186
 187 (* Basic_1: was pc3_t *)
 188 theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 189 #G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
 190
 191 theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 192 /3 width=3 by cpcs_trans, cpcs_sym/ qed-.
 193
 194 theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 195 /3 width=3 by cpcs_trans, cpcs_sym/ qed-.
 196
 197 lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
 198                   ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
 199                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 200 /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
 201
 202 lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
 203                   ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
 204                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 205 /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
 206
 207 (* Basic_1: was: pc3_wcpr0 *)
 208 lemma lpr_cpcs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
 209                      ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 210 #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
 211 /3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
 212 qed-.