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

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  10 (*       \ /        This file is distributed under the terms of the       *)
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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/
  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,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
  36 #G #L #k1 #k2 #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,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
  55 #a #G #L #W #T #k #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,s,d,e. ⇩[s, d, e] L ≡ K →
  63                      ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
  64                      ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
  65 #G #L #K #s #d #e #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 -d -e /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 lemma lpr_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
 110                      ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 111 /3 width=5 by lprs_cprs_conf, lpr_lprs/ qed-.
 112
 113 (* Basic_1: was only: pc3_pr0_pr2_t *)
 114 (* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
 115 lemma lpr_cpr_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
 116                     ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 117 /3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
 118
 119 (* Basic_1: was only: pc3_thin_dx *)
 120 lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
 121                  ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
 122 #G #L #V1 #V2 #HV12 #T1 #T2 #HT12
 123 elim (cpcs_inv_cprs … HV12) -HV12
 124 elim (cpcs_inv_cprs … HT12) -HT12
 125 /3 width=5 by cprs_flat, cprs_div/
 126 qed.
 127
 128 lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
 129                             ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
 130 /3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
 131
 132 lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
 133                     ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
 134 #a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
 135 /3 width=5 by cprs_div, cprs_bind/
 136 qed.
 137
 138 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.
 139 #a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
 140 /3 width=5 by cprs_div, cprs_bind/
 141 qed.
 142
 143 lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
 144                         ∀L2. L2 ⊑ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 145 #G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
 146 /3 width=5 by cprs_div, lsubr_cprs_trans/
 147 qed-.
 148
 149 (* Basic_1: was: pc3_lift *)
 150 lemma cpcs_lift: ∀G,L,K,s,d,e. ⇩[s, d, e] L ≡ K →
 151                  ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
 152                  ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
 153 #G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
 154 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
 155 elim (lift_total T d e) /3 width=12 by cprs_div, cprs_lift/
 156 qed.
 157
 158 lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
 159                   ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
 160 #G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
 161
 162 (* More inversion lemmas ****************************************************)
 163
 164 lemma cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
 165                         ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
 166 #a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
 167 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
 168 elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
 169 elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
 170 lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
 171 lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
 172 /4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
 173 qed-.
 174
 175 lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
 176                         ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
 177 #a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
 178 #HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
 179 qed-.
 180
 181 (* Main properties **********************************************************)
 182
 183 (* Basic_1: was pc3_t *)
 184 theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 185 #G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
 186
 187 theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 188 /3 width=3 by cpcs_trans, cpcs_sym/ qed-.
 189
 190 theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 191 /3 width=3 by cpcs_trans, cpcs_sym/ qed-.
 192
 193 lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
 194                   ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
 195                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 196 /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
 197
 198 lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
 199                   ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
 200                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 201 /3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
 202
 203 (* Basic_1: was: pc3_wcpr0 *)
 204 lemma lpr_cpcs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
 205                      ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 206 #G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
 207 /3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
 208 qed-.